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A note on the determinant formulas computation of generalized inverse matrix Padé approximation
Applied Mathematics and Computation 150 (2004) 865–873 www.elsevier.com/locate/amc
A note on the determinant formulas computation of generalized inverse matrix Pade approximation q Zhong-Yun Liu
1
Institute of Applied Mathematics, Changsha Communications University, Changsha, Hunan 410076, PR China
Abstract In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Pade approximation of type [n=2k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Determinant; Generalized inverse; Pade approximation; Schur complement
1. Introduction Let F ðzÞ be a given power series with matrix coefficient, i.e., F ðzÞ ¼ C0 þ C1 Z þ C2 z2 þ þ Cn zn þ ;
ðuvÞ
Ci ¼ ðCi
Þ 2 Cst ;
z 2 C; ð1Þ
st
where C consists of all s t complex matrices. A generalized inverse matrix Pade approximation (GMPA) of type [n=2k] for the given power series (1) is the rational function, see for example the Ref. [3] or [4]. q
Supported by the National Natural Science Foundation of China, No. 10271021. E-mail address: [email protected] (Z.-Y. Liu). 1 Supported by the Postdoctor Science Foundation of China.
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00316-3
866
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
RðzÞ ¼ P ðzÞ=QðzÞ;
ð2Þ
where P ðzÞ is a matrix polynomial and QðzÞ is a real scalar polynomial satisfying: ii(i) ofP ðzÞg 6 n, ofQðzÞg ¼ 2k, 2 i(ii) QðzÞ j kP ðzÞk , (iii) QðzÞF ðzÞ P ðzÞ ¼ Oðznþ1 Þ, where P ðzÞ ¼ ðpðuvÞ ðzÞÞ 2 Cst , the norm kP ðzÞk of P ðzÞ is defined as follows: kP ðzÞk ¼
s X t X
!ð1=2Þ jp
ðuvÞ 2
j
:
u¼1 v¼1
Ps In the GMPA based on the scalar products of matrices (A B ¼ u¼1 Pt [3], ðuvÞ bðuvÞ , A ¼ ðaðuvÞ Þ, B ¼ ðbðuvÞ Þ 2 Cst ) was introduced by Gu, by using a v¼1 a new type of generalized matrix inverse A 1 r ¼
1 A ¼ ; A kAk2
A 6¼ 0;
A 2 Cst ;
where A denotes the complex conjugate of A. The expression of GMPA posses the form of matrix numerator P ðzÞ and scalar denominator QðzÞ. For a given power series (1), if RðzÞ is a GMPA of type [n=2k], then Lð2kÞ lð2kÞ QðzÞ ¼ det ð2kÞT ð3Þ z 1 and
LðnÞ P ðzÞ ¼ det ðnÞT ~z
lðnÞ ; g
ð4Þ
where 2 ðmÞ
L
6 6 ¼6 6 4
0 Lð01Þ Lð02Þ .. .
Lð01Þ 0 Lð12Þ .. .
Lð02Þ Lð12Þ Lð23Þ .. .
.. .
Lð0;m 1Þ
Lð1;m 1Þ
Lð2;m 1Þ
ðm 1;mÞ T
3 Lð0;m 1Þ Lð1;m 1Þ 7 7 Lð2;m 1Þ 7 .. 7 5 . 0 Pv u 1
ð5Þ
and lðmÞ ¼ ðLð0;mÞ ; Lð1;mÞ ; . . . ; L Þ with LðuvÞ ¼ j¼0 Cjþuþn 2kþ1 CP v jþn 2k T ðnÞ ðuvÞ ðvuÞ ð2kÞ 2k 2k 1 n for v P u, L P¼ L for u < v; z P¼ ðz ; z ; . . . ; zÞ , ~z ¼ ðC0 z ; 1i¼0 n 1 T n Ci ziþn 1 ; . . . ; i¼0 Ci ziþ1 Þ , and g ¼ i¼0 Ci zi , respectively.
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
867
From the above review, we observe that the central point to construct a GMPA of type [n=2k] for the given power series (1) is how to compute two determinants (3) and (4). In order to compute determinants (3) and (4), we need the following wellknown result, see for instance [2]. Lemma 1 (Schur complement theorem). Let A be an n n complex matrix and be partitioned into 2 2 block matrix:
A11 A¼ A21
A12 : A22
If A11 2 Ckk is nonsingular, then detðAÞ ¼ detðA11 Þ detðA22 A21 A 1 11 A12 Þ. This paper is organized as follows: after reviewing some basic definition, notation and preliminaries in Section 1, we investigate the tridiagonal reduced process of a complex skew symmetric matrix in Section 2. Then we present the inverse of a skew symmetric tridiagonal matrix in Section 3 and end this paper by a conclusion remark in Section 4.
2. Tridiagonal reduction of a complex skew symmetric matrix In this section we consider the tridiagonal reduction of a complex skew symmetric matrix which is similar to one of a complex symmetric matrix (cf. [1]). T Let LðmÞ be an m m complex skew symmetric matrix such that LðmÞ ¼ LðmÞ , and consider the adaptation of the reduction stage for real skew symmetric matrices to the complex case. Similarly, the corresponding transformation should be
Q¼I 2
wwT ; wT w
w¼
v ; kvk2
v ¼ l ðlT lÞ
1=2
ek ;
where the sign is taken so that jvk j is maximal. The reduction stage proceeds as follows: T Lðtþ1Þ ¼ UðtÞ LðtÞ UðtÞ ;
t ¼ 1; 2; . . . ;
868
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
with Lð1Þ ¼ LðmÞ . At the beginning of the tth step we have 3 2 ðkÞ ðkþ1Þ bðtÞ LðtÞ ðkÞ 7 6 ðkþ1Þ ðkþ2Þ LðtÞ 2 Ckk 7 6 bðtÞ 0 bðtÞ 7 6 ðjÞ 7 6 .. .. .. LðtÞ ¼ 6 ; 7; bðtÞ 2 C . . . 7 6 j ¼ k þ 1; . . . ; m; ðmÞ 7 6 bðtÞ 5 4 kPm t þ 1 ðmÞ bðtÞ 0 where the leading sub-matrix is " # ðk 1Þ ðk 1Þ LðtÞ 2 Cðk 1Þðk 1Þ LðtÞ l ðkÞ LðtÞ ¼ : ; lT 0 l 2 Ck 1 Analogously to the transformation for the complex symmetric case in [1], we reduce the corresponding row and column using the transformation I R C I R C Pt ¼ PðtÞ PðtÞ UðtÞ , where PðtÞ and PðtÞ are real orthogonal, UðtÞ is complex orthogonal. For simplicity, we omit the index k as considered in a reduction stage. Let L ¼ A þ iB, AT ¼ A, BT ¼ B, where A, B 2 Rkk . 0 0 A a b B A0 ; B0 2 Rðk 1Þðk 1Þ ; A¼ ; B ¼ ; T T a 0 b 0 a; b 2 Rðk 1Þ : We first define PI to be the corresponding Householder transformation 0 BI y T BI ¼ PI BPI ¼ ; y ¼ kbk2 : y 0 Applying the same transformation to A, we get " # T a A0I A0I ¼ QI A0 QI T AI ¼ PI API ¼ ; T aT 0 a ¼ QI a
with PI ¼
Denoting 2
A00 6 IT AI ¼ 4 ~ a 0 aT
3 0 a~ a 7 0 x 5; x 0
2
B00 6 ~I T BI ¼ 4 b 0
b~ 0 y
QI
3 0 7 y 5: 0
We define P R to be the corresponding Householder transformation 2 3 a^ z A00R 0 6 T 7 AR ¼ PRT AI PR ¼ 4 ^ a 0 x 5; z ¼ ka k2 : z x 0
1
:
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
Applied to BI , we have 2 BR ¼
PRT BI PR "
PR ¼
QR
B00R
6 6 ¼ 6 b^T 4 0 #
b^ 0 y
0
869
3
7 7 ; y7 5 0
B00R ¼ QTR B00I QR
with
b^ ¼ QTR b~
: I2
Then the resulting transformed matrix is 2 00 3 v AR þ iB00R u 6 7 6 uT 0 l z 7 6 7; 6 vT l 0 x þ iy 7 4 5 z
ðx þ iyÞ
u; v 2 Ck 3 :
0
Therefore we define UC to be the complex orthogonal transformation 2 3 Ik 3 z 5; VC ¼ 1 ðx þ iyÞ ; VC UC ¼ 4 z x þ iy D 1 2
with D2 ¼ ðx þ iyÞ þ z2 , so that VCT VC ¼ I. Thus 3 2 ðk 3Þ v u Ltþ1 7 2 ðk 1Þ 6 6 T 0 l 0 7 7 4 Lðtþ1Þ 6 u ðkÞ Lðtþ1Þ ¼ 6 7 ðkÞ 6 vT l 0 bðkÞ 7 bðtþ1Þ 5 4 0
bðkÞ
bkðtþ1Þ 0
3 5
0
is desired. Note that we may encounter a breakdown or nearly breakdown in the reduction stage process for the case D ¼ 0 or D 0. When a breakdown or nearly breakdown occurs, some recovery transformation strategies must be selected, which can be referred to in [1].
3. Inverse of a skew symmetric tridiagonal matrix In the former section we describe the tridiagonal reduction process of a complex skew symmetric matrix, namely, there exists a complex orthogonal matrix P such that
870
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
2
0 6 ð2Þ 6 b 6 P T LðmÞ P ¼ 6 6 6 4
bð2Þ 0 .. .
3 bð3Þ .. .
..
.
bðmÞ
7 7 7 7 ¼ T ðmÞ 7 ðmÞ 7 b 5 0
and
T ð2kÞ QðzÞ ¼ det ð2kÞT z P
P T lð2kÞ : 1
ð6Þ
We remark that if n 6 2k, then P ðzÞ can be chosen as P ðzÞ ¼ zn 2k P ½2k=2k ðzÞ and Pn 2k 1 Ci zi , if n > 2k, then P ðzÞ can be selected as P ðzÞ ¼ zn 2k P ½2k=2k ðzÞ þ QðzÞ i¼0 where "
P
½2k=2k
with g ¼
Lð2kÞ ðzÞ ¼ det ð2kÞT ~z
P2k
i¼0
P
ð7Þ
Ci zi . Thus we need only compute P ½2k=2k ðzÞ and "
½2k=2k
# lð2kÞ ; g
T ð2kÞ ðzÞ ¼ det ð2kÞT ~z P
# P T lð2kÞ : g
In this section we consider the Inverse of a complex skew symmetric tridiagonal matrix T ð2kÞ . We note that when we construct a GMPA of type ½n=2k of a given power series F ðzÞ in (1), we always assume that Qð0Þ 6¼ 0. Therefore we easily obtain the following result: Lemma 2. Let P ðzÞ=QðzÞ be a GMPA of type ½n=2k of a given power series F ðzÞ in (1). Then detðLð2kÞ Þ and thus detðT ð2kÞ Þ are both not equal to zero. Although an explicit formulae for the elements of the inverse of a general tridiagonal matrix can be found, for example, in [5,6], due to its intricacies and the particularity of the skew symmetric tridiagonal matrix, we present an explicit formulae for the elements of the inverse of a skew symmetric tridiagonal matrix by a simple but tedious computation. Theorem 1. Let T ð2kÞ xi ¼ ei , for i ¼ 1; . . . ; 2k, where ei denotes the ð2kÞ-dimension unit vector with the ith entry 1. Then
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
871
T b2l 3 b3 b2k 1 b2k 3 b3 x1 ¼ 0; b12 ; 0; bb4 b3 2 ; 0; . . . ; 0; bb2l 1 ; 0; . . . ; 0; ; b b b b b 2l 2l 2 2 2k 2k 2 2 T x2 ¼ b12 ; 0; 0; 0; 0; . . . ; 0; 0; 0; . . . ; 0; 0 ; .. . 0 2i 1 zeroes 1T zfflfflfflffl}|fflfflfflffl{ b2iþ1 b b2l 3 b2iþ1 b b2k 3 b2iþ1 A x2i 1 ¼ @ 0; . . . ; 0; b12i ; 0; b2iþ2 ; . . . ; 2l 1 ; . . . ; 2k 1 ; b2i b2l b2l 2 b2i b2k b2k 2 b2i 0
1T zfflfflfflfflfflffl}|fflfflfflfflfflffl{ b2i 3 b3 b2i 3 b5 x2i ¼ @ bb2i 1 ; 0; bb2i 1 ; . . . ; b2ib2i 1 ; 0; b12i ; 0; . . . ; 0; 0 A ; b2i 2 2i b2i 2 b2 2i b2i 2 b4 2ðk iÞþ1 zeroes
.. . T x2k 1 ¼ 0; 0; 0; 0; 0; . . . ; 0; 0; 0; . . . ; 0; 0; b12k ; T b2k 3 b3 b2k 1 b2k 3 b5 b2k 1 1 x2k ¼ bb2k 1 ; 0; ; 0; . . . ; 0; ; 0; ; 0 : b2k b2k b2k 2 b4 b2k b2k 2 2k b2k 2 b2
1
Taking all xi together, for i ¼ 1; . . . ; 2k, we have gotten Tð2kÞ , where the subscripts are arranged in an increasing order. Thus we have completed the computation of the inverse of a complex skew symmetric tridiagonal matrix. In order to compute the determinants (6) and (7), according to Lemma 1, we also need to do detðT ð2kÞ Þ. Fortunately, since T ð2kÞ is a skew symmetric triagonal matrix, we can compute it directly. By using the expansion of a determinant, we easily have the following result: Theorem 2. Let 2 T
ð2kÞ
0 6 bð2Þ 6 6 ¼6 6 4
bð2Þ 0 .. .
3 bð3Þ .. .
..
.
bð2kÞ
7 7 7 7 7 ð2kÞ 5 b 0
be a skew symmetric tridiagonal matrix. Then detðT ð2kÞ Þ ¼ b22 b24 b22k .
4. Conclusion In this paper we consider the computation of the two determinants QðzÞ in P (3) and P ðzÞ in (4). Because zð2kÞ ¼ ðz2k ; z2k 1 ; . . . ; zÞT and ~zð2kÞ ¼ ðC0 z2k ; 1i¼0 P2k 1 T Ci ziþ2k 1 ; . . . ; i¼0 Ci ziþ1 Þ are two vectors with unknown variable, we can not handle the problems coming forth in this paper by a general method for the
872
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
computation of determinant. In order to compute QðzÞ and P ðzÞ, we must utilize the special block structure of the matrix of right side of (3) and (4). At the same time we note that the matrix Lð2kÞ in (5) is a nonsingular complex skew symmetric matrix. According to the above discussion, we have the following main result: Theorem 3. Let QðzÞ and P ½2k=2k ðzÞ be defined in the form (3) and (7), respectively. Then 1
T
QðzÞ ¼ b22 b24 b22k ð1 zð2kÞ P Tð2kÞ P T lð2kÞ Þ
ð8Þ
and T
1
P ½2k=2k ðzÞ ¼ b22 b24 b22k ðg ~zð2kÞ P Tð2kÞ P T lð2kÞ Þ; ð9Þ P P T 1 2k 1 T where zð2kÞ ¼ ðz2k ; z2k 1 ; . . . ; zÞ , ~zð2kÞ ¼ ðC0 z2k ; i¼0 Ci ziþ2k 1 ; . . . ; i¼0 Ci ziþ1 Þ , P2k i ð2kÞ ð0;2kÞ ð1;2kÞ ð2k 1;2kÞ T and g ¼ i¼0 Ci z , l ¼ ðL ;L ;...;L Þ , respectively. Proof. From the hypothesis, we have that there exist a complex orthogonal matrix P such that P T Lð2kÞ P ¼ T ð2kÞ : By taking P ~ P¼
1
;
we obtain that ð2kÞ ð2kÞ T L lð2kÞ ~ P~T ð2kÞT P ¼ ð2kÞT z 1 z P
P T lð2kÞ : 1
and ð2kÞ T L ~ T P ~zð2kÞ
ð2kÞ T lð2kÞ ~ P ¼ ð2kÞT g ~z P
Hence we yield that ð2kÞ T T QðzÞ ¼ det zð2kÞ P
P T lð2kÞ 1
P T lð2kÞ : g
and P ½2k=2k ¼ det
T ð2kÞ T ~zð2kÞ P
P T lð2kÞ g
:
By Lemma 1, we have got the desired results.
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
873
We end this paper by an algorithm to construct a (GMPA) of type [n=2k] for the given power series (1) which summarize the discussions throughout this paper. Algorithm. (The construction of a GMPA of type [n=2k]): step 1: to compute LðuvÞ , for u ¼ 0; . . . ; 2k 1, v ¼ 0; . . . ; 2k, step 2: to form Lð2kÞ and lð2kÞ , step 3: to find a complex orthogonal matrix P such that P T Lð2kÞ P ¼ T ð2kÞ ; i.e., tridiagonal reduction of complex skew symmetric matrix Lð2kÞ . ð2kÞ 1 step 4: to compute T and detðT ð2kÞ Þ. ð2kÞ ð2kÞ step 5: to form z and ~z , step 6: to compute QðzÞ and P ½2k=2k . step 7: to compute P ðzÞ.
References [1] I. Bar-on, V. Ryaboy, Fast diagonalization of large and dense complex symmetric matrices, with applications to quantum reaction dynamics, SIAM J. Sci. Comput. 18 (5) (1997) 1412–1435. [2] G. Golub, C. Van Loan, Matrix Computations, third ed., Johns Hopkins University Press, Baltimore, MD, 1996. [3] C. Gu, Generalized inverse matrix Pade approximation on the basis of scalar products, Linear Algebra Appl. 322 (2001) 141–167. [4] C. Gu, Thiele-type and Lagrange-type generalized inverse rational interpolation for rectangular complex matrices, Linear Algebra Appl. 295 (1999) 7–30. [5] R.K. Mallik, The inverse of a tridiagonal matrix, Linear Algebra Appl. 325 (2001) 109–139. [6] G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl. 13 (3) (1992) 707–728.
A note on the determinant formulas computation of generalized inverse matrix Pade approximation q Zhong-Yun Liu
1
Institute of Applied Mathematics, Changsha Communications University, Changsha, Hunan 410076, PR China
Abstract In this paper, the computation of two special determinants which appear in the construction of a generalized inverse matrix Pade approximation of type [n=2k] (described in [Linear Algebra Appl. 322 (2001) 141]) for a given power series is investigated. Here a common computational approach of determinant can not be used. The main tool to be used to do the two special determinants is the well-known Schur complement theorem. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Determinant; Generalized inverse; Pade approximation; Schur complement
1. Introduction Let F ðzÞ be a given power series with matrix coefficient, i.e., F ðzÞ ¼ C0 þ C1 Z þ C2 z2 þ þ Cn zn þ ;
ðuvÞ
Ci ¼ ðCi
Þ 2 Cst ;
z 2 C; ð1Þ
st
where C consists of all s t complex matrices. A generalized inverse matrix Pade approximation (GMPA) of type [n=2k] for the given power series (1) is the rational function, see for example the Ref. [3] or [4]. q
Supported by the National Natural Science Foundation of China, No. 10271021. E-mail address: [email protected] (Z.-Y. Liu). 1 Supported by the Postdoctor Science Foundation of China.
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00316-3
866
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
RðzÞ ¼ P ðzÞ=QðzÞ;
ð2Þ
where P ðzÞ is a matrix polynomial and QðzÞ is a real scalar polynomial satisfying: ii(i) ofP ðzÞg 6 n, ofQðzÞg ¼ 2k, 2 i(ii) QðzÞ j kP ðzÞk , (iii) QðzÞF ðzÞ P ðzÞ ¼ Oðznþ1 Þ, where P ðzÞ ¼ ðpðuvÞ ðzÞÞ 2 Cst , the norm kP ðzÞk of P ðzÞ is defined as follows: kP ðzÞk ¼
s X t X
!ð1=2Þ jp
ðuvÞ 2
j
:
u¼1 v¼1
Ps In the GMPA based on the scalar products of matrices (A B ¼ u¼1 Pt [3], ðuvÞ bðuvÞ , A ¼ ðaðuvÞ Þ, B ¼ ðbðuvÞ Þ 2 Cst ) was introduced by Gu, by using a v¼1 a new type of generalized matrix inverse A 1 r ¼
1 A ¼ ; A kAk2
A 6¼ 0;
A 2 Cst ;
where A denotes the complex conjugate of A. The expression of GMPA posses the form of matrix numerator P ðzÞ and scalar denominator QðzÞ. For a given power series (1), if RðzÞ is a GMPA of type [n=2k], then Lð2kÞ lð2kÞ QðzÞ ¼ det ð2kÞT ð3Þ z 1 and
LðnÞ P ðzÞ ¼ det ðnÞT ~z
lðnÞ ; g
ð4Þ
where 2 ðmÞ
L
6 6 ¼6 6 4
0 Lð01Þ Lð02Þ .. .
Lð01Þ 0 Lð12Þ .. .
Lð02Þ Lð12Þ Lð23Þ .. .
.. .
Lð0;m 1Þ
Lð1;m 1Þ
Lð2;m 1Þ
ðm 1;mÞ T
3 Lð0;m 1Þ Lð1;m 1Þ 7 7 Lð2;m 1Þ 7 .. 7 5 . 0 Pv u 1
ð5Þ
and lðmÞ ¼ ðLð0;mÞ ; Lð1;mÞ ; . . . ; L Þ with LðuvÞ ¼ j¼0 Cjþuþn 2kþ1 CP v jþn 2k T ðnÞ ðuvÞ ðvuÞ ð2kÞ 2k 2k 1 n for v P u, L P¼ L for u < v; z P¼ ðz ; z ; . . . ; zÞ , ~z ¼ ðC0 z ; 1i¼0 n 1 T n Ci ziþn 1 ; . . . ; i¼0 Ci ziþ1 Þ , and g ¼ i¼0 Ci zi , respectively.
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
867
From the above review, we observe that the central point to construct a GMPA of type [n=2k] for the given power series (1) is how to compute two determinants (3) and (4). In order to compute determinants (3) and (4), we need the following wellknown result, see for instance [2]. Lemma 1 (Schur complement theorem). Let A be an n n complex matrix and be partitioned into 2 2 block matrix:
A11 A¼ A21
A12 : A22
If A11 2 Ckk is nonsingular, then detðAÞ ¼ detðA11 Þ detðA22 A21 A 1 11 A12 Þ. This paper is organized as follows: after reviewing some basic definition, notation and preliminaries in Section 1, we investigate the tridiagonal reduced process of a complex skew symmetric matrix in Section 2. Then we present the inverse of a skew symmetric tridiagonal matrix in Section 3 and end this paper by a conclusion remark in Section 4.
2. Tridiagonal reduction of a complex skew symmetric matrix In this section we consider the tridiagonal reduction of a complex skew symmetric matrix which is similar to one of a complex symmetric matrix (cf. [1]). T Let LðmÞ be an m m complex skew symmetric matrix such that LðmÞ ¼ LðmÞ , and consider the adaptation of the reduction stage for real skew symmetric matrices to the complex case. Similarly, the corresponding transformation should be
Q¼I 2
wwT ; wT w
w¼
v ; kvk2
v ¼ l ðlT lÞ
1=2
ek ;
where the sign is taken so that jvk j is maximal. The reduction stage proceeds as follows: T Lðtþ1Þ ¼ UðtÞ LðtÞ UðtÞ ;
t ¼ 1; 2; . . . ;
868
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
with Lð1Þ ¼ LðmÞ . At the beginning of the tth step we have 3 2 ðkÞ ðkþ1Þ bðtÞ LðtÞ ðkÞ 7 6 ðkþ1Þ ðkþ2Þ LðtÞ 2 Ckk 7 6 bðtÞ 0 bðtÞ 7 6 ðjÞ 7 6 .. .. .. LðtÞ ¼ 6 ; 7; bðtÞ 2 C . . . 7 6 j ¼ k þ 1; . . . ; m; ðmÞ 7 6 bðtÞ 5 4 kPm t þ 1 ðmÞ bðtÞ 0 where the leading sub-matrix is " # ðk 1Þ ðk 1Þ LðtÞ 2 Cðk 1Þðk 1Þ LðtÞ l ðkÞ LðtÞ ¼ : ; lT 0 l 2 Ck 1 Analogously to the transformation for the complex symmetric case in [1], we reduce the corresponding row and column using the transformation I R C I R C Pt ¼ PðtÞ PðtÞ UðtÞ , where PðtÞ and PðtÞ are real orthogonal, UðtÞ is complex orthogonal. For simplicity, we omit the index k as considered in a reduction stage. Let L ¼ A þ iB, AT ¼ A, BT ¼ B, where A, B 2 Rkk . 0 0 A a b B A0 ; B0 2 Rðk 1Þðk 1Þ ; A¼ ; B ¼ ; T T a 0 b 0 a; b 2 Rðk 1Þ : We first define PI to be the corresponding Householder transformation 0 BI y T BI ¼ PI BPI ¼ ; y ¼ kbk2 : y 0 Applying the same transformation to A, we get " # T a A0I A0I ¼ QI A0 QI T AI ¼ PI API ¼ ; T aT 0 a ¼ QI a
with PI ¼
Denoting 2
A00 6 IT AI ¼ 4 ~ a 0 aT
3 0 a~ a 7 0 x 5; x 0
2
B00 6 ~I T BI ¼ 4 b 0
b~ 0 y
QI
3 0 7 y 5: 0
We define P R to be the corresponding Householder transformation 2 3 a^ z A00R 0 6 T 7 AR ¼ PRT AI PR ¼ 4 ^ a 0 x 5; z ¼ ka k2 : z x 0
1
:
Z.-Y. Liu / Appl. Math. Comput. 150 (2004) 865–873
Applied to BI , we have 2 BR ¼
PRT BI PR "
PR ¼
QR
B00R
6 6 ¼ 6 b^T 4 0 #
b^ 0 y
0
869
3
7 7 ; y7 5 0
B00R ¼ QTR B00I QR
with
b^ ¼ QTR b~
: I2
Then the resulting transformed matrix is 2 00 3 v AR þ iB00R u 6 7 6 uT 0 l z 7 6 7; 6 vT l 0 x þ iy 7 4 5 z
ðx þ iyÞ
u; v 2 Ck 3 :
0
Therefore we define UC to be the complex orthogonal transformation 2 3 Ik 3 z 5; VC ¼ 1 ðx þ iyÞ ; VC UC ¼ 4 z x þ iy D 1 2
with D2 ¼ ðx þ iyÞ þ z2 , so that VCT VC ¼ I. Thus 3 2 ðk 3Þ v u Ltþ1 7 2 ðk 1Þ 6 6 T 0 l 0 7 7 4 Lðtþ1Þ 6 u ðkÞ Lðtþ1Þ ¼ 6 7 ðkÞ 6 vT l 0 bðkÞ 7 bðtþ1Þ 5 4 0
bðkÞ
bkðtþ1Þ 0
3 5
0
is desired. Note that we may encounter a breakdown or nearly breakdown in the reduction stage process for the case D ¼ 0 or D 0. When a breakdown or nearly breakdown occurs, some recovery transformation strategies must be selected, which can be referred to in [1].
3. Inverse of a skew symmetric tridiagonal matrix In the former section we describe the tridiagonal reduction process of a complex skew symmetric matrix, namely, there exists a complex orthogonal matrix P such that
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2
0 6 ð2Þ 6 b 6 P T LðmÞ P ¼ 6 6 6 4
bð2Þ 0 .. .
3 bð3Þ .. .
..
.
bðmÞ
7 7 7 7 ¼ T ðmÞ 7 ðmÞ 7 b 5 0
and
T ð2kÞ QðzÞ ¼ det ð2kÞT z P
P T lð2kÞ : 1
ð6Þ
We remark that if n 6 2k, then P ðzÞ can be chosen as P ðzÞ ¼ zn 2k P ½2k=2k ðzÞ and Pn 2k 1 Ci zi , if n > 2k, then P ðzÞ can be selected as P ðzÞ ¼ zn 2k P ½2k=2k ðzÞ þ QðzÞ i¼0 where "
P
½2k=2k
with g ¼
Lð2kÞ ðzÞ ¼ det ð2kÞT ~z
P2k
i¼0
P
ð7Þ
Ci zi . Thus we need only compute P ½2k=2k ðzÞ and "
½2k=2k
# lð2kÞ ; g
T ð2kÞ ðzÞ ¼ det ð2kÞT ~z P
# P T lð2kÞ : g
In this section we consider the Inverse of a complex skew symmetric tridiagonal matrix T ð2kÞ . We note that when we construct a GMPA of type ½n=2k of a given power series F ðzÞ in (1), we always assume that Qð0Þ 6¼ 0. Therefore we easily obtain the following result: Lemma 2. Let P ðzÞ=QðzÞ be a GMPA of type ½n=2k of a given power series F ðzÞ in (1). Then detðLð2kÞ Þ and thus detðT ð2kÞ Þ are both not equal to zero. Although an explicit formulae for the elements of the inverse of a general tridiagonal matrix can be found, for example, in [5,6], due to its intricacies and the particularity of the skew symmetric tridiagonal matrix, we present an explicit formulae for the elements of the inverse of a skew symmetric tridiagonal matrix by a simple but tedious computation. Theorem 1. Let T ð2kÞ xi ¼ ei , for i ¼ 1; . . . ; 2k, where ei denotes the ð2kÞ-dimension unit vector with the ith entry 1. Then
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871
T b2l 3 b3 b2k 1 b2k 3 b3 x1 ¼ 0; b12 ; 0; bb4 b3 2 ; 0; . . . ; 0; bb2l 1 ; 0; . . . ; 0; ; b b b b b 2l 2l 2 2 2k 2k 2 2 T x2 ¼ b12 ; 0; 0; 0; 0; . . . ; 0; 0; 0; . . . ; 0; 0 ; .. . 0 2i 1 zeroes 1T zfflfflfflffl}|fflfflfflffl{ b2iþ1 b b2l 3 b2iþ1 b b2k 3 b2iþ1 A x2i 1 ¼ @ 0; . . . ; 0; b12i ; 0; b2iþ2 ; . . . ; 2l 1 ; . . . ; 2k 1 ; b2i b2l b2l 2 b2i b2k b2k 2 b2i 0
1T zfflfflfflfflfflffl}|fflfflfflfflfflffl{ b2i 3 b3 b2i 3 b5 x2i ¼ @ bb2i 1 ; 0; bb2i 1 ; . . . ; b2ib2i 1 ; 0; b12i ; 0; . . . ; 0; 0 A ; b2i 2 2i b2i 2 b2 2i b2i 2 b4 2ðk iÞþ1 zeroes
.. . T x2k 1 ¼ 0; 0; 0; 0; 0; . . . ; 0; 0; 0; . . . ; 0; 0; b12k ; T b2k 3 b3 b2k 1 b2k 3 b5 b2k 1 1 x2k ¼ bb2k 1 ; 0; ; 0; . . . ; 0; ; 0; ; 0 : b2k b2k b2k 2 b4 b2k b2k 2 2k b2k 2 b2
1
Taking all xi together, for i ¼ 1; . . . ; 2k, we have gotten Tð2kÞ , where the subscripts are arranged in an increasing order. Thus we have completed the computation of the inverse of a complex skew symmetric tridiagonal matrix. In order to compute the determinants (6) and (7), according to Lemma 1, we also need to do detðT ð2kÞ Þ. Fortunately, since T ð2kÞ is a skew symmetric triagonal matrix, we can compute it directly. By using the expansion of a determinant, we easily have the following result: Theorem 2. Let 2 T
ð2kÞ
0 6 bð2Þ 6 6 ¼6 6 4
bð2Þ 0 .. .
3 bð3Þ .. .
..
.
bð2kÞ
7 7 7 7 7 ð2kÞ 5 b 0
be a skew symmetric tridiagonal matrix. Then detðT ð2kÞ Þ ¼ b22 b24 b22k .
4. Conclusion In this paper we consider the computation of the two determinants QðzÞ in P (3) and P ðzÞ in (4). Because zð2kÞ ¼ ðz2k ; z2k 1 ; . . . ; zÞT and ~zð2kÞ ¼ ðC0 z2k ; 1i¼0 P2k 1 T Ci ziþ2k 1 ; . . . ; i¼0 Ci ziþ1 Þ are two vectors with unknown variable, we can not handle the problems coming forth in this paper by a general method for the
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computation of determinant. In order to compute QðzÞ and P ðzÞ, we must utilize the special block structure of the matrix of right side of (3) and (4). At the same time we note that the matrix Lð2kÞ in (5) is a nonsingular complex skew symmetric matrix. According to the above discussion, we have the following main result: Theorem 3. Let QðzÞ and P ½2k=2k ðzÞ be defined in the form (3) and (7), respectively. Then 1
T
QðzÞ ¼ b22 b24 b22k ð1 zð2kÞ P Tð2kÞ P T lð2kÞ Þ
ð8Þ
and T
1
P ½2k=2k ðzÞ ¼ b22 b24 b22k ðg ~zð2kÞ P Tð2kÞ P T lð2kÞ Þ; ð9Þ P P T 1 2k 1 T where zð2kÞ ¼ ðz2k ; z2k 1 ; . . . ; zÞ , ~zð2kÞ ¼ ðC0 z2k ; i¼0 Ci ziþ2k 1 ; . . . ; i¼0 Ci ziþ1 Þ , P2k i ð2kÞ ð0;2kÞ ð1;2kÞ ð2k 1;2kÞ T and g ¼ i¼0 Ci z , l ¼ ðL ;L ;...;L Þ , respectively. Proof. From the hypothesis, we have that there exist a complex orthogonal matrix P such that P T Lð2kÞ P ¼ T ð2kÞ : By taking P ~ P¼
1
;
we obtain that ð2kÞ ð2kÞ T L lð2kÞ ~ P~T ð2kÞT P ¼ ð2kÞT z 1 z P
P T lð2kÞ : 1
and ð2kÞ T L ~ T P ~zð2kÞ
ð2kÞ T lð2kÞ ~ P ¼ ð2kÞT g ~z P
Hence we yield that ð2kÞ T T QðzÞ ¼ det zð2kÞ P
P T lð2kÞ 1
P T lð2kÞ : g
and P ½2k=2k ¼ det
T ð2kÞ T ~zð2kÞ P
P T lð2kÞ g
:
By Lemma 1, we have got the desired results.
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873
We end this paper by an algorithm to construct a (GMPA) of type [n=2k] for the given power series (1) which summarize the discussions throughout this paper. Algorithm. (The construction of a GMPA of type [n=2k]): step 1: to compute LðuvÞ , for u ¼ 0; . . . ; 2k 1, v ¼ 0; . . . ; 2k, step 2: to form Lð2kÞ and lð2kÞ , step 3: to find a complex orthogonal matrix P such that P T Lð2kÞ P ¼ T ð2kÞ ; i.e., tridiagonal reduction of complex skew symmetric matrix Lð2kÞ . ð2kÞ 1 step 4: to compute T and detðT ð2kÞ Þ. ð2kÞ ð2kÞ step 5: to form z and ~z , step 6: to compute QðzÞ and P ½2k=2k . step 7: to compute P ðzÞ.
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