Matematics 4 — Sample problems
(1) Consider the matrix


7 2 3
A = 2 2 0 .
3 0 2
(a) Show that the matrix is positive definite.
(b) Calculate kAk2 .
(c) Calculate the condition number of A.
(2) Consider the matrix

10
A=6
0
(a) Calculate kAk1 and kAk∞ .
(b) Calculate kA−1 k2 . [Hint: Use the

6 0
10 0 .
0 9
formula κ(A) = kAk2 · kA−1 k2 .]
1 3
(3) Calculate condition number κ2 (A) of A =
.
2 −1
(4) 
Set up the energy
 functional F of the equation with the augmented matrix A =
10 6 0 18
 6 10 0 −2. Evaluate F (1, 2, −3).
0 0 9 18
(5) Consider the matrix

10 2 −1 0 3 1 0
 2 20 −1 −1 2 0 0 


0 1 0 1
−1 −1 7


A =  0 −1 0 12 5 0 2  .
3
2
1
5 30 1 6 


1
0
0
0 1 30 5 
0
0
1
2 6 5 20

(a) Use Gershgorin Theorem to estimate the condition number of A.
(b) Is A positive definite?


25 15 −5
(6) Let A =  15 18 0  .
−5 0 11
(a) Find Cholesky decomposition of A.
 
2

.
−1
(b) Using the decomposition, solve the equation Ax =
3


1 2 3

(7) Let A = 0 2 0 .
0 0 2
(a) Find the spectrum.
(b) Describe geometric and algebraic multiplicities of eigenvalues.
(c) Using Cayley–Hamilton Theorem, find the inverse of A.
2
(8) Suppose that A is a symmetric positive definite matrix with the condition number
κ = 10. Suppose that the initial error of the solution of Ax = b satisfies ke0 k = 1.
How many steps do you have to perform to achieve precision 0.0001 with
(a) the steepest descent method,
(b) the conjugate gradients method?
(9) How many solutions does the following boundary value problem has?
16u00 + 9π 2 u = −32,
u(−1) = u0 (1) = 0.
(10) How many solutions does the following boundary value problem have?
5πx
,
4u00 + π 2 u = sin
2
u(0) = u0 (5) = 0.
(11) How does the number of solutions of the following problem depend on λ?
π
π
u00 + λu = 2 sin x − 3 sin x,
3
2
u(0) = u(6) = 0.
(12) Find the minimal energy of the problem
x
−u00 + 3u = cos ,
2
u0 (0) = u0 (2π) = 0.
(13) Find the minimal energy of the problem
3
−u00 + u = sin x,
2
u(0) = u0 (π) = 0.
(14) Solve the initial value problem
y 00 + y 0 = 2x − 3y(0) = y 0 (0) = 1.
(15) Find the general solution of the equation
y 00 + 4y =
1
.
cos 2x
(16) Consider the boundary value problem
x
u00 + u = cos ,
2
0
u (0) = u(2π) = 0.
3
(a) Show that the problem has a unique solution.
(b) Is the solution stable?
(17) How many solutions does the following problem have? u00 + π 2 u = sin πx, u(0) =
u0 (2) = 0
(18) How many solutions does the following problem have? u00 +
u0 (3) = 0
π2
u
4
(19) How many solutions does the following problem have? u00 +
3 sin πx
, u(0) = u0 (3) = 0
6
= sin πx
, u(0) =
2
25 2
π u
36
= 2 sin 5πx
−
6
, u0 (0) = u0 (2) = 0
(20) u00 + π 2 u = cos 3πx
2
(21) How many solutions does the following problem have? u00 −u = 0, u(0)+u0 (0) = 1,
u(π) + u0 (π) = 0
(22) How does the number of solutions depend on the parameter λ? u00 + λu = 1,
u(0) = u0 (2) = 0
(23) How does the number of solutions depend on the parameter λ? u00 + λu = 1,
u0 (0) = u0 (2) = 0
(24) Show that the following problem has a unique solution. x3 u00 + 3x2 u0 − xu = x4 ,
u(2) = u(5) = 0
(25) Show that the following problem has a unique solution. u00 cos x − u0 sin x −
u tan x = ln tan x, u(π/6) = u0 (π/3) = 0