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Explicit inverse of a generalized Vandermonde matrix
Applied Mathematics and Computation 146 (2003) 643–651 www.elsevier.com/locate/amc
Explicit inverse of a generalized Vandermonde matrix Moawwad E.A. El-Mikkawy Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract In this paper the author gives an explicit closed form expression for the n n inðkÞ ðkÞ verse matrix ðVG ðnÞÞ1 of the generalized n n Vandermonde matrix VG ðnÞ by using the elementary symmetric functions. Symbolic and numerical results are presented. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Algorithms; Vandermonde matrix; Stirling matrix; Symmetric functions; MAPLE
1. Introduction There are many special types of matrices which are of great importance in many scientific and engineering work. For instance matrices of type tridiagonal [1,2], pentadiagonal [3], Pascal [4], Vandermonde [5,6] and others. This paper will be concerned with matrices of Vandermonde type. These types of matrices frequently appear in many applications. For example in curve fitting, interpolation, scattering and in the derivation of explicit Runge–Kutta and Runge–Kutta–Nystrom [7–9] numerical methods. In this paper we are going to study the possibility of obtaining an explicit closed form of the Vandermonde inverse matrix. The main results are presented in Sections 2 and 3.
E-mail address: [email protected] (M.E.A. El-Mikkawy). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00609-4
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M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
2. The main results Let us begin this section by giving the following definitions Definition 2.1 [5]. If the n parameters c1 ; c2 ; . . . ; cn are distinct, then the eleðnÞ mentary symmetric functions ri;j in c1 ; c2 ; . . . ; cj1 ; cjþ1 ; . . . ; cn are defined for j ¼ 1ð1Þn by ðnÞ
r1;j ¼ 1 n n X X ðnÞ ri;j ¼
n X
i1 Y
crm
for i ¼ 2ð1Þn
ð2:1Þ
ri1 ¼ri2 þ1 m¼1 ri1 6¼j
r1 ¼1 r2 ¼r1 þ1 r1 6¼j r2 6¼j
Definition 2.2 [10]. The n n Stirling matrix of the first kind is defined by 3 2 ð1Þ 0 0 0 r1;1 7 6 ð2Þ 6 rð2Þ r1;2 0 0 7 2;2 7 6 ð3Þ ð3Þ ð3Þ 6 r3;3 r2;3 r1;3 0 0 7 ð2:2Þ S1 ¼ 6 7 7 6 .. .. .. .. 7 6 4 . 0 5 . . . ðnÞ ðnÞ nþ1 ðnÞ nþ2 ðnÞ ð1Þ rn;n ð1Þ rn1;n r2;n r1;n ðnÞ
in which the ri;j are calculated at cr ¼ r, r ¼ 1ð1Þn. Denote by rðGÞ ¼ ðbij Þ; i; j ¼ 1ð1Þn, where bij are given by n ðnÞ
bij ¼ ri;j
ð2:3Þ
The elements of the n n matrix rnðGÞ in (2.3) may be calculated by using the following algorithm Algorithm 2.1. For n P 1 we may calculate the elements of the first column of in (2.3) as follows the n n matrix rðGÞ n ð1Þ
Set r1;1 ¼ 1 For i ¼ 2; 3; . . . ; n ðiÞ ði1Þ ri;1 ¼ ri1;1 ci For j ¼ i 1; i 2; . . . ; 2 ðiÞ
ði1Þ
ði1Þ
rj;1 ¼ rj1;1 ci þ rj;1
ð2:4Þ
Next j Next i The elements in the remaining n 1 columns of rðGÞ may be obtained by n symmetry by using
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651 ðnÞ
ri;k
ðnÞ ¼ ri;1
ck !c1
i ¼ 1ð1Þn; k ¼ 2ð1Þn
;
645
ð2:5Þ ðnÞ
The notation in (2.5) means that for specific i and k, ri;k may be obtained from ðnÞ the algebraic expression of ri;1 by replacing each ck by c1 in the expression of ðnÞ ri;1 . For example for n ¼ 4 we have 2
3 1 1 1 1 6 c1 þ c3 þ c4 c2 þ c1 þ c4 c2 þ c3 þ c1 7 c2 þ c3 þ c4 ðGÞ 7 r4 ¼ 6 4 c2 c3 þ c2 c4 þ c3 c4 c1 c3 þ c1 c4 þ c3 c4 c2 c1 þ c2 c4 þ c1 c4 c2 c3 þ c2 c1 þ c3 c1 5 c2 c3 c4 c1 c3 c4 c2 c1 c4 c2 c3 c1
ð2:6Þ
Unless otherwise stated, we shall be concerned with the case when cr ¼ r, r ¼ 1ð1Þn. For this case the matrix rðGÞ will be denoted by rðSÞ n n . The capital ðSÞ letters G and S stand for general and special respectively. For example r4 is given by 2
ðSÞ
r4
1 6 9 ¼6 4 26 24
1 8 19 12
1 7 14 8
3 1 6 7 7 11 5 6
ð2:7Þ
Definition 2.3. For the n þ 1 real values k; c1 ; c2 ; . . . ; cn we define a generalized ðkÞ ðkÞ Vandermonde matrix VG ðnÞ by VG ðnÞ ¼ ðVij Þ with Vij ¼ ckþj1 ; i; j ¼ 1ð1Þn. i For the special case when cr ¼ r, r ¼ 1ð1Þn this matrix will be denoted by ðkÞ VS ðnÞ. Qn Qn Qi1 ðkÞ It is known that detðVG ðnÞÞ ¼ ð m¼1 ckm Þ i¼2 j¼1 ðci cj Þ. Thus both ðkÞ ðkÞ VG ðnÞ and VS ðnÞ are non singular if and only if the n parameters c1 ; c2 ; . . . ; cn are distinct. ðkÞ ð0Þ For any k, the two matrices VG ðnÞ and VG ðnÞ satisfy ðkÞ
ð0Þ
VG ðnÞ ¼ DVG ðnÞ
ð2:8Þ
where D ¼ diagðck1 ; ck2 ; . . . ; ckn Þ
ð2:9Þ
For the matrix D in (2.9) we have k k D1 ¼ diagðck 1 ; c2 ; . . . ; cn Þ ðkÞ
ð2:10Þ ðkÞ
Therefore the inverse matrix ðVG ðnÞÞ1 of the matrix VG ðnÞ may be obtained ð0Þ once the inverse matrix ðVG ðnÞÞ1 is available.
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M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
In what follows we are going to show how can we obtain the matrix ð0Þ 1 ðVG ðnÞÞ . Suppose we have n distinct values c1 ; c2 ; . . . ; cn and consider the Lagrange polynomials L1 ðxÞ; L2 ðxÞ; . . . ; Ln ðxÞ of degree (n 1) defined by n Q
ðx cj Þ
j¼1 j6¼i
Li ðxÞ ¼ Q n
ð2:11Þ ðci cj Þ
j¼1 j6¼i
Consider also the polynomial wðxÞ of degree n defined by wðxÞ ¼
n Y ðx ci Þ
ð2:12Þ
i¼1
It is easy to show that Li ðxÞ ¼
ðx ci Þ1 wðxÞ d wðxÞjx¼ci dx
for i ¼ 1ð1Þn
ð2:13Þ
Moreover by using partial fractions together with (2.13) we obtain n X
Li ðxÞcmi ¼ xm
for m ¼ 0ð1Þn 1
ð2:14Þ
i¼1
Thus 2
L1 ðxÞ L2 ðxÞ .. .
3
2
1 c1 c21 .. .
7 6 6 7 6 6 7 6 6 7¼6 6 7 6 6 4 Ln1 ðxÞ 5 4 c1n1 Ln ðxÞ
1 c2 c22 .. .
1 cn1 c2n1 .. .
1 cn c2n .. .
31 2 7 7 7 7 7 5
3 1 6 x 7 6 . 7 6 .. 7 6 7 4 xn2 5
xn1 cn1 cn1 n1 n 3 3 2 2 1 1 6 x 7 6 x 7 7 7 6 6 ð0Þ ð0Þ T 1 6 .. 7 1 T 6 .. 7 ¼ ððVG ðnÞÞ Þ 6 . 7 ¼ ððVG ðnÞÞ Þ 6 . 7 4 xn2 5 4 xn2 5 c2n1
xn1 ð0Þ
ð2:15Þ
xn1
Now if ððVG ðnÞÞ1 ÞT ¼ ðwij Þ1 6 i; j 6 n then (2.15) and (2.11) together with (2.1) give
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651 n Q
Li ðxÞ ¼
n X j¼1
ðx cj Þ
j¼1 j6¼i
wij xj1 ¼ Q n
n P
¼ ðci cj Þ
j¼1 j6¼i n P
¼
647
ðnÞ
ð1Þjþ1 rj;i xnj
j¼1 n Q
ðci cj Þ
j¼1 j6¼i
nj ðnÞ rnjþ1;i xj1
ð1Þ
j¼1 n Q
ð2:16Þ ðci cj Þ
j¼1 j6¼i
Thus (2.16) yields nj ðnÞ
ð1Þ rnjþ1;i ; wij ¼ Q n ðci cj Þ
i; j ¼ 1ð1Þn
ð2:17Þ
j¼1 j6¼i
At this point it is now possible to formulate the following general result: ðkÞ
Algorithm 2.2. For any integer n P 1 the n n matrix ðVG ðnÞÞ1 in Definition 2.3 is given by ðkÞ
ðVG ðnÞÞ
1
¼ MðnÞ
ð2:18Þ
where the n n matrix MðnÞ is given by MðnÞ ¼ ðmij Þ;
i; j ¼ 1ð1Þn
mij ¼ ð1Þnþi
rniþ1;j ; n Q ckj ðcj ci Þ
with ðnÞ
i; j ¼ 1ð1Þn
ð2:19Þ
i¼1 i6¼j
The following are special cases of Algorithm 2.2 ð0Þ
1
Algorithm 2.3. For any integer n P 1, the n n matrix ðVS ðnÞÞ 2.3 is given by ð0Þ
ðVS ðnÞÞ1 ¼
1 V ðnÞ ðn 1Þ!
in Definition
ð2:20Þ
where the n n matrix V ðnÞ is a matrix whose entries are all integers and given by
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M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
V ðnÞ ¼ ðvij Þ;
i; j ¼ 1ð1Þn
with vij ¼ ð1Þ
iþj
n 1 ðnÞ rniþ1;j ; j1
i; j ¼ 1ð1Þn
ð2:21Þ
Note 2.1. Another approach to find the n n matrix V ðnÞ in (2.20) is given in [11]. This approach depends on writing V ðnÞ as the product of two matrices of the form T ðnÞU ðnÞ where T ðnÞ is an upper triangular matrix related to the matrix S1 in (2.2). Also the matrices T ðnÞ and U ðnÞ satisfy some recurrence relations. For more details see [11]. ðkÞ
Algorithm 2.4. For any integer n P 1, the n n matrix ðVS ðnÞÞ1 in Definition 2.3 is given by ðkÞ
ðVS ðnÞÞ
1
¼ W ðnÞ
ð2:22Þ
where the n n matrix W ðnÞ is given by W ðnÞ ¼ ðwij Þ;
i; j ¼ 1ð1Þn
with iþj
wij ¼
ð1Þ ðn 1Þ!
n1 j1
ðnÞ
rniþ1;j ; jk
i; j ¼ 1ð1Þn
ð2:23Þ
Note 2.2. From (2.23) it is clear that for any integer n P 1 and any k, the elðkÞ 1 ements of the n n matrix ðVS ðnÞÞ have a checkerboard sign pattern.
3. Numerical results By writing a MAPLE [12] program based on Algorithms 2.1 and 2.4, the following results are obtained as sample output for the following three cases Case 1: n ¼ 5 and k ¼ 0 In this case the MAPLE program gives 2 3 1 1 1 1 1 61 2 4 8 16 7 6 7 ð0Þ 6 VS ð5Þ ¼ 6 1 3 9 27 81 7 7 4 1 4 16 64 256 5 1 5 25 125 625
ð3:1Þ
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
and
2 ð0Þ
ðVS ð5ÞÞ1
5
10
10
5
107 6 59 6 13 6 1 6
39 2 49 4
61 6 41 6 11 6 1 6
6 77 6 12 6 71 ¼6 6 24 6 7 4 12 1 24
3 1 4
1
649
3
25 7 12 7 7 35 7 24 7 5 7 12 5 1 24
ð3:2Þ
Case 2: n ¼ 8 and k ¼ 1=2 For this case the program yields ð1=2Þ
VS
2 6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 4
ð8Þ 1 pffiffiffi 2 pffiffiffi 3
1 pffiffiffi 2 2 pffiffiffi 3 3
1 pffiffiffi 4 2 pffiffiffi 9 3
1 pffiffiffi 8 2 pffiffiffi 27 3
1 pffiffiffi 16 2 pffiffiffi 81 3
1 pffiffiffi 32 2 pffiffiffi 243 3
1 pffiffiffi 64 2 pffiffiffi 729 3
1 pffiffiffi 128 2 pffiffiffi 2187 3
2 8 32 128 512 2048 8192 32768 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 5 5 5 25 5 125 5 625 5 3125 5 15625 5 78125 5 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 6 6 6 36 6 216 6 1296 6 7776 6 46656 6 279936 6 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 7 7 7 49 7 343 7 2401 7 16087 7 117649 7 823543 7 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 2 16 2 128 2 1024 2 8192 2 65536 2 524288 2 4194304 2
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ð3:3Þ
and ð1=2Þ
ðVS
2
ð8ÞÞ1 8
6 6 481 6 35 6 6 349 6 6 36 6 6 329 6 90 ¼6 6 115 6 144 6 6 6 73 6 720 6 6 1 6 144 4 1 5040
pffiffiffi pffiffiffi 3 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 56 56 8 14 2 5 143 6 7 14 2 3 35 3 5 7 pffiffiffi pffiffiffi 7 pffiffiffi 691 pffiffiffi 2143 pffiffiffi pffiffiffi 7 621 363 141 5 7 3 6 103 2 2003 2 7 20 45 8 5 180 35 560 7 pffiffiffi pffiffiffi 7 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 797 1457 4891 187 527 469 7 18353 3 5 6 7 2 2 720 20 18 180 16 180 720 7 7 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 7 15289 268 10993 1193 2803 67 967 5 7 3 6 2 2 7 1440 15 288 90 480 45 2880 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 7 pffiffiffi pffiffiffi 7 179 71 179 2581 13 61 7 72 2 5 8 6 7 72 2 7 3 16 18 720 144 7 pffiffiffi pffiffiffi 7 pffiffiffi 209 pffiffiffi pffiffiffi pffiffiffi 239 149 391 61 49 23 7 5 7 3 6 2 2 7 720 240 144 720 240 720 1440 7 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 7 17 11 1 31 1 29 1 720 2 5 7 3 6 2 7 240 9 720 48 5040 720 5 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 1 1 1 1 1 1 1 720 5 1440 6 5040 7 20160 2 2 720 3 288 1440
ð3:4Þ
650
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
Case 3: n ¼ 5 and any k In this case the program gives ðkÞ
ðVG ð5ÞÞ1 c2 c3 c4 c5 c1 c3 c4 c5 ¼ ; ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1 ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2 c2 c1 c4 c5 c2 c3 c1 c5 ; ; ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 c2 c3 c4 c1 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5 c5 c4 c3 þ c5 c4 c2 þ c5 c2 c3 þ c2 c3 c4 ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1
c5 c4 c3 þ c5 c4 c1 þ c5 c1 c3 þ c1 c3 c4 c5 c4 c1 þ c5 c4 c2 þ c5 c2 c1 þ c2 c1 c4 ; ; ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2 ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 c5 c1 c3 þ c5 c2 c1 þ c5 c2 c3 þ c2 c3 c1 c1 c3 c4 þ c2 c1 c4 þ c2 c3 c1 þ c2 c3 c4 ; ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5 c5 c4 þ c5 c3 þ c5 c2 þ c4 c3 þ c4 c2 þ c2 c3 c5 c4 þ c5 c3 þ c5 c1 þ c4 c3 þ c4 c1 þ c1 c3 ; ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1 ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2
c5 c4 þ c5 c1 þ c5 c2 þ c4 c1 þ c4 c2 þ c2 c1 c5 c1 þ c5 c3 þ c5 c2 þ c1 c3 þ c2 c1 þ c2 c3 ; ; ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 c4 c1 þ c1 c3 þ c2 c1 þ c4 c3 þ c4 c2 þ c2 c3 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5 c5 þ c4 þ c3 þ c2 ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1
c5 þ c4 þ c3 þ c1 c5 þ c4 þ c1 þ c2 ; ; ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2 ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 c5 þ c1 þ c3 þ c2 c1 þ c4 þ c3 þ c2 ; k k ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þc4 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þc5 1 1 ; ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1 ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2
1 1 ; ; ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 1 ð3:5Þ ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5
Note that by putting k ¼ 0 and cr ¼ r, r ¼ 1ð1Þ5 in (3.5) we obtain (3.2).
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651
References [1] D.J. Evans, A recursive decoupling method for solving tridiagonal linear systems, International Journal of Computer Mathematics 33 (1990) 95–102. [2] M.E.A. El-Mikkawy, An algorithm for solving tridiagonal systems, Journal of Institute of Mathematics and Computer Sciences 4 (2) (1991) 205–210. [3] M.E.A. El-Mikkawy, An algorithm for solving pentadiagonal systems, Bulletin of the Faculty of Science, Assiut University 21 (2) (1992) 11–15. [4] M.E.A. El-Mikkawy, On solving linear systems of the Pascal type, Applied Mathematics and Computation 136 (2003) 195–202. [5] M.E.A. El-Mikkawy, An algorithm for solving Vandermonde systems, Journal of Institute of Mathematics and Computer Sciences 3 (3) (1990) 293–297. [6] W.P. Tang, G.H. Golub, The block decomposition of a Vandermonde matrix and its applications, BIT 21 (1981) 505–517. [7] M.E.A. El-Mikkawy, A better approach for the derivation of the embedded Runge–Kutta– Nystrom pair of orders 6 and 4, International Journal of Computer Mathematics 40 (1991) 245–249. [8] J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, Families of embedded Runge–Kutta– Nystrom formulae, IMA Journal of Numerical Analysis 7 (1987) 235–250. [9] J.R. Dormand, M.E.A. El-Mikkawy, P.J. Prince, High order embedded Runge–Kutta– Nystrom formulae, IMA Journal of Numerical Analysis 7 (1987) 423–430. [10] J.H. van Lint, R.M. Wilson, A Course in Combinatorics, Cambridge University Press, 1992. [11] C.M. Bender, D.C. Brody, B.K. Meister, Inverse of a Vandermonde matrix, submitted for publication. [12] MAPLE V RELEASE 3 Reference Manual.
Explicit inverse of a generalized Vandermonde matrix Moawwad E.A. El-Mikkawy Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract In this paper the author gives an explicit closed form expression for the n n inðkÞ ðkÞ verse matrix ðVG ðnÞÞ1 of the generalized n n Vandermonde matrix VG ðnÞ by using the elementary symmetric functions. Symbolic and numerical results are presented. Ó 2002 Elsevier Inc. All rights reserved. Keywords: Algorithms; Vandermonde matrix; Stirling matrix; Symmetric functions; MAPLE
1. Introduction There are many special types of matrices which are of great importance in many scientific and engineering work. For instance matrices of type tridiagonal [1,2], pentadiagonal [3], Pascal [4], Vandermonde [5,6] and others. This paper will be concerned with matrices of Vandermonde type. These types of matrices frequently appear in many applications. For example in curve fitting, interpolation, scattering and in the derivation of explicit Runge–Kutta and Runge–Kutta–Nystrom [7–9] numerical methods. In this paper we are going to study the possibility of obtaining an explicit closed form of the Vandermonde inverse matrix. The main results are presented in Sections 2 and 3.
E-mail address: [email protected] (M.E.A. El-Mikkawy). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00609-4
644
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
2. The main results Let us begin this section by giving the following definitions Definition 2.1 [5]. If the n parameters c1 ; c2 ; . . . ; cn are distinct, then the eleðnÞ mentary symmetric functions ri;j in c1 ; c2 ; . . . ; cj1 ; cjþ1 ; . . . ; cn are defined for j ¼ 1ð1Þn by ðnÞ
r1;j ¼ 1 n n X X ðnÞ ri;j ¼
n X
i1 Y
crm
for i ¼ 2ð1Þn
ð2:1Þ
ri1 ¼ri2 þ1 m¼1 ri1 6¼j
r1 ¼1 r2 ¼r1 þ1 r1 6¼j r2 6¼j
Definition 2.2 [10]. The n n Stirling matrix of the first kind is defined by 3 2 ð1Þ 0 0 0 r1;1 7 6 ð2Þ 6 rð2Þ r1;2 0 0 7 2;2 7 6 ð3Þ ð3Þ ð3Þ 6 r3;3 r2;3 r1;3 0 0 7 ð2:2Þ S1 ¼ 6 7 7 6 .. .. .. .. 7 6 4 . 0 5 . . . ðnÞ ðnÞ nþ1 ðnÞ nþ2 ðnÞ ð1Þ rn;n ð1Þ rn1;n r2;n r1;n ðnÞ
in which the ri;j are calculated at cr ¼ r, r ¼ 1ð1Þn. Denote by rðGÞ ¼ ðbij Þ; i; j ¼ 1ð1Þn, where bij are given by n ðnÞ
bij ¼ ri;j
ð2:3Þ
The elements of the n n matrix rnðGÞ in (2.3) may be calculated by using the following algorithm Algorithm 2.1. For n P 1 we may calculate the elements of the first column of in (2.3) as follows the n n matrix rðGÞ n ð1Þ
Set r1;1 ¼ 1 For i ¼ 2; 3; . . . ; n ðiÞ ði1Þ ri;1 ¼ ri1;1 ci For j ¼ i 1; i 2; . . . ; 2 ðiÞ
ði1Þ
ði1Þ
rj;1 ¼ rj1;1 ci þ rj;1
ð2:4Þ
Next j Next i The elements in the remaining n 1 columns of rðGÞ may be obtained by n symmetry by using
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651 ðnÞ
ri;k
ðnÞ ¼ ri;1
ck !c1
i ¼ 1ð1Þn; k ¼ 2ð1Þn
;
645
ð2:5Þ ðnÞ
The notation in (2.5) means that for specific i and k, ri;k may be obtained from ðnÞ the algebraic expression of ri;1 by replacing each ck by c1 in the expression of ðnÞ ri;1 . For example for n ¼ 4 we have 2
3 1 1 1 1 6 c1 þ c3 þ c4 c2 þ c1 þ c4 c2 þ c3 þ c1 7 c2 þ c3 þ c4 ðGÞ 7 r4 ¼ 6 4 c2 c3 þ c2 c4 þ c3 c4 c1 c3 þ c1 c4 þ c3 c4 c2 c1 þ c2 c4 þ c1 c4 c2 c3 þ c2 c1 þ c3 c1 5 c2 c3 c4 c1 c3 c4 c2 c1 c4 c2 c3 c1
ð2:6Þ
Unless otherwise stated, we shall be concerned with the case when cr ¼ r, r ¼ 1ð1Þn. For this case the matrix rðGÞ will be denoted by rðSÞ n n . The capital ðSÞ letters G and S stand for general and special respectively. For example r4 is given by 2
ðSÞ
r4
1 6 9 ¼6 4 26 24
1 8 19 12
1 7 14 8
3 1 6 7 7 11 5 6
ð2:7Þ
Definition 2.3. For the n þ 1 real values k; c1 ; c2 ; . . . ; cn we define a generalized ðkÞ ðkÞ Vandermonde matrix VG ðnÞ by VG ðnÞ ¼ ðVij Þ with Vij ¼ ckþj1 ; i; j ¼ 1ð1Þn. i For the special case when cr ¼ r, r ¼ 1ð1Þn this matrix will be denoted by ðkÞ VS ðnÞ. Qn Qn Qi1 ðkÞ It is known that detðVG ðnÞÞ ¼ ð m¼1 ckm Þ i¼2 j¼1 ðci cj Þ. Thus both ðkÞ ðkÞ VG ðnÞ and VS ðnÞ are non singular if and only if the n parameters c1 ; c2 ; . . . ; cn are distinct. ðkÞ ð0Þ For any k, the two matrices VG ðnÞ and VG ðnÞ satisfy ðkÞ
ð0Þ
VG ðnÞ ¼ DVG ðnÞ
ð2:8Þ
where D ¼ diagðck1 ; ck2 ; . . . ; ckn Þ
ð2:9Þ
For the matrix D in (2.9) we have k k D1 ¼ diagðck 1 ; c2 ; . . . ; cn Þ ðkÞ
ð2:10Þ ðkÞ
Therefore the inverse matrix ðVG ðnÞÞ1 of the matrix VG ðnÞ may be obtained ð0Þ once the inverse matrix ðVG ðnÞÞ1 is available.
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M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
In what follows we are going to show how can we obtain the matrix ð0Þ 1 ðVG ðnÞÞ . Suppose we have n distinct values c1 ; c2 ; . . . ; cn and consider the Lagrange polynomials L1 ðxÞ; L2 ðxÞ; . . . ; Ln ðxÞ of degree (n 1) defined by n Q
ðx cj Þ
j¼1 j6¼i
Li ðxÞ ¼ Q n
ð2:11Þ ðci cj Þ
j¼1 j6¼i
Consider also the polynomial wðxÞ of degree n defined by wðxÞ ¼
n Y ðx ci Þ
ð2:12Þ
i¼1
It is easy to show that Li ðxÞ ¼
ðx ci Þ1 wðxÞ d wðxÞjx¼ci dx
for i ¼ 1ð1Þn
ð2:13Þ
Moreover by using partial fractions together with (2.13) we obtain n X
Li ðxÞcmi ¼ xm
for m ¼ 0ð1Þn 1
ð2:14Þ
i¼1
Thus 2
L1 ðxÞ L2 ðxÞ .. .
3
2
1 c1 c21 .. .
7 6 6 7 6 6 7 6 6 7¼6 6 7 6 6 4 Ln1 ðxÞ 5 4 c1n1 Ln ðxÞ
1 c2 c22 .. .
1 cn1 c2n1 .. .
1 cn c2n .. .
31 2 7 7 7 7 7 5
3 1 6 x 7 6 . 7 6 .. 7 6 7 4 xn2 5
xn1 cn1 cn1 n1 n 3 3 2 2 1 1 6 x 7 6 x 7 7 7 6 6 ð0Þ ð0Þ T 1 6 .. 7 1 T 6 .. 7 ¼ ððVG ðnÞÞ Þ 6 . 7 ¼ ððVG ðnÞÞ Þ 6 . 7 4 xn2 5 4 xn2 5 c2n1
xn1 ð0Þ
ð2:15Þ
xn1
Now if ððVG ðnÞÞ1 ÞT ¼ ðwij Þ1 6 i; j 6 n then (2.15) and (2.11) together with (2.1) give
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651 n Q
Li ðxÞ ¼
n X j¼1
ðx cj Þ
j¼1 j6¼i
wij xj1 ¼ Q n
n P
¼ ðci cj Þ
j¼1 j6¼i n P
¼
647
ðnÞ
ð1Þjþ1 rj;i xnj
j¼1 n Q
ðci cj Þ
j¼1 j6¼i
nj ðnÞ rnjþ1;i xj1
ð1Þ
j¼1 n Q
ð2:16Þ ðci cj Þ
j¼1 j6¼i
Thus (2.16) yields nj ðnÞ
ð1Þ rnjþ1;i ; wij ¼ Q n ðci cj Þ
i; j ¼ 1ð1Þn
ð2:17Þ
j¼1 j6¼i
At this point it is now possible to formulate the following general result: ðkÞ
Algorithm 2.2. For any integer n P 1 the n n matrix ðVG ðnÞÞ1 in Definition 2.3 is given by ðkÞ
ðVG ðnÞÞ
1
¼ MðnÞ
ð2:18Þ
where the n n matrix MðnÞ is given by MðnÞ ¼ ðmij Þ;
i; j ¼ 1ð1Þn
mij ¼ ð1Þnþi
rniþ1;j ; n Q ckj ðcj ci Þ
with ðnÞ
i; j ¼ 1ð1Þn
ð2:19Þ
i¼1 i6¼j
The following are special cases of Algorithm 2.2 ð0Þ
1
Algorithm 2.3. For any integer n P 1, the n n matrix ðVS ðnÞÞ 2.3 is given by ð0Þ
ðVS ðnÞÞ1 ¼
1 V ðnÞ ðn 1Þ!
in Definition
ð2:20Þ
where the n n matrix V ðnÞ is a matrix whose entries are all integers and given by
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M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
V ðnÞ ¼ ðvij Þ;
i; j ¼ 1ð1Þn
with vij ¼ ð1Þ
iþj
n 1 ðnÞ rniþ1;j ; j1
i; j ¼ 1ð1Þn
ð2:21Þ
Note 2.1. Another approach to find the n n matrix V ðnÞ in (2.20) is given in [11]. This approach depends on writing V ðnÞ as the product of two matrices of the form T ðnÞU ðnÞ where T ðnÞ is an upper triangular matrix related to the matrix S1 in (2.2). Also the matrices T ðnÞ and U ðnÞ satisfy some recurrence relations. For more details see [11]. ðkÞ
Algorithm 2.4. For any integer n P 1, the n n matrix ðVS ðnÞÞ1 in Definition 2.3 is given by ðkÞ
ðVS ðnÞÞ
1
¼ W ðnÞ
ð2:22Þ
where the n n matrix W ðnÞ is given by W ðnÞ ¼ ðwij Þ;
i; j ¼ 1ð1Þn
with iþj
wij ¼
ð1Þ ðn 1Þ!
n1 j1
ðnÞ
rniþ1;j ; jk
i; j ¼ 1ð1Þn
ð2:23Þ
Note 2.2. From (2.23) it is clear that for any integer n P 1 and any k, the elðkÞ 1 ements of the n n matrix ðVS ðnÞÞ have a checkerboard sign pattern.
3. Numerical results By writing a MAPLE [12] program based on Algorithms 2.1 and 2.4, the following results are obtained as sample output for the following three cases Case 1: n ¼ 5 and k ¼ 0 In this case the MAPLE program gives 2 3 1 1 1 1 1 61 2 4 8 16 7 6 7 ð0Þ 6 VS ð5Þ ¼ 6 1 3 9 27 81 7 7 4 1 4 16 64 256 5 1 5 25 125 625
ð3:1Þ
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
and
2 ð0Þ
ðVS ð5ÞÞ1
5
10
10
5
107 6 59 6 13 6 1 6
39 2 49 4
61 6 41 6 11 6 1 6
6 77 6 12 6 71 ¼6 6 24 6 7 4 12 1 24
3 1 4
1
649
3
25 7 12 7 7 35 7 24 7 5 7 12 5 1 24
ð3:2Þ
Case 2: n ¼ 8 and k ¼ 1=2 For this case the program yields ð1=2Þ
VS
2 6 6 6 6 6 6 6 6 6 6 ¼6 6 6 6 6 6 6 6 6 4
ð8Þ 1 pffiffiffi 2 pffiffiffi 3
1 pffiffiffi 2 2 pffiffiffi 3 3
1 pffiffiffi 4 2 pffiffiffi 9 3
1 pffiffiffi 8 2 pffiffiffi 27 3
1 pffiffiffi 16 2 pffiffiffi 81 3
1 pffiffiffi 32 2 pffiffiffi 243 3
1 pffiffiffi 64 2 pffiffiffi 729 3
1 pffiffiffi 128 2 pffiffiffi 2187 3
2 8 32 128 512 2048 8192 32768 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 5 5 5 25 5 125 5 625 5 3125 5 15625 5 78125 5 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 6 6 6 36 6 216 6 1296 6 7776 6 46656 6 279936 6 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 7 7 7 49 7 343 7 2401 7 16087 7 117649 7 823543 7 pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi 2 2 16 2 128 2 1024 2 8192 2 65536 2 524288 2 4194304 2
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
ð3:3Þ
and ð1=2Þ
ðVS
2
ð8ÞÞ1 8
6 6 481 6 35 6 6 349 6 6 36 6 6 329 6 90 ¼6 6 115 6 144 6 6 6 73 6 720 6 6 1 6 144 4 1 5040
pffiffiffi pffiffiffi 3 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 56 56 8 14 2 5 143 6 7 14 2 3 35 3 5 7 pffiffiffi pffiffiffi 7 pffiffiffi 691 pffiffiffi 2143 pffiffiffi pffiffiffi 7 621 363 141 5 7 3 6 103 2 2003 2 7 20 45 8 5 180 35 560 7 pffiffiffi pffiffiffi 7 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 797 1457 4891 187 527 469 7 18353 3 5 6 7 2 2 720 20 18 180 16 180 720 7 7 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 7 15289 268 10993 1193 2803 67 967 5 7 3 6 2 2 7 1440 15 288 90 480 45 2880 pffiffiffi pffiffiffi pffiffiffi pffiffiffi 7 pffiffiffi pffiffiffi 7 179 71 179 2581 13 61 7 72 2 5 8 6 7 72 2 7 3 16 18 720 144 7 pffiffiffi pffiffiffi 7 pffiffiffi 209 pffiffiffi pffiffiffi pffiffiffi 239 149 391 61 49 23 7 5 7 3 6 2 2 7 720 240 144 720 240 720 1440 7 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 7 17 11 1 31 1 29 1 720 2 5 7 3 6 2 7 240 9 720 48 5040 720 5 p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi p ffiffi ffi 1 1 1 1 1 1 1 720 5 1440 6 5040 7 20160 2 2 720 3 288 1440
ð3:4Þ
650
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
Case 3: n ¼ 5 and any k In this case the program gives ðkÞ
ðVG ð5ÞÞ1 c2 c3 c4 c5 c1 c3 c4 c5 ¼ ; ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1 ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2 c2 c1 c4 c5 c2 c3 c1 c5 ; ; ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 c2 c3 c4 c1 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5 c5 c4 c3 þ c5 c4 c2 þ c5 c2 c3 þ c2 c3 c4 ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1
c5 c4 c3 þ c5 c4 c1 þ c5 c1 c3 þ c1 c3 c4 c5 c4 c1 þ c5 c4 c2 þ c5 c2 c1 þ c2 c1 c4 ; ; ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2 ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 c5 c1 c3 þ c5 c2 c1 þ c5 c2 c3 þ c2 c3 c1 c1 c3 c4 þ c2 c1 c4 þ c2 c3 c1 þ c2 c3 c4 ; ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5 c5 c4 þ c5 c3 þ c5 c2 þ c4 c3 þ c4 c2 þ c2 c3 c5 c4 þ c5 c3 þ c5 c1 þ c4 c3 þ c4 c1 þ c1 c3 ; ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1 ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2
c5 c4 þ c5 c1 þ c5 c2 þ c4 c1 þ c4 c2 þ c2 c1 c5 c1 þ c5 c3 þ c5 c2 þ c1 c3 þ c2 c1 þ c2 c3 ; ; ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 c4 c1 þ c1 c3 þ c2 c1 þ c4 c3 þ c4 c2 þ c2 c3 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5 c5 þ c4 þ c3 þ c2 ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1
c5 þ c4 þ c3 þ c1 c5 þ c4 þ c1 þ c2 ; ; ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2 ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 c5 þ c1 þ c3 þ c2 c1 þ c4 þ c3 þ c2 ; k k ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þc4 ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þc5 1 1 ; ; ðc2 c1 Þðc3 c1 Þðc4 c1 Þðc5 c1 Þck1 ðc2 c1 Þðc2 c3 Þðc2 c4 Þðc2 c5 Þck2
1 1 ; ; ðc2 c3 Þðc3 c1 Þðc3 c4 Þðc3 c5 Þck3 ðc2 c4 Þðc3 c4 Þðc4 c1 Þðc4 c5 Þck4 1 ð3:5Þ ðc2 c5 Þðc3 c5 Þðc4 c5 Þðc5 c1 Þck5
Note that by putting k ¼ 0 and cr ¼ r, r ¼ 1ð1Þ5 in (3.5) we obtain (3.2).
M.E.A. El-Mikkawy / Appl. Math. Comput. 146 (2003) 643–651
651
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