The Numerical Solution of
Boundary Value Problems
with an Essential Singularity
Winfried Auzinger
Othmar Koch
Jelena Petrickovic
Ewa Weinmüller
Technical Report
ANUM Preprint No. 3/03
Institute for Applied Mathematics
and Numerical Analysis
Contents
1 Introduction
5
2 Theoretical foundations
2.1 Problem class . . . . . . . . . . . . . . . . . . . .
2.2 Analytical properties . . . . . . . . . . . . . . . .
2.2.1 Linear systems . . . . . . . . . . . . . . .
2.2.2 Nonlinear case . . . . . . . . . . . . . . . .
2.3 Numerical approach . . . . . . . . . . . . . . . . .
2.3.1 Collocation methods for regular problems .
2.3.2 Collocation for α = 1 . . . . . . . . . . . .
2.3.3 Collocation for α > 1 . . . . . . . . . . . .
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3 Test problems
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4 Numerical tests based on the collocation solver sbvpcol
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4.1 Convergence results for the global error . . . . . . . . . . . . . 26
4.2 Convergence results for first and second derivatives . . . . . . 30
5 A posteriori error estimate and conditioning
5.1 A posteriori error estimate and conditioning in sbvp . . . . .
5.1.1 Conditioning of the collocation equations . . . . . . .
5.1.2 A posteriori error estimates . . . . . . . . . . . . . .
5.2 The backward Euler method . . . . . . . . . . . . . . . . . .
5.2.1 Backward Euler by means of sbvpcol . . . . . . . . .
5.2.2 An implementation of backward Euler with preconditioning . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Midpoint rule . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Midpoint rule by means of sbvpcol . . . . . . . . . .
5.3.2 An implementation of the midpoint rule with preconditioning . . . . . . . . . . . . . . . . . . . . . . . . .
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6 Error estimation based on a h-h/2 strategy
41
7 On the stability of the preconditioned midpoint rule
44
8 Lobatto distribution
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3
Appendix
Figures:
1. Figure 1.1
51
2. Figure 2.1
96
3. Figure 3.1
109
4. Figure 4.1
114
5. Figure 5.1
119
6. Figure 6.1
126
7. Figure 7.1
150
8. Figure 9.1
158
Tables:
1. Tables 1.0 - 1.43
52-95
2. Tables 2.1 - 2.12
97-108
3. Tables 3.1 - 3.4
110-113
4. Tables 4.1 - 4.4
115-118
5. Tables 5.1 - 5.6
120-125
6. Tables 6.1 - 6.23
127-149
7. Tables 7.1 - 7.4
151-154
8. Tables 8.1 - 8.3
155-157
9. Tables 9.1 - 9.3
159-161
4
Chapter 1
Introduction
Singular boundary value problems (BVPs) with a singularity of the first kind
frequently occur in applications, e.g. when a partial differential equation
(PDE) is reduced to an ordinary differential equation (ODE) in the presence
of symmetries. More difficult problems featuring a singularity of the second
kind (essential singularity) arise, for instance, when a boundary value ODE is
transformed from an infinite interval to a finite one. They are also very common in quantum physics, mechanics, investigations of ferromagnetic systems
(e.g. Ginzburg-Landau equations) etc.
In this work we present a study of certain numerical techniques – which
are well known to perform efficiently and accurately in the case of a singularity of the first kind – applied to problems with an essential singularity.
In particular, collocation methods are considered together with a posteriori
error estimates intended to provide the basis for a grid selection strategy.
We concentrate on the methods implemented in the MATLAB code sbvp.
This code was originally intended to solve regular BVPs and problems with
a singularity of the first kind. An extension to problems with an essential
singularity is in preparation, and our work provides important prerequisites
concerning the proper choice of algorithmic components.
This document is organized as follows:
• In Chapter 2, we recapitulate the analytical theory concerning the wellposedness of boundary value problems with an essential singularity,
and we give a short review of some relevant results from numerical
analysis. Here we note that, for the methods implemented in sbvp, a
fairly complete theoretical foundation exists for regular problems and
problems with a singularity of the first kind; however, hardly any results
are available in the presence of an essential singularity.
• This latter fact motivated our extensive numerical experiments, which
5
are based on a set of test examples specified in Chapter 3, for collocation
based on equidistant or Gaussian nodes.
• In Chapter 4, we present our numerical results obtained with the collocation solver sbvpcol (which is at the core of sbvp). Due to these
results, collocation seems to be a very accurate and reliable method for
problems with an essential singularity.
• In Chapter 5, we report the results obtained by direct application of
sbvp, which is based on sbvpcol and uses an a posteriori error estimate
implemented in sbvperr for global error estimation. This estimate is
based on a defect correction approach and uses the backward Euler
method as an auxiliary algorithmic component. Our results show that
this estimate is completely unreliable in the essentially singular case.
Furthermore, we observed that the condition numbers of the algebraic
systems representing the Euler equations are extremely large; an attempt to remedy this situation by preconditioning was not successful.
We have also considered the implicit midpoint rule as an alternative to
the backward Euler scheme. For this method, theoretical convergence
results are available which are consistent with our numerical results:
The convergence order is at least equal to the stage order and may
even be as high as the classical convergence order p = 2. Moreover,
an appropriate diagonal preconditioning yields optimal condition numbers for the corresponding algebraic system (not worse than for regular
problems).
If we consider the implicit midpoint rule as an auxiliary scheme for the
purpose of error estimation, the corresponding results are significantly
better but not really convincing.
• These results motivated us to test a classical, but computationally more
expensive error estimation strategy based on grid refinement (‘h-h/2
estimate’). The results presented in Chapter 6 show that this method
of error estimation is very successful and robust w.r.t. singularities of
the first or second kind.
• Chapter 7 is more theoretically inclined; here we have tried to prove
the ‘optimal preconditioning property’ observed for the midpoint rule
in Chapter 5, for a class of scalar model problems (cf. Example 1).
Here, only a partial theoretical explanation could be given.
• Finally, in Chapter 8 we report results obtained for collocation based on
Lobatto nodes (which are used in the standard MATLAB code bvp4c).
6
Our results show that this method cannot be successfully applied in
the presence of an essential singularity.
Our conclusion is that collocation methods as implemented in sbvp should
be seriously considered as candidates for numerically solving BVPs with an
essential singularity. Concerning a posteriori error estimation and mesh selection, strategies based on the auxiliary midpoint rule and h-h/2 refinement
should be included as options, where the latter is the most expensive but, at
the same time, seems to be the most reliable alternative.
7
Chapter 2
Theoretical foundations
2.1
Problem class
Consider the following boundary value problem for a system of ordinary
differential equations:
tα z 0 (t) = f (t, z(t)),
b(z(0), z(1)) = 0,
0 < t ≤ 1,
(2.1a)
(2.1b)
where α ≥ 1 and where z is required to be continuous, z ∈ C[0, 1] ∩ C 1 (0, 1].
The functions f and b are continuous nonlinear mappings on appropriate
domains. When α = 1 the problem (2.1) is said to have a singularity of
the first kind, while a singularity is of the second kind (or essential ) for α > 1.
More generally, we discuss nonlinear systems of the form
T (t)z 0 (t) = f (t, z(t)),
z ∈ C[0, 1] ∩ C 1 (0, 1],
b(z(0), z(1)) = 0,
0 < t ≤ 1,
(2.2a)
(2.2b)
(2.2c)
where
T (t) := diag(tα1 I1 , tα2 I2 , . . . , tαr Ir ),
and the Ik are unit matrices with either αk ≥ 1 for 1 ≤ k ≤ r, or αk ≥ 1 for
1 ≤ k ≤ r − 1 and αr = 0.
As a first step in the analysis of (2.2) we examine the solution structure of
linear systems
T (t)z 0 (t) = (M + A(t))z(t) + g(t),
8
0 < t ≤ 1,
(2.3a)
z ∈ C[0, 1] ∩ C 1 (0, 1],
B0 z(0) + B1 z(1) = γ,
(2.3b)
(2.3c)
where the matrix M ∈ Rn×n is a block upper

M11 M12 . . .
 0 M22 . . .

M :=  ..
..
 .
.
...
0
0 ...
triangular matrix,

M1r
M2r 

..  ,
. 
Mrr
(2.4)
and A, g ∈ C[0, 1]. Each matrix Mkk is a square matrix of the same size as Ik
which is assumed to be nonsingular when αk 6= 0 and has no eigenvalues that
are purely imaginary. When αr = 0 then Mrr = 0. Moreover, B0 , B1 ∈ Rp×n
are constant matrices and γ ∈ Rp is a constant vector. In general, p ≤ n.
2.2
Analytical properties
The analytical properties of (2.2) and (2.3) have been discussed in full detail
in [HW80b] and [HW79]. In these papers the Fredholm theory for linear
systems has been established and existence and smoothness results for nonlinear problems have been provided. In this section we recapitulate the most
important of these fundamental results.
2.2.1
Linear systems
We first examine
tα z 0 (t) = M z(t) + tα−ρ g(t),
0 < t ≤ 1,
α ≥ 1,
(2.5)
where M is an n×n matrix whose eigenvalues λj satisfy Re λj 6= 0, j = 1 . . . n.
The general solution of (2.5) reads
Z t
z(t) = Z(t)z(δ) + Z(t)
Z(s)−1 s−ρ g(s) ds,
(2.6)
δ
where
½
Z(t) =
exp[M (δ 1−α − t1−α )/(α − 1)],
(t/δ)M = exp [log(t/δ)M ],
α 6= 1,
α = 1,
is the fundamental solution matrix satisfying
tα Z 0 (t) − M Z(t) = 0,
0 < t ≤ 1,
9
Z(δ) = I,
(2.7)
for 0 < δ ≤ 1. Let
1
Q :=
2πi
1
P :=
2πi
Z
(λI − M )−1 dλ,
Γ−
(2.8)
Z
(λI − M )−1 dλ,
Γ+
where Γ− and Γ+ denote closed contours (oriented canonically) in the left
and right complex halfplane, respectively, such that each eigenvalue of M
is enclosed by either Γ− or Γ+ . The matrices Q and P are projections onto
the invariant subspaces of M associated with the eigenvalues having negative
and positive real parts, respectively.
Via the Jordan decomposition of M it is straightforward to obtain an explicit
representation of Z(t). This representation immediately yields
Lemma 2.2.1 Let α ≥ 1. Then for an arbitrary vector η, Zη ∈ C[0, 1] iff
Qη = 0.
For α > 1 we define1
 ρ−α
Rt
t Z(t) 0 QZ −1 (s)s−ρ g(s) ds


Rt

+ tρ−α Z(t) δ P Z −1 (s)s−ρ g(s) ds, 0 < t ≤ 1,
(Bρ g)(t) =



−M −1 g(0),
t = 0.
(2.9)
Concerning continuous solutions of (2.5) we have
Lemma 2.2.2 Let α > 1, ρ ≤ α and g ∈ C[0, 1]. Then, every z ∈ C[0, 1] ∩
C 1 (0, 1] which satisfies (2.5) has the form
z(t) = Z(t)P z(δ) + tα−ρ (Bρ g)(t).
(2.10)
Clearly, this means that the rank[Q] linearly independent conditions given by
Qz(0) = 0 have to be satisfied for z to be continuous, z ∈ C[0, 1]. Moreover,
we need to pose rank[P ] additional conditions at t = 1, P z(1) = P η, η ∈ Rn ,
for the uniqueness of z.
As a next step in the analysis we examine the system
T (t)z 0 (t) = M z(t) + g(t),
1
0 < t ≤ 1,
(2.11)
The analysis for systems with a singularity of the first kind, α = 1, can be found in
[HW76] and [HW78].
10
with matrices T (t) and M defined in §2.1. Let
M = D + U,
D = diag(M11 , M22 , . . . , Mrr ),
Z(t) = diag(Z1 (t), . . . , Zr (t)),
P = diag(P1 , . . . , Pr ),
Q = diag(Q1 , . . . , Qr )
and
½
R := I − P − Q =
where
½
Zk (t) =
diag(0, . . . , 0, Ir ), αr = 0,
0,
αr =
6 0,
exp [Mkk (δ 1−αk − t1−αk )/(αk − 1)], αk 6= 1,
exp [log(t/δ)Mkk ],
αk = 1,
(2.12)
and Pk , Qk are defined by (2.8) with M replaced by Mkk if αk 6= 0 and
Pk = Qk = 0 if αk = 0. In addition, define for 0 < δ ≤ 1
Z t
(B g)(t) = Z(t)
QZ −1 (s)T −1 (s)g(s) ds
0
Z t
Z t
−1
−1
+ Z(t)
P Z (s)T (s)g(s) ds + R
g(s) ds. (2.13)
δ
δ
Lemma 2.2.3 Let g ∈ C[0, 1]. Then any continuous solution of (2.11) must
satisfy
z(t) = Z(t)[P z(δ) + Rz(δ)] + (B[U z + g])(t)
= Z(t)(P + R)η + (B[U z + g])(t),
(2.14)
where η = (P + R)z(δ).
The most general linear equation that we shall consider is
T (t)z 0 (t) = (M + A(t))z(t) + g(t),
0 < t ≤ 1, z ∈ C[0, 1] ∩ C 1 (0, 1], (2.15)
where A, g ∈ C[0, 1] and
(I − R)A(0) = 0.
(2.16)
In order to show the existence and uniqueness of the solution of (2.15) a
contraction argument is used. Consequently, the following result holds.
Let p = rank[P + R], W be an n × p matrix which consists of linearly
independent columns of (P + R), and define
X(t) := Y (t)W,
11
where Y (t) is the unique solution of
T (t)Y 0 (t) = (M + A(t))Y (t), 0 < t ≤ 1, Y (t) ∈ C[0, 1] ∩ C 1 (0, 1],
RY (δ) = R,
P Y (δ) = P.
In addition, let ỹ be the unique particular solution of (2.15) subject to the
boundary conditions
Rỹ(δ) = P ỹ(δ) = 0.
Then we have
Theorem 2.2.1 Any solution of (2.15) has the form
z(t) = X(t)β + ỹ(t)
(2.17)
with a unique β ∈ Rp .
We now answer the question under which circumstances the solution of (2.15)
given by (2.17) satisfies the boundary conditions
B0 z(0) + B1 z(1) = γ.
(2.18)
Our aim is to establish conditions on B0 and B1 which lead to a Fredholm
alternative for (2.15) and (2.18), cf. (2.3c). To do this it is convenient to
introduce the differential expression
l(z) = T z 0 − (M + A)z
and associate with it the operator defined by
Lz = l(z)
for z ∈ D = {z ∈ C[0, 1], T z 0 ∈ C[0, 1], B0 z(0) + B1 z(1) = 0}. Then we have
Theorem 2.2.2 If
rank [B0 , B1 ] = k,
(2.19)
then L is Fredholm with index p − k. Furthermore, if L−1 exists, it is bounded.
In applications we are primarily interested in the case when L is Fredholm
with index zero. We therefore assume that B0 , B1 are p × n matrices, γ is a
p-vector and that (2.19) holds with k = p. On substitution of (2.17) in (2.18)
we find
Theorem 2.2.3 The problem (2.3) has a unique solution for all g ∈ C[0, 1]
and γ ∈ Rp iff the p × p matrix [B0 X(0) + B1 X(1)] is nonsingular.
12
Remark 1. The boundary conditions (2.3c) are equivalent to
B0 Rz(0) + B1 z(1) = γ̃
(2.20)
where
γ̃ = γ + B0 (M + R)−1 (I − R)g(0).
It turns out that the form (2.20) is advantageous for some numerical schemes
applied to the boundary value problem (2.3).
Remark 2. Moreover, it can be shown that the restriction that the solution
be continuous at t = 0 is not useful when constructing numerical schemes.
It turns out that the relation
Qz(0) = Q(M + R)−1 ((R − I)g(0) + Rz(0)))
(2.21)
is more satisfactory.
Equations (2.21) and (2.20) are the n linearly independent boundary
conditions which must be employed when the problem (2.3) is discretized
by a difference scheme. This and other related questions are discussed in
[HW79].
2.2.2
Nonlinear case
We now formulate smoothness results for nonlinear problems (2.2),
T (t)z 0 (t) = f (t, z(t)),
z ∈ C[0, 1] ∩ C 1 (0, 1],
b(z(0), z(1)) = 0.
0 < t ≤ 1,
We first make a number of assumptions.
(N1) Problem (2.2) has a solution z(t).
With this solution and some ρ > 0 we associate the spheres
sρ (z(t)) := {y ∈ Rn , |z(t) − y| ≤ ρ}
and the tube
Tρ := {(t, y), t ∈ [0, 1], y ∈ sρ (z(t))}.
13
(N2) For some ρ > 0, f (t, z(t)) is continuously differentiable with respect to
z, and ∂f
(t, z) is continuous on Tρ .
∂z
(N3) For all y ∈ sρ (z(0)), the matrix
M (y) :=
∂f
(0, y)
∂y
has the fixed block upper triangular structure introduced in (2.4). In
addition, the matrix
M := M (z(0))
satisfies all the conditions concerning the matrix M introduced in §2.1.
(N4) b(v, w) is a vector-valued function of dimension
p := rank[P + R]
which is continuously differentiable on sρ (z(0)) × sρ (z(1)).
(N5) The solution z(t) is isolated. This means that the linearized problem
T (t)u0 (t) − G(t)u(t) = 0,
u ∈ C[0, 1] ∩ C 1 (0, 1],
B0 u(0) + B1 u(1) = 0
with
G(t) =
∂f (t, z(t))
,
∂y
B0 =
∂b(z(0), z(1))
,
∂v
B1 =
∂b(z(0), z(1))
∂w
has only the trivial solution.
The requirement that the solution of (2.2) be continuous at t = 0 obviously
imposes the following restriction on the solution:
(I − R)f (0, z(0)) = 0.
(2.23)
(N3) implies that (2.23) can be inverted locally for (I − R)z(0), i.e.
(I − R)z(0) = ϕ(Rz(0)),
(2.24)
z(0) = ϕ(Rz(0)) + Rz(0) =: ψ(Rz(0)).
(2.25)
whence
14
This result is of value since it can be used to modify the boundary conditions
(2.2c). We substitute (2.25) into (2.2c) and obtain
b(ψ(Rz(0)), z(1)) = 0
, which involves Rz(0) rather than z(0). Hence, without loss of generality,
we may consider boundary conditions of the form
b(Rz(0), z(1)) = 0.
In the sequel, use will be made of the derivative of ψ(Rz) with respect to z.
Substitution of (2.24) into (2.23), differentiation with respect to Rz(0), and
use of (2.25) yield
∂ψ
= (M (ψ(Rz)) + R)−1 R.
∂z
Equation (2.23) immediately yields
Lemma 2.2.4 Let z satisfy (2.2a) and (2.2b). Then T z 0 ∈ C[0, 1] and
lim+ T (t)z 0 (t) = Rz 0 (0).
t→0
This lemma says that any component zi of z in a block associated with αk ≥ 1
satisfies limt→0+ tαk zi0 (t) = 0, while the components (if any) associated with
αr = 0 are in C 1 [0, 1]. This smoothness result can be extended if further
restrictions are imposed on the problem.
Lemma 2.2.5 Assume that
(i) all αk are integers,
(ii) the real parts of the eigenvalues of the matrix Mkk are negative whenever
αk = 1,
(iii) f ∈ C m (Tρ ).
Then
(i) z ∈ C m [0, 1] ∩ C m+1 (0, 1],
(ii) when αr 6= 0, (i.e. R = 0), and
∂k f
(0, z(0))
∂tk
z k (0) = 0,
= 0, k = 0, . . . , m, then
k = 0, . . . , m.
15
The situation m = ∞ is the one most frequently encountered in practice.
Remark. We now examine the structure of the solution when (2.2) is linear,
i.e.
T (t)z 0 (t) = (M + A(t))z(t) + g(t),
z ∈ C[0, 1] ∩ C 1 (0, 1],
B0 Rz(0) + B1 z(1) = γ,
0 < t ≤ 1,
see §2.1.
Let us denote by ỹ the unique particular solution of
T (t)ỹ 0 (t) = (M + A(t))ỹ(t) + g(t),
0 < t ≤ 1,
ỹ(t) ∈ C[0, 1] ∩ C 1 (0, 1],
subject to boundary conditions
Rỹ(0) = P ỹ(δ) = 0,
and let Y (t) satisfy
T (t)Y 0 (t) = (M + A(t))Y (t), 0 < t ≤ 1, Y (t) ∈ C[0, 1] ∩ C 1 (0, 1],
RY (0) = R,
P Y (δ) = P.
Here, Theorem 2.2.3 can be reformulated in the following way:
Theorem 2.2.4 The problem (2.3) has a unique solution for all g ∈ C[0, 1]
and γ̃ ∈ Rp iff the p × p matrix [B0 R + B1 Y (1)]W is nonsingular. The
solution in this case reads
y(t) = ỹ(t) + Y (t)W ([B0 R + B1 Y (1)]W )−1 (γ − B1 ỹ(1)).
16
2.3
Numerical approach
When solving singular problems with a singularity of the first kind, collocation methods work satisfactorily and provide a dependable high order
approximation for the solution of the underlying analytical problem. This
is not the case in general, when other standard high order methods are
applied to solve singular problems. Many of those methods suffer from
order reductions and become inefficient, see [HW77] for the discussion of
multistep methods or [HW85] for the analysis of explicit Runge-Kutta
schemes. Collocation methods show advantageous convergence properties,
see [HW78] and [Wei86] for the investigation of first and the second order
singular systems (with a singularity of the first kind), respectively.
The aim of this section is to recapitulate the convergence results for
collocation methods applied to regular problems and to a certain class of
singular problems. We also present the implementation of collocation in the
MATLAB code sbvp.
2.3.1
Collocation methods for regular problems
We consider the problem (2.1) and set α = 0,
z 0 (t) = f (t, z(t)),
b(z(0), z(1)) = 0.
0 < t ≤ 1,
(2.26a)
(2.26b)
For the numerical analysis we introduce the mesh
∆ = (τ0 , τ1 , . . . , τN ),
with τ0 = 0, τN = 1, hi := τi+1 − τi and h := max0≤i≤N hi . We denote the
corresponding grid vectors by
u∆ = (u0 , u1 , . . . , uN ) ∈ R(N+1)n .
The norm on the space of grid vectors is given by
ku∆ k∆ = max |uk |.
0≤k≤N
For a continuous function x ∈ C[0, 1], we denote by R∆ the pointwise projection onto the space of grid vectors,
R∆ (x) = (x(τ0 ), x(τ1 ), . . . , x(τN )).
17
For collocation, m equidistantly spaced points are inserted in each subinterval
Ji = [τi , τi+1 ]. This yields the (fine) grid2
∆m := {ti,j : ti,j = τi + jδi , i = 0, . . . , N − 1, j = 0, . . . , m + 1},
(2.27)
where
hi
.
m+1
We seek to approximate z by a collocating function p(t) := pi (t), t ∈ Ji ,
where pi is a polynomial of degree ≤ m satisfying
δi :=
p0i (ti,j ) = f (ti,j , pi (ti,j )), i = 0, . . . , N − 1, j = 1, . . . , m, (2.28a)
pi (τi ) = pi−1 (τi ),
i = 1, . . . , N − 1,
(2.28b)
b(p0 (0), pN −1 (1)) = 0.
(2.28c)
The degree m of the polynomial function is sometimes referred to as the stage
order (Runge-Kutta terminology). For any collocation method with the stage
order m, the convergence order is at least m. For a suitable (non-equidistant!)
choice of the collocation nodes ti,j , j = 1, . . . , m,, the convergence order of
the error in the mesh points, τk , 0 ≤ k ≤ N , can be improved (up to 2m for
Gaussian nodes, cf. [BS73]). This effect is known as superconvergence.
For the regular boundary value problem (2.26) the following convergence
result holds (see [AMR88] and [AKW02]):
Theorem 2.3.1 Let z(t) be an isolated solution of (2.26) with sufficiently
smooth data. Then for any collocation scheme of the form (2.28), there exist
constants ρ, h0 > 0 such that the following statements hold for all meshes ∆
with h ≤ h0 :
(i) There exists a unique solution p(t) of (2.28) in a tube of radius ρ around
z(t).
(ii) This solution can be computed by Newton’s method which converges
quadratically provided that the initial guess p[0] (t) is sufficiently close
to z(t).
(iii) The following error estimates hold:
kR∆ (p) − R∆ (z)k∆ = O(hm+ν ),
kp − zk∞ = O(hm+ν ),
kp(l) − z (l) k∞ = O(hm+1−l ), l = 1, . . . , m,
where ν = 0 if m is even and ν = 1 if m is odd.
2
For convenience, we denote τi by ti,0 ≡ ti−1,m+1 , i = 1, . . . , N .
18
(2.29a)
(2.29b)
(2.29c)
2.3.2
Collocation for α = 1
The convergence theory for the above collocation methods applied to a class
of linear singular boundary value problems of the form
M (t)
z(t) + g(t),
t
B0 z(0) + B1 z(1) = γ,
z 0 (t) =
0 < t ≤ 1,
(2.30a)
(2.30b)
where g ∈ C[0, 1], was presented in [HW78].
According to the theoretical and numerical results presented in [AKW02]
and [HW78] the situation in the singular case is more or less the same as in
the regular case. Theorem 2.3.1 also holds, but for a limited class of singular
problems. This means that certain restrictions have to be imposed, namely,
the real parts of the eigenvalues of M (0) are assumed to be nonpositive3 .
Furthermore, in the case of singular systems the superconvergence behavior
cannot be expected in general. In applications, where M (0) is diagonalizable,
the stage order hm can be uniformly improved by at least one power of h. In
the general case however, the convergence order may be negatively affected
by logarithmic terms occurring in the estimates (2.28) when m is odd (see
[HW78]).
Collocation methods were successfully implemented in standard codes
designed to solve regular boundary value problems. The best known are
the FORTRAN 90 code COLNEW, see [ACR78], [ACR81], and the standard
MATLAB code bvp4c. The applicability of these codes in the singular case
is somewhat limited, see [AKKW].
Recently, another MATLAB solver called sbvp4 has been developed at the
Department of Applied Mathematics and Numerical Analysis, Vienna University of Technology. This code is based on collocation, and its purpose is
to approximate solutions of nonlinear singular problems,
M (t)
z(t) + g(t, z(t)),
t−a
b(z(a), z(b)) = 0.
z 0 (t) =
a < t ≤ b,
(2.31a)
(2.31b)
The grid selection strategy is based on an estimate for the global error of the
collocation solution. The original idea for this error estimate was presented
3
The results for a general spectrum of M(0) in the context of second order systems can
be found in [Wei86].
4
The package is freely available from http : //www.math.tuwien.ac.at/∼ewa.
19
in [Ste78], see also [Zad76], and is closely related to the principle of Defect
Correction. In the current version of sbvp, a modified defect definition (see
[AKW02],[AKKW02]) is used. The advantage of this modification is to provide an asymptotically correct a posteriori error estimate for all grid points,
including collocation points5 . The proof of the asymptotical correctness of
the error estimate for the regular case was given in [AKW02], and this proof
was recently extended to the singular case where α = 1 in [AKW].
2.3.3
Collocation for α > 1
Analytical properties of singular systems with a singularity of the second
kind and convergence results for finite difference schemes in the context of
these problems have been discussed in [HW80b] and [HW79], respectively.
Theoretical results on the convergence properties for collocation schemes
applied to such problems are not yet available. This and the fact that
collocation works satisfactorily for problems with a singularity of the first
kind was the motivation for the present work.
The main goal was to carry out comprehensive numerical investigations in
order to complete, at least experimentally, the picture of the properties of
collocation methods when applied to problems with an essential singularity.
Moreover, we wanted to find out why the error estimate implemented in sbvp
fails in this case. A future goal is to develop a new module for our code which
would be suitable for the treatment of nonlinear boundary value problems
with an essential singularity, including an asymptotically correct a posteriori
error estimate as the basis for mesh adaptation.
5
The original version works satisfactory only in the mesh points.
20
Chapter 3
Test problems
In this section we list the model problems used in the numerical experiments.
• Example 1: We first consider the linear scalar problem (terminal value
problem)
1
et
t
z(t)
+
e
−
,
tα
tα
z(1) = e,
z 0 (t) =
(3.1a)
(3.1b)
with the exact solution z(t) = et . Here, M = 1, A(t) = 0 and T (t) = tα ,
which means that the only eigenvalue of M is positive. Consequently,
one condition, (3.1b), has to be posed in order to obtain a unique
solution of (3.1). This example can be found in [HW79].
• Example 2: Next we have a linear system


0 −1 0
0
1  0 0 −1 0 
 z(t)
z 0 (t) = β+2 
 0 0
0 −1 
t
ρ 0
0
0

0
0
0
0
β+1
1 
0
βt
0
0
+ β+2 
β+1

0
0
2βt
0
t
d(1/t) − ρ
0
0
3βtβ+1
µ
µ
4 −2
0
−2
2 −1
4
2
2
2
1
0
0 −1
1
0
¶
µ
z(0) =
¶
µ
z(1) =
21
0
0
1
1
(3.2a)


 z(t),

¶
,
(3.2b)
,
(3.2c)
¶
cf. [HW80b]. We choose the function d(t) = 1 and the constant ρ = 1.
In this case


0 −1 0
0
 0 0 −1 0 
,
M =
 0 0
0 −1 
1 0
0
0
r = 1 and T (t) = tβ+2 I1 , where I1 ∈ R× and β > 0. The eigenvalues
of M are
√
2
(λ1 , λ2 , λ3 , λ4 ) =
(1 + i, 1 − i, −1 + i, −1 − i),
2
and this means that according to the theory two initial conditions,
(3.2b), equivalent to Qz(0) = 0 are necessary for the solution to be continuous. The remaining two conditions, (3.2c), now yield the uniqueness of z. The exact solution of this problem is not known.
• Example 3: This problem is nonlinear,

− 1t (z1 (t) + z2 (t))

− 1t (2z2 (t) + z3 (t))
z 0 (t) = 
 13 z3 (t) + 12 z3 (t)z4 (t) + 1 z1 (t)z3 (t) − 3 z3 (t)
2t
2t
2t
t
0
¶
µ
¶
µ
0
1 0 0 0
z(0) =
,
0
0 1 0 0
µ
¶
µ
¶
1 0 0 1
−1
z(1) =
,
0 1 0 0
−1


,

(3.3a)
(3.3b)
(3.3c)
see [HW79]. Here, r = 3 and T (t) = diag(tI1 , t3 I2 , I3 ), where I1 ∈ R×
and I2 = I3 = 1. In the case of a nonlinear problem we need to
investigate the structure of fz (t, z(t)), where


−(z1 (t) + z2 (t))


−(2z2 (t) + z3 (t))

f (t, z(t)) = 
2
1
t
t
2
 z3 (t) + z3 (t)z4 (t) + z1 (t)z3 (t) − 3t z3 (t)  .
2
2
2
0
Consequently,

 
0 0
 
0 0
−2 −1 0  
+

0 1/2 0 
 
 0 0
0
0 0
0 0
−1 −1

 0
fz (t, z(t)) = 
 0

0
0
0
22
tz4 (t)
2
+
0
0
0
0
t2 z1 (t)
2
0
− 3t2
tz3 (t)
2
0




.


The eigenvalues of M are
1
(λ1 , λ2 , λ3 , λ4 ) = (−1, −2, , 0)
2
and therefore two initial conditions are necessary for the solution to be
continuous. For this problem the exact solution is not known either.
• Example 4: We consider the nonlinear problem


− tc2 z2 (t)
 −( c2 + 2 )z1 (t) + c2 z12 (t) + 2 z2 (t) 
t
c
t
t

z 0 (t) = 
 − 4 z3 (t) + c(1 − ctz4 (t))z4 (t)  ,
t
cz3 (t)
µ
¶
µ ¶
1 1 0 0
0
z(0) =
,
0 0 1 0
0
µ
¶
µ ¶
0 1 −c −1
0
z(1) =
.
1 0
0 −c
0
for a c ∈ R+ . This problem was discussed in [HW79].
For T (t) = diag(t2 I1 , tI2 , I3 ), with I1 ∈ R× and I2 ,
structure of the right-hand side is

−cz2 (t)
 −(c + 2t2 )z1 (t) + cz 2 (t) + 2tz2 (t)
1
c
f (t, z(t)) = 
 −4z3 (t) + ct(1 − ctz4 (t))z4 (t)
cz3 (t)
(3.4a)
(3.4b)
(3.4c)
I3 ∈ R, the


,

and therefore,
fz (t, z(t)) := M + A(t)

0 −c 0 0

0 0
 −c 0
=
 0
0 −4 0

0
0
0


0
0
0
0
0
0
c
0
 
2
 
0
− 2tc + 2cz1 (t) 2t 0

+
 
0
0 0 ct − 2c2 t2 z4 (t)
 
0
The eigenvalues of M read
(λ1 , λ2 , λ3 , λ4 ) = (c, −c, −4, 0).
On noting that P + Q = diag(I4 , 0), where I4 ∈ R× , it follows
immediately that the necessary condition for (3.4) to be well-posed,
(I − R)A(0) = 0, is satisfied if z1 (0) = 0, cf. (2.16).
The exact solution of (3.4) is again not known.
23




.


• Example 5: This problem is linear, cf. [HW80a],
µ
¶
1
0 −1
0
z (t) = 2
z(t),
−1 0
t
¡
¢
1 −2 z(0) = 0,
¡
¢
1 0 z(1) = 1.
(3.5a)
(3.5b)
(3.5c)
The exact solution is given by
z(t) = (z1 (t), z2 (t))T = (e
The matrix M ,
µ
M=
t−1
t
0 −1
−1 0
, −e
t−1
t
)T .
¶
has two nonzero eigenvalues
(λ1 , λ2 ) = (−1, 1),
and T (t) = t2 I1 where I1 ∈ R× .
• Example 6: The next problem,
¶µ
µ t ¶¶ µ
¶
µ
1
2e
2et
0 −1
0
z(t)−
+
, (3.6a)
z (t) = α
e−t
−e−t
−1 0
t
¡
¢
1 −2 z(0) = 0,
(3.6b)
¡
¢
1 0 z(1) = 2e,
(3.6c)
is a modification of Example 5 with the exact solution
z(t) = (z1 (t), z2 (t))T = (2et , e−t )T
and unchanged matrix M , and T (t) = tα I1 , I1 ∈ R2×2 .
• Example 7: The problem

z 0 (t) =
µ
µ
1 0
0 1
0 0
0 0
−z2 (t)

1
−z3 (t)

2
−z4 (t)
t
1 − e−z1 (t)/2
¶
µ
0 0
z(0) =
0 0
¶
µ
1 0
z(1) =
0 1
24


,

0
0
0
1
(3.7a)
¶
,
(3.7b)
.
(3.7c)
¶
can be found in [MR83]. This model is nonlinear, T (t) = t2 I1 , where
I1 ∈ R× and


0 −1 0
0


0 −1 0 
 0

.
M =
0
0 −1 

 0
1/2
0
0
0
The eigenvalues of M are
√
4
(λ1 , λ2 , λ3 , λ4 ) =
2
(1 + i, 1 − i, −1 + i, −1 − i).
2
The exact solution is unknown.
In order to use sbvp to solve the model problems numerically the boundary
conditions stated above in two sets need to be rewritten as a (formally)
coupled system of dimension n.
25
Chapter 4
Numerical tests based on the
collocation solver sbvpcol
In this section we will present various numerical results which have been
obtained in order to check the hypothesis that the convergence behavior of
collocation methods as stated in Theorem 2.3.1 is also valid for BVPs with
a singularity of the second kind.
Our investigations were based on the code sbvpcol, the collocation solver
underlying sbvp. In order to be able to call this routine we have to choose a
few options such as:
• ‘tau 0’ . . . given mesh
• ‘ColPts’ . . . specifies the distribution of collocation points, with two
available options which have been used in our investigations: equidistant
and gauss.
• ‘Basis’ . . . choice of basis for the internal representation of collocation
polynomials. Default is RungeKutta.
• ‘Degree’ . . . highest degree of basis polynomials. In the sequel we
denote this by m.
Any other unspecified options that concern the solution process are set to
default.
4.1
Convergence results for the global error
In Tables 1.0–9.3, the convergence results for collocation methods are given.
By err we denote the maximal global error, i.e. the difference between the
26
numerical solution obtained by sbvpcol and the exact solution, measured
in the discrete maximum norm, for those cases where the exact solution is
explicitly known (Examples 1, 5 and 6).
In all other examples, the unknown exact solution is replaced by a proper
approximation, namely a reference solution obtained by means of sbvpcol
using a very small step size.
Furthermore, p and const represent the empirical convergence rate and
error constant, respectively.
Note that for all problems tested, the convergence orders at the mesh points
(mesh) and at the collocation plus mesh points (coll) were both observed.
Now, we give a comprehensive overview of the numerical experiments.
• Example 1 ( (3.1) on page 21 )
This is a simple one-dimensional problem with a smooth solution.
We have tested three different types of singularity (α = 1, 2, 3, see
Tables 1.1–1.3), and also a regular version (α = 0, see Table 1.0).
For α = 1 this is a problem with a singularity of the first kind, for
which the code sbvp has already proved its efficiency (see [AKKW02]).
The other two choices, α = 2, 3, are crucial for our present study.
As it can be seen they show a slightly less advantageous behavior as
compared to the case α = 1.
In particular, superconvergence of Gauss methods at the mesh points
is not observed in the singular case, and the uniform convergence order
seems to be slightly deteriorating with increasing α.
For equidistant nodes, the uniform convergence order m (= stage order)
is observed throughout7 .
• Example 2 ( (3.2) on page 21 )
For this linear 4-dimensional problem we also consider three cases,
namely β = 0, 1, 2, each of them corresponding to a singularity of the
second kind. See Tables 2.1–2.3 and 2.7–2.9.
7
We have also performed some experiments with a (numerically) less smooth solution,
g(t) = sin(t10 ) cos(15t) instead of g(t) = et , for the analogous problem
z 0 (t) =
1
(z(t) − g(t)) + g 0 (t), z(1) = g(1),
tα
see Tables 8.1–8.3. In this case, high orders are better visible because the derivatives of
g(t) are larger in size, hence the global error level is also larger, and rounding errors do not
spoil the observable orders. Note that for this problems, the classical (super-) convergence
orders are observed for Gaussian points. This may possibly be attributed to the fact that
z(0) = z 0 (0) = . . . = z (10) (0) = 0.
27
Here, in all three cases considered, the superconvergence order of collocation at Gaussian points is approximately attained at the mesh points.
On the equidistant grid, the stage order m is obtained.
• Example 3 ( (3.3) on page 22 )
Here, no order reductions are encountered; see Tables 3.1 and 3.3.
For this example, sbvpcol reports a message about the maximum number of function evaluations being exceeded9 . This is probably caused
by convergence problems of the Newton iteration which is used to solve
the system of collocation equations.
• Example 4 ( (3.4) on page 23 )
According to [HW79], we can expect convergence only for a suitable
choice of the starting iterate. We have used the initial approximation
·µ
¶
µ
¶
¸
tg2 (− ct + 1) − c
tg2 − c
y0 =
g1 +
e t ; − g1 +
e t; 0 ; d
c
c
where (d, c) = (2, 1.5), and g1 , g2 are chosen as g1 = cec , g2 =
2 ec
− (d+1)c
.
c+1
Here, the results for the convergence order are not fully satisfactory,
in particular for the case of Gaussian collocation with m = 2 (see Tables 4.1 and 4.3).
Equidistant collocation performs satisfactorily for m = 2, 4, with a convergence order m. However, for m = 6 only an order 4 is observed (Table 4.3). The fact that no convergence order higher than 4 is observed
for any of the collocation methods applied to this problem suggests a
possible explanation for the order reductions. If the exact solution of
this test problem (which is unknown, unfortunately) is not sufficiently
smooth, we cannot expect an arbitrary convergence order for collocation schemes. Thus we conjecture that the order reductions down to
order 4 for Example 4 are caused by the fact that the exact solution
z(t) of (3.4) is no more than five times continuously differentiable.
In any case, global errors smaller than ≈ 1e−8 are not observed.
This effect may also be caused by an inappropriate choice of the reference solution (obtained with h = 1.5626e−03). However, it is not easy
to improve this reference solution.
In this example, difficulties in the Newton solution process were also
reported by sbvpcol (similarly as in Example 3).
9
‘MaxFunEvals’= 80000,‘MaxIter’= 50000, ‘TolX’= 1e−6
28
• Example 5 ( (3.5) on page 23 )
For this two-dimensional example the exact solution is explicitly known.
The test results are presented in Tables 5.1 and 5.5.
In all cases the full conventional orders are observed, including superconvergence at the meshpoints for Gaussian collocation.
• Example 6 ( (3.6) on page 24 )
According to our numerical results, see Tables 6.1–6.3 and 6.17–6.19
for three versions (α = 1, 2, 3), we can affirm the same behavior as for
Example 1. A convergence order higher than m + 1 is not observed,
and for larger values of α an order reduction becomes visible.
• Example 7 ( (3.7) on page 24 )
For this example, the same conclusions about the convergence order
as for Example 5 may be drawn (see Tables 7.1 and 7.3). (Note that
these results were obtained with the help of a reference solution with
h = 3.125e−03.)
Summarizing these results, we state that the stage order is obtained in all
cases. As in the case of a singularity of the first kind (α = 1), superconvergence of Gauss methods cannot be observed except in some special cases.
29
4.2
Convergence results for first and second
derivatives
For collocation applied to regular boundary value problems it is well known
that the first derivative is approximated (uniformly in t) with the same
asymptotic quality as the solution itself. For higher derivatives, the order of
this approximation gradually decreases (cf. e.g. Theorem 2.3.1 or [AMR88]).
For equidistant collocation nodes, for instance, we have
(k)
pcoll (t) − z (k) (t) = O(hm+1−k )
for k = 1, . . . , m, where m is the stage order of the method.
In this section, we investigate the approximation of the first and second
derivatives by the collocation method implemented in sbvp when applied to
problems with an essential singularity. This is motivated by the important
role which the approximation quality of the derivatives plays in the theory
of error estimates (cf. e.g. [AKW02]).
For some of our examples this numerical investigation is difficult or impossible. In particular, for the case where the exact solution is not known
(Examples (3.2), (3.3), (3.4), (3.7)), it was not possible to generate a reference solution sufficiently accurate for this purpose. Therefore these experiments were only performed for Examples 3.1, 3.5 and 3.6 (with exact
solution known). For these examples, the derivative of the global error was
obtained as the difference between the derivative of the polynomial (collocation) approximation, obtained using the MATLAB function polyder, and the
derivative of the exact solution.
• Example 1 ( (3.1) on page 21 )
Here, also for higher values of the parameter α, the asymptotic approximation quality of the first and second derivatives is nearly as good as
in the regular case α = 0, namely O(hm ) and O(hm−1 ), respectively;
see Tables 1.4–1.11. Slight order reductions which might(?) occur depending on α > 1 are hard to diagnose numerically.
Note that at the mesh points the polynomial from the interval left to
this point has been used.
• Example 5 ((3.5) on page 23)
For this 2-dimensional example, the results are very similar as for Example 1, see Tables 5.2–5.3.
• Example 6 ((3.6) on page 24)
30
Here, for higher values of α, the observed convergence orders are significantly smaller than m or m − 1, respectively, see Tables 6.4–6.11. For
α > 1, the amount of order reduction seems to be about O(1 − 1/α).
The conclusion is: An order reduction may occur in the approximation of
the derivatives which is, however, not severe for moderate values of α.
31
Chapter 5
A posteriori error estimate and
conditioning
Up to now we only have investigated the performance of the collocation solver
sbvpcol. Concerning the performance of the full code sbvp, however, another
important question is whether, in the case of an essential singularity, the a
posteriori error estimate implemented in sbvp produces an asymptotically
correct global error estimate.
Note that for regular problems or problems with a singularity of the
first kind, the asymptotic correctness of this estimate has been proven, see
[AKW02], [AKW]. For problems with an essential singularity, however, neither theoretical nor numerical results have been available so far.
For the algorithmic details of this estimate we refer to [AKW02]. Here we
only note that it is based on the idea of defect correction, and that a cheap
low order scheme (backward Euler) is applied twice to estimate the error of
sbvpcol.
As we shall see, this estimate does not work satisfactorily in the case of an
essential singularity. There is strong evidence that this is (at least partially)
caused by the ill-conditioning of the backward Euler equations in this case.
Therefore, we also pay attention to the conditioning of the various system
matrices involved, and we also shall try to apply preconditioning techniques
to improve conditioning where necessary.
32
5.1
A posteriori error estimate and conditioning in sbvp
5.1.1
Conditioning of the collocation equations
For solving a system of collocation equations, say F (p) = 0, sbvpcol applies
Newton’s method, where the Jacobian matrix DF corresponds to a linearized
version of the collocation equations. For numerical stability and for the
performance of Newton’s method, the condition number of DF ,
cond(DF ) = kDF k kDF −1 k,
is a crucial parameter.
In the sequel we will investigate these condition numbers and their asymptotical orders w.r.t. h, using the MATLAB functions cond and condest. The
best possible behavior we might expect is cond(DF ) = O(h−1 ), which holds
for a stable, correctly scaled discretization of a regular, first order boundary
value problem. (For a system of collocation equations, ‘correct scaling’ means
that the coefficients in the continuity and boundary conditions are scaled to
h−1 , which is the natural scaling occurring in the collocation equations; see
[AKKW02]).
Note that, by default, cond(A) returns the condition number of a matrix A
w.r.t. the 2-norm, while condest(A) computes a lower bound for the condition
number w.r.t. the 1-norm.
Our corresponding numerical results are discussed in the sequel.
• Example 1 ( (3.1) on page 21 )
The results given in Tables 1.12 and 1.13 show that the condition numbers of the matrices DF from sbvpcol have the optimal asymptotic
behavior O(h−1 ) for α = 0 and α = 1, For α > 1, the essential singularity causes them to decrease (approximately) like h−α , see Tables
1.14 and 1.15.
• Example 2 ( (3.2) on page 21 )
Also in this case, the order of cond(DF ) significantly decreases with
increasing β and is observed to be near −(β + 2), see Tables 2.4–2.6.
(Note that for all values β = 0, 1, 2, this is a problem with an essential
singularity.)
• Example 3 ( (3.3) on page 22 )
For this nonlinear example we observe cond(DF ) = O(h−3 ), which may
be expected due the t−3 -term present in the Jacobian, see Table 3.2.
33
• Example 4 ( (3.4) on page 23 )
We observe the same behavior as before: cond(DF ) ≈ h−α , where
α = 2 characterizes the degree of singularity; see Table 4.2.
• Example 5 ( (3.5) on page 23 )
Again we observe cond(DF ) ≈ h−α , with α = 2, see Table 5.4.
• Example 6 ( (3.6) on page 24 )
The results for this example can be found in Tables 6.12–6.15. The
influence of α on the condition number is again clearly visible.
These results can be interpreted in the following way: In all cases, the (negative) power of h which was observed in cond(DF ) corresponds to the highest
negative power, say −α, of t occurring in the right hand side of the corresponding singular problem, which implies kDF k = O(h−α ). But this means
that kDF −1 k = O(1), and in this sense the collocation methods applied to
these singular problems might be called stable, which is consistent with the
results from §4.1.
However, we have to be careful with this interpretation: Such a ‘stability
estimate’, even if it can be proven theoretically, does not yet imply that the
collocation method is convergent (with stage order m), as observed in §4.1.
Moreover, as we shall see in §5.3 for the case of the implicit midpoint rule,
there exist examples where the asymptotic conditioning can be improved by
a clever choice of a preconditioner.
5.1.2
A posteriori error estimates
We now describe our results concerning the performance of the a posteriori
error estimate implemented in sbvp (cf. [AKW02]) for problems with an
essential singularity.
An overview of these results can be seen from Figures 1.1, 2.1, . . . , 7.1
and 9.1, where the exact global error (‘exact’) is plotted together with its
estimate produced by the function sbvperr called by sbvp (a plot of the
solution is also provided) for default tolerances tol = 1e−12. In particular,
singularities of the first kind (where theory and numerical experience tells
us that sbvperr gives reliable results) are compared with essentially singular
situations.
• For Example 1, the corresponding results for α = 1, 2, 3 are displayed
in Figure 1.1.
34
In contrast to α = 1, the quality of the error estimate deteriorates near
the singular point for α = 2.
For α = 3, Figure 1.1 shows that the error estimate completely fails.
The results of the similar experiment done with a less smooth function
g(t) = cos(15t) are presented in Tables 9.1–9.3 and Figure 9.1.
• In Figure 2.1 we observe a similar effect for the essentially singular
Example 2, with β = 0, 1, 2.
• In Figures 3.1, 4.1, 5.1 and 7.1, plots of the error estimate provided by
sbvperr are presented for Examples 3, 4, 5 and 7. A similar, but less
drastic, behavior as before is observed.
• Figure 6.1 demonstrates a behavior of the error estimate for Example 6
similar to that presented in Figure 1.1.
These experimental results clearly show that the error estimate implemented
in sbvperr is useless for problems with an essential singularity.
This is possibly caused by the inability of the auxiliary scheme – the
backward Euler method – to cope with such a problem type. Therefore, in
the next section we consider the Euler scheme and investigate its convergence
and conditioning properties.
5.2
The backward Euler method
The backward Euler discretization of a boundary value problem for a first
order ODE system y 0 = F (t, y) is given by the simple difference scheme
ηi − ηi−1
= F (ti , ηi ),
hi
i = 1, . . . , N
(5.1)
on a grid {ti }, with step size hi = ti − ti−1 , plus boundary conditions. Its
classical convergence order is 1.
This can also be interpreted as a collocation method producing a
piecewise linear approximation on each interval [ti−1 , ti ]. Therefore, a slight
modification of the routine sbvpcol enabled us to perform experiments for
the backward Euler scheme, with a constant step size hi ≡ h. This is not
the most efficient way to implement (5.1) but it was convenient for a first test.
For a more thorough investigation of (5.1) we have concentrated on Example 1, with different choices of the parameter α. Although this is a simple
scalar terminal value problems, the above (negative!) results concerning error
estimation show that it is worthwhile to have a closer look at this example.
35
5.2.1
Backward Euler by means of sbvpcol
Tables 1.16 and 1.17 show numerical results for Example 1. For α = 0, 1, 2,
the conventional convergence order 1 is observed, but for α = 3 the method
is rapidly divergent (Table 1.16). This effect may be related to the illconditioning of the Euler system for α > 1, which can be clearly seen from
Table 1.17: While for α = 2 the ill-conditioning is present but not fatally
large, the condition number explodes for h → 0 in the case of α = 3.
Of course, a terminal value problem is a very special case of a boundary
value problem, and its numerical solution by a one-step scheme is a simple
iteration. Therefore the size of the condition number of the corresponding
bidiagonal (Euler) matrix may be not so significant as in the general case.
However, it is worthwhile to test preconditioning as a possible means to
improve the convergence properties.
5.2.2
An implementation of backward Euler with preconditioning
In sbvpcol, standard preconditioning is implemented for the continuity equations, which is also optimal for the special case of backward Euler for regular
situations or problems with a singularity of the first kind. For the essentially singular case the condition numbers are very large, and the question is
whether this can be improved by a proper scaling (depending on the value
of α). To investigate this question we have developed a direct MATLAB implementation for the backward Euler equations for the case of a linear scalar
boundary value problem (independent of sbvpcol), which enables us to test
different ways of preconditioning.
For Example 1, the backward Euler scheme yields the linear system
ηi − ηi−1
1
= α ηi + g(ti ),
h
ti
ηN = e,
with g(t) = et − et /tα .
{10, 20, 40, 80, 160}),
i = 1, . . . , N,
(5.2a)
(5.2b)
N is number of mesh points (N
36
∈
and h = 1/N . The matrix of this linear system reads (dimension N+1×N+1)
 1 1

− h h − t1α
0
...
0
0
0
1


1
− t1α . . .
− h1
0
0
0
 0

h
2
 .

..
..
..
..
 .

.
.
.
.
0
0
 .



1
1
A= 0
0
0
. . . h − tα
0
0
 (5.3)
N
−2


1
1
1
 0

− tα
0
0
0
...
−h
h


N −1

1
1
1 
0
0
...
0
−h
− tα 
 0
h
N
1
0
0
0
...
0
0
h
with a natural preconditioning for the boundary (end) condition already
included (last row). In Tables 1.18 and 1.20 the corresponding condition
numbers are given (with respect to the Euclidean and maximum norms) and
their size is in accordance with the results from Table 1.17.
In further series of experiments we have tested different versions of preconditioning this matrix, with the aim of improving its condition number and
the method’s asymptotic behavior for h → 0 for α > 1. A slight improvement
was obtained by diagonal (left)-preconditioning, i.e. premultiplication of A
with a diagonal matrix P with entries tα−1
. The corresponding condition
i
numbers are given in Tables 1.19 and 1.21. However, these numbers are still
very unsatisfactory.
Our conclusion from these experiments is simply the following: For problems with an essential singularity, the backward Euler scheme must be expected to lead to ill-conditioned systems and to behave unstably, with no
obvious way to improve this behavior by preconditioning.
For our simple Example 1, a natural explanation is the following: Since an
end condition is specified at the right endpoint, the ODE is integrated from
right to left and the backward Euler scheme is de facto an explicit method.
The corresponding iteration reads
ηi−1 := (1 − h/tαi )ηi − hg(ti ),
i = N, N −1, . . . ,
starting with ηN = e, where the factor (1 − h/tαi ) becomes large for α > 1
near t = 0. This hints at an unstable behavior.
Thus, a possible remedy could be to use the forward Euler scheme – but
only for this particular example, certainly not in general. However, instead we
consider another simple (symmetric!) difference scheme, namely the implicit
midpoint rule, which should behave more robustly and for which theoretical
convergence results are available, cf. [HW79].
37
5.3
Midpoint rule
The implicit midpoint rule corresponds to the difference scheme
µ
¶
ηi − ηi−1
ti−1 + ti ηi−1 + ηi
=F
,
, i = 1, . . . , N,
hi
2
2
(5.4)
and it is also equivalent to the collocation method of Gauss type with stage
order m = 1 (collocation at the midpoint of each collocation interval [ti−1 , ti ]).
Its classical (super-) convergence order is 2.
Again we restrict our considerations to our scalar model problem, Example 1.
5.3.1
Midpoint rule by means of sbvpcol
In Tables 1.22 and 1.23 the numerical results for the implicit midpoint rule
(realized using sbvpcol) are shown. Here, the full (super-) convergence order
2 is observed, despite the fact that the condition numbers are large and grow
with α. In general, however, we can only expect a convergence order 1 + γ,
where 0 < γ = γ(α) < 1, cf. [HW79].
If we intend to consider the midpoint rule as an alternative to the Euler
scheme for the purpose of error estimation, an appropriate preconditioning
may play a crucial role. Therefore, we have tested different ways of preconditioning.
5.3.2
An implementation of the midpoint rule with
preconditioning
In Tables 1.24–1.27 we present the condition numbers obtained for Example 1, using a direct implementation of the midpoint rule, with and without
preconditioning.
For Example 1, the midpoint rule with constant step size h = 1/N yields
the linear system
ηi−1 + ηi
ηi − ηi−1
=
+ g(ti−1/2 ),
h
2 ti−1/2 α
ηN = e,
t
i = 1, . . . , N,
(5.5a)
(5.5b)
where ti−1/2 := ti − h/2 and g(t) = et − teα . The corresponding system matrix
38
reads (end condition scaled to
 1
− h − 2t11 α h1 − 2t11 α
2
2

1
1

0
−
−
α
h
2t
3

2

..
..

.
.

B=
0
0


0
0



0
0

0
0
1/h as before)
...
0
0
...
0
..
.
0
0
..
.
0
0
..
.
...
...
1
h
1
−
...
2t(N − 3 ) α
2
− h1 − 2t 1 1 α
(N − )
...
0
2
1
h
−
1
2t(N − 1 ) α
2
1
h







.






(5.6)
The results obtained for the observed asymptotical order of the condition
numbers of B are presented in Tables 1.24 and 1.26, for the Euclidean and
maximum norm, respectively. We see that the condition numbers are large
(however, not so extremely large as for the implicit Euler scheme), with
negative orders depending on α.
For α > 1, the asymptotical behavior of the size of the entries in the i-th
row of the matrix B, for i → 0, i.e., for t → 0, is O(t−α ). Therefore a natural
idea is to ‘cancel’ this effect by (left)-preconditioning with factors O(tα−1 ),
such that all rows are scaled to O(1/h). If we multiply the i-th row by a
factor tα−1
(as above for the backward Euler scheme), we obtain the modified
i
matrix
 tα−1

tα−1
tα−1
2tα−1
1
1
−
.
.
.
0
0
− 1h − 2t1 1 α
h
2t 1 α


2
2
α−1


t
tα−1


0
− 2h − 2t2 3 α . . .
0
0


2


.
.
.
.
.


..
..
..
..
..


0

,
B =
0
0
...
0
0

α−1
α−1


tN −1
tN −1


0
0
...
− 2t 3 α
0
h


N− 2


α−1
α−1
α−1
α−1
tN
tN
tN
tN


0
0
.
.
.
−
−
−

h
2tN − 1 α
h
2tN − 1 α 
2
2
1
0
0
...
0
h
(5.7)
which evidently satisfies kB 0 k∞ = O(h−1 ). Now the question is how the norm
inverse B 0−1 behaves for h → 0. Such an estimate for kB 0−1 k is not easy to
obtain analytically; some preliminary results are given in Chapter 7.
Our corresponding numerical results concerning the norm of the inverse
0−1
B can be found in Table 1.25 (2-norm) and Table 1.27 (max-norm). These
results show that our preconditioner is optimal - for both norms we observe
39
kB 0−1 k = O(1) for h → 0. Together with kB 0 k = O(h−1 ) this is the optimal
estimate which may be expected, i.e., our preconditioner is optimal1 .
Due to our numerical results obtained for the midpoint rule, this method
appears to be a promising candidate for the purpose of a posteriori error estimation via defect correction, instead of the backward Euler scheme which
fails for α > 1. However, in the present version of sbvp, this way of error
estimation is not implemented. Therefore we content ourselves with a corresponding test for Example 1, since this is a terminal value problem which
can be solved by an initial value code available at our department.
This experiment is documented in Table 1.43. We see that – in contrast to
the backward Euler method – the quality of this error estimate is completely
satisfactory for α = 0 and α = 1, but begins to deteriorate for larger values
of α. It seems not to be really robust w.r.t. an essential singularity.
1
Obviously, our preconditioner does not provide a proper scaling for α = 0. Consequently, we can neglect the (unsatisfactory) results for this case.
40
Chapter 6
Error estimation based on a
h-h/2 strategy
In Chapter 5 we have seen that the a posteriori error estimation procedure
implemented in sbvp is completely unreliable for problems with an essential
singularity. The modified version based on the midpoint rule presented at the
end of Chapter 5 performed significantly better, but with decreasing quality
for larger values of α.
In this section we present our numerical results for a classical, but computationally more expensive procedure based on grid halving (‘h-h/2 estimate’).
The classical justification for the h-h/2 estimate is based on an asymptotic
representation for the global error,
zh = z + Chp + O(hp+1 ),
zh/2 = z + C(h/2)p + O(hp+1 ),
(6.1)
from which the estimate
zh − z =
1
(zh − zh/2 ) + O(hp+1 )
1 − 2−p
(6.2)
can be immediately derived. (Here, z denotes the exact solution, and zh , zh/2
are the collocation solutions obtained with step size h and h/2, respectively.)
For problems with an essential singularity, so far there exists no theoretical
justification for (6.1). However, our satisfactory numerical results for collocation methods reported in Chapter 4 motivate us to test the performance
of the h-h/2 estimate.
Thus, we compute two approximations using sbvpcol, with step sizes h
and h/2, respectively, and estimate the global error z − zh on the basis of
(6.2). Thus we compute
eest :=
1
(zh − zh/2 ),
1 − 2−p
41
compare this estimate with the true error etrue = zh − z, and observe the
asymptotical order of the difference, which should hopefully be higher than
the convergence order of the underlying method, i.e. the order of etrue . In
those cases where the exact solution z is not known, we again use a reference
solution obtained on a very fine mesh to obtain etrue as precisely as possible.
All numerical results to be discussed in the sequel are based on equidistant
grids and an even number of collocation points (m = 2, 4, 6). In the corresponding Tables, ‘err’ denotes the norm of the error of the error estimate,
i.e. err = keest − etrue k, and p0 denotes its observed order.
Here are our results for the seven test problems from Chapter 3:
• Example 1
The results are given in Tables 1.34–1.37, for α = 0, 1, 2, 3, respectively.
These numbers are to be compared with Tables 1.30–1.33 showing the
global errors ketrue k (with observed orders p = m).
We throughout observe uniform values p0 ≈ p + 1, or higher, and furthermore p0 ≈ p + 2 at the mesh points.
The asymptotic quality of this estimate is superior to the estimate
based on the midpoint rule discussed above, especially for higher values
of α.
We also have tested this example using a reference solution instead of
the exact solution, in order to test the effect of the resulting inaccuracy
in etrue . Tables 1.38–1.41 show that, on finer grids, this effect spoils
the orders which would be otherwise observed. The same has to be
expected for all examples where only a numerical reference solution is
available (Examples 2, 3, 4 and 7).
• Example 2
The results for β = 0, 1, 2 are given in Tables 2.10–2.12, based on a
reference solution obtained numerically. These are to be compared
with Tables 2.7–2.9 showing the corresponding global errors ketrue k.
Although these results are less ‘regular’ than for Example 1 concerning
the orders observed, the absolute values of the error estimate compared
to the errors show that the h-h/2 estimate yields reliable results.
• Example 3
Tables 3.3 and 3.4 show the global errors and the corresponding results
for the estimate, respectively (based on a numerical reference solution).
Our conclusions are the same as for Example 2.
42
• Example 4
See Tables 4.3 and 4.4; similar remarks as for Examples 2 and 3 apply.
Here, for m = 6 a significant order reduction is observed. Such an
effect is usually caused by a certain lack of smoothness of z, but the
use of reference solution may also play a role. We have not further
tried to analyze this point; anyway, the h-h/2 estimate again behaves
satisfactorily.
• Example 5
See Tables 5.5 and 5.6 (exact solution known). The observed orders are
acceptable, but their variation in the case of the estimate is somewhat
irregular. However, the performance of the estimate is satisfactory.
• Example 6
See Tables 6.16–6.19 for the global errors, and Tables 6.20–6.23 for the
error of the corresponding h-h/2 estimate (α = 0, 1, 2, 3, exact solution
known). We may draw similar conclusions as for Example 5.
• Example 7
See Tables 7.3 and 7.4, obtained using a reference solution. Again, the
orders observed for the error estimate are irregular, but, throughout,
the error of the estimate appears to be significantly smaller than the
error itself.
Summarizing, we may claim that the method of a posteriori error estimation by h-h/2 strategy should seriously be considered for implementation in
a further version of sbvp. Our tests have shown that this error estimate
behaves robustly and reliably for all problems tested featuring an essential
singularity.
43
Chapter 7
On the stability of the
preconditioned midpoint rule
This chapter is devoted to a theoretical explanation for the positive effect of
preconditioning observed for the midpoint rule in Example 1.
Let us consider the case of an essential singularity with α ≥ 2. We use a
slight but natural modification of our preconditioner: The rows of the matrix
B (cf. 5.6) are not premultiplied by tα−1
but by tα−1
i
i−1/2 . Numerically, this
makes hardly a difference: cf. Table 1.42, to be compared with Table 1.27,
which indicate that kB 0 −1 k = O(1) holds for both versions of B 0 . But the
structure of the modified matrix becomes simpler with the latter choice. We
thus consider
 α−1 1

(− h − 2t11 α )
t 1 α−1 ( h1 − 2t11 α )
...
0
t1
2
2
2
2




0
t 3 α−1 (− h1 − 2t13 α ) . . .
0
2


2


0
.
.
.
...
B =
.
..
..
..


α−1 1


1
0
0
.
.
.
t
(
−
)
α
1


2tN − 1
N− 2 h
2
0
0
...
1/h
In the sequel we try to derive a bound for the norm of the inverse B 0 −1 , in
order to prove kB 0 −1 k = O(1) for h → 0. Throughout, k · k denotes the
maximum norm for matrices.
1. First, we apply the following simple estimate, which can easily be proven
to be correct:
If D is an invertible matrix sufficiently ‘close’ to B 0 , in the sense that
kI − D−1 B 0 k ≤ m < 1
44
holds, then B 0 is also invertible and satisfies
kB 0−1 k ≤
1
kD−1 k.
1−m
(7.1)
We now apply this estimate with D = diag(B 0 ) (the diagonal of B 0 ). With
the denotation
sk :=
s0k :=
we have

D
−1
− s11

 0


 0

= .
 ..


 0








−1 0
I −D B =






Thus we have
with
where
2
h
1
+
2tk− 1
,
2
tα−1
k− 1
2
h
−
1
2tk− 1
,
2
0
0
... 0
0
− s12
0
... 0
0
− s13 . . . 0
0
s0
0

..
.
..
.
..
0
0
. . . 0 − s1N

0 


0 


0 


0 

0
0
... 0
h
0
0
and
tα−1
k− 1
0 − s11
.
. ..
..
.
0
0
0
0
0
0
0
0
0
0
0
..
.
0
0
0
s0
− s22
0
..
.
0
..
.
0
..
.
... 0
. . . ..
.
0
0
0
. . . 0 − sNN
0
0
0
0
... 0
0
s0
0
0







.






¯1¯
¯ ¯
kD−1 k = max{h, max ¯ ¯}
1≤k≤N sk
¯1¯ ³
´−1 ³
´−1
¯ ¯
max ¯ ¯ ≤ min |sk |
≤
min
|f (t; h, α)|
,
1≤k≤N sk
1≤k≤N
h/2≤t≤1−h/2
tα−1
1
tα + h/2
f (t; h, α) =
+
=
.
h
2t
th
45
(7.2)
(7.3)
An elementary calculation shows that, in the interval [h/2, 1−h/2] this function attains its minimum at tmin = (h/(2(α−1)))1/α , with
(f (tmin ))−1 = O(h1/α ).
Thus we have
kD−1 k = O(h1/α )
for h → 0.
(7.4)
Furthermore,
¯ s0 ¯
¯ ¯
kI − D B k = max ¯ k ¯ =
1≤k≤N sk
≤
−1
0
where
g(t; h, α) =
tα−1
h
tα−1
h
−
+
max |g(tk−1/2 ; h, α)| ≤
1≤k≤N
max
h/2≤t≤1−h/2
1
2t
1
2t
=
|g(t; h, α)|
tα − h2
.
tα + h2
It is easy to verify that g(t; h, α) is monotonously increasing for
t ∈ [h/2, 1 − h/2], and for α ≥ 2, |g(t; h, α)| takes its maximum at t = h/2,
with
1 − (h/2)α−1
α−1
|g(h/2; h, α)| =
).
α−1 = 1 − O(h
1 + (h/2)
Thus we have
kI − D−1 B 0 k = m
with m = 1 − O(hα−1 ).
(7.5)
Finally, (7.1) together with (7.4) and (7.5) yield
−1
kB 0 k = O(h1−α ) · O(h1/α ) = O(h−1+1/α )
for h → 0.
(7.6)
This is not what we had hoped for, namely a uniform O(1) estimate. For
arbitrary α ≥ 2, we only obtain the uniform bound kB 0 −1 k ≤ O(h−1 ). The
above estimate is obviously too crude.
2. To obtain a better estimate, we explicitly computed the inverse B 0 −1 . To
this end, we used the computer algebra system MAPLE to invert the 5 × 5
version of B 0 ,

t 1 α−1
t 1 α−1
1
2
(− h − 2t 1 ) ( 2 h − 2t11 )
0
0
0

2
2

α−1
α−1
t3
t3

0
(− 2 h − 2t13 ) ( 2 h − 2t13 )
0
0


2
2

t 5 α−1
t 5 α−1

1
2
2
−
)
(
− 2t15 )
0
0
0
(−

h
2t 5
h

2
2

tα−1
tα−1
7
7

1
2
2
0
0
0
(
)
(
−
− 2t17 )

h
2t 7
h
0
0
0
46
0
2
1/h
2







.





Its inverse reads

α−1 3h
α−1 5h
2α−1 h
1
2
− hα−1
M10 − 3α h2α−1 +2
− 5α h2α−1 +2
α−1 M1
α−1 M1
+2α−1

α−1 3h
α−1 5h
1
2

0
− 3α h2α−1 +2
− 5α h2α−1 +2
α−1 M2
α−1 M2


α−1 5h
2

0
0
− 5α h2α−1 +2
α−1 M3



0
0
0

0
0
where
Mkl
(2µ−1)α hα−1 −2α−1
µ=k (2µ−1)α hα−1 +2α−1 ,
1,
0
k≤l
k > l.
h
(7.7)
From this result it is easy to guess the structure of the inverse B 0 −1 for general
dimension, which can be verified to be correct. To obtain a sharp estimate
for kB 0 −1 k, it is now necessary to estimate terms of the form Mkl for arbitrary
k ≤ l. We tried to derive such estimates using asymptotic properties of the
Gamma function. However, a complete result including the desired estimate
for kB 0 −1 k could not be obtained so far.
47


α−1 7h
3
− 7α h2α−1 +2
hM24 
α−1 M2


α−1
7h
3
− 7α h2α−1 +2
hM34 
α−1 M3
,

α−1 7h
3
4 
− 7α h2α−1 +2
M
hM
α−1
4
4 
0
( Q
l
:=
α−1
7h
3
− 7α h2α−1 +2
hM14
α−1 M1
Chapter 8
Lobatto distribution
We now briefly report a few numerical experiments which have been obtained
using sbvp with collocation points of Lobatto type. In particular, we use the
3-stage scheme, where the endpoints and the midpoint of each collocation
interval are used as collocation points. This choice is motivated by the fact
that this is exactly the scheme which is used in the standard MATLAB code
bvp4c.
Note that the application of such a scheme to a singular problem is not
straightforward: The right-hand side of the given ODE cannot be evaluated
at t = 0, which is a collocation point. For test examples where the exact
solution is known the remedy is simple, namely we provide the value of z 0 (0)
instead. In the general case, Lemma 2.2.5 could provide this value.
In Tables 1.28 and 1.29 the respective results for Exampe 1 are given.
For α = 0 and α = 1 we obtain the classical convergence order m + 1 = 4,
but for α > 1 a slight order reduction is visible (Table 1.28). Moreover, the
condition number of the associated system matrix grows with α and becomes
very large (Table 1.29).
It is at least doubtful whether Lobatto methods are suitable candidates
for the solution of BVPs with an essential singularity. However, more test
runs will be necessary to get a clear picture.
48
Bibliography
[ACR78]
U. Ascher, J. Christiansen, and R.D. Russell, A collocation solver
for mixed order systems of boundary values problems, Math.
Comp. 33 (1978), 659–679.
[ACR81]
, Collocation software for boundary value ODEs, ACM
Transactions on Mathematical Software 7 (1981), no. 2, 209–
222.
[AKKW]
W. Auzinger, G. Kneisl, O. Koch, and E. Weinmüller,
A collocation code for boundary value problems in ordinary differential equations, To appear in Numer. Algorithms. Also available as ANUM Preprint Nr. 18/01 at
http://www.math.tuwien.ac.at/~inst115/preprints.htm.
[AKKW02]
,
A solution routine for singular boundary
value problems,
Techn. Rep. ANUM Preprint Nr.
1/02, Inst. for Appl. Math. and Numer. Anal., Vienna Univ. of Technology, Austria, 2002, Available at
http://www.math.tuwien.ac.at/~inst115/preprints.htm.
[AKW]
W. Auzinger, O. Koch, and E. Weinmüller, Analysis of a new error estimate for collocation methods applied to singular boundary
value problems., Submitted to SIAM J. Numer. Anal.
[AKW02]
, Efficient collocation schemes for singular boundary
value problems, Numer. Algorithms 31 (2002), 5–25.
[AMR88]
U. Ascher, R.M.M. Mattheij, and R.D. Russell, Numerical solution of boundary value problems for ordinary differential equations, Prentice-Hall, Englewood Cliffs, NJ, 1988.
[BS73]
C. de Boor and B. Swartz, Collocation at Gaussian points, SIAM
J. Numer. Anal. 10 (1973), 582–606.
49
[HW76]
F.R. de Hoog and R. Weiss, Difference methods for boundary
value problems with a singularity of the first kind, SIAM J. Numer. Anal. 13 (1976), 775–813.
[HW77]
, The application of linear multistep methods to singular
initial value problems, Math. Comp. 32 (1977), 676–690.
[HW78]
, Collocation methods for singular boundary value problems, SIAM J. Numer. Anal. 15 (1978), 198–217.
[HW79]
, The numerical solution of boundary value problems with
an essential singularity, SIAM J. Numer. Anal. 16 (1979), 637–
669.
[HW80a]
, An approximation theory for boundary value problems
on infinite intervals, Computing 24 (1980), 227–239.
[HW80b]
, On the boundary value problem for systems of ordinary
differential equations with a singularity of the second kind, SIAM
J. Math. Anal. 11 (1980), 41–60.
[HW85]
, The application of Runge-Kutta schemes to singular initial value problems, Math. Comp. 44 (1985), 93–103.
[MR83]
P.A. Markowich and C.A. Ringhofer, Collocation methods for
boundary value problems on “long” intervals, Math. Comp. 40
(1983), 123–150.
[Ste78]
H. J. Stetter, The defect correction principle and discretization
methods, Numer. Math. 29 (1978), 425–443.
[Wei86]
E. Weinmüller, Collocation for singular boundary value problems
of second order, SIAM J. Numer. Anal. 23 (1986), 1062–1095.
[Zad76]
P.E. Zadunaisky, On the estimation of errors propagated in the
numerical integration of ODEs, Numer. Math. 27 (1976), 21–39.
50
Figure 1.1 : plot of solution and error using sbvp for Example 1
1. a) Solution, α =1
b) Error, α =1
2. a) Solution, α =2
b) Error, α = 2
3. a) Solution, α =3
b) Error, α = 3
51
TABLE 1.0 : global error for Example 1
-α=01. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.11e-08
1.32e-09
8.25e-11
5.15e-12
3.22e-13
err
coll
2.18e-08
1.34e-09
8.31e-11
5.18e-12
3.23e-13
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
4.00
4.00
4.02
4.01
4.01
4.00
2.12e-04
2.11e-04
2.11e-04
2.11e-04
2.28e-04
2.22e-04
2.17e-04
2.14e-04
err
coll
2.10e-05
2.68e-06
3.38e-07
4.24e-08
5.31e-09
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
4.00
4.00
2.97
2.99
2.99
3.00
1.39e-03
1.39e-03
1.39e-03
1.37e-03
1.97e-02
2.06e-02
8.64e-01
2.14e-02
err
coll
6.87e-05
4.88e-06
3.25e-07
2.09e-08
1.33e-09
p
mesh
p
coll
const
mesh
const
coll
6.01
6.00
6.00
5.99
3.82
3.91
3.95
3.98
1.01e-05
9.98e-06
9.93e-06
9.57e-06
9.68e-04
1.10e-03
1.20e-03
1.29e-03
err
coll
1.64e-06
5.77e-08
1.91e-09
6.17e-11
1.96e-12
p
mesh
p
coll
const
mesh
const
coll
+8.01
+7.95
+2.46
–1.00
4.83
4.91
4.96
4.98
3.99e-08
3.66e-08
4.06e-13
2.78e-17
4.65e-05
5.24e-05
5.73e-05
6.09e-05
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.39e-07
8.68e-09
5.42e-10
3.39e-11
2.12e-12
3. sbvpcol, Gauss, m=3
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.57e-07
2.43e-09
3.79e-11
5.92e-13
9.33e-15
4. sbvpcol, Gauss, m=4
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.55e-10
6.02e-13
2.44e-15
4.44e-16
8.88e-16
52
TABLE 1.1 : global error for Example 1
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.53e-08
9.57e-10
5.98e-11
3.73e-12
2.35e-13
err
coll
1.63e-08
9.88e-10
6.08e-11
3.77e-12
2.41e-13
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
4.00
3.99
4.05
4.02
4.01
3.97
1.51e-04
1.54e-04
1.53e-04
1.45e-04
1.83e-04
1.69e-04
1.63e-04
1.34e-04
err
coll
2.01e-05
2.67e-06
3.37e-07
4.24e-08
5.31e-09
p
mesh
p
coll
const
mesh
const
coll
3.03
3.01
3.00
3.00
2.91
2.99
2.99
3.00
1.03e-02
9.73e-03
9.34e-03
9.32e-03
1.97e-02
2.07e-02
2.08e-02
2.08e-02
err
coll
1.34e-07
8.63e-09
5.46e-10
3.44e-11
2.16e-12
p
mesh
p
coll
const
mesh
const
coll
4.03
4.01
4.01
4.00
3.96
3.98
3.99
4.00
5.79e-04
5.55e-04
5.40e-04
5.32e-04
1.22e-03
1.30e-03
1.35e-03
1.45e-03
err
coll
6.35e-10
2.03e-11
6.43e-13
2.04e-14
2.22e-15
p
mesh
p
coll
const
mesh
const
coll
5.03
5.01
5.00
2.46
4.97
4.98
4.98
3.20
2.65e-05
2.53e-05
2.45e-05
3.51e-10
5.89e-05
6.14e-05
6.01e-05
2.50e-08
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
9.61e-06
1.18e-06
1.46e-07
1.82e-08
2.27e-09
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.42e-08
3.32e-09
2.05e-10
1.28e-11
7.97e-13
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.48e-10
7.60e-12
2.35e-13
7.33e-15
1.33e-15
53
TABLE 1.2 : global error for Example 1
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.29e-08
8.00e-10
5.00e-11
3.12e-12
1.95e-13
err
coll
1.40e-08
8.37e-10
5.11e-11
3.16e-12
1.96e-13
p
mesh
p
coll
const
mesh
const
coll
4.01
4.00
4.00
4.00
4.06
4.03
4.02
4.01
1.31e-04
1.28e-04
1.28e-04
1.28e-04
1.60e-04
1.48e-04
1.39e-04
1.35e-04
err
coll
3.63e-05
5.55e-06
8.59e-07
1.34e-07
2.10e-08
p
mesh
p
coll
const
mesh
const
coll
2.71
2.69
2.68
2.68
2.71
2.69
2.68
2.68
1.86e-02
1.75e-02
1.69e-02
1.69e-02
1.86e-02
1.75e-02
1.69e-02
1.69e-02
err
coll
1.69e-07
8.62e-09
5.46e-10
3.43e-11
2.15e-12
p
mesh
p
coll
const
mesh
const
coll
4.30
4.14
4.05
4.10
4.29
3.98
3.99
4.00
3.37e-03
2.08e-03
1.49e-03
1.85e-03
3.30e-03
1.30e-03
1.35e-03
1.40e-03
err
coll
1.10e-09
4.03e-11
1.55e-12
6.02e-14
2.22e-15
p
mesh
p
coll
const
mesh
const
coll
4.77
4.70
4.69
4.91
4.77
4.70
4.69
4.76
6.51e-05
5.25e-05
4.97e-05
1.34e-04
6.51e-05
5.25e-05
4.97e-05
6.90e-05
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.63e-05
5.55e-06
8.59e-07
1.34e-07
2.10e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.69e-07
8.56e-09
4.84e-10
2.92e-11
1.70e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.10e-09
4.03e-11
1.55e-12
6.02e-14
2.00e-15
54
TABLE 1.3 : global error for Example 1
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.11e-08
7.02e-10
4.38e-11
2.73e-12
1.72e-13
err
coll
1.24e-08
7.39e-10
4.50e-11
2.77e-12
1.73e-13
p
mesh
p
coll
const
mesh
const
coll
3.98
4.00
4.00
3.99
4.07
4.04
4.02
4.00
1.06e-04
1.13e-04
1.13e-04
1.07e-04
1.45e-04
1.32e-04
1.24e-04
1.15e-04
err
coll
6.83e-05
1.16e-05
2.01e-06
3.49e-07
6.09e-08
p
mesh
p
coll
const
mesh
const
coll
2.55
2.54
2.53
2.52
2.55
2.54
2.53
2.52
2.42e-02
2.34e-02
2.27e-02
2.18e-02
2.42e-02
2.34e-02
2.27e-02
2.18e-02
err
coll
1.56e-07
8.62e-09
5.46e-10
3.44e-11
2.15e-12
p
mesh
p
coll
const
mesh
const
coll
4.23
4.18
4.07
4.04
4.18
3.98
3.99
4.00
2.65e-03
2.28e-03
1.52e-03
1.34e-03
2.36e-03
1.30e-03
1.35e-03
1.41e-03
err
coll
2.12e-09
9.07e-11
3.92e-12
1.71e-13
5.99e-15
p
mesh
p
coll
const
mesh
const
coll
4.55
4.53
4.52
4.84
4.55
4.53
4.52
4.84
7.46e-05
7.11e-05
6.77e-05
2.76e-04
7.46e-05
7.11e-05
6.77e-05
2.76e-04
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
6.83e-05
1.16e-05
2.01e-06
3.49e-07
6.09e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.56e-07
8.32e-09
4.60e-10
2.75e-11
1.67e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.12e-09
9.07e-11
3.92e-12
1.71e-13
5.99e-15
55
TABLE 1.4 : convergence of first derivative for Example 1
-α=01. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.14e-07
2.65e-08
1.68e-09
1.08e-10
2.60e-11
err
coll
4.14e-07
2.65e-08
1.68e-09
1.08e-10
2.60e-11
p
mesh *
p
coll
const
mesh *
const
coll
3.96
3.98
3.96
2.05
3.96
3.98
3.96
2.05
3.80e-03
4.00e-03
3.70e-03
8.66e-07
3.80e-03
4.00e-03
3.70e-03
8.66e-07
err
coll
2.15e-03
5.52e-04
1.40e-04
3.52e-05
8.82e-06
p
mesh *
p
coll
const
mesh *
const
coll
1.99
1.98
1.99
2.00
1.99
1.98
1.99
2.00
2.10e-01
2.08e-01
2.16e-01
2.25e-01
2.10e-01
2.08e-01
2.16e-01
2.25e-01
err
coll
2.15e-05
2.76e-06
3.50e-07
4.40e-08
5.51e-09
p
mesh *
p
coll
const
mesh *
const
coll
2.96
2.98
2.99
3.00
2.96
2.98
2.99
3.00
1.96e-02
2.08e-02
2.16e-02
2.25e-02
1.96e-02
2.08e-02
2.16e-02
2.25e-02
err
coll
1.54e-07
9.86e-09
6.25e-10
3.99e-11
7.38e-12
p
mesh *
p
coll
const
mesh *
const
coll
3.96
3.98
3.98
2.26
3.96
3.98
3.97
2.43
1.40e-03
1.49e-03
1.49e-03
7.90e-07
1.40e-03
1.49e-03
1.49e-03
1.68e-06
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.15e-03
5.52e-04
1.40e-04
3.52e-05
8.82e-06
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.15e-05
2.76e-06
3.50e-07
4.40e-08
5.51e-09
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
1.54e-07
9.86e-09
6.25e-10
3.95e-11
8.22e-12
* solution computed at mesh points by left polynomial
56
TABLE 1.5 : convergence of first derivative for Example 1
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.14e-07
2.65e-08
1.67e-09
1.08e-10
2.60e-11
err
coll
4.14e-07
2.65e-08
1.67e-09
1.08e-10
9.57e-12
p
mesh *
p
coll
const
mesh *
const
coll
3.96
3.98
3.96
2.05
3.96
3.98
3.96
3.50
3.80e-03
4.00e-03
3.70e-03
8.66e-07
3.80e-03
4.00e-03
3.70e-03
4.80e-04
err
coll
2.16e-03
5.52e-04
1.40e-04
3.52e-05
8.82e-06
p
mesh *
p
coll
const
mesh *
const
coll
1.96
1.98
1.99
2.00
1.97
1.98
1.99
2.00
1.96e-01
2.08e-01
2.16e-01
2.25e-01
2.02e-01
2.08e-01
2.16e-01
2.25e-01
err
coll
2.16e-05
2.76e-06
3.50e-07
4.40e-08
5.51e-09
p
mesh *
p
coll
const
mesh *
const
coll
2.96
2.98
2.99
3.00
2.97
2.98
2.99
3.00
1.96e-02
2.08e-02
2.15e-02
2.25e-02
2.02e-02
2.08e-02
2.15e-02
2.25e-02
err
coll
1.54e-07
9.86e-09
6.25e-10
3.99e-11
7.38e-12
p
mesh *
p
coll
const
mesh *
const
coll
3.96
3.98
3.98
2.26
3.96
3.98
3.97
2.43
1.40e-03
1.49e-03
1.49e-03
7.90e-07
1.40e-03
1.49e-03
1.49e-03
1.68e-06
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.15e-03
5.52e-04
1.40e-04
3.52e-05
8.82e-06
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.15e-05
2.76e-06
3.49e-07
4.40e-08
5.51e-09
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
1.54e-07
9.86e-09
6.25e-10
3.95e-11
8.22e-12
* solution computed at mesh points by left polynomial
57
TABLE 1.6 : convergence of first derivative for Example 1
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.14e-07
2.93e-08
2.24e-09
1.75e-10
2.60e-11
err
coll
4.14e-07
2.93e-08
2.24e-09
1.75e-10
1.24e-11
p
mesh *
p
coll
const
mesh *
const
coll
3.82
3.71
3.67
2.75
3.82
3.71
3.67
3.81
2.74e-03
1.97e-03
1.70e-03
3.00e-05
2.74e-03
1.97e-03
1.70e-03
3.12e-05
err
coll
2.51e-03
7.44e-04
2.25e-04
6.88e-05
2.12e-05
p
mesh *
p
coll
const
mesh *
const
coll
1.75
1.73
1.71
1.70
1.75
1.73
1.71
1.70
1.41e-01
1.33e-01
1.24e-01
1.18e-01
1.41e-01
1.33e-01
1.24e-01
1.18e-01
err
coll
2.26e-05
2.76e-06
3.50e-07
4.40e-08
5.51e-09
p
mesh *
p
coll
const
mesh *
const
coll
3.03
2.98
2.99
3.00
3.03
2.98
2.99
3.00
2.42e-02
2.08e-02
2.16e-02
2.25e-02
2.42e-02
2.08e-02
2.16e-02
2.25e-02
err
coll
2.35e-07
1.69e-08
1.29e-09
9.89e-11
7.38e-12
p
mesh *
p
coll
const
mesh *
const
coll
3.79
3.72
3.70
3.58
3.79
3.72
3.70
3.74
1.45e-03
1.17e-03
1.09e-03
6.43e-04
1.45e-03
1.17e-03
1.09e-03
1.30e-03
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.51e-03
7.44e-04
2.25e-04
6.88e-05
2.12e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.26e-05
2.76e-06
3.50e-07
4.40e-08
5.51e-09
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.35e-07
1.69e-08
1.29e-09
9.89e-11
8.22e-12
* solution computed at mesh points by left polynomial
58
TABLE 1.7 : convergence of first derivative for Example 1
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
7.27e-07
6.17e-08
5.31e-09
4.59e-10
2.56e-11
err
coll
7.27e-07
6.17e-08
5.31e-09
4.59e-10
2.56e-11
p
mesh *
p
coll
const
mesh *
const
coll
3.56
3.54
3.53
4.17
3.56
3.54
3.53
4.17
2.64e-03
2.49e-03
2.40e-03
3.96e-02
2.64e-03
2.49e-03
2.40e-03
3.96e-02
err
coll
4.44e-03
1.47e-03
4.99e-04
1.72e-04
5.95e-05
p
mesh *
p
coll
const
mesh *
const
coll
1.59
1.56
1.54
1.53
1.59
1.56
1.54
1.53
1.73e-01
1.57e-01
1.46e-01
1.40e-01
1.73e-01
1.57e-01
1.46e-01
1.40e-01
err
coll
2.26e-05
2.76e-06
3.50e-07
4.40e-08
5.51e-09
p
mesh *
p
coll
const
mesh *
const
coll
2.96
2.98
2.99
3.00
3.03
2.98
2.99
3.00
1.96e-02
2.08e-02
2.15e-02
2.25e-02
2.42e-02
2.08e-02
2.16e-02
2.25e-02
err
coll
4.37e-07
3.70e-08
3.19e-09
2.76e-10
2.02e-11
p
mesh *
p
coll
const
mesh *
const
coll
3.56
3.54
3.53
3.77
3.56
3.54
3.53
3.77
1.59e-03
1.49e-03
1.44e-03
4.13e-03
1.59e-03
1.49e-03
1.44e-03
4.13e-03
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.44e-03
1.47e-03
4.99e-04
1.72e-04
5.95e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.15e-05
2.76e-06
3.49e-07
4.40e-08
5.51e-09
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.37e-07
3.70e-08
3.19e-09
2.76e-10
2.02e-11
* solution computed at mesh points by left polynomial
59
TABLE 1.8 : convergence of second derivative for Example 1
-α=01. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.35e-05
5.55e-06
7.00e-07
8.94e-08
3.08e-08
err
coll
4.35e-05
5.55e-06
7.00e-07
8.94e-08
1.69e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.97
2.99
2.97
1.54
2.97
2.99
2.97
2.41
4.06e-02
4.31e-02
4.01e-02
7.62e-05
4.06e-02
4.31e-02
4.01e-02
3.45e-03
err
coll
1.31e-01
6.68e-02
3.37e-02
1.70e-02
8.48e-03
p
mesh *
p
coll
const
mesh *
const
coll
0.98
0.99
1.00
0.99
0.98
0.99
1.00
0.99
1.25e+00
1.28e+00
1.33e+00
1.30e+00
1.25e+00
1.28e+00
1.33e+00
1.30e+00
err
coll
2.61e-03
6.66e-04
1.68e-04
4.22e-05
1.06e-05
p
mesh *
p
coll
const
mesh *
const
coll
1.97
1.99
1.99
2.00
1.97
1.99
1.99
2.00
2.44e-01
2.59e-01
2.59e-01
2.70e-01
2.44e-01
2.59e-01
2.59e-01
2.70e-01
err
coll
3.09e-05
3.95e-06
5.00e-07
6.35e-08
1.65e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.97
2.98
2.98
1.95
2.97
2.98
3.03
2.12
2.87e-02
3.01e-02
2.94e-02
3.24e-04
2.87e-02
3.01e-02
3.55e-02
6.50e-04
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
1.31e-01
6.68e-02
3.37e-02
1.70e-02
8.48e-03
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.61e-03
6.66e-04
1.68e-04
4.22e-05
1.06e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
3.09e-05
3.95e-06
5.00e-07
6.35e-08
1.65e-08
* solution computed at mesh points by left polynomial
60
TABLE 1.9 : convergence of second derivative for Example 1
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.35e-05
5.55e-06
7.00e-07
8.94e-08
3.08e-08
err
coll
4.35e-05
5.55e-06
7.00e-07
8.94e-08
1.67e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.97
2.99
2.97
1.54
2.97
2.99
2.97
2.42
4.07e-02
4.27e-02
4.01e-02
7.49e-05
4.07e-02
4.27e-02
4.01e-02
3.60e-03
err
coll
1.31e-01
6.68e-02
3.37e-02
1.69e-02
8.48e-03
p
mesh *
p
coll
const
mesh *
const
coll
0.98
0.99
1.00
1.00
0.98
0.99
1.00
1.00
1.25e+00
1.29e+00
1.33e+00
1.33e+00
1.25e+00
1.29e+00
1.33e+00
1.33e+00
err
coll
2.61e-03
6.66e-04
1.68e-04
4.22e-05
1.06e-05
p
mesh *
p
coll
const
mesh *
const
coll
1.97
1.99
1.99
2.00
1.97
1.99
1.99
2.00
2.44e-01
2.59e-01
2.59e-01
2.70e-01
2.44e-01
2.59e-01
2.59e-01
2.70e-01
err
coll
3.09e-05
3.95e-06
5.00e-07
6.35e-08
1.65e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.97
2.98
3.03
2.11
2.97
2.98
2.98
1.95
2.87e-03
3.00e-03
3.54e-02
6.50e-04
2.87e-03
3.00e-03
2.95e-03
3.24e-04
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
1.31e-01
6.68e-02
3.37e-02
1.69e-02
8.48e-03
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.61e-03
6.66e-04
1.68e-04
4.22e-05
1.06e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
3.09e-05
3.95e-06
5.00e-07
6.13e-08
1.42e-08
* solution computed at mesh points by left polynomial
61
TABLE 1.10 : convergence of second derivative for Example 1
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.35e-05
5.55e-06
7.01e-07
1.07e-07
3.08e-08
err
coll
4.35e-05
5.55e-06
7.01e-07
1.07e-07
1.67e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.97
2.98
2.71
1.79
2.97
2.98
2.71
2.68
4.06e-02
4.18e-02
1.54e-02
2.73e-04
4.06e-02
4.18e-02
1.54e-02
1.35e-02
err
coll
1.31e-01
6.68e-02
3.37e-02
1.69e-02
8.60e-02
p
mesh *
p
coll
const
mesh *
const
coll
0.98
0.99
1.00
0.97
0.98
0.99
1.00
0.97
1.24e+00
1.28e+00
1.33e+00
1.21e+00
1.24e+00
1.28e+00
1.33e+00
1.21e+00
err
coll
2.61e-03
6.66e-04
1.68e-04
4.22e-05
1.06e-05
p
mesh *
p
coll
const
mesh *
const
coll
1.97
1.98
1.99
2.00
1.97
1.98
1.99
2.00
2.44e-01
2.59e-01
2.59e-01
2.70e-01
2.44e-01
2.59e-01
2.59e-01
2.70e-01
err
coll
3.09e-05
3.95e-06
5.25e-07
7.88e-08
1.65e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.97
2.91
2.73
2.48
2.97
2.91
2.73
2.26
2.87e-02
2.44e-02
1.27e-02
4.08e-03
2.87e-02
2.44e-02
1.27e-02
1.57e-03
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
1.31e-01
6.68e-02
3.37e-02
1.69e-02
8.60e-02
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.61e-03
6.66e-04
1.68e-04
4.22e-05
1.06e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
3.09e-05
3.95e-06
5.25e-07
7.88e-08
1.42e-08
* solution computed at mesh points by left polynomial
62
TABLE 1.11 : convergence of second derivative for Example 1
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
5.56e-05
9.21e-06
1.56e-06
2.66e-07
3.02e-08
err
coll
5.56e-05
9.21e-06
1.56e-06
2.66e-07
3.02e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.59
2.56
2.55
3.13
2.59
2.56
2.55
3.13
2.16e-02
1.97e-02
1.90e-02
2.41e-01
2.16e-02
1.97e-02
1.90e-02
2.41e-01
err
coll
1.31e-01
7.28e-02
4.69e-02
3.09e-02
2.08e-02
p
mesh *
p
coll
const
mesh *
const
coll
0.85
0.63
0.60
0.57
0.85
0.63
0.60
0.57
9.27e-01
4.81e-01
4.29e-01
3.76e-01
9.27e-01
4.81e-01
4.29e-01
3.76e-01
err
coll
2.60e-03
6.65e-04
1.68e-04
4.22e-05
1.06e-05
p
mesh *
p
coll
const
mesh *
const
coll
1.97
1.98
1.99
2.00
1.97
1.98
1.99
2.00
2.43e-01
2.51e-01
2.59e-01
2.70e-01
2.43e-01
2.51e-01
2.59e-01
2.70e-01
err
coll
4.32e-05
7.14e-06
1.20e-06
2.06e-07
3.04e-08
p
mesh *
p
coll
const
mesh *
const
coll
2.60
2.57
2.55
2.76
2.60
2.57
2.55
2.76
1.72e-02
1.58e-02
1.46e-02
3.68e-02
1.72e-02
1.58e-02
1.46e-02
3.68e-02
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
1.31e-01
7.28e-02
4.69e-02
3.09e-02
2.08e-02
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
2.60e-03
6.65e-04
1.68e-04
4.22e-05
1.06e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh *
4.32e-05
7.14e-06
1.20e-06
2.06e-07
3.04e-08
* solution computed at mesh points by left polynomial
63
TABLE 1.12 : matrix condition estimates for Example 1
-α=01. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.05e+01
3.27e+01
5.77e+01
1.08e+02
2.09e+02
1.12e+01
2.24e+01
4.48e+01
8.98e+01
1.80e+02
ord.
condest
ord.
cond
const
condest
const
cond
-0.67
-0.82
-0.91
-0.95
-1.00
-1.00
-1.00
-1.00
4.34+00
2.79e+00
2.04e+00
1.67e+00
1.13e+00
1.11e+00
1.11e+00
1.12e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.83
-0.91
-0.96
-0.98
-1.01
-1.01
-1.00
-1.00
2.34e+00
1.85e+00
1.58e+00
2.83e+00
1.12e+00
1.14e+00
1.15e+00
1.16e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.75
-0.87
-0.93
-0.96
-1.01
-1.01
-1.00
-1.00
3.24e+00
2.28e+00
1.80e+00
1.55e+00
1.05e+00
1.06e+00
1.07e+00
1.09e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.67
-0.82
-0.91
-0.95
-1.00
-1.01
-1.01
-1.00
4.34e+00
2.79e+00
2.04e+00
1.67e+00
1.02e+00
1.01e+00
1.02e+00
1.05e+00
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.60e+01
2.86e+01
5.38e+01
1.04e+02
2.05e+02
1.15e+01
2.32e+01
4.65e+01
9.31e+01
1.86e+02
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.82e+01
3.06e+01
5.58e+01
1.06e+02
2.07e+02
2.15e+01
1.07e+01
4.32e+01
8.66e+01
1.73e+02
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.05e+01
3.27e+01
5.78e+01
1.08e+02
2.09e+02
1.03e+01
2.06e+01
4.14e+01
8.31e+01
1.66e+02
64
TABLE 1.13 : matrix condition estimates for Example 1
-α=11. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
9.70e+01
1.54e+02
2.68e+02
4.97e+02
9.53e+02
3.18e+01
5.62e+01
1.10e+02
2.20e+02
4.40e+02
ord.
condest
ord.
cond
const
condest
const
cond
-0.67
-0.80
-0.89
-0.94
-0.82
-0.97
-1.00
-1.00
2.09e+01
1.40e+01
1.01e+01
8.05e+00
4.80e+00
3.06e+00
2.78e+00
2.73e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.82
-0.90
-0.95
-0.97
-0.95
-1.00
-1.00
-1.00
6.84e+00
5.39e+00
4.55e+00
4.08e+00
2.27e+00
1.97e+00
1.92e+00
1.93e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.74
-0.85
-0.92
-0.96
-0.90
-0.99
-1.00
-1.00
1.79e+02
1.28e+02
9.91e+01
8.34e+01
4.61e+00
3.56e+00
3.38e+00
3.36e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.67
-0.80
-0.89
-0.94
-0.85
-0.98
-1.00
-1.00
3.84e+01
2.58e+01
1.86e+01
1.48e+01
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.55e+01
8.05e+01
1.50e+02
2.90e+02
5.70e+02
2.02e+01
3.90e+01
7.79e+01
1.56e+02
3.13e+02
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
9.75e+01
1.62e+02
2.92e+02
5.52e+02
1.07e+03
3.67e+01
6.85e+01
1.36e+02
2.72e+02
5.45e+02
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.78e+02
2.83e+02
4.93e+02
9.13e+02
1.75e+03
6.02e+01
1.08e+02
2.13e+02
4.26e+02
8.52e+02
65
8.59e+00
5.81e+00
5.33e+00
5.26e+00
TABLE 1.14 : matrix condition estimates for Example 1
-α=21. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.69e+03
8.25e+03
2.80e+04
1.02e+05
3.88e+05
1.16e+03
3.95e+03
1.53e+04
6.09e+04
2.44e+05
ord.
condest
ord.
cond
const
condest
const
cond
-1.62
-1.76
-1.86
-1.93
-1.76
-1.95
-1.99
-2.00
6.53e+01
4.21e+01
2.88e+01
2.19e+01
ord.
condest
ord.
cond
const
condest
-1.71
-1.88
-1.94
-1.97
-1.66
-1.99
-2.00
-2.00
2.76e+01
1.65e+01
1.34e+01
1.17e+01
ord.
condest
ord.
cond
const
condest
-1.49
-1.73
-1.90
-1.95
-1.38
-1.79
-2.00
-2.00
2.19e+02
1.07e+02
5.67e+01
4.61e+02
ord.
condest
ord.
cond
const
condest
const
cond
-1.84
-1.44
-1.85
-1.93
-1.52
-1.39
-2.00
-2.00
2.59e+02
8.69e+02
1.93e+02
1.35e+02
3.80e+02
5.60e+02
5.91e+01
5.76e+01
2.01e+01
1.13e+01
9.78e+00
9.47e+00
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.41e+03
4.61e+03
1.70e+04
6.49e+04
2.54e+05
8.93e+02
2.82e+03
1.12e+04
4.48e+04
1.80e+05
const
cond
1.97e+01
7.25e+00
6.92e+00
6.91e+00
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.74e+03
1.89e+04
6.27e+04
2.34e+05
9.02e+05
4.05e+03
1.06e+04
3.66e+04
1.46e+05
5.87e+05
const
cond
1.68e+02
4.95e+01
2.29e+01
2.26e+01
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.80e+04
6.47e+04
1.75e+05
6.30e+05
2.40e+06
1.25e+04
3.57e+04
9.33e+04
3.72e+05
1.49e+06
· condestDF:=condest(DF,1), cond:=cond(DF,2)
66
TABLE 1.15 : matrix condition estimates for Example 1
-α=31. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.02e+05
6.12e+05
4.07e+06
2.92e+07
2.21e+08
5.27e+04
3.30e+05
2.51e+06
1.99e+07
1.59e+08
ord.
condest
ord.
cond
const
condest
const
cond
-2.58
-2.73
-2.85
-2.92
-2.65
-2.93
-2.99
-3.00
2.69e+02
1.70e+02
1.12e+02
8.25e+01
1.18e+02
5.15e+01
4.11e+01
3.89e+01
ord.
condest
ord.
cond
const
condest
const
cond
-2.44
-2.56
-2.54
-2.85
-2.47
-2.49
-2.72
-3.00
4.18e+02
2.95e+02
3.25e+02
8.09e+01
2.36e+02
2.25e+02
9.47e+01
2.74e+01
ord.
condest
ord.
cond
const
condest
const
cond
-2.79
-2.52
-2.50
-2.52
-2.49
-2.49
-2.50
-2.55
1.73e+03
3.85e+03
4.26e+03
3.89e+03
2.16e+03
2.15e+03
2.12e+03
1.67e+03
ord.
condest
ord.
cond
const
condest
const
cond
-2.65
-2.49
-2.55
-2.51
-2.48
-2.49
-2.50
-2.50
1.64e+04
2.65e+04
2.12e+04
2.59e+04
1.32e+04
1.27e+04
1.24e+04
1.24e+04
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.16e+05
6.34e+05
3.74e+06
2.17e+07
1.56e+08
6.95e+04
3.85e+05
2.16e+06
1.42e+07
1.14e+08
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.06e+06
7.36e+06
4.23e+07
2.38e+08
1.36e+09
6.68e+05
3.75e+06
2.11e+07
1.19e+08
6.98e+08
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.35e+06
4.62e+07
2.60e+08
1.52e+09
8.66e+09
3.98e+06
2.22e+07
1.25e+08
7.06e+08
3.99e+09
67
TABLE 1.16 : global error for Example 1
– Backward Euler method1. sbvpcol, α=0, backward Euler, number of collocation points m= 1
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.22e-02
2.55e-02
1.26e-02
6.28e-03
3.13e-03
err
coll
5.22e-02
2.55e-02
1.26e-02
6.28e-03
3.13e-03
p
mesh
p
coll
const
mesh
const
coll
1.03
1.02
1.00
1.00
1.03
1.02
1.00
1.00
5.64e.01
5.37e-01
5.13e-01
5.10e-01
5.64e.01
5.37e-01
5.13e-01
5.10e-01
p
mesh
p
coll
const
mesh
const
coll
1.06
1.04
1.02
1.01
1.06
1.04
1.02
1.01
4.49e-01
4.23e-01
3.92e-01
3.79e-01
4.49e-01
4.23e-01
3.92e-01
3.79e-01
p
mesh
p
coll
const
mesh
const
coll
1.09
1.03
1.02
1.01
1.09
1.03
1.02
1.01
4.16e-01
3.55e-01
3.42e-01
3.18e-01
4.16e-01
3.55e-01
3.42e-01
3.18e-01
p
mesh
p
coll
const
mesh
const
coll
-3.04
-11.5
-15.6
-26.0
-3.04
-11.5
-15.6
-26.0
6.54e-03
6.27e-14
1.64e-20
2.45e-40
6.54e-03
6.27e-14
1.64e-20
2.45e-40
2. sbvpcol, α=1, backward Euler
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.95e-02
1.90e-02
9.27e-03
4.58e-03
2.28e-03
err
coll
3.95e-02
1.90e-02
9.27e-03
4.58e-03
2.28e-03
3. sbvpcol, α=2, backward Euler
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.40e-02
1.60e-02
7.81e-03
3.84e-03
1.91e-03
err
coll
3.40e-02
1.60e-02
7.81e-03
3.84e-03
1.91e-03
4. sbvpcol, α=3, backward Euler
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.20e+00
5.93e+01
1.73e+05
8.70e+09
5.99e+17
err
coll
7.20e+00
5.93e+01
1.73e+05
8.70e+09
5.99e+17
68
TABLE 1.17 : matrix condition estimates for Example 1
-Backward Euler method 1. sbvpcol, α=0, backward Euler, number of the collocation points m= 1
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.71e+01
2.96e+01
5.49e+01
1.05e+02
2.07e+02
1.43e+01
2.86e+01
5.73e+01
1.15e+02
2.29e+02
ord.
condest
ord.
cond
const
condest
const
cond
-0.80
-0.89
-0.94
-0.97
-1.00
-1.00
-1.00
-1.00
2.73e+00
2.07e+00
1.70e+00
1.50e+00
1.43e+00
1.44e+00
1.43e+00
1.43e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.78
-0.87
-0.93
-0.97
-0.97
-1.00
-1.00
-1.00
2.92e+00
1.46e+00
1.76e+00
1.53e+00
1.12e+00
1.04e+00
1.03e+00
1.02e+00
ord.
condest
ord.
cond
const
condest
const
cond
-3.51
-5.16
-7.09
-9.84
-3.90
-5.31
-7.27
-10.0
1.15e-01
8.45e-04
7.01e-07
4.02e-12
5.86e-02
8.69e-04
6.23e-07
3.41e-12
ord.
condest
ord.
cond
const
condest
const
cond
-12.0
-18.6
-28.9
-45.3
-12.2
-18.7
-29.3
-14.1
7.00e-07
1.55e-15
5.61e-32
3.50e-63
6.82e-07
2.24e-15
2.20e-32
1.38e-03
2. sbvpcol, α=1, backward Euler
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.75e+01
3.00e+01
5.50e+01
1.05e+02
2.05e+02
1.05e+01
2.06e+01
4.11e+01
8.22e+01
1.65e+02
3. sbvpcol, α=2, backward Euler
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.80e+02
4.35e+03
1.55e+05
2.10e+07
1.92e+10
4.70e+02
7.04e+03
2.79e+05
4.32e+07
4.54e+10
4. sbvpcol, α=3, backward Euler
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.32e+05
2.51e+09
1.00e+15
4.92e+23
2.08e+37
1.06e+06
4.94e+09
2.12e+15
1.43e+24
2.68e+28
· condestDF:=condest(DF,1), cond:=cond(DF,2)
69
TABLE 1.18 : matrix condition estimates for Example 1
- Backward Euler method -preconditioning 1/h 1. α=0, backward Euler, number of the collocation points m= 1
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(A)
cond(A)
1.24e+01
2.50e+01
5.03e+01
1.01e+02
2.03e+02
9.44e+00
1.83e+01
3.61e+01
7.15e+01
1.42e+02
ord.
condest
-1.02
-1.01
-1.00
-1.01
ord.
cond
const
condest
const
cond
-0.96
-0.98
-0.99
-0.99
1.19e+00
1.22e+00
1.24e+00
1.22e+00
1.04e+00
9.89e–01
9.49e–01
9.22e–01
ord.
cond
const
condest
const
cond
1.12e+00
1.20e+00
9.55e–01
9.74e–01
7.65e-01
7.23e-01
6.91e-01
6.70e-01
const
condest
const
cond
3.63e-02
3.20e-04
3.03e-07
1.86e-12
1.58e-02
2.61e-04
2.01e-07
1.14e-12
const
condest
const
cond
3.32e-07
7.64e-16
2.79e-32
1.75e-63
2.33e-07
7.84e-16
2.26e-32
1.50e-63
norm
A
1.88e+01
3.89e+01
7.89e+01
1.59e+02
3.19e+02
norm
inv(A)
5.02e-01
4.72e-01
4.57e-01
4.50e-01
4.46e-01
norm
A
1.83e+01
3.84e+01
7.85e+01
1.59e+02
3.19e+02
norm
inv(A)
3.75e-01
3.46e-01
3.32e-01
3.25e-01
3.22e-01
norm
A
9.11e+01
3.81e+02
1.56e+03
6.32e+03
2.54e+04
norm
inv(A)
1.65e+01
6.20e+00
6.16e+01
2.38e+03
6.26e+05
norm
A
9.90e+02
7.98e+03
6.40e+04
5.12e+05
4.10e+06
norm
inv(A)
3.74e+02
2.18e+05
1.17e+10
8.07e+17
4.75e+30
2. α=1, backward Euler
h
condest(A)
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.05e+01
2.05e+01
3.95e+01
7.95e+01
1.60e+02
6.87e+00
1.33e+01
2.61e+01
5.16e+01
1.03e+02
ord.
condest
-0.97
-0.95
-1.01
-1.00
-0.95
-0.97
-0.98
-0.99
3. α=2, backward Euler
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(A)
1.60e+02
2.00e+03
7.47e+04
1.03e+07
9.53e+09
cond(A)
1.50e+02
2.36e+03
9.62e+04
1.51e+07
1.59e+10
ord.
condest
-3.64
-5.22
-7.11
-9.85
ord.
cond
-3.98
-5.35
-7.29
-10.1
4. α=3, backward Euler
h
condest(A)
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.11e+05
1.25e+09
5.01e+14
2.46e+23
1.04e+37
3.70e+05
1.74e+09
7.49e+14
4.13e+23
1.95e+37
ord.
condest
-12.0
-18.6
-28.9
-45.3
ord.
cond
-12.2
-18.7
-29.0
-45.4
. condest:=condest(A,1), cond:=cond(A,2), norm(A) = norm(A,2)
70
TABLE 1.19 : matrix condition estimates for Example 1
- Backward Euler method -preconditioning tˆ(α-1) 1. α=0, backward Euler, number of the collocation points m= 1
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(A)
cond(A)
9.12e+01
3.72e+02
1.50e+03
6.04e+03
2.42e+04
5.62e+01
2.07e+02
7.87e+02
3.07e+03
1.21e+04
ord.
condest
-2.03
-2.01
-2.01
-2.00
ord.
cond
const
condest
const
cond
-1.88
-1.93
-1.96
-1.98
8.54e-01
8.92e-01
9.16e-01
9.30e-01
7.39e-01
6.37e-01
5.66e-01
5.23e-01
ord.
cond
const
condest
const
cond
norm
A
1.40e+02
5.72e+02
2.32e+03
9.33e+03
3.74e+04
norm
inv(A)
4.02e-01
3.61e-01
3.39e-01
3.29e-01
3.23e-01
norm
A
1.83e+01
7.65e–01 3.84e+01
1.06e+00 7.85e+01
6.94e–01 1.59e+02
6.53e-01 3.19e+02
norm
inv(A)
3.75e-01
3.46e-01
3.32e-01
3.25e-01
3.22e-01
2. α=1, backward Euler
h
condest(A)
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
9.50e+00
1.95e+01
3.95e+01
7.95e+01
1.60e+02
6.87e+00
1.33e+01
2.61e+01
5.16e+01
1.03e+02
ord.
condest
-1.04
-1.02
-1.01
-1.01
-0.95
-0.97
-0.98
-1.00
8.72e-01
9.23e-01
9.55e-01
9.55e-01
3. α=2, backward Euler
h
condest(A)
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.01e+02
1.05e+03
2.95e+04
2.93e+06
2.11e+09
1.09e+02
1.28e+03
3.85e+04
4.40e+06
3.37e+09
ord.
condest
ord.
cond
const
condest
const
cond
-3.37
-4.81
-6.63
-9.49
-3.55
-4.91
-6.84
-9.58
4.23e-02
5.76e-04
6.95e-07
2.53e-12
3.0e-02
5.23e-04
4.30e-07
2.57e-12
ord.
condest
ord.
cond
const
condest
const
cond
-10.8
-17.2
-27.7
-43.9
-11.0
-17.5
-27.8
-44.1
5.35e-07
2.85e-15
4.88e-32
5.67e-63
3.78e-07
1.45e-15
4.81e-32
3.45e-63
norm
A
1.67e+01
3.55e+01
7.40e+01
1.52e+02
3.09e+02
norm
inv(A)
6.50e+00
3.60e+01
5.20e+02
2.89e+04
1.09e+07
norm
A
1.63e+01
3.45e+01
7.22e+01
1.49e+02
3.05e+02
norm
inv(A)
2.42e+03
2.37e+06
2.08e+11
2.31e+19
2.18e+32
4. α=3, backward Euler
h
condest(A)
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.70e+04
6.78e+07
1.02e+13
2.17e+21
3.61e+34
3.94e+04
8.17e+07
1.50e+13
3.44e+21
6.66e+34
. condest:=condest(A,1), cond:=cond(A,2), norm(A) = norm(A,2)
71
TABLE 1.20 : matrix condition estimates based on max. norm for Example 1
- Backward Euler method - preconditioning 1/h 1. α=1, backward Euler, number of collocation points m= 1
2.
h
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.33e+00
1.57e+01
3.04e+01
5.99e+01
1.19e+02
ord.
cond
const
condest
-0.91
-0.95
-0.98
-0.99
1.01e+00
9.03e–01
8.23e–01
7.81e–01
ord.
cond
const
condest
-4.07
-5.38
-7.40
-10.2
2.30e-03
4.49e-04
2.65e-07
1.39e-12
ord.
cond
const
condest
-12.2
-18.9
-29.2
-45.6
4.45e-07
9.80e-16
2.51e-32
1.63e-63
norm
(A)
1.90e+01
3.90e+01
7.90e+01
1.59e+02
3.19e+02
norm
inv(A)
4.38e-01
4.02e-01
3.85e-01
3.76e-01
3.72e-01
norm
(A)
1.00e+02
4.00e+02
1.60e+03
6.40e+03
2.56e+04
norm
inv(A)
2.68e+00
1.12e+01
1.17e+02
4.92e+03
1.42e+06
norm
(A)
1.00e+03
8.00e+03
6.40e+04
5.12e+05
4.10e+06
norm
inv(A)
6.99e+02
4.09e+05
2.42e+10
1.88e+18
1.25e+31
α=2, backward Euler
h
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.68e+02
4.49e+03
1.87e+05
3.15e+07
3.63e+10
3. α=3, backward Euler
h
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.99e+05
3.28e+09
1.55e+15
9.60e+23
5.10e+37
72
TABLE 1.21 : matrix condition estimates based on max. norm for Example 1
- Backward Euler method -preconditioning tˆ(α-1) 1. α=1, backward Euler, number of collocation points m = 1
2.
h
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.33e+00
1.57e+01
3.04e+01
5.99e+01
1.19e+02
ord.
cond
const
condest
-0.91
-0.95
-0.98
-0.99
1.01e+00
9.03e–01
8.22e–01
7.81e–01
ord.
cond
const
condest
-3.66
-4.96
-6.93
-9.69
4.44e-02
8.92e-04
6.33e-07
3.55e-12
ord.
cond
const
condest
-11.1
-17.6
-27.9
-41.9
7.09e-07
2.30e-15
6.97e-32
1.47e-58
norm
(A)
1.90e+01
3.90e+01
7.90e+01
1.59e+02
3.19e+02
norm
inv(A)
4.38e-01
4.02e-01
3.85e-01
3.76e-01
3.72e-01
norm
(A)
1.90e+01
3.90e+01
7.90e+01
1.59e+02
3.19e+02
norm
inv(A)
1.06e+01
6.53e+01
1.01e+03
6.09e+04
2.50e+07
norm
(A)
1.90e+01
3.90e+01
7.90e+01
1.59e+02
3.19e+02
norm
inv(A)
4.32e+03
4.52e+06
4.39e+11
5.46e+19
5.79e+32
α=2, backward Euler
h
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.02e+02
2.55e+03
7.95e+04
9.68e+06
7.98e+09
3. α=3, backward Euler
h
cond(A)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.22e+04
1.76e+08
3.47e+13
8.69e+21
3.61e+34
73
TABLE 1.22 : global error for Example 1
- Midpoint rule 1. sbvpcol, α=0, Midpoint rule, number of collocation points m=1
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
8.34e-04
2.08e-04
5.21e-05
1.30e-05
3.26e-06
Err
Coll
3.13e-03
8.15e-04
2.08e-04
5.25e-05
1.32e-05
p
mesh
p
coll
const
mesh
const
coll
2.00
2.00
2.00
2.00
1.94
1.97
1.99
1.99
8.37e-02
8.34e-02
8.34e-02
8.33e-02
2.73e-01
2.98e-01
3.15e-01
3.25e-01
Err
Coll
1.34e-03
8.15e-04
2.08e-04
5.25e-05
1.32e-05
p
mesh
p
coll
const
mesh
const
coll
2.00
2.00
2.00
2.00
0.72
1.97
1.98
1.99
1.35e-01
1.34e-01
1.34e-01
1.34e-01
6.99e-03
2.98e-01
3.15e-01
3.25e-01
Err
Coll
3.11e-03
8.14e-04
2.08e-04
5.25e-05
1.32e-05
p
mesh
p
coll
const
mesh
const
coll
2.01
2.00
2.00
2.00
1.93
1.97
1.98
1.99
1.66e-01
1.61e-01
1.61e-01
1.61e-01
2.67e-01
2.97e-01
3.15e-01
3.25e-01
Err
Coll
3.11e-03
8.13e-04
2.08e-04
5.25e-05
1.32e-05
p
mesh
p
coll
const
mesh
const
coll
2.02
2.00
2.00
2.00
1.93
1.97
1.98
1.99
1.90e-01
1.83e-01
1.80e-01
1.81e-01
2.68e-01
2.96e-01
3.14e-01
3.25e-01
2. sbvpcol, α=1, Midpoint rule
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.34e-03
3.34e-04
8.35e-05
2.09e-05
5.21e-06
3. sbvpcol, α=2, Midpoint rule
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.62e-03
4.02e-04
1.01e-04
2.51e-05
6.29e-06
4. sbvpcol, α=3, Midpoint rule
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.83e-03
4.52e-04
1.13e-04
2.82e-05
7.05e-06
74
TABLE 1.23 : matrix condition estimates for Example 1
-Midpoint rule 1. sbvpcol, α=0, Midpoint rule, number of collocation points m=1
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.65e+01
2.92e+01
5.44e+01
1.05e+02
2.06e+02
1.42e+01
2.85e+01
5.72e+01
1.14e+02
2.29e+02
ord.
condest
ord.
cond
const
condest
-0.82
-0.90
-0.95
-0.97
-1.01
-1.00
-1.00
-1.00
2.47e+00
1.96e+00
1.64e+00
1.48e+00
ord.
condest
ord.
cond
const
condest
-0.81
-0.89
-0.94
-0.97
-0.98
-1.00
-1.00
-1.00
3.09e+00
2.42e+00
2.04e+00
3.48e+00
ord.
condest
ord.
cond
const
condest
-1.76
-1.86
-1.93
-1.96
-1.97
-2.00
-2.01
-2.00
3.99e+00
2.96e+00
2.35e+00
2.02e+00
ord.
condest
ord.
cond
const
condest
-2.67
-2.86
-2.92
-2.96
-2.61
-2.99
-3.00
-3.00
9.27e+00
5.22e+00
4.12e+00
3.50e+00
const
cond
1.39e+00
1.43e+00
1.43e+00
1.43e+00
2. sbvpcol, α=1, Midpoint rule
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.00e+01
3.51e+01
6.51e+01
1.25e+02
2.45e+02
1.09e+01
2.14e+01
4.30e+01
8.62e+01
1.73e+02
Const
cond
1.14e+00
1.08e+00
1.06e+00
1.07e+00
3. sbvpcol, α=2, Midpoint rule
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.32e+02
7.87e+02
2.87e+03
1.09e+04
4.25e+04
1.59e+02
6.21e+02
2.49e+03
9.99e+03
4.01e+04
const
cond
1.71e+00
1.55e+00
1.53e+00
1.54e+00
4. sbvpcol, α=3, Midpoint rule
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.29e+03
2.72e+04
1.97e+05
1.50e+06
1.16e+07
3.46e+03
2.11e+04
1.68e+05
1.34e+06
1.08e+07
75
const
cond
8.58e+00
2.69e+00
2.60e+00
2.59e+00
TABLE 1.24 : matrix condition estimates for Example 1
- Midpoint rule - preconditioning 1/h 1. α=0, Midpoint rule, number of collocation points m=1
h
condest(B)
cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.26e+01
2.53e+01
5.06e+01
1.01e+02
2.02e+02
9.68e+00
1.85e+01
3.62e+01
7.16e+01
1.42e+02
ord.
condest
ord.
cond
const
condest
const
cond
-1.00
-1.00
-1.00
-1.00
-0.93
-0.97
-0.98
-0.99
1.24e+00
1.27e+00
1.28e+00
1.26e+00
1.13e+00
1.02e+00
9.60e–01
9.44e–01
ord.
condest
ord.
cond
const
condest
const
cond
-0.83
-1.02
-1.01
-1.00
-0.91
-0.96
-0.98
-0.99
1.64e+00
9.23e–01
9.55e–01
9.74e–01
8.66e-01
7.48e-01
7.00e-01
6.71e-01
ord.
condest
ord.
cond
const
condest
const
cond
norm
B
1.98e+01
3.99e+01
7.99e+01
1.60e+02
3.20e+02
norm
inv(B)
4.89e-01
4.65e-01
4.53e-01
4.48e-01
4.45e-01
norm
B
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B)
3.56e-01
3.36e-01
3.27e-01
3.22e-01
3.20e-01
norm
B
2.84e+02
1.13e+03
4.53e+03
1.81e+04
7.25e+04
norm
inv(B)
2.91e-01
2.68e-01
2.58e-01
2.53e-01
2.51e-01
norm
B
5.66e+03
4.53e+04
3.62e+05
2.90e+06
2.32e+07
norm
inv(B)
2.54e-01
2.29e-01
2.18e-01
2.13e-01
2.11e-01
2. α=1, Midpoint rule
h
condest(B)
cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.10e+01
1.95e+01
3.95e+01
7.95e+01
1.60e+01
7.11e+00
1.34e+01
2.61e+01
5.16e+01
1.03e+02
3. α=2, Midpoint rule
h
condest(B)
cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.33e+01
3.81e+02
1.41e+03
5.69e+03
2.29e+04
8.25e+01
3.04e+02
1.17e+03
4.58e+03
1.82e+04
-2.19
-1.89
-2.01
-2.01
-1.88
-1.94
-1.97
-1.99
5.34e–01
1.34e+00
8.40e–01
8.63e–01
1.09e+00
9.00e–01
8.07e–01
7.60e–01
4. α=3, Midpoint rule
h
condest(B)
cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.30e+03
1.09e+04
8.92e+04
7.45e+05
5.82e+06
1.44e+03
1.04e+04
7.91e+04
6.18e+05
4.88e+06
ord.
condest
ord.
cond
const
condest
const
cond
-3.07
-3.04
-3.06
-2.97
-2.85
-2.93
-2.97
-2.98
1.12e+00
1.22e+00
1.12e+00
1.69e+00
2.02e+00
1.61e+00
1.40e+00
1.30e+00
. condest:=condest(B,1), cond:=cond(B,2), norm(B) = norm(B,2)
76
TABLE 1.25 : matrix condition estimates for Example 1
- Midpoint rule -preconditioning tˆ(α-1) 1. α=0, Midpoint rule, number of collocation points m=1
h
condest(B’)
cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.03e+02
3.76e+02
1.55e+03
6.06e+03
2.42e+04
5.80e+01
2.09e+02
7.91e+02
3.07e+03
1.21e+04
ord.
condest
ord.
cond
const
condest
const
cond
-1.87
-2.04
-1.97
-2.00
-1.85
-1.92
-1.96
-1.98
1.40e+00
8.25e–01
1.09e+00
9.57e–01
8.20e-01
6.64e-01
5.80e-01
5.27e-01
ord.
condest
ord.
cond
const
condest
const
cond
-0.83
-1.02
-1.01
-1.01
-0.91
-0.96
-0.98
-1.00
1.64e+00
9.23e–01
9.55e–01
9.55e–01
8.66e-01
7.51e-01
6.94e-01
6.53e-01
ord.
condest
ord.
cond
const
condest
const
cond
-1.02
-1.08
-0.94
-1.01
-0.97
-1.00
-1.00
-1.00
9.03e–01
7.56e–01
1.30e+00
9.51e–01
1.03e+00
9.45e–01
9.45e–01
9.53e–01
ord.
condest
ord.
cond
const
condest
const
cond
-1.17
-1.09
-1.05
-1.04
-1.10
-1.08
-1.06
-1.04
2.27e+00
2.90e+00
3.34e+00
3.46e+00
2.42e+00
2.58e+00
2.83e+00
3.01e+00
norm
B’
1.47e+02
5.87e+02
2.35e+03
9.39e+03
3.76e+04
norm
inv(B’)
3.95e-01
3.56e-01
3.37e-01
3.27e-01
3.22e-01
norm
B’
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B’)
3.56e-01
3.36e-01
3.27e-01
3.22e-01
3.20e-01
norm
B’
2.87e+01
5.71e+01
1.14e+02
2.28e+02
4.56e+02
norm
inv(B’)
3.36e-01
3.32e-01
3.31e-01
3.32e-01
3.32e-01
norm
B’
5.67e+01
1.13e+02
2.27e+02
4.54e+02
9.08e+02
norm
inv(B’)
5.67e+01
5.79e–01
6.11e–01
6.37e–01
6.56e–01
2. α=1, Midpoint rule
h
condest(B’)
cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.10e+01
1.95e+01
3.95e+01
7.95e+01
1.60e+02
7.11e+00
1.34e+01
2.61e+01
5.16e+01
1.03e+02
3. α=2, Midpoint rule
h
condest(B’)
cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
9.54e+00
1.94e+01
4.11e+01
7.86e+01
1.58e+02
9.64e+00
1.89e+01
3.78e+01
7.56e+01
1.58e+02
4. α=3, Midpoint rule
h
condest(B’)
cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.35e+01
7.53e+01
1.60e+02
3.31e+02
6.81e+02
3.06e+01
6.57e+01
1.39e+02
2.89e+02
5.95e+02
. condest:=condest(B’,1), cond:=cond(B’,2), norm(B’) = norm(B’,2)
77
TABLE 1.26 : matrix condition estimates based on max. norm for Example 1
–Midpoint rule - preconditioning 1/h 1. α=1, Midpoint rule, number of collocation points m=1
2.
h
cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.17e+00
1.55e+01
3.02e+01
5.96e+01
1.18e+02
ord.
cond
const
cond
-0.92
-0.96
-0.98
-0.99
9.74e-01
8.68e-01
8.11e-01
7.94e-01
ord.
cond
const
cond
-2.05
-1.95
-1.97
-1.99
1.04e+00
1.43e+00
1.32e+00
1.23e+00
ord.
cond
const
cond
-2.88
-2.93
-2.96
-2.98
3.00e+00
2.59e+00
2.29e+00
2.10e+00
norm
(B)
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B)
4.08e-01
3.87e-01
3.77e-01
3.73e-01
3.70e-01
norm
(B)
4.00e+02
1.60e+03
6.40e+03
2.56e+04
1.02e+05
norm
inv(B)
3.29e-01
3.06e-01
2.95e-01
2.89e-01
2.87e-01
norm
(B)
8.00e+03
6.40e+04
5.12e+05
4.10e+06
3.28e+07
norm
inv(B)
2.85e-01
2.62e-01
2.50e-01
2.44e-01
2.41e-01
α=2, Midpoint rule
h
Cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.32e+02
4.90e+02
1.89e+03
7.40e+03
2.93e+04
3. α=3, Midpoint rule
h
Cond(B)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.28e+03
1.68e+04
1.28e+05
9.98e+05
7.89e+06
78
TABLE 1.27 : matrix condition estimates based on max. norm for Example 1
–Midpoint rule -preconditioning tˆ(α-1) 1. α=1, Midpoint rule, number of collocation points m=1
2.
h
Cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.17e+00
1.55e+01
3.02e+01
5.96e+01
1.18e+02
ord.
cond
const
cond
-0.92
-0.96
-0.98
-0.99
9.73e-01
8.68e-01
8.11e-01
7.94e-01
ord.
cond
const
cond
-1.00
-1.00
-1.00
-1.01
1.70e+00
1.70e+00
1.70e+00
1.66e+00
ord.
cond
const
cond
-1.16
-1.13
-1.10
-1.08
3.09e+00
3.37e+00
3.78e+00
4.20e+00
norm
(B’)
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B’)
4.08e-01
3.87e-01
3.77e-01
3.73e-01
3.70e-01
norm
(B’)
4.00e+01
8.00e+01
1.60e+02
3.20e+02
6.40e+02
norm
inv(B’)
4.24e-01
4.25e-01
4.25e-01
4.26e-01
4.27e-01
norm
(B’)
8.00e+01
1.60e+02
3.20e+02
6.40e+02
1.28e+03
norm
inv(B’)
5.57e-01
6.23e-01
6.81e-01
7.30e-01
7.69e-01
α=2, Midpoint rule
h
Cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.70e+01
3.40e+01
6.80e+01
1.36e+02
2.73e+02
3. α=3, Midpoint rule
h
Cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.46e+01
9.96e+01
2.18e+02
4.67e+02
9.84e+02
79
TABLE 1.28 : global error for Example 1
- Lobatto points 1. sbvpcol, α=0, Lobatto points, number of collocation points m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.41e-07
8.75e-09
5.45e-10
3.40e-11
2.12e-12
err
coll
6.56e-07
4.26e-08
2.71e-09
1.71e-10
1.08e-11
p
mesh
p
coll
const
mesh
const
coll
4.01
4.01
4.00
4.00
3.95
3.97
3.99
3.99
1.46e-03
1.43e-03
1.41e-03
1.39e-03
5.78e-03
6.27e-03
6.60e-03
6.80e-03
err
coll
6.55e-07
4.26e-08
2.71e-09
1.71e-10
1.08e-11
p
mesh
p
coll
const
mesh
const
coll
4.03
4.02
4.01
4.01
3.94
3.97
3.99
3.99
2.73e-03
2.63e-03
2.55e-03
2.50e-03
5.75e-03
6.26e-03
6.60e-03
6.80e-03
err
coll
6.63e-07
5.14e-08
4.00e-09
3.13e-10
2.45e-11
p
mesh
p
coll
const
mesh
const
coll
3.69
3.68
3.68
3.67
3.69
3.68
3.68
3.67
3.25e-03
3.18e-03
3.12e-03
3.07e-03
3.25e-03
3.18e-03
3.12e-03
3.07e-03
err
coll
1.28e-06
1.09e-07
9.41e-09
8.17e-10
7.06e-11
p
mesh
p
coll
const
mesh
const
coll
3.55
3.54
3.52
3.53
3.55
3.54
3.52
3.53
4.54e-03
4.35e-03
4.18e-03
4.30e-03
4.54e-03
4.35e-03
4.18e-03
4.30e-03
2. sbvpcol, α=1, Lobatto points
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.54e-07
1.55e-08
9.59e-10
5.95e-11
3.70e-12
3. sbvpcol, α=2, Lobatto points
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
6.63e-07
5.14e-08
4.00e-09
3.13e-10
2.45e-11
4. sbvpcol, α=3, Lobatto points
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.28e-06
1.09e-07
9.41e-09
8.17e-10
7.06e-11
80
TABLE 1.29 : matrix condition estimates for Example 1
- Lobatto points 1. sbvpcol, α=0, Lobatto points, number of collocation points m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(DF) cond(DF)
1.81e+01
3.06e+01
5.57e+01
1.06e+02
2.07e+02
1.16e+01
2.32e+01
4.65e+01
9.31e+01
1.86e+02
ord.
condest
ord.
cond
Const
condest
const
cond
-0.75
-0.87
-0.93
-0.96
-1.00
-1.00
-1.00
-1.00
3.19e+00
2.27e+00
1.80e+00
1.55e+00
ord.
condest
ord.
cond
Const
condest
const
cond
-0.75
-0.85
-0.92
-0.96
-0.91
-0.99
-1.00
-1.00
5.52e+00
4.04e+00
3.17e+00
2.67e+00
1.34e+00
1.07e+00
1.02e+00
1.01e+00
ord.
condest
ord.
cond
const
condest
const
cond
-2.18
-2.09
-2.04
-2.02
-2.08
-2.06
-2.04
-2.03
4.82e+01
6.40e+01
7.54e+01
8.29e+01
3.96e+01
4.16e+01
4.44e+01
4.73e+01
ord.
condest
ord.
cond
const
condest
const
cond
-4.02
-4.01
-4.00
-4.00
-4.02
-4.01
-4.00
-4.00
15.6e+01
1.65e+01
1.68e+01
1.68e+01
9.81e+01
1.03e+01
1.05e+01
1.05e+01
norm
DF
2.11e+01
1.16e+00 4.27e+01
1.15e+00 8.58e+01
1.16e+00 1.72e+02
1.17e+00 3.44e+02
norm
inv(DF)
5.48e-01
5.43e-01
5.42e-01
5.42e-01
5.42e-01
2. sbvpcol, α=1, Lobatto points
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(DF) cond(DF)
3.09e+01
5.19e+01
9.37e+01
1.77e+02
3.44e+02
1.09e+01
2.06e+01
4.08e+01
8.15e+01
1.63e+02
norm
DF
2.63e+01
5.25e+01
1.05e+02
2.10e+02
4.20e+02
norm
inv(DF)
4.16e-01
3.92e-01
3.88e-01
3.88e-01
3.89e-01
norm
DF
4.43e+02
1.78e+03
7.12e+03
2.85e+04
1.14e+05
norm
inv(DF)
1.07e+01
1.12e+01
1.17e+01
1.20e+01
1.22e+01
norm
DF
8.68e+03
6.94e+04
5.56e+05
4.44e+06
3.56e+07
norm
inv(DF)
1.19e+01
2.42e+01
4.86e+01
9.72e+01
1.95e+02
3. sbvpcol, α=2, Lobatto points
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(DF) cond(DF)
7.33e+02
3.33e+03
1.41e+04
5.83e+04
2.36e+05
4.72e+02
1.99e+03
8.30e+03
3.42e+04
1.39e+05
4. sbvpcol, α=3, Lobatto points
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
condest(DF) cond(DF)
1.65e+05
2.69e+06
4.31e+07
6.91e+08
1.11e+10
1.03e+05
1.68e+06
2.70e+07
4.32e+08
6.92e+09
. condest:=condest(DF,1), cond:=cond(DF,2), norm(DF) = norm(DF,2)
81
TABLE 1.30 : global error for Example 1
-α=0-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.78e-04
6.95e-05
1.74e-05
4.34e-06
1.09e-06
err
coll
2.83e-04
7.02e-05
1.75e-05
4.35e-06
1.09e-06
p
mesh
p
coll
const
mesh
const
coll
2.00
2.00
2.00
2.00
2.01
2.01
2.00
2.00
2.79e-02
2.78e-02
2.78e-02
2.78e-02
2.92e-02
2.87e-02
2.83e-02
2.81e-02
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
4.00
4.00
4.02
4.01
4.01
4.00
2.12e-04
2.11e-04
2.11e-04
2.11e-04
2.28e-04
2.22e-04
2.17e-04
2.14e-04
p
mesh
p
coll
const
mesh
const
coll
6.03
6.01
6.00
5.05
6.04
6.05
6.03
4.86
7.77e-07
7.53e-07
7.36e-07
5.35e-08
8.38e-07
8.53e-07
8.17e-07
3.19e-08
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.11e-08
1.32e-09
8.25e-11
5.15e-12
3.22e-13
err
coll
2.18e-08
1.34e-09
8.31e-11
5.18e-12
3.23e-13
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.19e-08
1.82e-10
2.82e-12
4.42e-14
1.33e-15
err
coll
1.28e-08
1.95e-10
2.94e-12
4.51e-14
1.55e-15
82
TABLE 1.31 : global error for Example 1
-α=1-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.01e-04
5.04e-05
1.26e-05
3.15e-06
7.86e-07
err
coll
2.10e-04
5.14e-05
1.27e-05
3.16e-06
7.89e-07
p
mesh
p
coll
const
mesh
const
coll
2.00
2.00
2.00
2.00
2.03
2.01
2.01
2.00
1.99e-02
2.02e-02
2.01e-02
2.01e-02
2.26e-02
2.15e-02
2.10e-02
2.06e-02
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
4.00
3.99
4.05
4.02
4.01
3.97
1.51e-04
1.54e-04
1.53e-04
1.45e-04
1.83e-04
1.69e-04
1.63e-04
1.34e-04
p
mesh
p
coll
const
mesh
const
coll
6.10
6.03
5.98
5.19
6.16
6.11
6.05
4.67
6.32e-07
5.68e-07
5.12e-07
5.75e-08
7.92e-07
7.46e-07
6.46e-07
1.44e-08
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.53e-08
9.57e-10
5.98e-11
3.73e-12
2.35e-13
err
coll
1.63e-08
9.88e-10
6.08e-11
3.77e-12
2.41e-13
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
9.18e-09
1.33e-10
2.04e-12
3.24e-14
8.88e-16
err
coll
1.11e-08
1.55e-10
2.24e-12
3.40e-14
1.33e-15
83
TABLE 1.32 : global error for Example 1
-α=2-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.69e-04
4.21e-05
1.05e-05
2.63e-06
6.58e-07
err
coll
1.78e-04
4.34e-05
1.07e-05
2.65e-06
6.60e-07
p
mesh
p
coll
const
mesh
const
coll
2.01
2.00
2.00
2.00
2.03
2.02
2.01
2.01
1.73e-02
1.68e-02
1.69e-02
1.68e-02
1.92e-02
1.85e-02
1.78e-02
1.74e-02
p
mesh
p
coll
const
mesh
const
coll
4.01
4.00
4.00
4.00
4.06
4.03
4.02
4.01
1.31e-04
1.28e-04
1.28e-04
1.28e-04
1.60e-04
1.48e-04
1.39e-04
1.35e-04
err
coll
p
mesh
p
coll
const
mesh
const
coll
1.28e-10
1.93e-12
2.80e-14
2.22e-15
6.24
6.03
6.02
3.57
6.04
6.05
6.11
3.66
6.38e-07
4.80e-07
4.65e-07
5.28e-10
5.58e-07
5.58e-07
6.38e-07
7.09e-10
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.29e-08
8.00e-10
5.00e-11
3.12e-12
1.95e-13
err
coll
1.40e-08
8.37e-10
5.11e-11
3.16e-12
1.96e-13
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
8.45e-09
1.12e-10
1.71e-12
2.64e-14
2.22e-15
84
TABLE 1.33 : global error for Example 1
-α=3-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.46e-04
3.69e-05
9.21e-06
2.30e-06
5.75e-07
err
coll
1.57e-04
3.81e-05
9.37e-06
2.32e-06
5.78e-07
p
mesh
p
coll
const
mesh
const
coll
1.98
2.00
2.00
2.00
2.05
2.02
2.01
2.01
1.39e-02
1.49e-02
1.48e-02
1.47e-02
1.76e-02
1.62e-02
1.57e-02
1.53e-02
p
mesh
p
coll
const
mesh
const
coll
3.98
4.00
4.00
3.99
4.07
4.04
4.02
4.00
1.06e-04
1.13e-04
1.13e-04
1.07e-04
1.45e-04
1.32e-04
1.24e-04
1.15e-04
p
mesh
p
coll
const
mesh
const
coll
6.40
5.91
6.04
2.18
6.40
6.17
6.11
2.27
6.50e-07
3.31e-07
4.29e-07
9.65e-12
6.49e-07
6.38e-07
5.55e-07
1.32e-11
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.11e-08
7.02e-10
4.38e-11
2.73e-12
1.72e-13
err
coll
1.24e-08
7.39e-10
4.50e-11
2.77e-12
1.73e-13
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
7.70e-09
9.13e-11
1.52e-12
2.31e-14
5.11e-15
err
coll
7.70e-09
1.22e-10
1.70e-12
2.46e-14
5.11e-15
85
TABLE 1.34 : Error of error estimate based on h-h/2 for Example 1
-α=0-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
8.29e-08
5.18e-09
3.24e-10
2.02e-11
1.27e-12
err
coll
3.22e-06
3.94e-07
4.88e-08
6.06e-09
7.56e-10
p`
mesh
p`
coll
const
mesh
const
coll
4.00
4.00
4.00
4.00
3.03
3.01
3.01
3.00
8.29e-04
8.30e-04
8.30e-04
8.25e-04
3.45e-03
3.30e-03
3.21e-03
3.16e-03
p`
mesh
p`
coll
const
mesh
const
coll
6.01
6.00
6.01
8.22
5.11
5.07
5.04
5.03
1.21e-06
1.20e-06
1.20e-06
5.61e-04
5.14e-06
4.90e-06
4.59e-06
4.52e-06
p`
mesh
p`
coll
const
mesh
const
coll
+7.93
+7.60
–1.25
–3.18
+7.11
+6.93
+2.12
–2.15
9.80e-10
6.26e-10
6.31e-18
2.95e-20
4.14e-09
3.22e-09
1.48e-13
1.07e-18
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.87e-08
2.91e-10
4.54e-12
7.07e-14
2.37e-16
err
coll
1.49e-07
4.34e-09
1.29e-10
3.93e-12
1.20e-13
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
4.02e-12
1.65e-14
8.46e-17
2.01e-16
1.83e-15
err
coll
3.01e-11
2.18e-13
1.79e-15
4.12e-16
1.83e-15
86
TABLE 1.35 : Error of error estimate based on h-h/2 for Example 1
-α=1-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.12e-07
1.51e-08
9.42e-10
5.70e-11
3.62e-12
err
coll
4.71e-06
6.12e-07
7.61e-08
9.48e-09
1.19e-09
p`
mesh
p`
coll
const
mesh
const
coll
3.81
4.00
4.05
3.98
2.94
3.01
3.00
2.99
1.38e-03
2.40e-03
2.85e-03
2.11e-03
4.14e+00
5.03e+00
4.95e+00
4.71e+00
p`
mesh
p`
coll
const
mesh
const
coll
5.97
5.99
5.78
5.84
4.75
4.89
4.95
5.01
3.06e-06
3.15e-06
2.03e-06
2.39e-06
4.03e-06
4.90e-06
5.50e-06
6.60e-06
p`
mesh
p`
coll
const
mesh
const
coll
+7.94
+7.24
+2.12
–4.84
+6.81
+7.53
+4.13
–4.37
2.52e-09
9.44e-10
2.24e-14
9.39e-23
3.56e-09
9.68e-09
8.30e-12
4.78e-22
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
4.89e-08
7.82e-10
1.23e-11
2.25e-13
3.94e-15
err
coll
1.50e-07
5.56e-09
1.87e-10
6.07e-12
1.88e-13
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.02e-11
4.15e-14
2.75e-16
6.34e-17
1.82e-15
err
coll
3.18e-11
2.85e-13
1.54e-15
8.81e-17
1.83e-15
87
TABLE 1.36 : Error of error estimate based on h-h/2 for Example 1
-α=2-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.28e-07
2.06e-08
1.41e-09
8.44e-11
5.39e-12
err
coll
5.13e-06
6.73e-07
8.60e-08
1.07e-08
1.35e-09
p`
mesh
p`
coll
const
mesh
const
coll
3.99
3.87
4.07
3.97
2.93
2.97
3.00
2.99
3.23e-03
2.20e-03
4.61e-03
3.00e-03
4.37e-03
4.89e-03
5.56e-03
5.35e-03
p`
mesh
p`
coll
const
mesh
const
coll
5.86
6.61
5.65
6.48
3.84
5.03
4.94
5.02
5.69e-06
1.63e-05
2.17e-06
2.18e-05
1.40e-06
7.30e-06
6.14e-06
7.63e-06
err
coll
p`
mesh
p`
coll
const
mesh
const
coll
3.49e-13
3.17e-15
7.05e-18
1.84e-15
+7.87
+7.53
–0.02
–1.93
+5.88
+6.79
+8.81
–8.03
4.77e-09
2.97e-09
4.55e-16
2.29e-18
1.20e-09
4.26e-09
2.86e-07
1.52e-27
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
9.82e-08
1.70e-09
1.73e-11
3.45e-13
3.88e-15
err
coll
9.82e-08
6.88e-09
2.11e-10
6.86e-12
2.11e-13
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
2.05e-11
8.77e-14
4.76e-16
4.83e-16
1.84e-15
88
TABLE 1.37 : Error of error estimate based on h-h/2 for Example 1
-α=3-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.74e-07
2.89e-08
1.81e-09
1.13e-10
7.07e-12
err
coll
5.62e-06
7.02e-07
9.02e-08
1.15e-08
1.44e-09
p`
mesh
p`
coll
const
mesh
const
coll
4.31
4.00
4.00
4.00
3.00
2.96
2.98
2.99
1.18e-02
4.57e-03
4.62e-03
4.62e-03
5.62e-03
5.01e-03
5.27e-03
5.60e-03
p`
mesh
p`
coll
const
mesh
const
coll
3.80
8.59
5.98
6.19
4.56
5.01
4.86
5.01
2.25e-06
1.72e-03
7.48e-06
1.37e-05
3.81e-06
7.08e-06
5.12e-06
7.93e-06
p`
mesh
p`
coll
const
mesh
const
coll
+6.07
+9.70
+2.10
–4.42
+6.59
+6.75
+4.30
–4.18
2.26e-09
3.54e-07
4.71e-14
6.75e-22
3.25e-09
4.10e-09
2.50e-11
1.54e-21
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.61e-07
1.16e-08
3.00e-11
4.76e-13
6.50e-15
err
coll
1.61e-07
6.81e-09
2.11e-10
7.30e-12
2.26e-13
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
3.38e-11
5.05e-13
6.03e-16
1.41e-16
3.01e-15
err
coll
3.38e-11
3.52e-13
3.26e-15
1.66e-16
3.01e-15
89
TABLE 1.38 : Error of error estimate based on h-h/2 using reference solution
for Example 1
-α=01. c) sbvpcol, equidistant, m=2, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
5.35e-05
4.94e-06
1.90e-06
1.71e-06
1.70e-06
err
coll
4.60e-04
5.49e-05
8.04e-06
2.46e-06
1.79e-06
p`
mesh
p`
coll
const
mesh
const
coll
3.44
1.38
0.15
0.01
3.07
2.77
1.71
0.46
5.80e-04
3.34e-05
2.60e-06
1.76e-06
3.85e-03
2.56e-03
2.79e-04
8.79e-06
p`
mesh
p`
coll
const
mesh
const
coll
6.00
5.78
2.64
0.12
5.11
5.06
4.78
2.38
1.20e-06
8.75e-07
1.28e-09
1.21e-12
5.14e-06
4.84e-06
2.72e-06
3.47e-09
p`
mesh
p`
coll
const
mesh
const
coll
+8.24
+2.00
–0.05
+1.41
+7.12
+7.36
–1.33
+1.37
1.22e-09
2.14e-13
2.98e-15
1.73e-13
4.19e-09
5.80e-09
8.32e-17
1.46e-13
2. c) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.87e-08
2.92e-10
5.32e-12
8.57e-13
7.87e-13
err
coll
1.49e-07
4.34e-09
1.30e-10
4.71e-12
9.04e-13
3. c) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
4.02e-12
1.33e-14
3.30e-15
3.42e-15
1.28e-15
err
coll
3.01e-11
2.16e-13
1.31e-15
3.30e-15
1.28e-15
· Exact solution is the reference solution for step size h = 1.5625e-03
90
TABLE 1.39 : Error of error estimate based on h-h/2 using reference solution
for Example 1
-α=11. c) sbvpcol, equidistant, m=2, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.28e-04
9.39e-06
1.74e-06
1.26e-06
1.23e-06
err
coll
2.26e-04
6.63e-05
1.03e-05
2.41e-06
1.38e-06
p`
mesh
p`
coll
const
mesh
const
coll
3.77
2.43
0.46
0.04
1.78
2.69
2.09
0.80
1.74e-03
2.74e-04
4.51e-06
1.41e-06
7.70e-04
2.77e-03
7.94e-04
2.22e-05
p`
mesh
p`
coll
const
mesh
const
coll
5.97
5.92
4.02
0.47
4.75
4.89
4.82
3.13
3.06e-06
2.88e-06
5.50e-08
2.92e-12
4.03e-06
4.88e-06
4.26e-06
3.94e-08
p`
mesh
p`
coll
const
mesh
const
coll
+7.99
+4.64
+0.34
+1.38
+6.81
+10.4
–1.92
+0.70
2.60e-09
2.50e-11
3.27e-15
5.88e-14
3.58e-09
5.09e-07
3.90e-18
5.53e-15
2. c) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
4.89e-08
7.83e-10
1.29e-11
7.95e-13
5.74e-13
err
coll
1.50e-07
5.56e-09
1.88e-10
6.63e-12
7.56e-13
3. c) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.02e-11
4.01e-14
1.61e-15
1.27e-15
4.86e-16
err
coll
3.18e-11
2.83e-13
2.11e-16
8.00e-16
4.93e-16
· Exact solution is the reference solution for step size h = 1.5625e-03
91
TABLE 1.40 : Error of error estimate based on h-h/2 using reference solution
for Example 1
-α=21. c) sbvpcol, equidistant, m=2, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
2.57e-04
1.84e-05
1.74e-06
1.08e-06
1.03e-06
err
coll
2.57e-04
8.21e-05
1.11e-05
2.35e-06
1.20e-06
p`
mesh
p`
coll
const
mesh
const
coll
3.80
3.40
0.69
0.07
1.65
2.89
2.23
0.97
3.58e-03
2.06e-03
7.24e-06
1.32e-06
8.04e-04
4.51e-03
1.15e-03
3.49e-05
p`
mesh
p`
coll
const
mesh
const
coll
5.86
6.58
4.44
0.77
3.84
5.02
4.85
3.42
5.69e-06
1.54e-05
1.80e-07
7.03e-12
1.40e-06
7.27e-06
5.07e-06
9.70e-08
err
coll
p`
mesh
p`
coll
const
mesh
const
coll
3.49e-13
2.94e-15
6.73e-16
4.76e-17
+7.89
+7.50
–0.96
+4.28
+5.87
+6.89
+2.13
+3.82
4.85e-09
2.84e-09
6.44e-17
1.34e-10
1.20e-09
4.93e-09
2.46e-13
2.70e-11
2. c) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
9.82e-08
1.70e-09
1.78e-11
8.21e-13
4.80e-13
err
coll
9.82e-08
6.88e-09
2.12e-10
7.34e-12
6.84e-13
3. c) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
2.05e-11
8.63e-14
4.76e-16
9.27e-16
4.76e-17
· Exact solution is the reference solution for step size h = 1.5625e-03
92
TABLE 1.41 : Error of error estimate based on h-h/2 using reference solution
for Example 1
-α=31. c) sbvpcol, equidistant, m=2, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.27e-03
1.14e-04
2.14e-06
9.72e-07
9.03e-07
err
coll
1.27e-03
1.14e-04
1.22e-05
2.30e-06
1.08e-06
p`
mesh
p`
coll
const
mesh
const
coll
3.48
5.74
1.14
0.11
3.48
3.23
2.40
1.09
1.41e-02
3.24e-01
2.28e-05
1.30e-06
1.41e-02
1.00e-02
1.81e-03
4.74e-05
p`
mesh
p`
coll
const
mesh
const
coll
3.80
8.57
5.09
1.07
4.56
5.01
4.78
3.59
2.25e-06
1.67e-03
1.21e-06
1.74e-11
3.81e-06
7.05e-06
4.38e-06
1.64e-07
p`
mesh
p`
coll
const
mesh
const
coll
+6.08
+9.24
+3.35
–5.73
+6.58
+6.76
+5.85
–6.26
2.28e-09
1.84e-07
8.69e-13
1.01e-23
3.24e-09
4.12e-09
6.33e-10
1.65e-24
2. c) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
1.61e-07
1.16e-08
3.04e-11
8.90e-13
4.23e-13
err
coll
1.61e-07
6.81e-09
2.12e-10
7.71e-12
6.39e-13
3. c) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
3.38e-11
5.00e-13
8.25e-16
8.11e-17
4.31e-15
err
coll
3.38e-11
3.53e-13
3.26e-15
5.64e-17
4.31e-15
· Exact solution is the reference solution for step size h = 1.5625e-03
93
TABLE 1.42 : matrix condition estimates based on max. norm for Example 1
–Midpoint rule -preconditioning (t-h/2)ˆ(α-1) -
1. α=1, Midpoint rule, number of the collocation points m=1
2.
h
cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.17e+00
1.55e+01
3.02e+01
5.96e+01
1.18e+02
ord.
cond
const
cond
-0.92
-0.96
-0.98
-0.99
9.73e-01
8.68e-01
8.11e-01
7.94e-01
ord.
cond
const
cond
-0.96
-0.97
-0.99
-0.99
9.84e-01
9.65e-01
9.02e-01
8.94e-01
ord.
cond
const
cond
-0.98
-1.00
-1.00
-1.00
1.87e+00
1.80e+00
1.78e+00
1.78e+00
norm
(B’)
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B’)
4.08e-01
3.87e-01
3.77e-01
3.73e-01
3.70e-01
norm
(B’)
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B’)
4.54e-01
4.41e-01
4.34e-01
4.30e-01
4.29e-01
norm
(B’)
2.00e+01
4.00e+01
8.00e+01
1.60e+02
3.20e+02
norm
inv(B’)
8.98e-01
8.90e-01
8.88e-01
8.87e-01
8.87e-01
α=2, Midpoint rule
h
Cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
9.07e+00
1.77e+01
3.47e+01
6.89e+01
1.37e+02
3. α=3, Midpoint rule
h
Cond(B’)
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.80e+01
3.56e+01
7.10e+01
1.42e+02
2.84e+02
94
TABLE 1.43 : Error estimate based on Midpoint rule for Example 1
1. α=0, Midpoint rule, number of the collocation points m=1
h
2.00e-01
1.00e-01
5.00e-02
2.50e-02
err exact
3.39e-07
2.11e-08
1.32e-09
8.25e-11
p exact
err estimate
4.00
4.00
4.00
4.00
3.39e-07
2.11e-08
1.32e-09
8.25e-11
p exact
4.02
4.00
4.00
4.00
err estimate
2.43e-07
1.52e-08
9.55e-10
5.97e-11
p exact
4.03
4.01
4.00
4.00
p estimate
4.00
4.00
4.00
4.00
errexc-errest
p
2.44e-10
3.80e-12
5.48e-14
4.66e-15
+6.00
+6.12
+3.56
−0.82
p estimate
4.00
3.99
4.00
4.00
errexc-errest
2.51e-08
7.61e-10
2.35e-11
7.27e-13
p
5.04
5.02
5.01
5.02
err estimate
2.06e-07
1.28e-08
7.96e-10
4.99e-11
p estimate
4.02
4.00
4.00
4.00
errexc-errest
7.17e-08
3.26e-09
1.24e-10
4.45e-12
p
4.46
4.72
4.80
4.80
p exact
4.07
3.98
4.00
4.00
err estimate
1.78e-07
1.09e-08
6.98e-10
4.36e-11
p estimate
4.02
3.97
4.00
4.00
errexc-errest
1.50e-07
5.35e-09
2.26e-10
1.02e-11
p
4.81
4.56
4.47
4.55
p exact
4.44
4.02
4.00
4.00
err estimate
1.44e-07
1.01e-08
6.26e-10
3.91e-11
p estimate
3.84
4.01
4.00
4.00
errexc-errest
1.80e-07
8.26e-09
3.85e-10
1.77e-11
p
4.45
4.42
4.44
4.43
p exact
4.41
4.38
4.16
4.00
err estimate
1.38e-07
9.01e-09
5.68e-10
3.57e-11
p estimate
3.93
3.99
3.99
4.00
errexc-errest
2.22e-07
1.18e-08
5.48e-10
2.62e-11
p
4.23
4.43
4.39
4.38
1. α=1, Midpoint rule
h
2.00e-01
1.00e-01
5.00e-02
2.50e-02
err exact
2.47e-07
1.53e-08
9.57e-10
5.98e-11
3. α=2, Midpoint rule
h
2.00e-01
1.00e-01
5.00e-02
2.50e-02
err exact
2.11e-07
1.29e-07
8.00e-10
5.00e-11
4. α=3, Midpoint rule
h
2.00e-01
1.00e-01
5.00e-02
2.50e-02
err exact
1.85e-07
1.11e-08
7.02e-10
4.38e-11
4. α=4, Midpoint rule
h
2.00e-01
1.00e-01
5.00e-02
2.50e-02
err exact
2.22e-07
1.02e-08
6.30e-10
3.93e-11
4. α=4, Midpoint rule
h
2.00e-01
1.00e-01
5.00e-02
2.50e-02
err exact
2.86e-07
1.34e-08
6.42e-10
3.58e-11
95
Figure 2.1 : plot of solution and error using sbvp for Example 2
1. a) Solution, β=0
b) Error, β=0
2. a) Solution, β=1
b) Error, β=1
3. a) Solution, β=2
b) Error, β=2
96
TABLE 2.1 : global error for Example 2
-β=0*1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.02e-02
1.26e-02
7.01e-05
2.19e-06
1.17e-07
err
coll
3.02e-02
1.26e-02
7.01e-05
2.24e-06
1.34e-07
p
mesh
p
coll
const
mesh
const
coll
1.26
7.49
5.00
4.23
1.26
7.49
4.97
4.06
5.50e–01
7.00e+07
7.23e+03
2.42e+02
5.50e–01
7.00e+07
6.37e+03
1.19e+02
err
coll
3.12e-01
3.63e-02
5.26e-04
7.16e-05
8.53e-06
p
mesh
p
coll
const
mesh
const
coll
3.10
6.72
3.68
4.00
3.10
6.11
2.88
3.07
3.97e+02
2.01e+07
2.76e+02
1.09e+03
3.97e+02
3.22e+06
2.13e+01
4.97e+01
err
coll
4.38e-02
3.50e-03
7.11e-05
4.36e-06
2.78e-07
p
mesh
p
coll
const
mesh
const
coll
3.65
5.90
5.98
5.65
3.64
5.62
4.03
3.97
1.93e+02
1.69e+05
2.26e+05
5.13e+04
1.93e+02
7.19e+04
2.01e+02
1.58e+02
err
coll
2.03e-02
1.90e-03
9.35e-06
3.58e-07
1.21e-08
p
mesh
p
coll
const
mesh
const
coll
3.41
8.33
6.70
7.17
3.41
7.67
4.71
4.89
5.31e+01
1.32e+08
3.21e+05
2.57e+06
5.31e+01
1.79e+07
3.23e+02
7.28e+01
2. sbvpcol, Gauss, m =2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.12e-01
3.63e-02
3.44e-04
2.68e-05
1.67e-06
3. sbvpcol, Gauss, m =3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.38e-02
3.50e-03
5.83e-05
9.21e-07
1.84e-08
4. sbvpcol, Gauss, m =4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.03e-02
1.90e-03
5.89e-06
5.65e-08
3.91e-10
(*) Exact solution is a reference solution for step size h = 3.125e-003
97
TABLE 2.2 : global error for Example 2
-β=1*1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.27e-03
1.23e-04
2.44e-06
1.42e-07
8.45e-09
err
coll
5.27e-03
1.23e-04
2.55e-06
1.65e-07
9.19e-09
p
mesh
p
coll
const
mesh
const
coll
5.42
5.66
4.10
4.07
5.42
5.59
3.95
4.16
1.42e+03
2.83e+03
9.06e+00
7.97e+00
1.42e+03
2.31e+03
5.50e+00
1.39e+01
err
coll
3.61e-02
9.51e-04
1.04e-04
1.28e-05
1.51e-06
p
mesh
p
coll
const
mesh
const
coll
5.25
4.59
3.94
4.07
5.25
3.12
3.01
3.08
6.36e+03
8.80e+02
8.18e+01
1.43e+02
6.36e+03
1.38e+01
6.99e+00
9.34e+00
err
coll
2.70e-03
8.08e-05
5.71e-06
3.55e-07
2.34e-08
p
mesh
p
coll
const
mesh
const
coll
5.85
5.44
5.78
5.97
5.06
3.82
4.01
3.92
1.91e+03
5.62e+02
1.95e+03
4.55e+03
3.11e+02
7.60e+00
1.50e+01
1.03e+01
err
coll
6.58e-04
1.28e-05
3.99e-07
1.33e-08
4.29e-10
p
mesh
p
coll
const
mesh
const
coll
6.37
7.51
7.58
7.91
5.69
5.00
4.91
4.95
1.56e+03
4.71e+04
6.18e+04
2.55e+05
3.21e+02
4.05e+01
2.93e+01
3.55e+01
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.61e-02
9.51e-04
3.96e-05
2.58e-06
1.53e-07
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.70e-03
4.68e-05
1.08e-06
1.96e-08
3.12e-10
4. sbvpcol, Gauss, m =4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
6.58e-04
7.93e-06
4.34e-08
2.26e-10
9.40e-13
(*) Exact solution is a reference solution for step size h = 3.125e-003
98
TABLE 2.3 : global error for Example 2
-β=2*1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.15e-03
1.40e-05
6.81e-07
4.21e-08
2.50e-09
err
coll
2.15e-03
1.40e-05
7.60e-07
4.94e-08
2.76e-09
p
mesh
p
coll
const
mesh
const
coll
7.22
4.36
4.01
4.07
7.22
4.20
3.94
4.16
3.53e+04
6.71e+00
1.83e+00
2.39e+00
3.53e+04
4.11e+00
1.58e+00
4.15e+00
err
coll
3.10e-03
3.09e-04
4.16e-05
5.22e-06
6.09e-07
p
mesh
p
coll
const
mesh
const
coll
4.11
3.76
3.94
4.04
3.32
2.89
2.99
3.10
4.37e+01
1.14e+01
2.69e+01
4.21e+01
6.56e+00
1.80e+00
2.61e+00
4.10e+00
err
coll
1.90e-03
2.97e-05
1.84e-06
1.19e-07
7.56e-09
p
mesh
p
coll
const
mesh
const
coll
7.44
5.42
5.92
5.88
6.00
4.02
3.95
3.97
5.21e+04
1.23e+02
7.78e+02
6.58e+02
1.89e+03
5.02e+00
3.94e+00
4.30e+00
err
coll
8.12e-05
3.09e-06
1.11e-07
3.38e-09
1.13e-10
p
mesh
p
coll
const
mesh
const
coll
6.34
7.18
7.62
7.87
4.72
4.79
5.03
4.91
1.80e+02
2.20e+03
1.14e+04
3.36e+04
4.23e+00
5.34e+00
1.32e+01
7.33e+00
2. sbvpcol, Gauss, m =2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.10e-03
1.79e-04
1.32e-05
8.60e-07
5.22e-08
3. sbvpcol, Gauss, m =3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.90e-03
1.10e-05
2.56e-07
4.23e-09
7.17e-11
4. sbvpcol, Gauss, m =4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
8.12e-05
9.98e-07
6.88e-09
3.47e-11
1.48e-13
(*) Exact solution is a reference solution for step size h = 3.125e-03
99
TABLE 2.4 : matrix condition estimates for Example 2
- β=01. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.21e+04
1.88e+05
6.23e+05
2.24e+06
8.42e+06
1.95e+04
7.30e+04
2.90e+05
1.17e+06
4.68e+06
ord.
condest
ord.
cond
const
condest
const
cond
-1.60
-1.73
-1.84
-1.91
-1.90
-1.99
-2.01
-2.00
1.58e+03
1.04e+03
6.95e+02
5.17e+02
ord.
condest
ord.
cond
const
condest
-1.34
-1.79
-1.92
-1.96
-1.56
-2.01
-2.01
-2.01
1.95e+03
5.13e+02
3.07e+02
2.62e+02
4.92e+02
1.28e+02
1.28e+02
5.23e+02
ord.
condest
ord.
cond
const
condest
const
cond
-1.31
-1.35
-1.88
-1.94
-1.39
-1.80
-2.01
-2.01
1.07e+04
9.16e+03
1.32e+03
1.04e+03
ord.
condest
ord.
cond
const
condest
-1.29
-1.32
-1.45
-1.94
-1.41
-1.43
-2.01
-2.01
4.19e+04
3.89e+04
2.41e+04
2.82e+03
2.43e+02
1.88e+02
1.78e+02
1.83e+02
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.26e+04
1.08e+05
3.71e+05
1.41e+06
5.48e+06
1.79e+04
5.28e+04
2.13e+05
8.58e+05
3.45e+06
const
cond
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.17e+05
5.38e+05
1.38e+06
5.09e+06
1.95e+07
7.62e+04
2.00e+05
6.96e+05
2.80e+06
1.13e+07
3.08e+03
9.18e+02
4.20e+02
4.26e+02
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.20e+05
2.01e+06
5.00e+06
1.36e+07
5.20e+07
2.48e+05
6.57e+05
1.77e+06
7.12e+06
2.86e+07
· condestDF:=condest(DF,1), cond:=cond(DF,2)
100
const
cond
9.67e+03
9.04e+03
1.08e+03
1.08e+03
TABLE 2.5 : matrix condition estimates for Example 2
- β=11. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.31e+06
1.90e+07
1.22e+08
7.36e+08
6.37e+09
1.60e+06
1.14e+07
8.98e+07
7.22e+08
5.81e+09
ord.
condest
ord.
cond
const
condest
-2.52
-2.68
-2.59
-3.11
-2.83
-2.97
-3.01
-3.01
9.94e+03
6.18e+03
8.57e+03
8.80e+02
ord.
condest
ord.
cond
const
condest
-2.47
-2.57
-2.54
-2.73
-2.64
-3.02
-3.02
-3.01
1.19e+04
9.03e+03
1.01e+04
4.31e+03
ord.
condest
ord.
cond
const
condest
-2.66
-2.57
-2.53
-2.52
-2.63
-2.57
-2.80
-3.01
7.52e+04
9.98e+04
1.13e+05
1.23+05
ord.
condest
ord.
cond
const
condest
-2.71
-2.58
-2.53
-2.52
-2.62
-2.57
-2.54
-2.74
1.49e+03
6.19e+05
7.27e+05
7.83e+05
const
cond
2.34e+03
1.52e+03
1.37e+03
1.36e+03
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.54e+06
1.96e+07
1.16e+08
6.74e+08
4.47e+09
1.26e+06
7.86e+06
6.36e+07
5.15e+08
4.15e+09
const
cond
2.89e+03
9.36e+02
9.35e+02
9.61e+02
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.46e+07
2.19e+08
1.30e+09
7.51e+09
4.30e+10
1.24e+07
7.66e+07
4.55e+08
3.16e+09
2.54e+10
const
cond
2.92e+04
3.49e+04
1.51e+04
5.92e+03
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.13e+08
1.39e+09
8.31e+09
4.81e+10
2.75e+11
7.41e+07
4.55e+08
2.70e+09
1.57e+10
1.05e+11
· condestDF:=condest(DF,1), cond:=cond(DF,2)
101
const
cond
1.78e+05
2.08e+05
2.32e+05
9.40e+04
TABLE 2.6 : matrix condition estimates for Example 2
- β=21. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.67e+08
1.53e+09
2.35e+10
2.87e+11
3.70e+12
7.18e+07
9.37e+08
1.45e+10
2.33e+11
3.76e+12
ord.
condest
ord.
cond
const
condest
-3.20
-3.94
-3.61
-3.69
-3.71
-3.95
-4.01
-4.01
1.06e+05
1.14e+04
3.84e+04
2.74e+04
ord.
condest
ord.
cond
const
condest
-3.63
-3.68
-3.64
-3.62
-3.78
-3.69
-3.64
-3.82
5.08e+04
4.42e+04
5.15e+04
5.64e+04
ord.
condest
ord.
cond
const
condest
-3.73
-3.66
-3.63
-3.61
-3.75
-3.67
-3.63
-3.62
8.28e+05
1.03e+06
1.45e+07
1.25e+06
ord.
condest
ord.
cond
const
condest
-3.71
-3.65
-3.62
-3.61
-3.72
-3.66
-3.63
-3.61
9.62e+06
1.16e+07
1.28e+07
1.35e+07
const
cond
1.41e+04
6.81e+03
5.50e+03
5.39e+03
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.18e+08
2.70e+09
3.46e+10
4.31e+11
5.29e+12
8.13e+07
1.12e+09
1.45e+10
1.81e+11
2.55e+12
const
cond
1.34e+04
1.76e+04
2.11e+04
9.77e+03
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.50e+09
5.99e+10
7.59e+11
9.38e+12
1.15e+14
1.63e+09
2.20e+10
2.79e+11
3.47e+12
4.25e+13
const
cond
2.91e+05
3.69e+05
5.24e+06
4.55e+05
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.95e+10
6.49e+11
8.15e+12
1.00e+14
1.23e+15
1.67e+10
2.19e+11
2.78e+12
3.44e+13
4.21e+14
· condestDF:=condest(DF,1), cond:=cond(DF,2)
102
const
cond
3.18e+06
3.78e+06
4.25e+06
4.54e+06
TABLE 2.7 : global error for Example 2
- β=0-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.25e-01
3.35e-02
1.84e-03
4.34e-04
8.65e-05
err
coll
2.25e-01
3.35e-02
1.89e-03
4.50e-04
9.15e-05
p
mesh
p
coll
const
mesh
const
coll
2.75
4.19
2.08
2.33
2.75
4.15
2.07
2.30
1.25e+02
9.37e+03
4.00e+00
1.17e+01
1.25e+02
8.31e+03
3.93e+00
1.06e+01
p
mesh
p
coll
const
mesh
const
coll
1.26
7.49
5.00
4.23
1.26
7.49
4.97
4.06
5.50e–01
7.00e+07
7.23e+03
2.42e+02
5.50e–01
7.00e+07
6.37e+03
1.19e+02
p
mesh
p
coll
const
mesh
const
coll
2.96
7.58
8.55
6.55
2.96
7.58
8.55
6.50
2.49e+01
2.53e+07
9.02e+08
1.43e+05
2.49e+01
2.53e+07
9.02e+08
1.26e+03
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.02e-02
1.26e-02
7.01e-05
2.19e-06
1.17e-07
err
coll
3.02e-02
1.26e-02
7.01e-05
2.24e-06
1.34e-07
3. a) sbvpcol, equidistant, m=6, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.73e-02
3.51e-03
1.84e-05
4.92e-08
5.25e-10
err
coll
2.73e-02
3.51e-03
1.84e-05
4.92e-08
5.45e-10
· Exact solution is a reference solution for step size h = 3.125e-03
103
TABLE 2.8 : global error for Example 2
- β=1-
1. a) sbvpcol, equidistant, m=2, Error
H
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.51e-02
7.21e-03
1.85e-03
4.44e-04
8.90e-05
err
coll
2.51e-02
7.21e-03
1.85e-03
4.44e-04
8.90e-05
p
mesh
p
coll
const
mesh
const
coll
1.80
1.96
2.06
2.32
1.80
1.96
2.06
2.32
1.58e+00
2.58e+00
3.68e+00
1.15e+01
1.58e+00
2.58e+00
3.68e+00
1.15e+01
2. a) sbvpcol, equidistant, m=4, Error
H
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.27e-03
1.23e-04
2.44e-06
1.42e-07
8.45e-09
err
coll
5.27e-03
1.23e-04
2.55e-06
1.65e-07
9.19e-09
p
mesh
p
coll
const
mesh
const
coll
5.42
5.66
4.10
4.07
5.42
5.59
3.95
4.16
1.42e+03
2.83e+03
9.06e+00
7.97e+00
1.42e+03
2.31e+03
5.50e+00
1.39e+01
3. a) sbvpcol, equidistant, m=6, Error
H
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.65e-03
8.24e-06
3.76e-08
4.40e-10
6.52e-12
err
coll
1.65e-03
8.24e-06
3.76e-08
4.88e-10
7.84e-12
p
mesh
p
coll
const
mesh
const
coll
7.65
7.78
6.42
6.08
7.65
7.78
6.27
5.96
7.30e+04
1.08e+05
7.22e+02
1.61e+02
7.30e+04
1.08e+05
4.14e+02
1.07e+02
· Exact solution is a reference solution for step size h = 3.125e-03
104
TABLE 2.9 : global error for Example 2
- β=2-
1. a) sbvpcol, equidistant, m=2, Error
H
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.05e-02
1.23e-02
3.03e-03
7.21e-04
1.44e-04
err
coll
5.05e-02
1.23e-02
3.03e-03
7.21e-04
1.44e-04
p
mesh
p
coll
const
mesh
const
coll
2.04
2.02
2.07
2.32
2.04
2.02
2.07
2.32
5.51e+00
5.24e+00
6.32e+00
1.89e+01
5.51e+00
5.24e+00
6.32e+00
1.89e+01
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.15e-03
1.40e-05
6.81e-07
4.21e-08
2.50e-09
err
coll
2.15e-03
1.40e-05
7.60e-07
4.94e-08
2.76e-09
P
mesh
p
coll
const
mesh
const
coll
7.22
4.36
4.01
4.07
7.22
4.20
3.94
4.16
3.53e+04
6.71e+00
1.83e+00
2.39e+00
3.53e+04
4.11e+00
1.58e+00
4.15e+00
3. a) sbvpcol, equidistant, m=6, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.21e-04
7.01e-07
7.33e-09
9.05e-11
1.41e-12
err
coll
1.21e-04
7.01e-07
7.33e-09
1.07e-10
1.57e-12
p
mesh
p
coll
const
mesh
const
coll
7.43
6.58
6.34
6.00
7.43
6.58
6.10
6.09
3.21e+03
2.56e+02
1.05e+02
2.40e+01
3.21e+03
2.56e+02
4.37e+01
4.06e+01
· Exact solution is a reference solution for step size h = 3.125e-03
105
TABLE 2.10 : Error of error estimate based on h-h/2 for Example 2
- β=0-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.73e-02
1.31e-02
1.76e-05
1.09e-06
4.93e-07
err
coll
3.73e-02
1.31e-02
1.17e-04
1.75e-05
3.07e-06
p`
mesh
p`
coll
const
mesh
const
coll
1.51
9.54
4.02
1.14
1.51
6.81
2.74
2.51
1.21e+00
3.36e+10
4.84e+01
1.61e-04
1.21e+00
9.39e+06
2.89e+00
1.05e+00
p`
mesh
p`
coll
const
mesh
const
coll
2.73
7.30
8.38
6.13
2.73
7.30
8.08
4.07
3.07e+00
2.76e+06
1.48e+08
7.79e+03
3.07e+00
2.76e+06
4.92e+07
1.15e+00
p`
mesh
p`
coll
const
mesh
const
coll
5.09
6.90
9.55
8.10
5.09
6.90
9.55
9.43
1.54e+02
3.48e+04
6.18e+08
1.10e+06
1.54e+02
3.48e+04
6.18e+08
3.71e+08
2. b) sbvpcol, equidistant, m=4, Error of error
H
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.77e-03
8.73e-04
5.53e-06
1.66e-08
2.36e-10
err
coll
5.77e-03
8.73e-04
5.53e-06
2.04e-08
1.21e-09
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.26e-03
3.71e-05
3.11e-07
4.16e-10
1.51e-12
err
coll
1.26e-03
3.71e-05
3.11e-07
4.16e-10
6.01e-13
· Exact solution is a reference solution for step size h = 3.1250e-03 (m=2,4)
· Exact solution is a reference solution for step size h = 3.9063e-04 (m=6)
106
TABLE 2.11 : Error of error estimate based on h-h/2 for Example 2
- β=1-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.11e-03
8.76e-05
5.75e-06
6.09e-08
4.39e-07
err
coll
1.08e-03
8.76e-05
5.75e-06
6.09e-08
4.39e-07
p`
mesh
p`
coll
const
mesh
const
coll
+3.66
+3.93
+6.56
–2.85
+3.66
+3.93
+6.56
–2.85
5.11e+00
1.13e+01
1.87e+05
2.31e–13
5.11e+00
1.13e+01
1.87e+05
2.31e–13
p`
mesh
p`
coll
const
mesh
const
coll
4.78
8.62
8.46
4.11
7.23
4.76
3.85
4.65
1.32e+01
1.32e+06
7.20e+05
3.89e–03
1.39e+03
8.64e–01
2.91e–02
9.55e–01
p`
mesh
p`
coll
const
mesh
const
coll
7.10
8.26
8.58
10.7
7.10
8.26
9.03
5.90
2.26e+02
7.13e+03
2.38e+04
2.25e+08
2.26e+02
7.13e+03
1.23e+05
1.35e–01
2. b) sbvpcol, equidistant, m=4, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.20e-04
8.01e-06
2.03e-08
5.77e-11
3.34e-12
err
coll
8.20e-05
5.46e-07
2.01e-08
1.40e-09
5.60e-11
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.78e-05
1.29e-07
4.23e-10
1.10e-12
6.74e-16
err
coll
1.78e-05
1.29e-07
4.23e-10
8.11e-13
1.36e-14
· Exact solution is a reference solution for step size h = 3.1250e-03 (m=2,4)
· Exact solution is a reference solution for step size h = 3.9063e-04 (m=6)
107
TABLE 2.12 : Error of error estimate based on h-h/2 for Example 2
- β=2-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.11e-04
6.68e-06
1.12e-06
7.73e-07
7.52e-07
Err
coll
4.11e-04
6.68e-06
1.12e-06
7.73e-07
7.52e-07
p`
mesh
p`
coll
const
mesh
const
coll
5.94
2.58
0.53
0.04
5.94
2.58
0.53
0.04
3.61e+02
1.51e–02
8.03e–06
9.24e–07
3.61e+02
1.51e–02
8.03e–06
9.24e–07
p`
mesh
p`
coll
const
mesh
const
coll
7.51
6.19
8.62
5.81
7.51
6.98
3.88
4.73
4.57e+03
8.78e+01
6.95e+05
3.06e+00
4.57e+03
9.36e+02
1.01e–01
4.22e–01
p`
mesh
p`
coll
const
mesh
const
coll
7.44
8.29
11.3
5.79
7.44
8.29
7.17
6.62
4.93e+01
1.95e+05
4.59e+07
1.35e–03
4.93e+01
1.95e+05
1.02e+01
8.98e–01
2. b) sbvpcol, equidistant, m=4, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.42e-04
7.77e-07
1.06e-08
2.69e-11
4.79e-13
err
coll
1.42e-04
7.77e-07
6.16e-09
4.19e-10
1.58e-11
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.79e-06
1.03e-08
3.30e-11
1.29e-14
2.32e-16
err
coll
1.79e-06
1.03e-08
3.30e-11
2.29e-13
2.33e-15
· Exact solution is the reference solution for step size h = 3.1250e-03 (m=2,4)
· Exact solution is the reference solution for step size h = 3.9063e-04 (m=6)
108
Figure 3.1 : plot of solution and error using sbvp for Example 3
109
TABLE 3.1 : global error for Example 3
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.39e-17
1.06e-18
2.83e-20
1.63e-21
9.37e-23
err
coll
1.39e-17
1.06e-18
3.06e-20
1.74e-21
9.95e-23
p
mesh
p
coll
3.71
5.23
4.12
4.12
3.71
5.11
4.14
4.13
err
coll
1.01e-16
1.35e-17
1.44e-18
1.92e-19
2.23e-20
p
mesh
p
coll
2.90
3.73
4.04
4.08
2.90
3.22
2.91
3.10
err
coll
2.77e-17
9.98e-19
8.70e-20
6.38e-21
4.15e-22
p
mesh
p
coll
5.27
4.42
5.79
5.94
4.80
3.52
3.77
3.94
err
coll
2.56e-18
2.12e-19
5.57e-21
2.16e-22
6.96e-24
p
mesh
p
coll
3.03
7.04
7.13
7.67
3.59
5.25
4.69
4.95
const
mesh
const
coll
7.13e-14
6.81e-12
1.11e-13
1.13e-13
7.13e-14
1.38e-13
7.46e-15
7.07e-15
const
mesh
const
coll
8.07e-14
7.18e-14
1.85e-13
2.13e-13
8.07e-14
2.24e-14
8.89e-15
1.78e-14
const
mesh
const
coll
5.17e-12
3.99e-13
6.25e-11
1.25e-10
1.73e-12
3.79e-14
9.49e-14
2.03e-13
const
mesh
const
coll
1.15e-15
1.91e-10
2.61e-10
2.81e-09
9.96e-15
1.44e-12
1.82e-13
5.76e-13
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.01e-16
1.35e-17
1.02e-18
6.16e-20
3.64e-21
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.77e-17
7.17e-19
3.36e-20
6.09e-22
9.89e-24
4. sbvpcol, Gauss , m =4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.07e-18
1.31e-19
9.95e-22
7.11e-24
3.49e-26
· Exact solution is a reference solution for step size h = 3.125e-003
110
TABLE 3.2 : matrix condition estimates for Example 3
1. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.20e+04
4.28e+05
3.16e+06
2.42e+07
1.89e+08
2.01e+04
1.53e+05
1.20e+06
9.56e+06
7.62e+07
ord.
condest
ord.
cond
const
condest
const
cond
-2.79
-2.88
-2.94
-2.97
-2.93
-2.98
-2.99
-3.00
1.01e+02
7.60e+01
6.19e+01
5.44e+01
ord.
condest
ord.
cond
const
condest
-2.78
-2.88
-2.94
-2.97
-2.93
-2.98
-2.99
-3.00
3.68e+02
2.69e+02
2.20e+02
1.93e+02
8.39e+01
7.24e+01
6.88e+01
6.74e+01
ord.
condest
ord.
cond
const
condest
const
cond
-2.71
-2.83
-2.91
-2.95
-2.90
-2.97
-2.99
-3.00
3.23e+03
2.24e+03
1.69e+03
1.40e+03
ord.
condest
ord.
cond
const
condest
-1.05
-4.39
-2.88
-2.94
-2.87
-2.96
-2.99
-3.00
7.11e+05
3.20e+01
8.37e+03
6.54e+03
2.39e+01
2.04e+01
1.94e+01
1.90e+01
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.22e+05
1.52e+06
1.12e+07
8.61e+07
6.74e+08
7.12e+04
5.42e+05
4.27e+06
3.39e+07
2.71e+08
const
cond
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.65e+06
1.08e+07
7.70e+07
5.78e+08
4.47e+09
4.48e+05
3.34e+06
2.62e+07
2.08e+08
1.66e+09
4.22e+03
4.55e+02
4.25e+02
4.14e+02
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.93e+06
1.64e+07
3.43e+08
2.52e+09
1.93e+10
1.90e+06
1.39e+07
1.08e+08
8.59e+08
6.85e+09
· condestDF:=condest(DF,1), cond:=cond(DF,2)
111
const
cond
2.57e+03
1.92e+03
1.77e+03
1.71e+03
TABLE 3.3 : global error for Example 3
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.78e-16
2.42e-17
5.73e-18
1.29e-18
2.55e-19
err
coll
1.78e-16
2.42e-17
5.73e-18
1.34e-18
2.68e-19
p
mesh
p
coll
const
mesh
const
coll
2.88
2.08
2.15
2.34
2.88
2.08
2.10
2.32
1.34e-13
1.22e-14
1.58e-14
3.70e-14
1.34e-13
1.22e-14
1.31e-14
3.53e-14
p
mesh
p
coll
const
mesh
const
coll
3.71
5.23
4.12
4.12
3.71
5.11
4.14
4.13
7.13e-14
6.81e-12
1.11e-13
1.13e-13
7.13e-14
1.38e-13
7.46e-15
7.07e-15
p
mesh
p
coll
const
mesh
const
coll
6.19
6.72
6.71
6.17
6.19
6.72
6.48
6.15
4.31e-12
2.08e-11
2.02e-11
1.94e-12
4.31e-12
2.08e-11
8.71e-12
2.03e-12
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.39e-17
1.06e-18
2.83e-20
1.63e-21
9.37e-23
err
coll
1.39e-17
1.06e-18
3.06e-20
1.74e-21
9.95e-23
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
2.77e-18
3.79e-20
3.61e-22
3.45e-24
4.77e-26
err
coll
2.77e-18
3.79e-20
3.61e-22
4.04e-24
5.70e-26
· Exact solution is a reference solution for step size h = 3.125e-03
112
TABLE 3.4 : Error of error based on h-h/2 for Example 3
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.14e-17
3.93e-18
1.02e-19
7.64e-21
1.72e-21
err
coll
3.14e-17
3.93e-18
1.02e-19
4.80e-20
7.90e-21
p
mesh
p
coll
const
mesh
const
coll
3.00
5.27
3.74
2.15
3.00
5.27
1.08
2.60
3.13e-14
2.84e-11
9.80e-14
9.61e-17
3.13e-14
2.84e-11
5.55e-18
4.30e-15
p
mesh
p
coll
const
mesh
const
coll
2.81
5.61
8.39
5.93
2.81
6.60
5.44
4.64
1.86e-16
8.13e-13
2.26e-08
4.94e-13
1.86e-16
1.59e-11
2.18e-13
6.48e-15
p
mesh
p
coll
const
mesh
const
coll
6.75
6.58
9.17
8.92
6.75
6.58
8.14
6.53
1.82e-13
1.09e-13
1.58e-09
1.05e-09
1.82e-13
1.09e-13
3.49e-11
3.08e-14
2. b) sbvpcol, equidistant, m=4, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.86e-19
4.07e-20
8.32e-22
2.49e-24
4.06e-26
err
coll
2.86e-19
4.07e-20
4.19e-22
9.64e-24
3.87e-25
3. b) sbvpcol, equidistant, m=6, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.26e-20
3.03e-22
3.18e-24
5.51e-27
2.33e-29
err
coll
3.26e-20
3.03e-22
3.18e-24
1.13e-26
1.22e-28
· Exact solution is the reference solution for step size h =1/2560
113
Figure 4.1 : plot of solution and error using sbvp for Example 4
114
TABLE 4.1 : global error using reference solution for Example 4
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.73e-05
1.00e-04
2.19e-05
1.31e-06
8.24e-08
err
coll
1.48e-04
1.00e-04
2.19e-05
1.31e-06
8.24e-08
p
mesh
p
coll
const
mesh
const
coll
–0.37
+2.19
+4.06
+3.99
+0.57
+2.19
+4.06
+3.99
3.28e–05
7.11e–02
7.09e+01
5.13e+01
6.30e–05
7.11e–02
7.09e+01
5.13e+01
p
mesh
p
coll
const
mesh
const
coll
+1.44
–0.24
+2.54
–0.50
+1.98
+0.41
+2.52
–0.48
3.62e-03
2.39e-05
6.70e-01
1.15e-06
2.94e-02
2.62e-04
6.30e-01
1.25e-06
p
mesh
p
coll
const
mesh
const
coll
–2.23
+3.78
+4.03
+4.07
+1.71
+3.78
+4.03
+4.07
5.55e–08
3.63e+00
9.26e+00
1.08e+01
7.40e–07
3.63e+00
9.26e+00
1.08e+01
p
mesh
p
coll
const
mesh
const
coll
3.16
3.65
4.04
4.07
3.16
3.65
4.04
4.07
5.10e–01
2.22e+00
9.42e+00
1.10e+01
5.16e–01
2.22e+00
9.42e+00
1.10e+01
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.33e-04
4.92e-05
5.82e-05
1.00e-05
1.42e-05
err
coll
3.05e-04
7.72e-05
5.82e-05
1.02e-05
1.42e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
9.34e-06
4.37e-05
3.18e-06
1.94e-07
1.15e-08
err
coll
1.43e-04
4.37e-05
3.18e-06
1.94e-07
1.15e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.56e-04
3.99e-05
3.19e-06
1.94e-07
1.15e-08
err
coll
3.57e-04
3.99e-05
3.19e-06
1.94e-07
1.15e-08
· Exact solution is the reference solution for step size h =1.5626e-03
115
TABLE 4.2 : matrix condition estimates for Example 4
1. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
8.91e+09
2.61e+09
1.45e+11
3.21e+11
1.17e+12
2.47e+09
7.26e+08
6.98e+10
1.00e+11
3.78e+11
ord.
condest
ord.
cond
const
condest
const
cond
+1.77
–6.49
–0.45
–1.87
+1.76
–6.59
–0.52
–1.92
5.25e+11
9.31e+00
4.52e+10
8.77e+07
1.43e+11
1.96e+00
1.02e+10
2.26e+07
ord.
condest
ord.
cond
const
condest
const
cond
-3.75
-1.08
-3.28
-2.42
-3.68
-1.09
-3.30
-2.43
1.98e+05
5.91e+08
1.75e+05
7.72e+06
8.72e+04
2.04e+08
5.86e+04
2.64e+06
ord.
condest
ord.
cond
const
condest
const
cond
–5.60
+0.83
–1.74
–1.97
–5.58
+0.78
–1.79
–1.99
1.93e+04
4.44e+12
3.31e+08
1.26e+08
6.19e+03
1.15e+12
8.86e+07
3.64e+07
ord.
condest
ord.
cond
const
condest
const
cond
+0.96
+0.77
–1.72
–1.95
+0.92
+0.68
–1.79
–1.99
1.70e+13
9.76e+12
1.01e+09
3.61e+08
4.29e+12
2.01e+12
2.26e+08
9.24e+07
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.12e+09
1.52e+10
3.21e+10
3.13e+11
1.68e+12
4.17e+08
5.34e+09
1.14e+10
1.12e+11
6.04e+11
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.63e+09
3.69e+11
2.08e+11
6.97e+11
2.72e+12
2.35e+09
1.12e+11
6.56e+10
2.27e+11
9.05e+11
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.88e+12
9.69e+11
5.68e+11
1.87e+12
7.22e+12
5.07e+11
2.66e+11
1.67e+11
5.77e+11
2.30e+12
· condestDF:=condest(DF,1), cond:=cond(DF,2)
116
TABLE 4.3 : global error using reference solution for Example 4
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.07e-02
2.45e-03
6.81e-04
1.94e-04
4.70e-05
err
coll
1.07e-02
2.45e-03
6.81e-04
1.94e-04
4.70e-05
p
mesh
p
coll
const
mesh
const
coll
2.13
1.85
1.81
2.05
2.13
1.85
1.81
2.05
1.43e+00
6.18e–01
5.38e–01
1.55e+00
1.43e+00
6.18e–01
5.38e–01
1.55e+00
p
mesh
p
coll
const
mesh
const
coll
–0.37
+2.19
+4.06
+3.99
0.57
2.19
4.06
3.99
3.28e–05
7.11e–02
7.09e+01
5.13e+01
6.30e–05
7.11e–02
7.09e+01
5.13e+01
p
mesh
p
coll
const
mesh
const
coll
1.62
3.66
4.04
4.08
1.87
3.66
4.04
4.08
5.18e–03
2.33e+00
9.44e+00
1.11e+01
1.10e–02
2.33e+00
9.44e+00
1.11e+01
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.73e-05
1.00e-04
2.19e-05
1.31e-06
8.24e-08
err
coll
1.48e-04
1.00e-04
2.19e-05
1.31e-06
8.24e-08
3. a) sbvpcol, equidistant, m=6, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.24e-04
4.03e-05
3.19e-06
1.94e-07
1.15e-08
err
coll
1.48e-04
4.03e-05
3.19e-06
1.94e-07
1.15e-08
· Exact solution is the reference solution for step size h =1.5626e-03
117
TABLE 4.4 : Error of error estimate based on h-h/2 for Example 4
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.92e-04
9.20e-05
3.22e-05
1.15e-04
1.83e-05
err
coll
2.92e-04
9.20e-05
3.22e-05
1.15e-04
1.83e-05
p
mesh
p
coll
const
mesh
const
coll
+1.67
+1.51
−1.84
+2.65
+1.67
+1.51
−1.84
+2.65
1.36e−02
8.59e−03
3.70e−08
1.27e+01
1.36e−02
8.59e−03
3.70e−08
1.27e+01
p
mesh
p
coll
const
mesh
const
coll
−1.51
+8.90
+6.66
+0.91
+0.82
+8.90
+6.66
+0.91
3.27e−07
1.13e+07
2.92e+03
3.29e+08
3.52e−04
1.13e+07
2.92e+03
3.29e+08
p
mesh
p
coll
const
mesh
const
coll
3.47
4.72
4.09
4.16
2.32
4.72
4.09
4.16
1.26e−01
5.47e+00
5.20e−01
7.06e−01
3.99e−03
5.47e+00
5.20e−01
7.06e−01
2. b) sbvpcol, equidistant, m=4, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.06e-05
3.00e-05
6.30e-08
6.23e-10
3.33e-10
err
coll
5.03e-05
3.00e-05
6.30e-08
6.23e-10
3.33e-10
3. b) sbvpcol, equidistant, m=6, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.29e-05
3.88e-06
1.47e-07
8.61e-09
4.82e-10
err
coll
1.93e-05
3.88e-06
1.47e-07
8.61e-09
4.82e-10
· Exact solution is the reference solution for step size h =1.5626e-03
118
Figure 5.1 : plot of solution and error using sbvp for Example 5
119
TABLE 5.1 : global error for Example 5
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.10e-04
7.83e-06
8.44e-08
4.60e-09
2.69e-10
err
coll
1.10e-04
7.83e-06
9.65e-08
5.21e-09
3.00e-10
p
mesh
p
coll
const
mesh
const
coll
3.81
6.54
4.20
4.10
3.81
6.34
4.21
4.11
7.13e–01
2.50e+03
4.45e–01
2.88e–01
7.13e–01
1.39e+03
5.37e–01
3.58e–01
err
coll
2.69e-03
1.63e-04
1.01e-05
1.30e-06
1.62e-07
p
mesh
p
coll
const
mesh
const
coll
4.05
4.85
4.01
4.01
4.05
4.02
2.95
3.00
3.03e+01
3.32e+02
1.52e+01
1.49e+01
3.03e+01
2.74e+01
5.38e–01
6.66e–01
err
coll
2.33e-04
1.59e-05
3.66e-07
2.27e-08
1.43e-09
p
mesh
p
coll
const
mesh
const
coll
3.87
6.65
5.71
5.89
3.87
5.44
4.00
3.99
1.73e+00
7.09e+03
2.24e+02
5.00e+02
1.73e+00
1.93e+02
9.37e–01
8.90e–01
err
coll
8.28e-05
1.10e-06
1.48e-08
5.02e-10
1.57e-11
p
mesh
p
coll
const
mesh
const
coll
6.24
7.02
7.35
7.72
6.24
6.22
4.88
5.00
1.43e+02
1.50e+03
5.16e+03
2.50e+03
1.43e+02
1.35e+02
9.73e–01
1.65e+00
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.69e-03
1.63e-04
5.66e-06
3.50e-07
2.17e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.33e-04
1.59e-05
1.59e-07
3.04e-09
5.11e-11
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
8.28e-05
1.10e-06
8.44e-09
5.15e-11
2.45e-13
120
TABLE 5.2 : convergence of first derivative for Example 5
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.14e-02
1.62e-03
4.07e-05
2.66e-06
1.56e-07
err
coll
1.14e-02
1.62e-03
4.07e-05
2.66e-06
1.56e-07
p
mesh
p
coll
const
mesh
const
coll
2.81
5.31
3.94
4.09
2.81
5.31
3.94
4.09
7.45e+00
1.33e+04
8.29e+01
1.61e+02
7.45e+00
1.33e+04
8.29e+01
1.61e+02
err
coll
1.59e-01
2.58e-02
5.21e-03
1.21e-03
2.86e-04
p
mesh
p
coll
const
mesh
const
coll
2.62
2.31
2.11
2.08
2.62
2.31
2.11
2.08
6.68e+01
2.60e+01
1.23e+01
2.61e+00
6.68e+01
2.60e+01
1.23e+01
2.61e+00
err
coll
2.77e-02
3.80e-03
3.38e-04
3.59e-05
4.11e-06
p
mesh
p
coll
const
mesh
const
coll
2.86
3.49
3.24
3.13
2.86
3.49
3.24
3.13
2.02e+01
1.33e+02
5.20e+01
3.19e+01
2.02e+01
1.33e+02
5.20e+01
3.19e+01
err
coll
1.65e-02
5.57e-04
2.18e-05
1.23e-06
6.70e-08
p
mesh
p
coll
const
mesh
const
coll
4.89
4.68
4.14
4.20
4.89
4.68
4.14
4.20
1.28e+03
6.74e+02
9.43e+01
1.23e+02
1.28e+03
6.74e+02
9.43e+01
1.23e+02
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.59e-01
2.58e-02
5.21e-03
1.21e-03
2.86e-04
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.77e-02
3.80e-03
3.38e-04
3.59e-05
4.11e-06
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.65e-02
5.57e-04
2.18e-05
1.23e-06
6.70e-08
121
TABLE 5.3 : convergence of second derivative for Example 5
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
9.52e-01
2.25e-01
1.58e-02
2.20e-03
2.73e-04
err
coll
9.52e-01
2.25e-01
1.58e-02
2.20e-03
2.73e-04
p
mesh
p
coll
const
mesh
const
coll
2.08
3.83
2.84
3.01
2.08
3.83
2.84
3.01
1.15e+02
2.16e+04
5.63e+02
1.19e+03
1.15e+02
2.16e+04
5.63e+02
1.19e+03
err
coll
4.89e+00
2.15e+00
1.09e+00
5.36e–01
2.63e–01
p
mesh
p
coll
const
mesh
const
coll
1.18
0.98
1.03
1.03
1.18
0.98
1.03
1.03
73.8
41.1
48.4
48.1
73.8
41.1
48.4
48.1
err
coll
1.37e+00
4.35e–01
1.27e–01
3.11e–02
7.49e–03
p
mesh
p
coll
const
mesh
const
coll
1.65
1.77
2.03
2.05
1.65
1.77
2.03
2.05
6.10e+01
8.79e+01
2.30e+02
2.52e+02
6.10e+01
8.79e+01
2.30e+02
2.52e+02
err
coll
1.47e+00
1.36e–01
1.24e–02
1.72e–03
2.03e–04
p
mesh
p
coll
const
mesh
const
coll
3.43
3.46
2.84
3.09
3.43
3.46
2.84
3.09
3.96e+03
4.32e+03
4.42e+02
1.32e+03
3.96e+03
4.32e+03
4.42e+02
1.32e+03
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.89e+00
2.15e+00
1.09e+00
5.36e–01
2.63e–01
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.37e+00
4.35e–01
1.27e–01
3.11e–02
7.49e–03
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.47e+00
1.36e–01
1.24e–02
1.72e–03
2.03e–04
122
TABLE 5.4 : matrix condition estimates for Example 5
1. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.41e+03
1.27e+04
5.61e+04
1.90e+05
4.56e+05
2.04e+03
6.22e+03
2.45e+04
9.77e+04
3.91e+05
ord.
condest
ord.
cond
const
condest
const
cond
-1.53
-2.14
-1.76
-1.26
-1.61
-1.98
-2.00
-2.00
1.31e+02
2.09e+01
8.43e+01
7.51e+02
4.99e+01
1.66e+01
1.55e+01
1.53e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.59
-1.47
-1.41
-1.38
-1.39
-1.37
-1.84
-2.00
1.55e+02
2.19e+02
1.10e+02
3.12e+02
1.22e+02
1.29e+02
2.30e+02
1.12e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.53
-1.64
-1.50
-1.43
-1.52
-1.42
-1.39
-1.94
8.02e+02
2.61e+02
9.59e+02
1.31e+03
3.66e+02
5.01e+02
5.66e+02
5.03e+01
ord.
condest
ord.
cond
const
condest
const
cond
-0.08
-2.82
-1.61
-1.49
-1.52
-1.51
-1.42
-1.54
8.48e+04
2.29e+01
2.01e+03
3.48e+03
1.15e+03
1.15e+03
1.65e+03
9.82e+02
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
5.97e+03
1.79e+04
4.97e+04
1.32e+05
3.44e+05
2.98e+03
7.80e+03
2.01e+04
7.19e+04
2.88e+05
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.69e+04
7.75e+04
2.41e+05
6.81e+05
1.83e+06
1.22e+04
3.51e+04
9.39e+04
2.45e+05
9.40e+05
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.02e+05
1.08e+05
7.63e+05
2.33e+06
6.52e+06
3.77e+04
1.08e+05
3.08e+05
8.23e+05
2.39e+06
· condestDF:=condest(DF,1), cond:=cond(DF,2)
123
TABLE 5.5 : global error for Example 5
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.71e-03
3.51e-04
5.00e-05
1.25e-05
3.12e-06
err
coll
4.71e-03
3.51e-04
5.29e-05
1.29e-05
3.17e-06
p
mesh
p
coll
const
mesh
const
coll
3.75
2.81
2.00
2.00
3.75
2.73
2.04
2.02
2.62e+01
1.60e+00
8.02e–02
8.08e–02
2.62e+01
8.30e+00
9.86e–02
8.98e–02
p
mesh
p
coll
const
mesh
const
coll
3.81
6.54
4.20
4.10
3.81
6.34
4.21
4.11
7.13e–01
2.50e+03
4.45e–01
2.88e–01
7.13e–01
1.39e+03
5.37e–01
3.58e–01
p
mesh
p
coll
const
mesh
const
coll
6.83
8.62
6.34
6.50
6.83
8.62
6.34
6.16
1.98e+02
4.29e+04
9.28e+00
1.91e+01
1.98e+02
4.29e+04
9.28e+00
4.22e+00
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.10e-04
7.83e-06
8.44e-08
4.60e-09
2.69e-10
err
coll
1.10e-04
7.83e-06
9.65e-08
5.21e-09
3.00e-10
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
2.94e-05
2.59e-07
6.57e-10
8.13e-12
8.98e-14
err
coll
2.94e-05
2.59e-07
6.57e-10
8.13e-12
1.14e-13
124
TABLE 5.6 : Error of error estimate based on h-h/2 for Example 5
2. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.10e-03
1.07e-04
9.05e-08
6.72e-09
4.21e-10
err
coll
1.10e-03
1.07e-04
1.57e-06
2.03e-07
2.56e-08
p`
mesh
p`
coll
const
mesh
const
coll
3.37
10.2
3.75
4.00
3.37
6.08
2.95
2.99
1.07e+00
2.00e+09
9.24e−02
2.73e−01
1.07e+00
8.72e+03
8.47e−02
1.01e−01
p`
mesh
p`
coll
const
mesh
const
coll
1.10
7.95
6.56
5.96
1.10
7.86
4.96
4.73
1.24e−05
1.01e+04
6.08e+01
4.38e+00
1.24e−05
7.66e+03
1.75e−01
6.31e−02
p`
mesh
p`
coll
const
mesh
const
coll
7.61
8.50
7.73
8.36
7.61
8.50
7.73
6.81
2.91e+01
4.15e+02
2.43e+01
3.89e+01
2.91e+01
4.15e+02
2.43e+01
4.53e−01
2. b) sbvpcol, equidistant, m=4, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
9.88e-07
4.61e-07
1.87e-09
1.98e-11
3.18e-13
err
coll
9.88e-07
4.61e-07
1.99e-09
6.41e-11
2.42e-12
3. b) sbvpcol, equidistant, m=6, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.15e-07
3.66e-09
1.01e-11
4.78e-14
1.45e-16
err
coll
7.15e-07
3.66e-09
1.01e-11
4.78e-14
4.24e-16
125
Figure 6.1 : plot of solution and error using sbvp for Example 6
1. a) Solution, α =1
b) Error, α =1
2. a) Solution, α =2
b) Error, α = 2
3. a) Solution, α =3
b) Error, α = 3
126
TABLE 6.1 : global error for Example 6
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.69e-08
2.31e-09
1.45e-10
9.05e-12
5.62e-13
err
coll
3.71e-08
2.32e-09
1.45e-10
9.06e-12
5.62e-13
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
4.00
4.01
4.00
4.00
4.00
4.01
3.69e-04
3.69e-04
3.69e-04
3.88e-04
3.70e-04
3.73e-04
3.72e-04
3.90e-04
err
coll
5.63e-05
6.99e-06
8.71e-07
1.09e-07
1.36e-08
p
mesh
p
coll
const
mesh
const
coll
3.00
3.00
3.00
3.00
3.00
3.00
3.00
3.00
5.76e-02
5.67e-02
5.62e-02
5.59e-02
5.76e-02
5.67e-02
5.62e-02
5.59e-02
err
coll
2.69e-07
1.73e-08
1.09e-09
6.87e-11
4.31e-12
p
mesh
p
coll
const
mesh
const
coll
4.08
4.04
4.02
4.01
3.96
3.98
3.99
4.00
1.41e-03
1.24e-03
1.15e-03
1.10e-03
2.45e-03
2.61e-03
2.69e-03
2.81e-03
err
coll
1.45e-09
4.50e-11
1.40e-12
4.44e-14
1.24e-14
p
mesh
p
coll
const
mesh
const
coll
5.01
5.00
5.00
1.94
5.01
5.00
5.00
1.84
1.49e-04
1.46e-04
1.32e-04
2.22e-10
1.49e-04
1.46e-04
1.32e-04
1.39e-10
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.63e-05
6.99e-06
8.71e-07
1.09e-07
1.36e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.17e-07
6.90e-09
4.19e-10
2.58e-11
1.60e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.45e-09
4.50e-11
1.40e-12
4.44e-14
1.15e-14
127
TABLE 6.2 : global error for Example 6
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.25e-08
2.03e-09
1.27e-10
7.94e-12
4.89e-13
err
coll
3.27e-08
2.04e-09
1.27e-10
7.95e-12
4.90e-13
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
3.99
4.02
4.00
4.00
4.00
4.02
3.26e-04
3.26e-04
3.25e-04
3.54e-04
3.28e-04
3.28e-04
3.26e-04
3.56e-04
err
coll
2.08e-04
3.23e-05
5.06e-06
7.94e-07
1.25e-07
p
mesh
p
coll
const
mesh
const
coll
2.68
2.68
2.67
2.67
2.68
2.68
2.67
2.67
9.96e-02
9.91e-02
9.59e-02
9.57e-02
9.96e-02
9.91e-02
9.59e-02
9.57e-02
err
coll
3.62e-07
1.73e-08
1.09e-09
6.87e-11
4.31e-12
p
mesh
p
coll
const
mesh
const
coll
4.39
4.42
3.94
4.04
4.39
3.98
3.99
4.00
8.89e-03
9.74e-03
1.65e-03
2.56e-03
8.89e-03
2.61e-03
2.69e-03
2.81e-03
err
coll
6.40e-09
2.37e-10
9.17e-12
3.60e-13
1.29e-14
p
mesh
p
coll
const
mesh
const
coll
4.76
4.69
4.67
4.80
4.76
4.69
4.67
4.80
3.64e-04
3.01e-04
2.80e-04
4.99e-04
3.64e-04
3.01e-04
2.80e-04
4.99e-04
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.08e-04
3.23e-05
5.06e-06
7.94e-07
1.25e-07
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.62e-07
1.73e-08
8.06e-10
5.25e-11
3.18e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
6.40e-09
2.37e-10
9.17e-12
3.60e-13
1.29e-14
128
TABLE 6.3 : global error for Example 6
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.74e-08
1.82e-09
1.14e-10
7.11e-12
4.43e-13
err
coll
3.74e-08
1.83e-09
1.14e-10
7.11e-12
4.43e-13
p
mesh
p
coll
const
mesh
const
coll
4.36
4.00
4.00
4.00
4.36
4.00
4.00
4.00
8.61e-04
2.91e-04
2.92e-04
2.96e-04
8.50e-04
2.93e-04
2.94e-04
2.96e-04
err
coll
3.81e-04
6.63e-05
1.16e-05
2.04e-06
3.58e-07
p
mesh
p
coll
const
mesh
const
coll
2.52
2.51
2.51
2.51
2.52
2.51
2.51
2.51
1.27e-01
1.24e-01
1.22e-01
1.20e-01
1.27e-01
1.24e-01
1.22e-01
1.20e-01
err
coll
3.11e-07
1.73e-08
1.09e-09
6.87e-11
4.31e-12
p
mesh
p
coll
const
mesh
const
coll
4.39
4.11
4.02
4.01
4.17
3.98
3.99
4.00
7.63e-03
3.29e-03
2.37e-03
2.26e-03
4.60e-03
2.61e-03
2.69e-03
2.81e-03
err
coll
1.20e-08
5.23e-10
2.29e-11
1.01e-12
4.35e-14
p
mesh
p
coll
const
mesh
const
coll
4.52
4.51
4.50
4.53
4.52
4.51
4.50
4.53
4.03e-04
3.90e-04
3.76e-04
4.31e-04
4.03e-04
3.90e-04
3.76e-04
4.31e-04
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.81e-04
6.63e-05
1.16e-05
2.04e-06
3.58e-07
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.11e-07
1.48e-08
8.59e-10
5.27e-11
3.27e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.20e-08
5.23e-10
2.29e-11
1.01e-12
4.35e-14
129
TABLE 6.4 : convergence of first derivative for Example 6
-α=01. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
8.46e-07
5.39e-08
3.40e-09
2.22e-10
2.23e-11
error
coll *
9.17e-08
5.83e-09
3.68e-10
2.31e-11
9.57e-12
p
mesh
p
coll *
const
mesh
const
coll*
3.97
3.99
3.94
3.32
3.97
3.99
3.99
1.27
7.91e-03
8.31e-03
6.87e-03
4.58e-04
8.66e-04
8.99e-04
9.13e-04
6.11e-09
error
coll *
4.20e-05
5.35e-06
6.75e-07
8.47e-08
1.06e-08
p
mesh
p
coll *
const
mesh
const
coll*
1.97
1.99
1.99
2.00
2.97
2.99
2.99
3.00
4.11e-01
4.29e-01
4.39e-01
4.45e-01
3.95e-02
4.11e-02
4.21e-02
4.28e-02
error
coll *
2.69e-07
1.73e-08
1.09e-09
6.87e-11
5.13e-12
p
mesh
p
coll *
const
mesh
const
coll*
2.97
2.99
2.99
3.00
3.96
3.98
3.99
3.742
4.11e-02
4.28e-02
4.38e-02
4.44e-02
2.47e-03
2.61e-03
1.70e-04
9.12e-04
error
coll *
1.27e-09
4.07e-11
1.84e-12
3.29e-12
6.88e-12
p
mesh
p
coll *
const
mesh
const
coll*
+4.96
+4.47
–0.83
–1.06
2.90e-03
3.02e-03
2.92e-03
3.40e-06
1.17e-04
2.63e-05
8.39e-14
3.09e-14
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
4.38e-03
1.11e-03
2.81e-04
7.05e-05
1.77e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
4.37e-05
5.56e-06
7.01e-07
8.81e-08
1.10e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
3.11e-07
1.98e-08
1.25e-09
7.97e-11
1.48e-11
+3.97
+3.98
+3.97
+2.43
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
130
TABLE 6.5 : convergence of first derivative for Example 6
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
7.99e-07
5.10e-08
3.22e-09
2.18e-10
1.85e-11
error
coll *
2.11e-07
1.46e-08
9.98e-10
6.79e-11
8.37e-12
p
mesh
p
coll *
const
mesh
const
coll*
3.97
3.98
3.88
3.56
3.86
3.87
3.88
3.02
7.45e-03
7.78e-03
5.38e-03
1.31e-03
1.52e-03
1.58e-03
1.63e-03
3.78e-05
error
coll *
1.17e-03
2.92e-04
7.30e-05
1.82e-05
4.55e-06
p
mesh
p
coll *
const
mesh
const
coll*
1.98
1.99
1.99
2.00
2.00
2.00
2.00
2.00
4.13e-01
4.29e-01
4.39e-01
4.45e-01
1.17e-01
1.18e-01
1.17e-01
1.17e-01
error
coll *
2.80e-06
3.17e-07
3.76e-08
4.58e-09
5.64e-10
p
mesh
p
coll *
const
mesh
const
coll*
2.97
2.99
2.99
3.00
3.14
3.08
3.04
3.02
4.11e-02
4.28e-02
4.38e-02
4.44e-02
3.87e-03
3.18e-03
2.78e-03
2.59e-03
error
coll *
9.51e-08
5.91e-09
3.68e-10
2.29e-11
6.67e-12
p
mesh
p
coll *
const
mesh
const
coll*
3.97
3.98
3.97
2.61
4.01
4.00
4.01
1.78
2.90e-03
3.02e-03
2.92e-03
7.46e-06
9.67e-04
9.58e-04
9.71e-04
5.57e-08
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
4.38e-03
1.11e-03
2.81e-04
7.05e-05
1.77e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
4.37e-05
2.80e-06
7.01e-07
8.81e-08
1.10e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
3.11e-07
1.98e-08
1.25e-09
7.97e-11
1.30e-11
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
131
TABLE 6.6 : convergence of first derivative for Example 6
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.02e-06
1.60e-07
1.26e-08
9.99e-10
5.57e-11
error
coll *
2.34e-07
1.84e-08
1.45e-09
1.15e-10
8.65e-12
p
mesh
p
coll *
const
mesh
const
coll*
3.66
3.67
3.65
4.16
3.67
3.67
3.65
3.74
9.15e-03
9.60e-03
9.01e-03
8.42e-02
1.09e-03
1.11e-03
1.00e-03
1.51e-03
error
coll *
6.82e-03
2.15e-03
6.77e-04
2.14e-04
6.76e-05
p
mesh
p
coll *
const
mesh
const
coll*
1.69
1.68
1.68
1.68
1.67
1.66
1.66
1.66
6.22e-01
6.15e-01
6.06e-01
6.00e-01
3.18e-01
3.14e-01
3.12e-01
3.11e-01
error
coll *
1.88e-05
1.71e-06
1.52e-07
2.01e-08
2.37e-09
p
mesh
p
coll *
const
mesh
const
coll*
3.09
2.99
2.99
3.00
3.46
3.50
2.92
3.08
5.90e-02
4.28e-02
4.38e-02
4.45e-02
5.38e-02
6.07e-02
7.12e-03
1.47e-02
error
coll *
5.93e-07
4.39e-08
3.41e-09
2.67e-10
1.95e-11
p
mesh
p
coll *
const
mesh
const
coll*
3.75
3.69
3.68
3.78
3.76
3.69
3.67
3.77
7.35e-03
6.08e-03
5.83e-03
9.04e-03
3.38e-03
2.76e-03
2.62e-03
4.03e-03
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
1.28e-02
3.99e-03
1.24e-03
3.89e-04
1.22e-04
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
4.75e-05
5.56e-06
7.01e-07
8.81e-08
1.10e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
1.29e-06
9.57e-08
7.41e-09
5.78e-10
1.95e-11
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
132
TABLE 6.7 : convergence of first derivative for Example 6
-α=31. sbvpcol, equidistant, m=4
error
coll *
4.53e-07
4.00e-08
3.55e-09
3.13e-10
2.74e-11
p
mesh
p
coll *
const
mesh
const
coll*
3.52
3.51
3.51
3.58
3.50
3.50
3.50
3.52
1.30e-02
1.28e-02
1.28e-02
1.71e-02
1.43e-03
1.42e-03
1.44e-03
1.54e-03
error
coll *
1.28e-02
4.49e-03
1.58e-03
5.58e-04
1.97e-04
p
mesh
p
coll *
const
mesh
const
coll*
1.53
1.52
1.52
1.51
1.51
1.51
1.50
1.50
7.97e-01
7.69e-01
7.58e-01
7.38e-01
4.16e-01
4.09e-01
4.05e-01
4.02e-01
error
coll *
1.59e-05
p
mesh
p
coll *
const
mesh
const
coll*
1.00e-01
error
mesh
4.37e-05
5.00e-02
2.50e-02
1.25e-02
6.25e-03
5.56e-06
7.01e-07
8.81e-08
1.10e-08
1.44e-06
1.64e-07
2.02e-08
2.51e-09
2.97
2.99
2.99
3.00
3.47
3.13
3.02
3.01
4.12e-02
4.28e-02
4.38e-02
4.44e-02
4.64e-02
1.69e-02
1.15e-02
1.08e-02
error
coll *
1.12e-06
9.77e-08
8.58e-09
7.57e-10
6.56e-11
p
mesh
p
coll *
const
mesh
const
coll*
3.53
3.52
3.51
3.54
3.52
3.51
3.50
3.53
8.20e-03
7.92e-03
7.67e-03
8.64e-03
3.70e-03
3.60e-03
3.51e-03
3.91e-03
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
3.96e-06
3.46e-07
3.03e-08
2.66e-09
2.23e-10
2. sbvpcol, Gauss, m =2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.34e-02
8.09e-03
2.81e-03
9.86e-04
3.46e-04
3. sbvpcol, Gauss, m =3
h
4. sbvpcol, Gauss, m =4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.43e-06
2.11e-07
1.84e-08
1.62e-09
1.40e-10
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
133
TABLE 6.8 : convergence of second derivative for Example 6
-α=01. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
8.72e-05
1.11e-05
1.40e-06
1.80e-07
4.02e-08
error
coll *
1.04e-05
1.33e-06
1.68e-07
2.31e-08
1.77e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.97
2.99
2.96
2.16
2.97
2.99
2.86
0.38
8.18e-02
8.55e-02
7.69e-02
2.38e-03
9.64e-03
1.02e-02
6.34e-03
1.24e-07
error
coll *
1.51e-01
7.69e-02
3.88e-02
1.95e-02
9.78e-03
p
mesh
p
coll *
const
mesh
const
coll*
0.98
0.99
0.99
1.00
0.97
0.99
0.99
1.00
1.40e+00
1.48e+00
1.51e+00
1.53e+00
2.52e+00
2.60e+00
2.64e+00
2.66e+00
error
coll *
2.61e-03
6.66e-04
1.68e-04
4.23e-05
1.06e-05
p
mesh
p
coll *
const
mesh
const
coll*
1.97
1.99
1.99
2.00
1.97
1.99
1.99
2.00
4.90e-01
5.20e-01
5.20e-01
5.41e-01
2.44e-01
2.59e-01
2.59e-01
2.71e-01
error
coll *
2.93e-05
3.74e-06
4.72e-07
6.02e-08
1.90e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.97
2.98
2.98
1.95
2.97
2.98
2.97
1.67
5.83e-02
6.07e-02
5.94e-02
6.48e-04
2.73e-02
2.85e-02
2.73e-02
8.94e-05
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.64e-01
1.34e-01
6.75e-02
3.39e-02
1.70e-02
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
5.25e-03
1.34e-03
3.37e-04
8.46e-05
2.12e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
6.22e-05
7.93e-06
1.00e-06
1.27e-07
3.29e-08
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
134
TABLE 6.9 : convergence of second derivative for Example 6
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
8.72e-05
1.11e-05
1.40e-06
1.85e-07
3.38e-08
error
coll *
1.04e-05
1.33e-06
1.68e-07
2.42e-08
2.10e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.97
2.99
2.93
2.45
2.97
2.98
2.80
0.20
8.19e-02
8.53e-02
6.84e-02
8.44e-03
9.74e-03
1.02e-02
5.09e-03
5.89e-08
error
coll *
1.51e-01
7.69e-02
3.88e-02
1.95e-02
9.78e-03
p
mesh
p
coll *
const
mesh
const
coll*
0.98
0.99
0.99
1.00
0.97
0.99
0.99
1.00
2.52e+00
2.60e+00
2.64e+00
2.66e+00
1.40e+00
1.48e+00
1.51e+00
1.53e+00
error
coll *
2.61e-03
6.66e-04
1.68e-04
4.23e-05
1.06e-05
p
mesh
p
coll *
const
mesh
const
coll*
1.97
1.99
1.99
2.00
1.97
1.99
1.99
2.00
4.90e-01
5.20e-01
5.20e-01
5.41e-01
2.44e-01
2.59e-01
2.59e-01
2.71e-01
error
coll *
2.93e-05
3.74e-06
4.72e-07
6.02e-08
1.59e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.97
2.98
2.98
2.19
2.97
2.98
2.97
1.92
5.83e-02
6.07e-02
5.94e-02
1.88e-03
2.73e-02
2.85e-02
2.73e-02
2.72e-04
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.64e-01
1.34e-01
6.75e-02
3.39e-02
1.70e-02
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
5.25e-03
1.34e-03
3.37e-04
8.46e-05
2.12e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
6.22e-05
7.93e-06
1.00e-06
1.27e-07
2.78e-08
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
135
TABLE 6.10 : convergence of second derivative for Example 6
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
1.48e-04
2.33e-05
3.64e-06
5.75e-07
6.58e-08
error
coll *
4.22e-05
6.76e-06
1.07e-06
1.71e-07
2.15e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.67
2.68
2.66
3.12
2.64
2.66
2.65
2.99
6.85e-02
7.15e-02
6.72e-02
5.17e-01
1.84e-02
1.96e-02
1.85e-02
8.43e-02
error
coll *
2.63e-02
1.64e-02
1.02e-02
6.36e-02
3.97e-02
p
mesh
p
coll *
const
mesh
const
coll*
0.72
0.72
0.71
0.70
0.68
0.68
0.68
0.68
1.63e+00
1.58e+00
1.54e+00
1.48e+00
1.27e+00
1.27e+00
1.26e+00
1.25e+00
error
coll *
2.61e-03
6.66e-04
1.68e-04
4.23e-05
1.06e-05
p
mesh
p
coll *
const
mesh
const
coll*
1.97
1.99
1.99
2.00
1.97
1.99
1.99
2.00
4.90e-01
5.20e-01
5.20e-01
5.41e-01
2.44e-01
2.59e-01
2.59e-01
2.71e-01
error
coll *
8.13e-05
1.21e-05
1.87e-06
2.92e-07
4.24e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.76
2.70
2.69
2.79
2.75
2.69
2.68
2.78
6.94e-02
5.84e-02
5.62e-02
8.77e-02
4.57e-02
3.83e-02
3.69e-02
5.76e-02
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
3.06e-01
1.85e-01
1.13e-01
6.89e-02
4.24e-02
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
5.25e-03
1.34e-03
3.37e-04
8.46e-05
2.12e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
1.21e-04
1.80e-05
2.77e-06
4.29e-07
6.20e-08
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
136
TABLE 6.11 : convergence of second derivative for Example 6
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.86e-04
4.95e-05
8.63e-06
1.51e-06
2.51e-07
error
coll *
8.44e-05
1.49e-05
2.63e-06
4.63e-07
7.79e-08
p
mesh
p
coll *
const
mesh
const
coll*
2.53
2.52
2.52
2.59
2.50
2.50
2.50
2.57
9.67e-02
9.43e-02
9.35e-02
1.26e-01
2.70e-02
2.68e-02
2.71e-02
3.59e-02
error
coll *
4.81e-01
3.30e-01
2.29e-01
1.59e-01
1.12e-01
p
mesh
p
coll *
const
mesh
const
coll*
0.58
0.55
0.54
0.53
0.54
0.53
0.52
0.51
1.97e+00
1.85e+00
1.75e+00
1.67e+00
1.68e+00
1.62e+00
1.56e+00
1.52e+00
error
coll *
2.61e-03
6.66e-04
1.68e-04
4.23e-05
1.06e-05
p
mesh
p
coll *
const
mesh
const
coll*
1.97
1.99
1.99
2.00
1.97
1.99
1.99
2.00
4.90e-01
5.20e-01
5.20e-01
5.41e-01
2.44e-01
2.59e-01
2.59e-01
2.71e-01
error
coll *
1.54e-04
2.66e-05
4.64e-06
8.16e-07
1.40e-07
p
mesh
p
coll *
const
mesh
const
coll*
2.54
2.53
2.51
2.54
2.53
2.52
2.51
2.54
7.85e-02
7.52e-02
7.21e-02
8.18e-02
5.21e-02
5.03e-02
4.86e-02
5.52e-02
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
5.24e-01
3.52e-01
2.39e-01
1.65e-01
1.14e-01
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
5.25e-03
1.34e-03
3.37e-04
8.46e-05
2.12e-05
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
error
mesh
2.26e-04
3.88e-05
6.74e-06
1.18e-06
2.02e-07
* error computed only in collocations points (‘ mesh ‘ – mesh and coll. and mesh )
137
TABLE 6.12 : matrix condition estimates for Example 6
-α=01. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.95e+01
1.34e+02
2.42e+02
4.57e+02
8.89e+02
3.17e+01
6.15e+01
1.22e+02
2.42e+02
4.83e+02
ord.
condest
ord.
cond
const
condest
const
cond
-0.75
-0.85
-0.92
-0.96
-0.95
-0.98
-0.99
-1.00
1.42e+01
1.03e+01
8.09e+00
6.85e+00
3.53e+00
3.23e+00
3.11e+00
3.06e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.88
-0.94
-0.96
-0.98
-0.97
-0.99
-0.99
-1.00
8.47e+00
7.27e+00
6.47e+00
6.00e+00
3.48e+00
3.28e+00
3.19e+00
3.15e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.81
-0.89
-0.94
-0.97
-0.96
-0.99
-1.00
-1.00
1.94e+01
8.71e+00
7.25e+00
6.41e+00
3.34e+00
3.11e+00
3.00e+00
2.97e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.74
-0.85
-0.92
-0.96
-0.96
-0.98
-0.99
-1.00
1.42e+01
10.3e+00
8.08e+00
6.85e+00
3.25e+00
3.02e+00
2.92e+00
2.88e+00
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.45e+01
1.19e+02
2.27e+02
4.43e+02
8.75e+02
3.23e+01
6.30e+01
1.25e+02
2.49e+02
4.97e+02
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.19e+01
1.26e+02
2.34e+02
4.50e+02
8.82e+02
3.06e+01
5.96e+01
1.18e+02
2.35e+02
4.70e+02
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.95e+01
1.34e+02
2.42e+02
4.57e+02
8.89e+02
2.98e+01
5.78e+01
1.14e+02
2.28e+02
4.55e+02
· condestDF:=condest(DF,1), cond:=cond(DF,2)
138
TABLE 6.13 : matrix condition estimates for Example 6
-α=11. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.28e+02
3.62e+02
6.31e+02
1.17e+03
2.24e+03
4.90e+01
9.28e+01
1.82e+02
3.63e+02
7.26e+02
ord.
condest
ord.
cond
const
condest
const
cond
-0.67
-0.80
-0.89
-0.94
-0.91
-0.99
-1.00
-1.00
4.91e+01
3.30e+01
2.38e+01
1.89e+01
6.07e+00
4.80e+00
4.59e+00
4.54e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.82
-0.90
-0.95
-0.97
-0.98
-1.00
-1.00
-1.00
1.76e+01
1.39e+01
1.17e+01
1.05e+01
3.59e+00
3.40e+00
3.36e+00
3.36e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.74
-0.85
-0.92
-0.96
-0.93
-0.99
-1.00
-1.00
4.12e+01
2.96e+01
2.29e+01
1.92e+01
6.80e+00
5.61e+00
5.47e+00
5.45e+00
ord.
condest
ord.
cond
const
condest
const
cond
-0.67
-0.80
-0.89
-0.94
-0.27
-0.99
-1.00
-1.00
8.41e+01
5.66e+01
4.08e+01
3.24e+01
7.44e+01
8.78e+00
8.43e+00
8.34e+00
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.17e+02
2.07e+02
3.87e+02
7.47e+02
1.47e+03
3.43e+01
6.76e+01
1.35e+02
2.70e+02
5.41e+02
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.25e+02
3.75e+02
6.75e+02
1.28e+03
2.48e+03
5.77e+01
1.10e+02
2.18e+02
4.36e+02
8.73e+02
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
3.91e+02
6.21e+02
1.08e+03
2.00e+03
3.84e+03
1.40e+02
1.69e+02
3.34e+02
6.67e+02
1.33e+03
· condestDF:=condest(DF,1), cond:=cond(DF,2)
139
TABLE 6.14 : matrix condition estimates for Example 6
-α=21. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.41e+03
1.66e+04
5.61e+04
2.04e+05
7.76e+05
2.04e+03
6.22e+03
2.45e+04
9.77e+04
3.91e+05
ord.
condest
ord.
cond
const
condest
const
cond
-1.91
-1.76
-1.86
-1.93
-1.61
-1.98
-2.00
-2.00
5.45e+01
8.49e+01
5.79e+01
4.39e+01
4.99e+01
1.66e+01
1.55e+01
1.53e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.59
-1.47
-1.41
-1.38
-1.39
-1.37
-1.84
-2.00
1.55e+02
2.19e+02
2.72e+02
3.08e+02
1.22e+02
1.29e+02
2.30e+01
1.12e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.53
-1.64
-1.50
-1.43
-1.52
-1.42
-1.39
-1.94
8.02e+02
5.75e+02
9.59e+02
1.31e+03
3.66e+02
5.01e+02
5.66e+02
5.03e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.29
-1.61
-1.61
-1.49
-1.52
-1.51
-1.42
-1.54
5.17e+03
2.02e+03
2.01e+03
3.48e+03
1.15e+03
1.15e+03
1.65e+03
9.82e+02
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
5.97e+03
1.79e+04
4.97e+04
1.32e+05
3.44e+05
2.98e+03
7.80e+03
2.01e+04
7.19e+04
2.88e+05
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
2.69e+04
7.75e+04
2.41e+05
6.81e+05
1.83e+06
1.22e+04
3.51e+04
9.39e+04
2.45e+05
9.40e+05
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.02e+05
2.50e+05
7.63e+05
2.33e+06
6.52e+06
3.77e+04
1.08e+05
3.08e+05
8.23e+05
2.39e+06
· condestDF:=condest(DF,1), cond:=cond(DF,2)
140
TABLE 6.15 : matrix condition estimates for Example 6
-α=31. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.87e+05
2.72e+06
1.53e+07
8.63e+07
4.90e+08
1.82e+05
9.77e+05
5.36e+06
3.27e+07
2.61e+08
ord.
condest
ord.
cond
const
condest
const
cond
-2.48
-2.49
-2.50
-2.50
-2.42
-2.46
-2.61
-3.00
1.61e+03
1.56e+03
1.52e+03
1.49e+03
6.90e+02
6.25e+02
3.55e+02
6.40e+01
ord.
condest
ord.
cond
const
condest
const
cond
-2.56
-2.53
-2.52
-2.51
-2.53
-2.52
-2.52
-2.51
1.85e+03
2.01e+03
2.09e+03
2.16e+03
8.49e+02
8.73e+02
8.95e+02
9.13e+02
ord.
condest
ord.
cond
const
condest
const
cond
-2.62
-2.55
-2.53
-2.52
-2.56
-2.54
-2.52
-2.52
1.68e+04
2.03e+04
2.20e+04
2.31e+04
7.39e+03
8.00e+03
8.40e+03
8.71e+03
ord.
condest
ord.
cond
const
condest
const
cond
-2.68
-2.57
-2.54
-2.52
-2.58
-2.55
-2.53
-2.52
8.47e+04
1.18e+05
1.34e+05
1.43e+05
4.05e+04
4.45e+04
4.78e+04
5.03e+04
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.70e+05
3.95e+06
2.28e+07
1.31e+08
7.47e+08
2.89e+05
1.67e+06
9.61e+06
5.49e+07
3.13e+08
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
6.92e+06
4.24e+07
2.49e+08
1.44e+09
8.24e+09
2.71e+06
1.60e+07
9.30e+07
5.35e+08
3.06e+09
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.10e+07
2.63e+08
1.57e+09
9.12e+09
5.25e+10
1.55e+07
9.29e+07
5.45e+08
3.15e+09
1.81e+10
· condestDF:=condest(DF,1), cond:=cond(DF,2)
141
TABLE 6.16 : global error for Example 6
-α=0-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.25e-03
3.12e-04
7.81e-05
1.95e-05
4.88e-06
err
coll
1.25e-03
3.12e-04
7.81e-05
1.95e-05
4.88e-06
p
mesh
p
coll
const
mesh
const
coll
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
1.25e-01
1.25e-01
1.25e-01
1.25e-01
1.25e-01
1.25e-01
1.25e-01
1.25e-01
p
mesh
p
coll
const
mesh
const
coll
4.02
4.01
4.00
4.00
4.02
4.01
4.00
4.00
9.84e-04
9.62e-04
9.53e-04
9.51e-04
9.84e-04
9.62e-04
9.53e-04
9.51e-04
p
mesh
p
coll
const
mesh
const
coll
6.02
6.01
6.01
5.47
6.02
6.01
6.01
5.47
3.43e-06
3.36e-06
3.38e-06
7.66e-07
3.43e-06
3.36e-06
3.38e-06
7.66e-07
2. a) sbvpcol, equidistant, m=4, Error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
6.06e-05
3.73e-06
2.32e-07
1.45e-08
9.06e-10
err
coll
6.06e-05
3.73e-06
2.32e-07
1.45e-08
9.06e-10
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
5.29e-08
8.15e-10
1.27e-11
1.97e-13
4.44e-15
err
coll
5.29e-08
8.15e-10
1.27e-11
1.97e-13
4.44e-15
142
TABLE 6.17 : global error for Example 6
-α=1-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.86e-04
1.22e-04
3.05e-05
7.62e-06
1.91e-06
err
coll
4.86e-04
1.22e-04
3.05e-05
7.63e-06
1.91e-06
p
mesh
p
coll
const
mesh
const
coll
1.99
2.00
2.00
2.00
1.99
2.00
2.00
2.00
4.80e-02
4.86e-02
4.87e-02
4.88e-02
4.79e-02
4.87e-02
4.89e-02
4.89e-02
p
mesh
p
coll
const
mesh
const
coll
4.17
3.95
3.99
4.00
4.17
3.95
3.99
4.00
4.55e-04
3.35e-04
3.64e-04
3.69e-04
4.55e-04
3.35e-04
3.64e-04
3.69e-04
p
mesh
p
coll
const
mesh
const
coll
6.16
5.96
6.04
4.59
6.16
5.96
6.04
4.59
1.56e-06
1.19e-06
1.42e-06
2.49e-08
1.56e-06
1.19e-06
1.42e-06
2.49e-08
2. a) sbvpcol, equidistant, m=4, Error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
2.52e-05
1.40e-06
9.00e-08
5.65e-09
3.53e-10
err
coll
2.52e-05
1.40e-06
9.00e-08
5.65e-09
3.53e-10
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
2.19e-08
3.07e-10
4.93e-12
7.47e-14
3.11e-15
err
coll
2.19e-08
3.07e-10
4.93e-12
7.47e-14
3.11e-15
143
TABLE 6.18 : global error for Example 6
-α=2-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.29e-04
1.07e-04
2.67e-05
6.68e-06
1.67e-06
err
coll
4.28e-04
1.07e-04
2.68e-05
6.69e-06
1.67e-06
p
mesh
p
coll
const
mesh
const
coll
2.00
2.00
2.00
2.00
2.00
2.00
2.00
2.00
4.30e-02
4.31e-02
4.29e-02
4.28e-02
4.26e-02
4.31e-02
4.30e-02
4.29e-02
p
mesh
p
coll
const
mesh
const
coll
4.00
4.00
3.99
4.02
4.00
4.00
4.00
4.02
3.26e-04
3.26e-04
3.25e-04
3.54e-04
3.28e-04
3.28e-04
3.26e-04
3.56e-04
p
mesh
p
coll
const
mesh
const
coll
6.55
6.06
5.99
3.17
6.55
6.06
6.00
3.17
2.54e-06
1.29e-06
1.12e-06
4.53e-10
2.54e-06
1.28e-06
1.14e-06
4.53e-10
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.25e-08
2.03e-09
1.27e-10
7.94e-12
4.89e-13
err
coll
3.27e-08
2.04e-09
1.27e-10
7.95e-12
4.90e-13
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e-01
6.250e-02
3.125e-02
err
mesh
2.71e-08
2.90e-10
4.34e-12
6.82e-14
7.55e-15
err
coll
2.71e-08
2.90e-10
4.36e-12
6.82e-14
7.55e-15
144
TABLE 6.19 : global error for Example 6
-α=3-
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.55e-04
9.60e-05
2.40e-05
5.99e-06
1.50e-06
err
coll
5.55e-04
9.62e-05
2.40e-05
5.99e-06
1.50e-06
p
mesh
p
coll
const
mesh
const
coll
2.53
2.00
2.00
2.00
2.53
2.00
2.00
2.00
1.88e-01
3.88e-02
3.84e-02
3.84e-02
1.87e-02
3.88e-02
3.86e-02
3.84e-02
p
mesh
p
coll
const
mesh
const
coll
4.36
4.00
4.00
4.00
4.36
4.00
4.00
4.00
8.61e-04
2.91e-04
2.92e-04
2.96e-04
8.50e-04
2.93e-04
2.94e-04
2.96e-04
p
mesh
p
coll
const
mesh
const
coll
6.32
6.58
6.39
1.00
6.32
6.58
6.37
1.01
3.16e-06
4.57e-06
3.04e-06
9.82e-13
3.16e-06
4.57e-06
2.93e-06
1.05e-12
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.74e-08
1.82e-09
1.14e-10
7.11e-12
4.43e-13
err
coll
3.74e-08
1.83e-09
1.14e-10
7.11e-12
4.43e-13
3. a) sbvpcol, equidistant, m=6, Error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
3.96e-08
4.97e-10
5.19e-12
6.20e-14
3.11e-14
err
coll
3.96e-08
4.97e-10
5.19e-12
6.28e-14
3.11e-14
145
TABLE 6.20 : Error of error estimate based on h-h/2 for Example 6
-α=0-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.13e-07
1.96e-08
1.23e-09
7.66e-11
4.77e-12
err
coll
3.13e-07
1.96e-08
1.23e-09
7.66e-11
4.77e-12
p`
mesh
p`
coll
const
mesh
const
coll
4.00
4.00
4.00
4.00
4.00
4.00
4.00
4.00
3.12e-03
3.13e-03
3.14e-03
3.20e-03
3.12e-03
3.13e-03
3.14e-03
3.20e-03
p`
mesh
p`
coll
const
mesh
const
coll
5.99
6.00
6.00
5.61
5.99
6.00
6.00
5.61
3.90e-06
3.92e-06
3.82e-06
1.34e-06
3.90e-06
3.92e-06
3.82e-06
1.34e-06
p`
mesh
p`
coll
const
mesh
const
coll
+8.00
+5.34
−0.29
−1.50
+8.00
+5.34
−0.29
−1.42
2.99e-09
7.56e-11
6.13e-16
2.16e-17
2.99e-09
7.56e-11
6.13e-16
2.76e-18
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
6.11e-08
9.59e-10
1.50e-11
2.37e-13
4.86e-15
err
coll
6.11e-08
9.59e-10
1.50e-11
2.37e-13
4.86e-15
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.17e-11
4.58e-14
1.13e-15
1.38e-15
3.91e-15
err
coll
1.17e-11
4.58e-14
1.13e-15
1.38e-15
3.69e-15
146
TABLE 6.21 : Error of error estimate based on h-h/2 for Example 6
-α=1-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.70e-07
3.17e-08
1.75e-09
9.49e-11
5.09e-12
err
coll
1.90e-07
9.41e-08
1.59e-08
2.27e-09
2.96e-10
p`
mesh
p`
coll
const
mesh
const
coll
4.17
4.18
4.20
4.22
1.01
2.56
2.81
2.94
8.44e-03
8.62e-03
9.52e-03
1.02e-02
1.94e-06
2.04e-04
5.10e-04
8.84e-04
p`
mesh
p`
coll
const
mesh
const
coll
8.89
6.22
6.04
5.34
9.36
7.00
3.45
4.80
4.19e-04
1.02e-05
7.13e-06
1.02e+06
2.00e+03
2.19e-05
1.37e-08
5.77e-07
p`
mesh
p`
coll
const
mesh
const
coll
+11.1
+4.89
−0.07
+0.12
12.1
3.69
0.12
0.06
4.11e-07
7.10e-11
2.37e-15
4.04e-15
2.54e+02
6.76e−12
4.75e−15
3.72e−15
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
8.80e-07
1.85e-09
2.48e-11
3.77e-13
9.30e-15
err
coll
8.80e-07
1.34e-09
1.05e-11
9.64e-13
3.47e-14
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.83e-10
8.11e-14
2.74e-15
2.88e-15
2.64e-15
err
coll
1.83e-10
4.03e-14
3.12e-15
2.88e-15
2.76e-15
147
TABLE 6.22 : Error of error estimate based on h-h/2 for Example 6
-α=2-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.59e-07
4.79e-08
3.38e-09
2.11e-10
1.31e-11
err
coll
1.59e-07
1.69e-07
2.03e-08
2.49e-09
3.09e-10
p`
mesh
p`
coll
const
mesh
const
coll
+1.73
+3.83
+4.00
+4.01
−0.09
+3.06
+3.02
+3.01
8.64e-06
4.55e-03
8.63e-03
9.01e-03
1.29e-07
1.62e-03
1.42e-03
1.35e-03
p`
mesh
p`
coll
const
mesh
const
coll
9.67
9.63
3.90
8.33
10.1
5.22
4.71
4.92
1.14e-03
1.08e-03
7.22e-09
1.54e-03
1.76e+04
1.78e−06
6.14e−07
1.10e−06
p`
mesh
p`
coll
const
mesh
const
coll
+5.90
+11.1
−0.03
−1.98
+5.90
+11.2
+0.65
−2.73
8.13e-09
1.09e-05
9.72e-16
4.40e-18
8.13e−09
7.57e+02
1.09e−14
4.05e−21
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.40e-06
1.72e-09
2.17e-12
1.45e-13
4.51e-16
err
coll
1.40e-06
1.28e-09
3.42e-11
1.31e-12
4.32e-14
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.36e-10
2.27e-12
1.04e-15
1.06e-15
4.18e-15
err
coll
1.36e-10
2.27e-12
9.91e-16
6.31e-16
4.18e-15
148
TABLE 6.23 : Error of error estimate based on h-h/2 for Example 6
-α=3-
1. b) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.76e-05
9.08e-06
4.77e-09
3.00e-10
1.86e-11
err
coll
5.76e-05
8.96e-06
2.15e-08
2.54e-09
3.10e-10
p`
mesh
p`
coll
const
mesh
const
coll
2.66
10.9
3.99
4.01
2.68
8.70
3.08
3.03
2.66e−02
1.36e+09
1.18e−02
1.29e−02
2.78e−02
1.87e+06
1.88e−03
1.51e−03
p`
mesh
p`
coll
const
mesh
const
coll
−1.07
+4.88
+4.49
+15.8
−1.07
+4.88
+4.49
+11.1
1.33e−08
5.05e−05
2.28e−05
8.85e+08
1.33e−08
5.05e−05
2.28e−05
1.08e+01
p`
mesh
p`
coll
const
mesh
const
coll
+5.56
+5.18
+4.76
−1.33
+5.56
+5.18
+5.25
−1.82
5.87e-09
3.43e-09
1.44e-09
6.71e-17
5.87e-09
3.43e-09
1.89e-05
6.40e-19
2. b) sbvpcol, equidistant, m=4, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
2.79e-08
5.86e-08
2.00e-09
8.86e-11
1.58e-15
err
coll
2.79e-08
5.86e-08
2.00e-09
8.86e-11
4.07e-14
3. b) sbvpcol, equidistant, m=6, Error of error
h
5.000e-01
2.500e-01
1.250e+01
6.250e-02
3.125e-02
err
mesh
1.24e-10
2.62e-12
7.24e-14
2.68e-15
6.72e-15
err
coll
1.24e-10
2.62e-12
7.24e-14
1.90e-15
6.72e-15
149
Figure 7.1 : plot of solution and error using sbvp for Example 7
150
TABLE 7.1 : global error for Example 7
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.06e-17
4.26e-18
1.65e-19
6.22e-21
3.74e-22
err
coll
2.06e-17
4.26e-18
1.65e-19
6.22e-21
3.84e-22
p
mesh
p
coll
const
mesh
const
coll
2.27
4.69
4.73
4.06
2.27
4.69
4.73
4.02
3.85e-15
5.34e-12
6.31e-12
3.27e-13
3.85e-15
5.34e-12
6.31e-12
2.75e-13
err
coll
7.84e-17
1.64e-17
1.13e-18
1.88e-19
2.24e-20
p
mesh
p
coll
const
mesh
const
coll
2.26
4.00
3.64
4.00
2.26
3.85
2.59
3.07
1.43e-14
2.64e-12
6.88e-13
3.44e-12
1.43e-14
1.69e-12
1.56e-14
1.30e-13
err
coll
1.49e-17
2.74e-18
1.32e-19
1.18e-20
8.06e-22
p
mesh
p
coll
const
mesh
const
coll
2.44
4.66
5.32
5.33
2.44
4.37
3.49
3.87
4.09e-15
3.18e-12
3.65e-11
3.76e-11
4.09e-15
1.33e-12
5.10e-14
2.80e-13
err
coll
1.73e-17
8.09e-19
3.32e-20
9.47e-22
4.15e-23
p
mesh
p
coll
const
mesh
const
coll
4.42
5.62
5.88
7.34
4.42
4.61
5.13
4.51
4.53e-13
1.68e-11
4.35e-11
2.70e-08
4.53e-13
7.94e-13
5.55e-12
3.67e-13
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.84e-17
1.64e-17
1.02e-18
8.19e-20
5.10e-21
3. sbvpcol, Gauss, m =3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.49e-17
2.74e-18
1.08e-19
2.70e-21
6.72e-23
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.73e-17
8.09e-19
1.64e-20
2.78e-22
1.70e-24
· Exact solution is a reference solution for step size h = 3.125e-03
151
TABLE 7.2 : matrix condition estimates for Example 7
1. sbvpcol, equidistant, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.80e+04
5.39e+04
1.47e+05
5.25e+05
1.98e+06
5.59e+03
2.05e+04
8.02e+04
3.20e+05
1.28e+06
ord.
condest
ord.
cond
const
condest
const
cond
-1.58
-1.44
-1.84
-1.91
-1.87
-1.97
-1.99
-2.00
4.69e+02
7.15e+02
1.65e+02
1.20e+02
7.52e+01
5.58e+01
5.12e+01
5.01e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.72
-1.85
-1.92
-1.96
-1.95
-1.99
-2.00
-2.00
1.43e+02
9.54e+01
7.36e+01
6.22e+01
4.34e+01
3.80e+01
3.68e+01
3.66e+01
ord.
condest
ord.
cond
const
condest
const
cond
-1.31
-1.83
-1.89
-1.63
-1.68
-1.98
-2.00
-2.00
2.21e+03
4.66e+02
3.81e+02
1.16e+03
3.17e+02
1.29e+02
1.22e+02
1.20e+02
ord.
condest
ord.
cond
const
condest
const
cond
-1.29
-1.12
-1.84
-2.22
-1.26
-1.92
-1.99
-2.00
8.66e+03
1.46e+04
1.02e+03
1.93e+02
3.02e+03
1.55e+03
3.13e+02
3.07e+02
2. sbvpcol, Gauss, m=2
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
7.41e+03
2.43e+04
8.77e+04
3.32e+05
1.29e+06
3.84e+03
1.48e+04
5.88e+04
2.35e+05
9.42e+05
3. sbvpcol, Gauss, m=3
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
4.53e+04
1.13e+05
4.01e+05
1.48e+06
4.59e+06
1.52e+04
4.88e+04
1.93e+05
7.69e+05
3.08e+06
4. sbvpcol, Gauss, m=4
h
condestDF
cond
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
1.70e+05
4.17e+05
9.05e+05
3.25e+06
1.51e+07
5.43e+04
1.30e+05
4.91e+05
1.96e+06
7.82e+06
·
condestDF:=condest(DF,1), cond:=cond(DF,2), Max. number of function evaluations
exceeded
152
TABLE 7.3 : global error for Example 7
1. a) sbvpcol, equidistant, m=2, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.59e-16
1.64e-17
3.71e-18
9.27e-19
1.85e-19
err
coll
1.59e-16
1.64e-17
4.09e-18
9.70e-19
1.96e-19
p
mesh
p
coll
const
mesh
const
coll
3.28
2.14
2.00
2.33
3.28
2.00
2.08
2.31
3.02e-13
1.00e-15
5.96e-15
2.52e-14
3.02e-13
6.60e-15
8.72e-15
2.42e-14
p
mesh
p
coll
const
mesh
const
coll
2.27
4.69
4.73
4.06
2.27
4.69
4.73
4.02
3.85e-15
5.34e-12
6.31e-12
3.27e-13
3.85e-15
5.34e-12
6.31e-12
2.75e-13
p
mesh
p
coll
const
mesh
const
coll
5.06
6.31
6.66
6.24
5.06
6.31
6.66
6.22
5.05e-12
2.22e-10
7.75e-10
1.22e-10
5.05e-12
2.22e-10
7.75e-10
1.14e-10
2. a) sbvpcol, equidistant, m=4, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.06e-17
4.26e-18
1.65e-19
6.22e-21
3.74e-22
err
coll
2.06e-17
4.26e-18
1.65e-19
6.22e-21
3.84e-22
3. a) sbvpcol, equidistant, m=6, Error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.44e-17
1.34e-18
1.67e-20
1.66e-22
2.20e-24
err
coll
4.44e-17
1.34e-18
1.67e-20
1.66e-22
2.23e-24
· Exact solution is a reference solution for step size h = 3.125e-03
153
TABLE 7.4 : Error of error estimate based on h-h/2 for Example 7
1. a) sbvpcol, equidistant, m=2, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.85e-17
2.95e-18
9.93e-20
2.73e-21
1.08e-21
err
coll
4.85e-17
2.95e-18
2.57e19
3.79e-20
6.31e-21
p
mesh
p
coll
const
mesh
const
coll
4.04
4.89
5.19
1.34
4.04
3.52
2.76
2.58
5.26e-13
6.83e-12
2.03e-11
9.77e-19
5.26e-13
1.12e-13
6.86e-15
3.13e-15
p
mesh
p
coll
const
mesh
const
coll
4.06
3.35
7.31
2.77
4.06
3.35
7.31
1.07
6.86e-14
8.07e-15
6.27e-09
1.59e-18
6.86e-14
8.07e-15
6.27e-09
9.23e-22
p
mesh
p
coll
const
mesh
const
coll
0.83
7.08
9.52
4.18
0.83
7.08
9.52
7.98
4.52e-19
6.18e-11
4.99e-07
3.38e-17
4.52e-19
6.18e-11
4.99e-07
5.89e-10
2. a) sbvpcol, equidistant, m=4, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.92e-18
3.54e-19
1.24e-20
8.35e-24
5.21e-25
err
coll
5.92e-18
3.54e-19
1.24e-20
8.35e-24
2.85e-24
3. a) sbvpcol, equidistant, m=6, Error of error
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
6.74e-20
3.80e-20
2.81e-22
3.83e-25
2.11e-26
err
coll
6.74e-20
3.80e-20
2.81e-22
3.83e-25
1.31e-28
· Exact solution is the reference solution for step size h = 3.90625e-004
154
TABLE 8.1 : global error for g(t)=sin(t10)cos(15t)
-α=1-
1. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.50e-03
1.88e-04
1.08e-05
6.63e-07
4.13e-08
err
coll
3.69e-03
1.89e-04
1.09e-05
6.67e-07
4.14e-08
p
mesh
p
coll
const
mesh
const
coll
4.22
4.11
4.03
4.01
4.30
4.11
4.03
4.01
5.83e+01
4.23e+01
3.09e+01
2.78e+01
7.33e+01
4.13e+01
3.12e+01
2.86e+01
err
coll
2.28e-02
5.10e-03
7.88e-04
1.08e-04
2.28e-02
p
mesh
p
coll
const
mesh
const
coll
4.35
4.13
4.04
4.01
2.16
2.69
2.86
2.94
6.77e+01
3.42e+01
2.47e+01
2.17e+01
3.30e+00
1.63e+01
3.06e+01
4.24e+01
err
coll
5.20e-03
4.41e-04
2.82e-05
1.77e-06
1.11e-07
p
mesh
p
coll
const
mesh
const
coll
5.83
5.99
6.00
6.00
3.56
3.96
3.99
4.00
5.37e+01
8.87e+01
9.07e+01
9.01e+01
1.89e+01
6.35e+01
7.02e+01
7.15e+01
err
coll
4.33e-04
1.31e-05
4.43e-07
1.40e-08
4.71e-10
p
mesh
p
coll
const
mesh
const
coll
9.49
6.65
7.90
5.49
5.04
4.88
4.98
4.90
1.47e+03
2.99e-01
2.98e+01
7.77e-04
4.81e+01
2.97e+01
4.23e+01
2.96e+01
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.03e-02
1.47e-04
8.41e-06
5.12e-07
3.18e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
8.00e-05
1.41e-06
2.21e-08
3.45e-10
5.40e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.75e-07
6.61e-10
6.56e-12
2.75e-14
6.11e-16
155
TABLE 8.2 : global error for g(t)=sin(t10)cos(15t)
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.40e-03
1.86e-04
1.07e-05
6.56e-07
4.09e-08
err
coll
3.67e-03
1.86e-04
1.08e-05
6.60e-07
4.10e-08
p
mesh
p
coll
const
mesh
const
coll
4.19
4.11
4.03
4.01
4.30
4.11
4.03
4.01
5.31e+01
4.19e+01
3.07e+01
2.76e+01
7.39e+01
4.10e+01
3.07e+01
2.83e+01
err
coll
2.28e-02
5.10e-03
7.88e-04
1.08e-04
1.41e-05
p
mesh
p
coll
const
mesh
const
coll
4.34
4.13
4.03
4.01
2.16
2.69
2.86
2.94
6.39e+01
3.43e+01
2.36e+01
2.11e+01
3.30e+00
1.63e+01
3.06e+01
4.24e+01
err
coll
5.20e-03
4.41e-04
2.82e-05
1.77e-06
1.11e-07
p
mesh
p
coll
const
mesh
const
coll
5.82
5.99
6.00
6.00
3.56
3.96
3.99
4.00
5.17e+01
8.72e+01
8.84e+01
9.03e+01
1.89e+01
6.35e+01
7.02e+01
7.15e+01
err
coll
4.33e-04
1.31e-05
4.43e-07
1.40e-08
4.71e-10
p
mesh
p
coll
const
mesh
const
coll
9.73
6.50
7.89
5.45
5.04
4.88
4.98
4.90
2.64e+03
1.66e–01
27.6e+01
6.39e–04
4.81e+01
2.97e+01
4.23e+01
2.96e+01
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.90e-03
1.43e-04
8.13e-06
4.96e-07
3.08e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.87e-05
1.39e-06
2.19e-08
3.43e-10
5.36e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.89e-07
5.75e-10
6.34e-12
2.68e-14
6.11e-16
156
TABLE 8.3 : global error for g(t)=sin(t10)cos(15t)
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
3.30e-03
1.84e-04
1.06e-05
6.49e-07
4.04e-08
err
coll
3.65e-03
1.84e-04
1.06e-05
6.53e-07
4.05e-08
p
mesh
p
coll
const
mesh
const
coll
4.17
4.11
4.03
4.01
4.31
4.11
4.03
4.01
4.84e+01
4.14e+01
3.04e+01
2.75e+01
7.48e+01
4.08e+01
3.03e+01
2.79e+01
err
coll
2.28e-02
5.10e-03
7.88e-04
1.08e-04
1.41e-05
p
mesh
p
coll
const
mesh
const
coll
4.29
4.15
4.03
4.01
2.16
2.69
2.86
2.94
5.24e+01
3.47e+01
2.20e+01
2.04e+01
3.30e+00
1.63e+01
3.06e+01
4.24e+01
err
coll
5.20e-03
4.41e-04
2.82e-05
1.77e-06
1.11e-07
p
mesh
p
coll
const
mesh
const
coll
5.81
5.99
5.99
6.00
3.56
3.96
3.99
4.00
5.03e+01
8.62e+01
8.64e+01
9.00e+01
1.89e+01
6.35e+01
7.02e+01
7.15e+01
err
coll
4.33e-04
1.31e-05
4.43e-07
1.40e-08
4.71e-10
p
mesh
p
coll
const
mesh
const
coll
9.77
6.52
7.88
5.56
5.04
4.88
4.98
4.90
2.96e+03
1.76e–01
2.61e+01
9.87e–04
4.81e+01
2.97e+01
4.23e+01
2.96e+01
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.70e-02
1.38e-04
7.78e-06
4.77e-07
2.96e-08
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.77e-05
1.38e-06
2.18e-08
3.42e-10
5.35e-12
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
4.98e-07
5.68e-10
6.16e-12
2.61e-14
5.55e-16
157
Figure 9.1 : plot of solution and error using sbvp for Example 1
1. a) Solution, α =1
b) Error, α =1
2. a) Solution, α =2
b) Error, α = 2
3. a) Solution, α =3
b) Error, α = 3
158
TABLE 9.1 : global error for g(t)=cos(15t)
-α=11. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.57e-03
1.08e-04
6.67e-06
4.15e-07
2.60e-08
err
coll
1.70e-03
1.08e-04
6.67e-06
4.17e-07
2.60e-08
p
mesh
p
coll
const
mesh
const
coll
3.89
4.02
4.01
4.00
3.97
4.02
4.00
4.00
1.23e+01
1.84e+01
1.74e+01
1.70e+01
1.60e+01
1.84e+01
1.70e+01
1.72e+01
err
coll
2.84e-02
3.46e-03
4.29e-04
5.33e-05
6.63e-06
p
mesh
p
coll
const
mesh
const
coll
3.88
4.01
3.99
4.00
3.02
3.03
3.01
3.01
1.23e+02
1.81e+02
1.71e+02
1.74e+02
2.98e+01
3.06e+01
2.81e+01
2.82e+01
err
coll
2.56e-03
1.64e-04
1.03e-05
6.44e-07
4.02e-08
p
mesh
p
coll
const
mesh
const
coll
3.75
3.94
3.99
4.00
3.99
4.00
4.00
4.00
1.17e+01
2.11e+01
2.47e+01
2.59e+01
2.52e+01
2.59e+01
2.62e+01
2.63e+01
err
coll
1.84e-04
5.82e-06
1.82e-07
5.71e-09
1.78e-10
p
mesh
p
coll
const
mesh
const
coll
5.90
5.98
5.99
6.00
4.98
5.00
5.00
5.00
8.15e+01
1.03e+02
1.10e+02
1.12e+02
1.77e+01
1.84e+01
1.85e+01
1.90e+01
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.62e-02
1.08e-03
6.83e-05
4.29e-06
2.68e-07
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.12e-03
1.56e-04
1.02e-05
6.42e-07
4.02e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.03e-04
1.73e-06
2.74e-08
4.30e-10
6.73e-12
159
TABLE 9.2 : global error for g(t)=cos(15t)
-α=21. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.59e-03
1.07e-04
6.60e-06
4.11e-07
2.57e-08
err
coll
1.69e-03
1.07e-04
6.60e-06
4.12e-07
2.57e-08
p
mesh
p
coll
const
mesh
const
coll
3.90
4.02
4.01
4.00
3.97
4.02
4.00
4.00
1.27e+01
1.83e+01
1.72e+01
1.69e+01
1.61e+01
1.83e+01
1.69e+01
1.70e+01
err
coll
7.75e-02
1.00e-03
1.09e-03
1.12e-04
1.13e-05
p
mesh
p
coll
const
mesh
const
coll
2.95
3.18
3.29
3.31
2.95
3.18
3.29
3.31
6.97e+01
1.39e+02
2.06e+02
2.29e+02
6.97e+01
1.39e+02
2.06e+02
2.29e+02
err
coll
7.31e-03
4.83e-04
2.24e-05
1.44e-06
8.34e-08
p
mesh
p
coll
const
mesh
const
coll
3.92
4.43
3.96
4.11
3.92
4.43
3.96
4.11
6.04e+01
2.81e+02
5.01e+02
9.39e+02
6.04e+01
2.81e+02
5.01e+02
9.39e+02
err
coll
5.18e-04
1.25e-05
3.19e-07
7.90e-09
1.95e-10
p
mesh
p
coll
const
mesh
const
coll
5.37
5.30
5.34
5.34
5.37
5.30
5.34
5.34
1.21e+02
9.74e+01
1.13e+02
1.16e+02
1.21e+02
9.74e+01
1.13e+02
1.16e+02
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.75e-02
1.00e-03
1.09e-03
1.12e-04
1.13e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
7.31e-03
4.83e-04
2.24e-05
1.44e-06
8.34e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
5.18e-04
1.25e-05
3.19e-07
7.90e-09
1.95e-10
160
TABLE 9.3 : global error for g(t)=cos(15t)
-α=31. sbvpcol, equidistant, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
2.78e-03
1.38e-04
6.52e-06
4.06e-07
2.54e-08
err
coll
2.78e-03
1.38e-04
6.52e-06
4.07e-07
2.54e-08
p
mesh
p
coll
const
mesh
const
coll
4.34
4.41
4.01
4.00
4.34
4.41
4.00
4.00
6.10e+01
7.49e+01
1.70e+01
1.67e+01
6.10e+01
7.49e+01
1.67e+01
1.68e+01
err
coll
1.05e-01
2.43e-02
4.14e-03
6.04e-04
8.14e-05
p
mesh
p
coll
const
mesh
const
coll
2.11
2.57
2.76
2.89
2.11
2.57
2.76
2.89
1.37e+01
5.32e+01
1.10e+02
1.91e+02
1.37e+01
5.32e+01
1.10e+02
1.91e+02
err
coll
1.09e-02
3.08e-04
2.09e-05
1.29e-06
8.05e-08
p
mesh
p
coll
const
mesh
const
coll
5.14
3.88
4.01
4.00
5.14
3.88
4.01
4.00
1.52e+03
3.49e+01
5.61e+01
5.38e+01
1.52e+03
3.49e+01
5.61e+01
5.38e+01
err
coll
1.01e-03
4.32e-05
1.58e-06
5.36e-08
1.74e-09
p
mesh
p
coll
const
mesh
const
coll
4.53
4.77
4.88
4.94
4.53
4.77
4.88
4.94
3.40e+01
6.94e+01
1.06e+02
1.37e+02
3.40e+01
6.94e+01
1.06e+02
1.37e+02
2. sbvpcol, Gauss, m=2
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.05e-01
2.43e-02
4.14e-03
6.04e-04
8.14e-05
3. sbvpcol, Gauss, m=3
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.09e-02
3.08e-04
2.09e-05
1.29e-06
8.05e-08
4. sbvpcol, Gauss, m=4
h
1.00e-01
5.00e-02
2.50e-02
1.25e-02
6.25e-03
err
mesh
1.01e-03
4.32e-05
1.58e-06
5.36e-08
1.74e-09
161