www.elsevier.com/locate/laa
Generalized inverse matrix Padé approximation
on the basis of scalar products
聻
Chuanqing Gu
Department of Mathematics, Shanghai University, 99 Qi Xiang Road, Shanghai 200436, People’s Republic of China
Received 14 November 1999; accepted 15 July 2000 Submitted by R.A. Brualdi
Abstract
A new type of generalized matrix inverse is used to define the generalized inverse matrix Padé approximants (GMPA). GMPA is introduced on the basis of scalar product of matrices, with the form of matrix numerator and scalar denominator. It is different from the existing matrix Padé approximants in that it does not need multiplication of matrices in the construction process. Some algebraic properties are discussed. The representations of GMPA are provided with the following three forms: (i) the explicit determinantal formulas for the denominator sca-lar polynomials and the numerator matrix polynomials; (ii)ε-algorithm expression; (iii) Thi-ele-type continued fraction expression. The equivalence relations above three representations are proposed. © 2001 Elsevier Science Inc. All rights reserved.
AMS classification: 65D05; 41A21; CR: G.1.2
Keywords: Scalar product of matrices; Generalized inverse; Matrix Padé approximation; Algebraic
prop-erties; Determinantal formula; Thiele-type continued fraction;ε-Algorithm
1. Introduction
Letf (z)be a given power series with matrix coefficients, i.e.,
f (z)=c0+c1z+c2z2+ · · · +cnzn+ · · ·,
ci =(ci(uv))∈C
s×t, z∈C, (1.1)
聻 The work is supported by the National Natural Science Foundation of China (19871054).
E-mail address: guchqing@guomai.sh.cn (C. Gu).
0024-3795/01/$ - see front matter2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 0 0 ) 0 0 2 3 0 - 5
whereCs×t consists of alls×tmatrices with their elements in the complex planC. A (right) matrix Padé approximant off (z)is an expression of the formU (z)V (z)−1, such that
f (z)V (z)−U (z)=R(z), (1.2) whereU (z)andV (z)are matrix polynomials of degree at most m and n, respectively, whose expansion agrees withf (z)up to and including the termzm+n.R(z)in (1.2) is referred to as the residual of the approximant. A left matrix Padé approximant of
f (z)can be similarly defined.
The definition of a Padé approximant can be made more formal in a variety of ways. Typically,U (z)andV (z)ares×spolynomial matrices, andV (z)is further restricted by the condition that the constant term,V (0), is invertible (cf. [5,7,27]). Labahn and Cabay called such approximants matrix Padé fractions, which were con-sistent with the scalar (p=1) case (cf. [18]). They introduced and developed the notion of a matrix power series remainder sequence and its corresponding cofactor sequence in [25]. An algorithm for constructing these sequences was presented. Xu and Bultheel considered some possible definitions of matrix Padé approximants for a power series with rectangular matrix coefficients in [24]. They had to consider left and right approximants on account of the noncommutativity of the matrix mul-tiplication. Depending on the normalization of the denominator, they defined type I (constant term is the unit matrix) and type II (by conditions on the leading coeffi-cient) approximants. A uniform approach was given by Beckermann and Labahn for different concepts of matrix-type Padé approximants, such as descriptions of vector and matrix Padé approximants along with generalizations of simultaneous and Her-mite Padé approximants. In [3], they introduced the definition for a power HerHer-mite Padé approximant (PHPA) which takes right-hand (left-hand) and rectangular matrix Padé approximant, matrix Hermite Padé approximant and matrix simultaneous Padé approxmant as its special case (see [3, Example 2.1–2.4]).
Various vector rational interpolants were introduced by Graves-Morris (cf. [20– 22]). The problem here is to approximate a number of functions by rational functions with a common denominator. Graves-Morris and Jenkins [22] at first presented an axiomatic approach which uniquely define vector Padé approximants and estab-lished their algebraic structure without reference to matrix-valuedC-fractions. It is important for vector-valued rational approximation problems. However, in con-trast to our GMPA approach, Graves-Morris and Roberts extended their approach from vector Padé approximants to matrix Padé approximants by exploiting an iso-morphism between vectors and matrices by means of Clifford algebra represen-tation. By using modified Euclidean and Kronecker algorithms, they established the interconnections between vector Padé approximation and matrix Padé approx-imation in [23]. In this respect, Antoulas [1] gave a recursive method of updat-ing the partial realization to include further Markov parameters, so makupdat-ing a ma-jor generalization of the generalized Euclidean algorithm. Bultheel and Van Ba-rel [8] formulated a generalization of the Euclidean algorithm to treat the case of
matrix Padé approximation which reduces to Kronecker’s algorithm in the scalar case.
Motivated by Graves-Morris’s Thiele-type vector-valued rational interpolants [20] and Graves-Morris and Jenkins’ axiomatic approach to vector-valued rational inter-polants [22], we discussed matrix-valued rational interinter-polants in Thiele-type form and the determinantal formula form in [10] and bivariate matrix-valued rational interpolants in Thiele-type form in [11]. We defined the reciprocal of a matrix as the generalized inverse of the matrix (2.4), which is shown to be successful in matrix continued fraction interpolation (cf. [10–12]), but we do not analyse the basis of the definition and do not discuss matrix-valued Padé approximation problems, which is distinct from interpolation in definitions and algorithms.
In this paper, we present an axiomatic definition to the matrix Padé approx-imants (GMPA), which is an extension and improvement of [22] in the case of matrices and analyse the connection between the definition and scalar product of matirces. Some algebraic properties are discussed. The expression of GMPA is of the form of matrix numerator and scalar denominator. We obtain three efficient al-gorithms, where the determinantal formulas and Thiele-type formulas are distinct from interpolation form [10]. As forε-algorithm, the interpolation problems [10] do not deal with it. In this way, the representations of GMPA are provided with the following three forms: (i) the explicit determinantal formulas for the denom-inator scalar polynomials and the numerator matrix polynomials (Section 4); (ii)
ε-algorithm expression (Section 5); (iii) the expressin of convergents of Thiele-type continued fractions (Section 6). The equivalence relations above three representa-tions are proposed. Uniqueness is discussed in detail in Section 3. Given results show that arbitrary two GMPA of the same type may only differ by a scalar poly-nominal factor. The result also holds for vector-valued Padé approximants [22]. An existence theorem is given in the case of determinantal formulas in Section 4, but holds for other forms. A simple proof of Wynn’s identity for GMPA is given in Section 5.
As compared to the existing matrix Padé approximants (cf. [3,5,7–8,23–25,27]), GMPA do not need multiplication of matrices in the construction process, and hence, we do not have to define left-handed and right-handed approximants. It may be useful in the noncommutativity problems of the matrix multiplication. Second, the exis-tence condition of GMPA is relaxed if onlyQ(0)of scalar denominator polynomial is not equal to zero. So, it can be applied to singular matrices. Third, the construction of GMPA can be simplified in the computation because it only computes the reciprocals of matrices instead of usual matrix inverse in the case ofε-algorithm expression and Thiele-type continued fraction expression. On the other hand, the method of GMPA possesses the degree constraint (2.8) and divisibility constratint (2.9), which are due to the construction process of GMPA. The construction constraints imply that our method does not construct matrix Padé approximants of type [m/n] when n is odd. Thus, it may not be as effective as the existing matrix Padé approximation in some cases.
2. Definition
LetEa=(a1, a2, . . . , ad),bE=(b1, b2, . . . , bd),a,E bE∈Cd. The scalar product of vectoraEandbEis given byaE· Eb=Pdi=1aibi. The following definition is a natural extension from vector to matrix, which is different from trAof matrix A in the case of complex matrix.
Definition 2.1. LetA=(aij), B=(bij), A, B∈Cs×t. The scalar product of ma-trices A and B is defined by
A·B = s X i=1 t X j=1 aijbij, (2.1)
where Euclidean norm kAk = Xs i=1 t X j=1 |aij|2 1/2 . (2.2)
From (2.1) and (2.2), it is held
A·A∗= s X i=1 t X j=1 |aij|2= kAk2, (2.3)
whereA∗denotes the complex conjugate of A. Lemma 2.2 is given by Definition 2.1.
Lemma 2.2. LetA, B, D∈Cs×t, b∈C. Then hold:
(i) A·B =B·A;
(ii) (A+B)·D=A·D+B·D; (iii) (bA)·B =b(A·B);
(iv) A·A>0, A=0 if and only ifA·A=0.
On the basis of(2.2)and(2.3),the generalized inverse of matrix A is defined as
A−r1= 1 A = A∗ kAk2, A /=0, A∈C s×t (2.4) and A−r1= 1 A = A kAk2, A /=0, A∈R s×t, (2.5)
where the generalized inverseA−r1denotes the reciprocal of matrix A. Lemma 2.3. LetA, B∈Cs×t, A, B /=0 andb∈R, b /=0. Then hold:
b A =
1
Proof. As bothA, B /=0, it is easy to deriveb/A=1/C fromA=bC. By the left-hand side of (2.6), we get from Definition 2.1 thatbA∗/kAk2=C∗/kCk2. Then
A∗= kAk 2 bkCk2C∗ or A= kAk2 bkCk2C. Thus, kAk2=A·A∗= kAk 4kCk2 b2kCk4 .
It follows thatb2= kAk2/kCk2, and soA=bCholds. Lemma 2.4. LetA∈Cs×t, A /=0 andb∈R, b /=0. Then hold:
(i) (A−r1)−r1=A; (ii) (bA)−r1= 1bA−r1.
Proof. The result of (ii) is evident. We only prove (i). In fact, suppose(A−r1)−r1=
1/A−r1=b/B. By (2.6), we haveB=bA−r1=b/A. Using (2.6) again,A=b/B
holds.
Lemma 2.4 shows that we need not compute each inverse in the construction process of GMPA in theε-algorithm form and Thiele-type continued fraction form. Definition 2.5. A matrix-valued polynomialN (x)=(auv(x))∈Cs×t is said to be of degree m and denoted byN{N (x)} =m, if N{auv(x)}6m foru=1,2, . . . , s,
v=1,2, . . . , tandN{auv(x)} =mfor someu, v(16u6s,16v6t).
Definition 2.6. A GMPA of type [n/2k]for the given power series (1.1) is the rational function
R(z)=P (z)/Q(z) (2.7)
defined thatP (z)is a matrix polynomial andQ(z)is a real scalar polynomial satis-fying:
(i) N{P (z)}6n,N{Q(z)} =2k, (2.8)
(ii) Q(z)|kP (z)k2, (2.9)
(iii) Q(z)f (z)−P (z)=O(zn+1), (2.10) whereP (z)=(p(uv)(z))∈Cs×t, the normkP (z)kofP (z)is as in (2.2).
Lemma 2.7. LetA(z)∈Cs×tbe a matrix polynomial, A(z) /=0 andB(z)be a real scalar polynomial. If using(2.4)to rational function
B(z) A(z) = B(z)A∗(z) kA(z)k2 = P (z) Q(z),
thenQ(z)is a real polynomial,and holds:
(i) Q(z)|kP (z)k2;
(ii)N{Q(z)} =2k, k∈N. Proof. As
kP (z)k2=B2(z)kA∗(z)k2=B2(z)Q(z), Q(z)= kA(z)k2,
the conclusion is evident.
Note that if P0(z)/Q0(z)is the irreducible form of P (z)/Q(z)in Lemma 2.7,
P0(z)/Q0(z)may not satisfy divisibility (2.9). Lemma 2.7 explains that the general-ized inverse for matrices (2.4) gives rise to the scalar denominator of rational fraction GMPA which divides the square of the norm of numerator. It is shown that Definition 2.6 for GMPA depends on the properties of scalar product of matrices. Example 2.8 illustrates that (2.4) is efficient in matrix continued fraction interpolation problems as compared with usual matrix inverse.
Suppose the interpolation point set U= {zi =xi, i=0,1, . . . , n;xi ∈R}. By making use of (2.4), we recrusively defined the nth convergent of Thiele-type con-tinued fractions in [10,12]: R(n0)(x)=B0(x0)+ x−x0 B1(x0x1)+ · · · + x−xn−1 Bn(x0x1· · ·xn) with B0(xi)=A(xi), i=0,1, . . . , n, B1(x0x1)=(x1−x0)/(B0(x1)−B0(x0)), (2.11) Bl(x0x1· · ·xl)=(xl−xl−1)/(Bl−1(x0· · ·xl−2xl) −Bl−1(x0· · ·xl−1)), l>2.
Example 2.8. Using algorithm (2.11), find a rational interpolantR2(z)=P (z)/
Q(z)for the data
A(z0)= 2 0 0 −i , A(z1)= 1 0 1 i , A(z2)= 0 i 1 0 at pointsz0= −1, z1=0, z2=1.
Solution 1. Using the generalized inverse (2.4), we get that
R2(z)= 2 0 0 −i + 1 z+1 6 −1 0 1 −2i + z 1 11 −17 12i 5 −2i = 1 11z2+6z+7 5z2−12z+7 12z(z+1)i 5z2+12z+7 (−13z2+6z+7)i
Solution 2. Using usual inverseA−1=adjA/detA, we get that R2(z)= 2 0 0 −i + z+1 −1 0 −i/2 −2i + z 5 −4i −1 2i = 1 −z2+6z+3 −7z2+4z+3 4z(z+1)i (z+1)(3+z) (−z2−2z+3)i
and also find thatR2(zi)=A(zi), i =0,1,2.
But for higher order matrices the method of Solution 1 is superior to the method of Solution 2 in matrix continued fraction interpolation problems.
3. Uniqueness and algebraic properties
We discuss uniqueness and some algebraic properties of GMPA in this section. Lemma 3.1. Let(Pi(z), Qi(z))be two different GMPA of type[n/2k], i=1,2. Then hold
P1(z)Q2(z)=P2(z)Q1(z). (3.1)
Proof. From Definition 2.6, we get that
U (z)=P1(z)Q2(z)−P2(z)Q1(z)=(Q2(z)−Q1(z))O(zn+1), (3.2) then
N{U}6n+2k, order{U}>n+1. (3.3)
From (2.9) it follows that
kU (z)k2=kP1(z)k2Q22(z)+ kP2(z)k2Q21(z)
−Q1(z)Q2(z)(P1∗(z)·P2(z)+P1(z)·P2∗(z)) =Q1(z)Q2(z)Q(z)
for some real scalar polynomialQ(z). From (3.2) and (3.3), we find that N{Q}62n, order{Q}>2n+2,
which are contradictory unlessU (z)=0. Let
Mn,2k= {(P , Q):P /Qis the GMPA of type[n/2k]},
and
u=min{N{P}:(P , Q)∈Mn,2kor(P , Q)∈Nn,2k},
v=min{N{Q}:(P , Q)∈Mn,2kor(P , Q)∈Nn,2k}.
Note that(P , Q)∈Nn,2k means thatQ|kPk2 and degree conditions (2.8) may not hold as compared with(P , Q)∈Mn,2k.
Lemma 3.2.
(i)There exists unique(P ,˜ Q)˜ ∈Mn,2kor(P ,˜ Q)˜ ∈Nn,2kso that
N{ ˜P} =u, N{ ˜Q} =v. (3.4)
(ii)For any(P , Q)∈Mn,2k,there exists a scalar polynomialβ(z)so that
P (z)=β(z)P (z),˜ Q(z)=β(z)Q(z).˜ (3.5) Proof. By definition of u andv, it is known that there exists(Pi(z), Qi(z))∈Mn,2k, or(Pi(z), Qi(z))∈Nn,2k, i=1,2, so that
N{P1} =u, N{Q1}>v, N{P2}>u, N{Q2} =v.
From Lemma 3.1, it follows that N{P1} +N{Q2} =N{P2} +N{Q1}.
Then,N{P2} =u,N{Q1} =v.So (i) holds.
For any(P , Q)∈Mn,2kthere exists some scalar polynomialβi(i =1,2)so that
P =β1P˜+α1, N{α1}6N{ ˜P}, (P /˜=0), (3.6)
Q=β2Q˜ +α2, N{α2}<N{ ˜Q}. (3.7)
FromPQ˜ = ˜P Qby Lemma 3.1, it is derived from (3.6) and (3.7) that
(β1−β2)P˜Q˜ =α2P˜−α1Q.˜ (3.8)
By coefficient comparison from both terms of (3.8), we obtained that
β1=β2=β. Thus
α1=P −βP ,˜ α2=Q−βQ.˜ (3.9)
Using (2.10) in (3.9), it follows from
α2f −α1=(Qf −P )−β(Qf˜ − ˜P )=O(zn+1)
that(α1, α2)∈Mn,2kor(α1, α2)∈Nn,2k. By the definition ofuandv,we find that
α1=α2=0. So (3.5) holds.
Lemma 3.2 implies that for any(P , Q)∈Mn,2k, they may only differ by a sca-lar polynomial factor between(P , Q)and(P ,˜ Q)˜ . Therefore,R(z)=P (z)/Q(z)is unique in the sense.
Theorem 3.3 (Uniqueness). IfR(z)=P (z)/Q(z)is a GMPA of type[n/2k],then it is unique.
Let(P , Q)∈Mn,2k be as[n/2k]f for the given power series (1.1).
Example 3.4. Let f (z)=ezA= 1 0 0 1 + 0 1 0 −2 z+ 0 −1 0 2 z2+ · · · (3.10) Find[2/2]f, where A= 0 1 0 2 .
Solution. By the determinantal formulas of GMPA (see (4.1) and (4.2) in Section 4), we get[2/2]f =P2(z)/Q2(z), where P2(z)=25(z+1) z+1 z 0 1−z =βP (z),˜ (3.11) Q2(z)=25(z+1)2=βQ(z),˜ β =25(z+1). (3.12) Note that(P2(z), Q2(z))∈M2,2,but(P (z),˜ Q(z))˜ ∈N2,2. In fact,Q(z)˜ =z+1, k ˜P (z)k2=3z2+2,Q(z)˜ |k ˜P (z)k2does not hold.
Theorem 3.5. LetR(z)=P (z)/Q(z) be a [n/2k]f and if the coefficients in the power series(1.1)satisfy
ci =cTi, i=0,1,2. . . , (3.13) where T denotes the transposition of a matrix, thenR(z)=RT(z).
Proof. From Definition (2.6),Q(z)f (z)−P (z)=O(zn+1), then
Q(z)fT(z)−PT(z)=O(zn+1). (3.14) By (3.13), we can derive thatfT(z)=f (z). Thus, it is known from (3.14) that
Q(z)f (z)−PT(z)=O(zn+1).
The divisibility condition, thatQ(z)|kPT(z)k2 is easily verified because ofQ(z)|
kP (z)k2. The degree condition is evident. So we obtain [n/2k]f(z)=R(z)=
PT(z)/Q(z). By Theorem 3.3,RT(z)=R(z)holds.
Theorem 3.6. Let[n/2k]f(z)=P (z)/Q(z)andg(z)=fr−1(z), f (0) /=0,where
z∈Rfor(1.1), P (0) /=0,andfr−1(z)is defined as(2.4). Then
Proof. By Definition 2.6, Q(z)f (z)−P (z)=O(zn+1). According to the usual multiplication of matrices, we have
Q(z)P∗(z)f (z)−P∗(z)P (z)=O(zn+1). (3.16) From the conditionf (0) /=0, it is known thatfr−1(z)exists. From (3.16), we get that P∗(z)P (z)fr−1(z)−Q(z)P∗(z)=O(zn+2k+1). That is fr−1(z)−Q(z)P∗(z)/kP (z)k2=O(zn+2k+1). So we have kP (z)k2g(z)−Q(z)P∗(z)=O(zn+2k+1).
The divisibility conditionkP (z)k2|kQ(z)P∗(z)k2is easily verified. As for the de-gree, we find that
N{Q(z)P∗(z)}6n+2k,N{kP (z)k2} =2n.
Example 3.7. Letg(z)=f (z)−r1in the power series (3.10). Thenf (0) /=0. Find [4/4]g.
Solution. By (3.11) and (3.12), we get that
Q(z)g= kP (z)k2=625(z+1)2(3z2+1), P (z)g=Q(z)P∗(z)=625(z+1)3 z+1 z 0 1−z , whereP (z)g, Q(z)gsatisfy: (i) Q(z)gg(z)−P (z)g=O(z5), (ii) N{P (z)g} =4,N{Q(z)g} =4, (iii) kP (z)gk2=625(z+1)3Q(z)g, Q(z)g|kP (z)gk2. So[4/4]g=P (z)g/Q(z)g.
Theorem 3.8. Letf (z)be given by(1.1), z∈Rand
g(z)=z−m " f (z)− mX−1 i=0 cizi # =cm+cm+1z+cm+2z2+ · · · If m>1, n−m>2k−1, (3.17)
then[n−m/2k]g(z)exists and
[n−m/2k]g(z)=z−m ( [n/2k]f(z)− mX−1 i=0 cizi ) .
Proof. Let[n/2k]f(z)=P (z)/Q(z). By Definition 2.6,
Q(z)f (z)−P (z)=O(zn+1). (3.18) By virtue of (3.17) and (3.18), we derive that
Q(z) mX−1 i=0 cizi−P (z)=Q(z)f (z)−P (z)−Q(z) ∞ X i=m cizi =O(zn+1)+O(zm)=O(zm). Define P1(z)= P (z)−Q(z) mX−1 i=0 cizi ! z−m. Then by (3.17), N{P1(z)}6n−m. From (3.18), we obtain that
Q(z) f (z)− mX−1 i=0 cizi ! − P (z)−Q(z) mX−1 i=0 cizi ! =O(zn+1). (3.19)
Multiplyingz−mto the both sides of (3.19), we find that
Q(z)g(z)−P1(z)=O(zn−m+1). By virtue of kP1(z)k2= kP1(z)k2+Q2(z) mX−1 i=0 cizi !2 −Q(z) × " P (z)· mX−1 i=0 c∗izi ! +P∗(z)· mX−1 i=0 cizi #) z−2m
andQ(z)|kP (z)k2, we obtainQ(z)|kP1(z)k2. Hence[n−m/2k]g(z)exists and
[n−m/2k]g(z)=z−m ( [n/2k]f(z)− mX−1 i=0 cizi ) .
4. Determinantal formulas of GMPA
Theorem 4.1. LetR(z)=P (z)/Q(z)be a GMPA of type[n/2k]for the given pow-er spow-eries(1.1). Then hold:
Q(z)=det 0 L01 L02 · · · L0,2k−1 L0,2k L10 0 L12 · · · L1,2k−1 L1,2k L20 L21 0 · · · L2,2k−1 L2,2k .. . ... ... . .. ... ... L2k−1,0 L2k−1,1 L2k−1,2 · · · 0 L2k−1,2k z2k z2k−1 z2k−2 · · · z 1 (4.1) and P (z) =det 0 L01 L02 · · · L0,n−1 L0,n L10 0 L12 · · · L1,n−1 L1,n L20 L21 0 · · · L2,n−1 L2,n .. . ... ... . .. ... ... Ln−1,0 Ln−1,1 Ln−1,2 · · · 0 Ln−1,n c0zn 1 P i=0 cizi+n−1 2 P i=0 cizi+n−2 · · · nP−1 i=0 cizi+1 n P i=0 cizi , (4.2) where Lij= j−Xi−1 l=0 cl+i+n−2k+1·c∗j−l+n−2k = s X u=1 t X v=1 jX−i−1 l=0 cl(uv)+i+n−2k+1c˜j(uv)−l+n−2k , j >i (4.3) Lij = −Lij, j < i, (4.4) wherec∗l =(c˜(uv)l )is the complex conjugate matrix ofcl.
The proof of (4.1) was given by Chuanqing [13], which is an extension of that of Graves-Morris and Jenkins [22], from the vector case to the matrix case, but the proof of (4.2) is at first given.
Proof. (i)n=2k. We express(P (z), Q(z))as
Q(z)=Q0+Q1z+ · · · +Q2kz2k, (4.5)
P (z)=P0+P1z+ · · · +Pnzn, Pi ∈Cs×t (4.6) and define the Maclaurin section off (z)by
Then, Definition 2.6 implies that
P (z)= [Q(z)f (z)]n0= [Gn(z)Q(z)]n0. (4.7) Definitions (2.8) and (2.9) imply that kP (z)k2/Q(z)is a polynomial of degree 2n−2kat most, and so that
h
kP (z)k2/Q(z)
i2n+1
2n−2k+1=0. (4.8)
Define a scalar polynomialF (z)as follows:
F (z)=(P∗(z)−G∗n(z)Q(z))·(P (z)−Gn(z)Q(z))
=kP (z)k2+Q2(z)kGn(z)k2
−Q(z)(P∗(z)·Gn(z)+G∗n(z)·P (z)). (4.9) According to Definitions (2.9) and (2.10), we derive thatF (z)/Q(z)=O(z2n+2), which leads to
F (z)/Q(z)2n+1
2n−2k+1=0. From (4.9) and (4.8), we obtain that
h
−P∗(z)·Gn(z)−Gn∗(z)·P (z)+Q(z)kGn(z)k2 i2n+1
2n−2k+1=0. (4.10) By (4.7) we find that (4.10) represents 2k+1(n=2k)linear equations forQ0, Q1,
. . . , Q2kof (4.5), which may be expressed as
2k X j=0 LijQ2k−j =0, i=0,1, . . . ,2k−1, (4.11) 2k X j=0 L2k,jQ2k−j =0, (4.12)
where the coefficients ofQ2k−jin (4.11) areLij, as given by (4.3) and (4.4). Eqs. (4.11) and (4.5) form a system of 2k+1 non-homogeneous equations for
Q0, Q1, . . . , Q2k, as expressed by 0 L01 L02 · · · L0,2k−1 L0,2k L10 0 L12 · · · L1,2k−1 L1,2k L20 L21 0 · · · L2,2k−1 L2,2k .. . ... ... . .. ... ... L2k−1,0 L2k−1,1 L2k−1,2 · · · 0 L2k−1,2k z2k z2k−1 z2k−2 · · · z 1 Q2k Q2k−1 Q2k−2 .. . Q1 Q0
= 0 0 0 .. . 0 Q(z) . (4.13) Solving (4.13), we obtain (4.1).
From (4.6), (4.7) andn=2k, we derive that
P (z)=c0Q0+(c1Q0+c0Q1)z+ · · · + Xn j=0 cn−jQj zn = n X i=0 cizi ! Q0+ n X i=0 cizi+1 ! Q1 + · · · +(c0zn−1+c1zn)Qn−1+(c0zn)Qn. (4.14) Using (4.13) in (4.14), we obtain (4.2) in the case ofn=2k.
(ii)n62k. Define
Di =0, i=0,1, . . . ,2k−n−1;
Di =ci−2k+n, i=2k−n,2k−n+1, . . . ,2k.
By means of the coefficients{Di|i=0,1, . . . ,2k}, we constructQ(z)as in (4.1) andP[2k/2k](z)of the type[2k/2k]as in (4.2). The numerator polynomial is defined by
P (z)=zn−2kP[2k/2k](z). (4.15) Then,P (z)/Q(z)is the[n/2k]f required.
(iii)n >2k. Let ˜ f (z)= ∞ X i=n−2k cizi+2k−n.
By means of the coefficient{cn−2k+i}, i=0,1, . . . ,2k, we construct Q(z)as in (4.1), andP[2k/2k](z)of type[2k/2k]as in (4.2). The numerator polynomial is de-fined by
P (z)=zn−2kP[2k/2k](z)+Q(z) n−X2k−1
i=0
cizi. (4.16)
Example 4.2[26]. Let f (z)= 10 01 0 0 + 11 00 1 0 z+ 10 00 1 0 z2. Find[2/2]f.
Solution. By (4.1) and (4.2), we get
Q(z)=det −03 30 42 z2 z 1 =3(2z2−4z+3), P (z)=det −03 30 42 c0z2 c0z+c1z2 c0+c1z+c2z2 =3 z 2−z+3 0 z(3−4z) 2z2−4z+3 z(3−z) 0 , where(P (z), Q(z))∈M2,2satisfy: (i)N{P} =2,N{Q} =2, (ii)kPk2=27(3z2+2)Q, Q|kPk2, (iii) Q(z)f (z)−P (z)=O(z3).
To calculate higher-order determinantal formulas (4.1) and (4.2), we introduce Cayley’s theorem.
Lemma 4.3 (see [19]). Let A be a square matrix of even dimension. LetR, C de-note the anti-symmetric matrices formed by altering only the rth row,column of A,
respectively. Then
detA=PfR·PfC, (4.17)
where PfRdenotes the Pfaffian formula of R.
According to(4.17),we obtain the following result in[14](also see[19]). Theorem 4.4. Define Pfaffian formulas,respectively,
ϕ1(z)=det L12 L13 · · · L1,2k+1 z2k L23 · · · L2,2k+1 z2k−1 . .. ... ... L2k,2k+1 z 1 ,
ϕ2(z)=det L12 L13 · · · L1,2k+1 2k P i=0 cizi L23 · · · L2,2k+1 2kP−1 i=0 cizi+1 . .. ... ... L2k,2k+1 1 P i=0 cizi+2k−1 c0z2k . Then hold: (i) ϕ1(z)=QPf(z), Q(z)=QPf(z)QPf(0); (4.18) (ii) ϕ2(z)=PPf(z), P (z)=PPf(z)PPf(0), (4.19) whereQ(z)is as in(4.1)andP (z)is as in(4.2).
Example 4.5. Find[4/4]f =P (z)/Q(z)for the given power series (1.1).
Solution. Making use of Pfaffian formulas (4.18) for the denominator polynomial of GMPA and Pfaffian formulas (4.19) for the numerator polynomial of GMPA, we get that, respectively, Q(z)=det 0 G1 2H12 2H13+G2 2H14+2H23 −G1 0 G2 2H23 2H24+G3 −2H12 −G2 0 G3 2H34 −2H13+G2 −2H23 −G3 0 G4 z4 z3 z2 z 1 =QPf(z)QPf(0), whereGi = kcik2, Hij =ci ·c∗j, QPf(z)=det G1 2H12 2H13+G2 2H14+2H24 z4 G2 2H23 2H24+G3 z3 G3 2H34 z2 G4 1 1 =det a b c d z4 e f g z3 h j z2 k z 1
=(ah−bf +ce)+(de+j a−gb)z+(ak−gc+df )z2 +(j c−bk−hd)z3+(gh+ek−jf )z4, QPf(0)=(ah−bf +ce) and P (z) = det 0 G1 2H12 2H13+G2 2H14+2H23 −G1 0 G2 2H23 2H24+G3 −2H12 −G2 0 G3 2H34 −2H13−G2 −2H23 −G3 0 G4 c0z4 c0z3+c1z4 2 P i=0 cizi+2 3 P i=0 cizi+1 4 P i=0 cizi =PPf(z)PPf(0), (4.20) where PPf(z)=(ah−bf+ce)(c0+c1z+c2z2+c3z3+c4z4) +(de+j a−gb)(c0z+c1z2+c2z3+c3z4) +(ak−gc+df )(c0z2+c1z3+c2z4) +(j c−bk−hd)(c0z3+c1z4)+(gh+ek−jf )c0z4, PPf(0)=(ah−bf+ce)c0.
Notice thatP (z)=PPf(z)PPf(0)using usual multiplication operation of matrices in (4.20). Let H (0,2k,2k−1) = L00 L01 L02 · · · L0,2k−1 L0,2k L10 0 L11 · · · L1,2k−1 L1,2k L20 L21 0 · · · L2,2k−1 L2,2k .. . ... ... . .. ... ... L2k−1,0 L2k−1,1 L2k−1,2 · · · 0 L2k−1,2k .
Theorem 4.6 (Existence). Let[n/2k]f =P (z)/Q(z),whereP (z)andQ(z)be giv-en by(4.1)and(4.2),respectively, andn=2k.Then [n/2k]f exists if and only if
Q(0)= detH (0,2k−1,2k−1) /=0.
H (0,2k,2k−1) Q2k Q2k−1 .. . Q1 Q0 = 0, then H (0,2k−1,2k−1) Q2k Q2k−1 .. . Q1 = −Q0 L0,2k L1,2k .. . L2k−1,2k−1 . (4.21)
IfQ(0)=Q0= detH (0,2k−1,2k−1) /=0,it means that non-homogeneous equations (4.21) exist as a unique solutionQ0, Q1, . . . , Q2k forQ(z). From (4.14), it also means that (4.21) exists as a unique solutionP0, P1, . . . , Pn forP (z)in the case ofn=2k. Hence,[n/2k]f =P (z)/Q(z)exists.
Let[n/2k]f =P (z)/Q(z)exist, then it implies from (4.21) that
rankH (0,2k−1,2k−1)= rankH (0,2k,2k−1). (4.22) IfQ(0)= detH (0,2k−1,2k−1)=0, it holds from (4.22) that rankH (0,2k,
2k−1) <2k. Hence, it follows from (4.1) thatQ(z)≡0, which is contradictory to definition (2.7) of GMPA. Example 4.7[26]. Let w(z)= 1 0 0 1 + 1/2 1/4 0 0 z2+ 0 0 1/4 1/8 z3.
Note that[2/2]w=P2(z)/Q2(z)does not exist because
Q2(0)= det 0 0 0 0 =0,
but[3/2]w =P3(z)/Q3(z)exists because
Q3(0)= det 0 5/16 −5/16 0 =25/256=/ 0,
whereQ3(z)is constructed by the coefficients{c1, c2, c3}.
5. ε-Algorithm expression of GMPA
By making use of the generalized inverse (2.4), we define the matrix valuedε -algorithm by
ε0(j )= j X i=0 cizi, j =0,1,2, . . . , (5.2) εk(j )+1=ε(jk−+11)+(εk(j+1)−εk(j ))−r1, j, k>0 (5.3) for the given power series (1.1).
By usual construction, it involves the two-dimensional array called the ε-table (see [4]).
Theorem 5.1 (Identification theorem). The matrix-valuedε-algorithm,as expressed by(5.1)–(5.3)with the generalized inverse(2.4),construct GMPA,is identified by
ε2(j )k = [j+2k/2k]f, j, k>0. (5.4)
Proof. For zeroth columnk=0, the proof is obvious. From (5.1)–(5.3) and (2.4) it is derived that
ε1(j )=(ε0(j+1)−εj0)−r1=1/cj+1zj+1=H0(j )(z)/zj+1, (5.5) whereH0(j )(z)=cj∗+1/kcj+1k2,N{H0(j )} =0.
For the second column, by (5.3) it is obtained that
ε2(j )=
j X i=0
Cizi+Cj+1zj+1+zj+2/(Cj−+12−Cj−+11z). (5.6) It holds from (5.6) that
ε2(j )=O(zj)(z→ ∞). (5.7)
Letg(z)=Cj−+12−Cj−+11z. ThenN{kg(z)k2} =2 and (5.6) becomes
ε2(j )=(kg(z)k2ε(j0+1)+zj+2g∗(z))/kg(z)k2=P2(z)/Q2(z), (5.8) whereQ2(z)= kg(z)k2. It is known from (5.6) and (5.7) that
N{P2} =j+2, N{Q2} =2.
From
kP2k2=Q22kε0(j+1)k2+Q2z2(j+1)+Q2(ε(j0+1)∗·G∗+ε0(j+1)·G), we getQ2(z)|kP2(z)k2. Hence, by above discussion, we have shown that
ε2(j )=P2(z)/Q2(z)= [j+2/2]f.
Make the following inductive hypotheses, each true forj =0,1,2, . . . , k >2:
(i) ε2(j )k−1=O(z−j−1), ε2(j )k =O(zj)(z→ ∞), (5.9)
where
H2(j )k−2(z)∈Cs×t,N{H2(j )k−2}62k−2,
(iii) ε2(j )k = [j+2k/2k]f. (5.11)
To prove the theorem, it is necessary to show that (5.9)–(5.11) hold withk+1. As (i) stands, it is obtained from (5.3) that
ε2(j )k+1=O(z−j−2)+O((ε2(jk+1))−r1)=O(z−j−2)+O(z−j−1)
=O(z−j−1) (z→ ∞), (5.12)
ε2(j )k+2=O(zj+1)−O((ε(j )2k+1)−r1)=O(zj)(z→ ∞),
so (5.9) is proved fork+1. As it stands, (i) and (ii) are reduced fractions, suppose that
ε2(jk+1)=S(z)/T (z), ε(j )2k =U (z)/V (z). (5.13) Define the matrix polynomialF (z)by
F (z)=S(z)V (z)−U (z)T (z)=zj+2k+1Fc(z), (5.14) we can prove thatV (z)T (z)|kFc(z)k2[13]. Note that
N{S}6j+2k+1, N{T} =2k, N{U}6j+2k, N{V} =2k. (5.15) From (5.14) and (5.15), we getN{Fc}62k,N{V T} =4k. Then, it is derived that
M(z)=V (z)T (z)/Fc(z)=V (z)T (z)Fc∗(z)/kFc(z)k2 is a matrix polynomial andNM>2k.It follows that
(ε(j2k+1)−ε2(j )k)−r1=1/(S(z)/T (z)−U (z)/V (z))
=V (z)T (z)/Fc(z)zj+2k+1
=M(z)/zj+2k+1. (5.16) Substituting (5.16) intoε2(j )k+1, we get from (5.16) that
ε2(j )k+1=ε2(j )k−1+(ε2(jk+1)−ε(j )2k)−r1
=H2(jk−+12)(z)/zj+2k−1+M(z)/zj+2k+1
=H2(j )k (z)/zj+2k+1, (5.17) whereH2(j )k ∈Cs×t,N{H2(j )k }62k.
Hence (5.10) is proved fork+1. By means of (5.17) and (5.3), we have
ε2(j )k+2=ε(j2k+1)+zj+2k+2
.
H2(jk+1)(z)−zH2(j )k (z)
LetG(z)=H2(jk+1)(z)−zH2(j )k (z). Then
N{G}62k+1. (5.19)
Letx =ξ be any of 2kzero ofT (z)forε(j2k+1)=S(t)/T (z)in (5.13), it is de-duced from
S(ξ )/T (ξ )=ε2(jk+1)(ξ )=ε2(jk+−22)(ξ )+(ε2(jk+−21)(ξ )−ε2(jk+−11)(ξ ))−r1
that
ε2(jk+2−1)(ξ )=ε2(jk+1−1)(ξ ). (5.20) Similarly, we can deduce that
ε2(j )k+1(ξ )=ε2(jk−+11)(ξ ), ε2(jk++11)(ξ )=ε2(jk+−21)(ξ ).
Then, it follows from (5.20) that
ε2(jk++11)(ξ )=ε2(j )k+1(ξ ). (5.21) Substituting (5.21) into (5.18), we have
ε2(j )k+2(ξ )=S(ξ )/ T (ξ )+1 ε(j2k++11)(ξ )−ε2(j )k+1(ξ )
=S(ξ )/T (ξ )+ξ(j+2k+2)/G(ξ ). (5.22) Note thatT (0) /=0 by the Existence Theorem 4.6, that is to say,ξ /=0. Hence, it is known that both terms of the right-hand side of (5.22) have poles atz=ξ. Above discussion shows that ifz=ξ is arbitrary zero ofT (z), then holdsG(ξ )=0. From (5.15) and (5.19), suppose that
G(z)=T (z)G0(z), G0(z)∈Cs×t and by (5.19), it holds N{G0}61. (5.23) Then, (5.18) becomes ε2(j )k+2=S(z)/T (z)+zj+2k+2/G(z) =(T (z)kG0(z)k2S(z)+zj+2k+2T (z)G∗0(z))/T2(z)kG0(z)k2 =P2k+2(z)/Q2k+2(z), (5.24) whereQ2k+2=T2(z)kG0(z)k2. From (5.24), kP2k+2k2=kSk2kG0k2Q2k+2+z2(j+2k+2)Q2k+2 +Q2k+2zj+2k+2(S∗·G∗0+S·G0), we get Q2k+2(z)|kP2k+2k2. (5.25)
The reduced form of (5.24) is
ε2(j )k+2=P (z)/Q(z)=(S(z)kG0(z)k2
+zj+2k+2G∗(z))/T (z)kG0(z)k2, (5.26) whereQ(z)=T (z)kG0(z)k2. It follows that
N{Q} =2k+2. (5.27)
By means of (5.12), it is shown from (5.27) that
N{P} =N{ε2(j )k+2} +N{Q} =j+2k+2. (5.28) From (5.25)–(5.28), it is proved that
ε2(j )k+2=P (z)/Q(z)= [j+2k+2/2k+2]f.
Example 5.2. Letf (z)=c0+c1z+c2z2be the same as Example 4.2. Find[2/2]f byε-algorithm. Solution. By (5.1)–(5.3) and (2.4), ε0(0)=c0, ε(01)=c0+c1z, ε0(2)=c0+c1z+c2z2, ε1(0)= 1 3z 11 00 1 0 , ε1(1)= 1 2z2 10 00 1 0 , ε2(0)= 1 2z2−4z+3 z 2−z+3 0 z(3−4z) 2z2−4z+3 z(3−z) 0 = [2/2]f. Now suppose N =ε(j2k+−12), W =ε2(jk−1), C=ε(j )2k, E=ε2(jk+1), S=ε2(jk−+12)
as bonafide entries in theε-table, and
nw=ε(j )2k−1, ne=ε(j2k+−11), sw=ε2(jk++11), se=ε2(j )k+1,
which occur in column of odd index. According to theε- algorithm (4.1)–(4.3), they are located as follows:
N nw ne W C E sw se S (5.29)
Lemma 5.3. The identity
(N−C)−r1+(S−C)−r1=(E−C)−r1+(W −C)−r1 (5.30) holds provided that the quantities involved are well defined by using(5.1)–(5.3)and the generalized inverse(2.4).
Proof. As
(nw−ne)+(se−sw)=(se−ne)+(nw−sw) (5.31)
it follows from (5.1)–(5.3) that (5.31) holds, then (5.30) holds by (5.29). By virtue of Lemma 5.3 and Theorem 5.1, we obtain the following result. Theorem 5.4. Wynn identify of GMPA
([j+2k−1/2k]f − [j+2k/2k]f)−r1 +([j+2k+1/2k]f − [j +2k/2k]f)−r1 = ([j +2k−1/2k−2]f − [j +2k/2k]f)−1r +([j+2k+1/2k+2]f − [j+2k/2k]f)−r1
holds provided that the GMPAs involved are well defined by using(5.1)–(5.3)and the generalized inverse(2.4).
6. Continued fraction expression of GMPA
By making use of the generalized inverse (2.4), we construct Thiele-type matrix-valued continued fraction as
H (z)=B0+ z B1+ z B2+ · · · + z Bn+ · · ·
with matrix elementsBi ∈Cs×t, z∈R. The nth convergent ofH (z)is defined by
Rn(z)=B0+ z B1 + z B2+ · · · + z Bn (6.1) and it is evaluated by backward recursion.
As the result of [16], recursive coefficient algorithm of (6.1) is given as follows for the given power series (1.1).
Coefficient algorithm:
A0(z)=f (z), B0=B0(0)=A0(0),
A1(z)=(Df (z))−r1= [Df (z)]∗/kDf (z)k2, B1=B1(0)=A1(0),
k>2, Bk=k(DAk−1(z))−r1|z=0=k(DAk−1(z))∗/kDAk−1(z)k2|z=0,
Ak(z)=Bk(z)+Ak−2(z),
Theorem 6.1 (Identification theorem). If zero divisors are not encountered in the construction ofRn(z)in(6.1)by using coefficient algorithm with the generalized inverse(2.4),then hold:
Rn(z)= [n/2k]f =
[2k/2k]f, n=2k, k=0,1,2, . . . ,
[2k+1/2k]f, n=2k+1, k=0,1,2, . . . (6.2)
Proof. The proof is recursive. Forn=2k, k=0, it holds
R0(z)=B0=B0/1=P0(z)/Q0(z)= [0/0]f. Forn=2k+1, k=0, we get that
R1(z)=B0+ z B1 = B0+ zB1∗ kB1k2 = B0kB1k2+zB1∗ kB1k2 = P1(z) Q1(z) , (6.3) whereN{P1} =1,N{Q1} =0, Q1|kP1k2. Then R1(z)= [1/0]f. (6.4) Forn=2k, k=1, R2(z)=B0+ z B1+ z B2 = B0+ z S1(z) , (6.5)
whereS1(z)=B1+z/B2= ˜P1(z)/Q˜1(z). By the proof of (6.4), we have N{ ˜P1} =1, N{ ˜Q1} =0, Q˜1|k ˜P1k2.
Letk ˜P1k2= ˜Q1g1,thenN{g1} =2. We get that
R2(z)=B0+ z S1(z) =B0+ zQ˜1P˜1∗ k ˜P1k2 = B0g1+zP˜1∗ g1 = P2(z) Q2(z) , (6.6)
whereQ2(z)=g1. ThenN{P2} =2,N{Q2} =2. From
kP2k2= kB0k2g12+z2k ˜P1k2+zg1(B0∗· ˜P1∗+B0· ˜P1), it is heldQ2|kP2k2. HenceR2(z)= [2/2]f. Forn−1=2korn−1=2k+1, k=0,1,2, . . .let Rn(z)=Pn−1(z)/Qn−1(z)= [n−1/2k]f = ( [2k/2k]f n=2k, k=0,1,2, . . . [2k+1/2k]f n=2k+1, k=0,1,2, . . . (6.7) Forn=2k, k=0,1,2, . . .let Rn(z)=B0+ z B1 + z B2+ · · · + z Bn =B0+ z Sn(z) , (6.8) where
Sn(z)=B1+ z B2 + z B3+ · · · + z Bn = ˜ Pn−1(z) ˜ Qn−1(z) . (6.9)
Using the inductive hypothesis (6.7), we obtain that
N{ ˜Pn−1} =n−1=2k, N{ ˜Qn−1} =2k, Q˜n−1|k ˜Pn−1k2. (6.10) From (6.10), letk ˜Pn−1k2= ˜Qn−1gn−1. ThenN{gn−1} =2k.
Substituting (6.9) into (6.8), we get that
Rn(z)=B0+ z Sn(z) = B0+ zQ˜n−1P˜n∗−1 k ˜Pn−1k2 =B0gn−1+zP˜n∗−1 gn−1 = Pn(z) Qn(z) , (6.11)
whereQn(z)=gn−1. Then, we obtain from (6.10) that
N{Pn} =n, N{Qn} =2k. (6.12) For kPnk2= kB0k2g2n−1+z 2k ˜P∗ n−1k 2+zg n−1(B0∗· ˜Pn−1+B0· ˜Pn∗−1) we get Qn|kPnk2. (6.13)
Hence it follows from (6.11)–(6.13) that
Rn(z)=Pn(z)/Qn(z)= [n/2k]f = [2k/2k]f, n=2k, k=0,1,2, . . . As forn=2k+1, k=0,1,2, . . . ,the proof is similar to the case ofn=2k.
It is known from Theorem 6.1 that each convergentRn(z)in (6.1) is a corre-sponding[n/2k]f and is of the sequences[0/0]f,[1/0]f,[2/2]f,[3/2]f,[4/4]f,
[5/4]f, . . .
Example 6.2. Let f (z)=C0+C1z+C2z2 be the same as Example 4.2. Find [2/2]f by the convergent of continued fraction (6.1).
Solution. By coefficient algorithm of (6.1) and using (2.4), we get
R2(z)= 10 01 0 0 + z 1 3 11 00 1 0 + z 1 2 −−14 00 −1 0 = 1 2z2−4z+3 z 2−z+3 0 z(3−4z) 2z2−4z+3 z(3−z) 0 = [2/2]f,
where A0(z)=f (z), B0=B0(0)=A0(0)= 10 01 0 0 , A1(z)=(Df (z))−r1= 1 8z2+8z+3 1+12z 00 1+2z 0 , B1=B1(0)=A1(0)= 1 3 11 00 1 0 , B2=B2(0)=2(DA1(z))−r1|z=0 = 1 4 (8z2+8z+3)2 [(8z2+8z+1)+8(2z+1)2] −2(8z 2+8z+1) 0 −8(2z+1) 0 −2(8z2+8z+1) 0 z=0 = −1 2 14 00 1 0 .
We take notice that[2/2]f in Example 4.2 is the same as that of in Example 5.2 and in Example 6.2. It illustrates that the generalized inverse (2.4), which is on the basis of the scalar product of matrices (2.1), is successful in the matrix-valued ratio-nal approximation and interpolation. On the other hand, Lemma 2.4 shows that we need not compute each inverse in the construction process of GMPA in some case.
Acknowledgement
The author would like to thank the referees for their corrections and many valu-able suggestions.
References
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