A Concise Formula for Two Dimensional Divided Differences with

Journal of Mathematical Extension
Vol. 7, No. 4, (2013), 63-69
A Concise Formula for Two Dimensional
Divided Differences with Multiple Knots
R. Roozegar∗
Yazd University
A. A. Jafari
Yazd University
Abstract. The well-known formulas for divided differences in the case
of one and two dimensional with multiple knots have played an important role in applied mathematics, particularly in numerical analysis,
polynomial and lagrange interpolations. In this article we obtain a new
formula for two dimensional divided differences in the case of multiple
knots. This formula has a simpler form than the known formula given
in literature. It is friendly using for computations and analysis.
AMS Subject Classification: 41A63; 65D05
Keywords and Phrases: Divided difference, multiple knots, one and
two dimensional divided differences
1.
Introduction
The divided difference has appeared to be a powerful tool in certain areas
of applied mathematics, as numerical analysis, calculus of finite differences,
interpolation theory and probability theory. Two good sources are the texts by
Gelfond [2] and Milne-Thomson [4].
Let f : R → C be a differentiable function, where C is the complex number. Assume that s0 , s1 , ..., sm are distinct real numbers subject to s0 < s1 < · · · < sm ;
such points are known as knots. The divided differences of f are recursively
given by the following formula,[3] and [7]:
[f ; s0 ] = f (s0 ),
∗
Received: June 2013; Accepted: December 2013
Corresponding author
63
64
R. ROOZEGAR AND A. A. JAFARI
[f ; s1 , . . . , sm ] − [f ; s0 , s1 , . . . , sm−1 ]
if sm 6= s0 .
sm − s0
From the recursive formulas, it is clear that if s = s0 , then
[f ; s0 , s1 , . . . , sm ] =
[f ; s, s0 ] =
f (s) − f (s0 )
.
s − s0
If repetitions are permitted in the arguments and the function f is smooth
enough, then
lim [f ; s, s0 ]
s→s0
=
lim
s→s0
f (s) − f (s0 )
s − s0
0
= f (s0 ).
This gives the definition of first-order divided difference with repeated points,
0
[f ; s0 , s0 ] = f (s0 ).
In general, let s0 6 s1 6 · · · 6 sm ; the divided differences with repeated points
obey the following recursive formula:
( [f ;s ,··· ,s ]−[f ;s ,s ,··· ,s
1
m
0 1
m−1 ]
, if sm 6= s0,
sm −s0
(1)
[f ; s0 , s1 , · · · , sm ] =
(m)
f
(s0 )
,
if sm = s0.
m!
When s0 , s1 , · · · , sm are all distinct, it follows inductively from (1) that
[f ; s0 , s1 , · · · , sm ] =
m
X
f (sk )
,
v (1) (sk )
k=0
(2)
Qm
d
where v(s) = j=0 (s − sj ) and v (1) (s) = ds
v(s), [5].
We recall that (2) is the m-th order divided differences of f on the knots
s0 , s1 , ..., sm .
Let us introduce the following notations. Let N = {1, 2, . . .}, N0 = N ∪ {0},
m, n ∈ N0 , and let k0 , k1 , . . . , km , l0 , l1 , . . . , ln ∈ N so that k0 + k1 + · · · + km =
M +1 and l0 +l1 +· · ·+ln = N +1. Also let α = max{k0 −1, k1 −1, . . . , km −1}
and β = max{l0 − 1, l1 − 1, . . . , ln − 1}; I, J ⊆ R be intervals, I × J be a
bidimensional interval and Dα,β (I × J) be the set of all real valued bivariate
∂f i+j
functions f with the property that exists ∂s
i ∂tj on I ×J, where i ∈ {0, 1, . . . , α}
and j ∈ {0, 1, . . . , β}.
In the following, the one dimensional divided differences with multiple knots is
(k ) (k )
(k )
denoted by [f ; s0 0 , s1 1 , . . . , smm ] where s0 , s1 , . . . , sm are distinct and
(kj )
sj
= sj , sj , . . . , sj .
|
{z
}
kj times
AACONCISE
FORMULA
FORFOR
TWOTWO
DIMENSIONAL
...
CONCISE
FORMULA
DIMENSIONAL
... 65
A CONCISE
FORMULA
FOR TWO
DIMENSIONAL
...
A CONCISE FORMULA FOR TWO DIMENSIONAL ...
65
65
65
We
for] for
distinct
knotsknots
only. only.
We use
use[f;
[fs;;0ss,0s,,1ss, 1. .,, ... ,..s..m,, ss] m
distinct
We
use
[f
knots only.
0which
1 are available
m ] for distinct
The
formulas
in
the
literature
for onefor
dimensional
divided divided
The
formulas
which
are
available
in the
the
literature
one dimensional
dimensional
We use
[f ; s0 , which
s1 , . . . ,are
sm ]available
for distinct
knots
only.
The
formulas
in
literature
for complicated.
one
differences
with
multiple
knots
are
computationally
rather
It is divided
differences
with
multiple
knots are
are
computationally
rather
complicated.
It is
is
The formulas
which
are available
in computationally
the literature forrather
one dimensional
divided
differences
with
multiple
knots
complicated.
It
the
ratio
ofoftwo
determinants,
namely,
the
ratio
two
determinants,
namely,
differences
with
multiple
knots
are
computationally
rather
complicated.
It
is
the ratio of two determinants, namely,
the ratio of two determinants, namely,
(k0 ) (k
(k )
(k10)), . . (k
(km )
, s1(k
. , 1s)mm ) s(k
(k (k
) )(k1 )(k )
(km )(k ) (W f)(s
(W0 ff )(s
)(s
s(k
)
mm )(3)
0 0 ) ,, s
1 1 ) ,, .. .. .. ,, s
m
[f(s,
t);t);
s0 s0(k
,00s)1, s(k, .11.)., ,.s.m
](ks m
=) ]s = (W
m
[f
(s,
(3)
.
,
s
(k
)
(k
)
(km )) ,
1 (k
1m)
(k0 ) (k
m ] = (W
(k100)), 0. .,(k
(k
) ),
[f (s, t); s00(k0 ), s11(k1 ), . . . , sm
(3)
1s)
m
s
,
.
.
.
,
s
m
,(ss)(s
.
,
)
m
(km ) s V (s0 V f
0
1
s(k
s(k
mm ) ) ,
0 0 ) ,, s
1 1 ) ,, .. .. .. ,, s
[f (s, t); s0 , s1 , . . . , sm ]s = V (s1(k
(3)
m
(k0 ) 1
(k1 )
(km ))
0
V
(s
,
s
,
.
.
.
,
s
)
m
0
1
where
where
(k0 ) (k(k)1 ) (k ) (km )
where
(km )
(4)
(W(W
f)(s )(s
,(k
s 0 ,s.(k. 1.1 ), ,s.m. . ,)s(k
)
(4)
where
mm ) )
0 10 ) ,, s
1
(W ff0)(s
(4)
(k0 ) 1
(k1 ), . . . , sm
0
(km )
(4)
(W f )(s0 , s1 , . . . , sm )


 1 s
2 2
M −1
· · ·· · ·
sM−1
f(s0 ) f (s0 )

0s0 s0 s2
0
1
s
M
−1
0
0
s0 · · ·
s0M −1M −2
f (s0 )
 1 s
M−2

· · ·· · ·
(M −(M
1)s−
f´(s0 ) f´´
 001 1s100 2s2s
0s2
0s01)sM
(s0 ))


00

0 −2
0
1
2s
·
·
·
(M
−
1)s
f
(s
0

0
M −2
´ 0
 · ·· 0·· · · ···1· · 2s · · ·· · ·
· · · −· 1)s
···
·
·
·
(M
f


0

0
···
·(s
·· ··0 )
M−k0 M −k
(k00−1)


 0··0·· ·· 0··0·· ·· 0 0 · · ··· ·· ··(M (M
(k
−1)
− 1)(M
− 2).−
. . 2).
. .(M
f−k
(sf0(k
) 00 −1)
·−
·.(M
·k0 +−1)s
·
· · (s0 )
0 + 1)sM


−
1)(M
.
.
.
k
0
0

− k0 + 1)s00M −k0· · · f (k0
0 · · ·· · · (M − 1)(M − ·2).

0)
 · ··00·· · · ···00· ·
−1)
=
,· (s
· · .. ..·..·.(M
=


·
·
·
·
·
·
(M
−
1)(M
−
2).
.(M
−
k
+
1)s
f
(s0 )
0

0
0
=
·
·
·
·
·
·
·
·
·
·
·
·
·
·
·
2
M−1

 1·1· · sms· · · sms2 · · ·· · ·
M −1
sm sM
f(sm ) f (s
=

· ·−1
·
· ·m
· ))
m
 1 sm
m
m
s2m
···
sm
M−2
´

m

2
M
−1
M
−2
0
1
2s
·
·
·
(M
−
1)s
f
(sm ) ff´´(s
m
m


1
s
s
s
0
1
2s
·
·
·
(M
−
1)s
(s
)
M
−2

m
m
m
m
m
m
(M
1)sm
f´(s

m)
m· · ·· · ·

M −2
 · ··00·· · · ···11· · 2s
··· −
···
·
·
·
·
·
·
·
·
·
2s
(M
−
1)s
f
(s
m
m

 ··· ···
· · k· m+ 1)sM−km Mf −k
··· )
(km −1) (k −1)
 0·0· · 0·0· · 0 0 · · ··· ·· ·(M
− 1)(M
− 2).−
. . 2).
. .(M
−
(s
)
m
m···
m
· .(M
· · m − kmm+ 1)sM
(M
−
1)(M
.
.
.
f
(sm ))
·
−k
(k
−1)
m
m
f (km
(s
0
0
0
· · · (M − 1)(M − 2). . . . .(M − km + 1)sm
m
−km
m −1)
f
(sm )
0
0
0
· · · (M − 1)(M − 2). . . . .(M − km + 1)sM
m
and
and
(k0 ) (k(k)1 ) (k ) (km )
and
m)
V (sV0 (s
,(ks 0 ) , (k
· · 1· ), sm ) (km
(5)
)
(5)
and
m ))
V (s00(k100 ),, ss11(k11 ),, ·· ·· ·· ,, ss(k
(5)
m
(km )
V (s0 , s1 , · · · , sm )
(5)



M
sM

 1 1 s0s0 s20 s220 · · · · · ·
0
sM
0
s
·
·
·
s
1
s


M−1
0
 0 1 1 s 2s0 s02 · · ·

M −1
Ms0 M s0M

10 2s
2s00 ·· ·· ··
 00


0 −1
1
M
s
0
· · · · · ·

−1

· · · M s· 0M
 ·· 0·· ·· ·· 1·· ·· 2s0· · · ·· ·· ··


· ·0· ·· M−k0 +1M −k

 0
+1
0
1).−. . 1).
. .(M
k0· +
·−
· 2)s
M−
(M
. .. ..−.(M
.(M
k00 +
+ 2)s
2)sM
 ·00· · 0 ·00· · 0 00 · · · ·· ·· ··M(M
 0 +1 
−k
0
M
(M
−
1).
.
−
k
 ,0 +1 
0
−k
· ···0· · · · ·· 0· ·
,
==
· ·· −
· k0 + 2)s0M
0 · · · ·· ·· ·· M (M − 1). .· .· .· .(M


,
0
=
·
·
·
·
·
·
·
·
M
 1 sm s2 2 · · ·

,
M
sm s· M
= ·1· · s· m
· · mss2m ·· ·· ··
··


m
1
s
s
M−1 m
 0

m
m

M
−1
2
Msm M sM −1
2s
 001 1s11m 2sm

smm· · · ·· ·· ··

m
M
s
2s
m


m

M
−1
· · · M s· · ·
· ···0· · · · ·· 1· · 2sm· · · · · ·



m·
 ··· ···

·
·
·
·
·
M−km +1
M
−k
+1
−
1).−. .1).
. .(M
−.(M
km· +
2)s
 0·0· · 0 ·0· · 0 0 · · · · ·M(M

m
m
·
·
·
M
(M
.
.
.
−
k
+
2)s
m + 2)sM
m −km +1
0
0
0
· · · M (M − 1). . . . .(M − km
M −km +1
0
0
0
· · · M (M − 1). . . . .(M − km + 2)sm
m
is
determinant
of Vandermonde,
[5]. [5].
is the
thegeneralized
generalized
determinant
of Vandermonde,
Vandermonde,
is
the
generalized
determinant
of
[5].
Similarly,
the
order
divided
differences
of theoffunction
f(s, .) :f (s, .) :
Similarly,
theN-th
N -th
-th
order
divided
differences
the
is the generalized
determinant
of Vandermonde,
[5].function
the
N
order
differences
.) :
l the function
ln
1t n is defined
JSimilarly,
→
R,
s s∈∈
I,I,with
respect
todivided
the
knots
tl00 , tl11tll,00.,. t.llof
,of
by ff (s,
n
J
→
R,
with
respect
to
the
knots
,
.
.
.
,
t
is defined
defined
by:
Similarly,
the
N
-th
order
divided
differences
the
function
(s, by
.)
J → R, s ∈ I, with respect to the knots t00l0 , t11l11 , . . . , tnlnlnn is
J → R, s∈ I, with respect to the
knots
t
,
t
,
.
.
.
,
t
is
defined
 (W f)0(t(l10 ) , t(l(l1 )0,).n. .(l, 1t)(ln ) ) (lnby
)

n
(l0 )(l (l
)
1
(l
)
(W0ff )) (t
(t1(l

n
nn) )
(l
(ln=) 
0 0 ) ,, tt(l
1 1 ) ,, .. .. .., ,, tt(l
f(s,
t);t);
t0 t(l, 00t)1) , t(l
, .11.)).,,.t.n. , t(l
(W
n
(s,
(l0 ) 1
(l
(ln )) ,
0
ff (s,
1n))
(l
(l
)
n )  = (l0 )
1
n
0
1
(l
(l
t); t (l ), t (l ), . . . , t t n ) t =
(t0,0). . .(l,,11tt)1)n, . .,).. ,. t.(l
, ntn) ) ,
t)1(l
V (t(WV,f(t
nn) )
t = 0V
1 , . . . , tn
0 0 ) ,, tt(l
,
f (s, t); t00 0 , t11 1 , . . . , tn(l
(t
n
(l
))
(l
)
(l
)
1
0
1
0
t
V (t0 , t1 , . . . , tn n )









,
,
,








66
R. ROOZEGAR AND A. A. JAFARI
(l )
(l )
(l )
(l )
(l )
(l )
where (W f ) (t0 0 , t1 1 , . . . , tn n ) and V (t0 0 , t1 1 , . . . , tn n ) are defined as above.
Soltani and Roozegar [6] give a new and compact formula for one dimensional
divided differences with multiple knots. They use this formula to find the distribution of randomly weighted averages. This formula comes as a theorem in
next section.
In this article we present a concise and compact formula for two dimensional
divided differences with multiple knots. The advantages of our formula are
in its formulation and computation. It can be readily used in calculus of divided differences, Hermite interpolation polynomial and bivariate Lagrange interpolation polynomials. As an illustration, we derive two dimensional divided
1
.
differences for the fw (s, t) = (w−s)(w−t)
2.
Main Results
In this section we give our formula for two dimensional divided differences
with multiple knots, Theorem 2.2. First we present a compact formula for one
dimensional divided differences, [6].
Theorem 2.1. Let f ∈ Dα (I), where Dα (I) is the set of all real functions f ,
α-times differentiable on I. Then
(k0 )
(k1 )
[f ; s0
, s1
We let
Qm
(k0 )
m)
, . . . , s(k
]
m
i=0, i6=j (si
=
(W f )(s0
(k )
(k1 )
, s1
(k )
(k )
, . . . , smm )
(k )
V (s0 0 , s1 1 , . . . , smm )
m
X
(−1)M +kj −1 dkj −1
f (sj )
=
( Qm
).
k
−1
ki
j
(kj − 1)! ds
i=0, i6=j (si − sj )
j=0
j
− sj )ki = 1 when m = 0.
Now we notice to the following theorem which states the main theorem. The
main difference between the following formula and the result obtained by Pop
and Barbosu [5] is in its formulation, computation and application. No closed
form expression for two dimensional divided differences in the case of multiple
knots expressed in [5]. That formula is based on some long and complicated
determinants and is used numerically in application. For instance, the formula
in [5] is not applicable to find the distribution of two dimensional randomly
weighted averages in [6], but the following result is a user-friendly formula for
applicability and analysis.
Theorem 2.2. Let f ∈ Dα,β (I × J). Then the following equalities
i
i
hh
(k ) (k )
(l ) (l )
n)
m)
; t0 0 , t1 1 , . . . , t(l
f (s, t); s0 0 , s1 1 , . . . , s(k
m
n
s
t
67
A CONCISE FORMULA FOR TWO DIMENSIONAL ...
hh
=
(l )
(l )
n)
f (s, t); t0 0 , t1 1 , . . . , t(l
n
i
(k0 )
t
m X
n
X
(−1)M +N +ki +lj −2
=
; s0
(k1 )
, s1
m)
, . . . , s(k
m
i
s
∂ ki +lj −2
(ki − 1)!(lj − 1)! ∂ski −1 ∂tlj −1
j
i
i=0 j=0
f (si , tj )
Qn
Qm
kq
lr
r=0,r6=j (tr − tj )
q=0,q6=i (sq − si )
!
.
Qm
Qn
hold. We let q=0,q6=i (sq − si )kq = 1 when m = 0 and r=0,r6=j (tr
− tj )lr = 1 when n = 0.
Proof. For m = n = 0, k0 = M + 1 and l0 = N + 1, it follows from the
Definition 2.1. that


1 ∂M


f (s0 , t),
f (s, t); s0 , s0 , . . . , s0  =
|
{z
}
M ! ∂sM
0
(M +1) times
s
and


1 ∂N


f (s, t0 ).
f (s, t); t0 , t0 , . . . , t0  =
|
{z
}
N ! ∂tN
0
(N +1) times
t
Therefore,
hh
(k0 )
f (s, t); s0
i
(l )
s
; t0 0
1 ∂M
(l0 )
f
(s
,
t);
t
0
0
M ! ∂sM
0
t
M
1
∂
(l0 )
f (s0 , t); t0
M ! ∂sM
0
t
i
=
t
=
=
1
∂ M +N
f (s0 , t0 ).
N
M !N ! ∂sM
0 ∂t0
For general m, n, by Theorem 2.1. we have
hh
(k0 )
f (s, t); s0
(k1 )
, s1
m)
, . . . , s(k
m
i
(l )
s
(l )
n)
; t0 0 , t1 1 , . . . , t(l
n
i
t
68
R. ROOZEGAR AND A. A. JAFARI
"
#
m
X
f (si , t)
(−1)M +ki −1 ∂ ki −1
(l0 ) (l1 )
(ln )
( Qm
=
); t0 , t1 , . . . , tn
kq
(ki − 1)! ∂ski i −1
q=0,q6=i (sq − si )
i=0
#t
"
m
X
(−1)M +ki −1 ∂ ki −1
f (si , t)
(l ) (l )
); t0 0 , t1 1 , . . . , tn(ln )
=
( Qm
ki −1
kq
(k
−
1)!
(s
−
s
)
∂s
i
q
i
q=0,q6=i
i
i=0
t


ki −1
f (si ,tj )
∂
m
n
(Q
kq )
X
(s
−s
)
(−1)M +ki −1 X (−1)N +lj −1 ∂ lj −1 ∂ski i −1 m
q
i
q=0,q6=i
=
) ,
( Qn
lj −1
lr
(k
−
1)!
(l
−
1)!
(t
−
t
)
i
j
r
j
∂t
r=0,r6
=
j
i=0
j=0
j
which gives the result. Definition 2.3. The (m, n)-th order divided difference of the function f ,
where f ∈ Dα,β (I × J), with respect to the distinct knots (si , tj ) ∈ I × J,
i ∈ {0, 1, . . . , m} and j ∈ {0, 1, . . . , n} is defined by
#
"
(k ) (k )
(k )
s0 0 , s1 1 , . . . , smm
f;
(l ) (l )
(l )
t0 0 , t 1 1 , . . . , t n n
=
m X
n
X
(−1)M +N +lj +ki −2
i=0 j=0
∂ ki +lj −2
(ki − 1)!(lj − 1)! ∂ski −1 ∂tlj −1
i
j
f (si , tj )
Qn
).
kq
lr
(s
−
s
q
i)
q=0,q6=i
r=0,r6=j (tr − tj )
( Qm
The following corollary has given by [1].
Corollary 2.4. If f (x, y) = g(x)h(y), (∀)(x, y) ∈ [a, b]×[c, d] where g : [a, b] →
R, h : [c, d] → R, the following equality
x , x , . . . , xm
f; 0 1
= [g; x0 , x1 , . . . , xm ][h; y0 , y1 , . . . , yn ],
y0 , y 1 , . . . , yn
holds.
Now, we present an example using the Theorem 2.2, Corollary 2.1 and the work
of Soltani and Roozegar [6].
1
and for given real numbers s0 , s1 , . . . , sm ,
Example 2.5. Let fw (s, t) = (w−s)(w−t)
t0 , t1 , . . . , tn , w and integers ki > 1, i = 0, 1, . . . , m; li > 1, i = 0, 1, . . . , n then
#
"
m
n
(k )
(k ) (k )
Y
Y
1
1
s0 0 , s1 1 , . . . , smm
=
f;
.
(ln )
(l0 ) (l1 )
kj
(w
−
s
)
(w
−
ti )li
t0 , t 1 , . . . , tn
j
j=0
i=0
A CONCISE FORMULA FOR TWO DIMENSIONAL ...
69
Acknowledgement
The authors express their sincere thanks to the referees for valuable comments
and to the Editor-in-Chief and Manager for his time and kind efforts. The
authors are also thankful to the Yazd University for supporting this research.
References
[1] D. Barbosu, Two dimensional divided differences revisited, Creative Math.
& Inf., 17 (2008), 1-7.
[2] A. O. Gelfond, Calculus of Finite Differences, Hindustan publishing corporation, Delhi-7, India, 1971.
[3] S. Karlin, C. A. Micchelli, and Y. Rinott, Multivariate splines: A probabilistic perspective. J. Multivariate Anal., 20(1) (1986), 69-90.
[4] L. M. Milne-Thomson, The Calculus of Finite Differences, Mcmillan, New
York, 1965.
[5] O. T. Pop and D. Barbosu, Two dimensional divided differences with
multiple knots, An. Şt. Univ. Ovidius Constanţa, 17 (2009), 181-190.
[6] A. R. Soltani and R. Roozegar, On distribution of randomly ordered uniform incremental weighted averages: Divided difference approach. Statist.
Probab. Lett., 82(5) (2012), 1012-1020.
[7] A. Xu and C. Wang, Some extensions of fa‘a di brunos formula with
divided differences. Comput. Math. Appl., 59(6) (2010), 2047-2052.
Rasool Roozegar
Department of Statistics
Assistant Professor of Statistics
Yazd University
P.O. Box 89195-741
Yazd, Iran
E-mail: [email protected]
Ali Akbar Jafari
Department of Statistics
Assistant Professor of Statistics
Yazd University
P.O. Box 89195-741
Yazd, Iran
E-mail: [email protected]