- Email: [email protected]
Generalized Hermite formula for the two-sided Lagrange–Sylvester interpolation
Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
www.elsevier.com/locate/cam
Generalized Hermite formula for the two-sided Lagrange–Sylvester interpolation Yong-Jian Hua;∗ , Zheng-Hong Yangb , Gong-Ning Chena a
b
Department of Mathematics, Beijing Normal University, Beijing 100875, China Department of Mathematics, P.O. Box 71, East Campus, China Agriculture University, Beijing 100083, China Received 20 June 2003; received in revised form 28 February 2004
Abstract The classical Hermite formula of the scalar polynomial interpolation is generalized to the setting of two-sided Lagrange–Sylvester interpolation for matrix polynomials. The formula then is applied to a description of the set of all matrix polynomial solutions to that two-sided interpolation. c 2004 Elsevier B.V. All rights reserved. MSC: 65F05; 15A09 Keywords: Lagrange–Sylvester interpolation; Matrix polynomial; Lagrange formula; Hermite formula; Vandermonde-type matrix; McMillan degree
1. Introduction The two-sided Lagrange–Sylvester interpolation (TSLSI) under consideration is formulated in the following manner. It is required to =nd all rational p × q matrix-valued functions F(z) and/or p × q matrix polynomials F(z), satisfying simultaneously (1) For N distinct points c1 ; : : : ; cN in the complex plane C, nonzero row vectors z1 ; : : : ; zN in C1×p and row vectors w1 ; : : : ; wN in C1×q , zi F(ci ) = wi
(i = 1; : : : ; N ):
(1.1)
Project supported by National Natural Science Foundation of China (No. 10271018) and Tianyuan Youth Foundation of China (No. TY10126009). ∗ Corresponding author. E-mail address: [email protected] (Y.-J. Hu). c 2004 Elsevier B.V. All rights reserved. 0377-0427/$ - see front matter doi:10.1016/j.cam.2004.03.018
346
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
(2) For M distinct points d1 ; : : : ; dM in the complex plane C, nonzero column vectors y1 ; : : : ; yM in Cq×1 and column vectors v1 ; : : : ; vM in Cp×1 , F(dj )yj = vj
(j = 1; : : : ; M ):
(1.2)
(3) For all i; j such that ij = ci = dj , zi F (ij )yi = ij ;
(1.3)
where ij ’s are prescribed complex numbers. With the TSLSI, there are associated two tangential Lagrange–Sylvester interpolation problems: one is to =nd all p × q matrix-valued functions F(z) satisfying only Eq. (1.1), the other is to =nd all p × q matrix-valued functions F(z) satisfying only Eq. (1.2), and they will be called the left-sided Lagrange–Sylvester interpolation (LTLSI) and the right-sided Lagrange–Sylvester interpolation (RTLSI), respectively. For the sake of simplicity, we restrict ourselves here within the matrix polynomial case. Of particular importance are the questions of =nding a matrix polynomial solution with the minimal possible complexity measured by the polynomial degree or McMillan degree, and of describing all matrix polynomial solutions when the TSLSI is solvable. The present paper aims to give a direct generalization of the classical Hermite formula for polynomial interpolation to the case of the TSLSI with a complete interpolation data, and to give a parametrization of the set of all matrix polynomial solutions to the TSLSI on the basis of the generalized Hermite formula. This investigation is motivated by the recent papers [3–5,7,8] on the vector polynomial interpolation and the Nevanlinna–Pick interpolation. Heinig and Al-Musallam gave direct generalizations of classical Lagrange and Hermite formulas in [7,8], respectively, for vector polynomial interpolation, and their applications to the structured matrices. However, they did not oJer the generalized Lagrange or Hermite formula in the setting of two-sided (bi-tangential) interpolation problems. In [3–5], Chen and his collaborators presented the block-Hankel vector approach to deal with many kinds of nontangential Nevanlinna–Pick interpolations for matrix-valued functions, in which the classical Hermite formula plays an essential role. Also, applying the same approach to the tangential Nevanlinna–Pick interpolation, we need necessarily a generalization of the classical Lagrange or Hermite formula adapted to the indicated problem (see [9] for details) into the case of the TSLSI. We remark that the TSLSI, together with more intricate versions thereof, has been considered in the monograph [1] for the rational matrix-valued functions. The main strategy there is to reduce the solution of them to the inversion of structured matrices which is assumed to be a standard problem. From the corresponding formulas for the solutions of these interpolation problems even the classical Lagrange and Hermite formulas do not emerge as special cases (see the introduction part of [8] for further details). However, Ball and Kang [2] deal explicitly with the matrix polynomial solutions with the minimal possible McMillan degree via the methods of [1]. The problem considered in [2] is more general than that considered by the authors, in that, the interpolation conditions are allowed to be of possible incompleteness and to have higher multiplicity in terms of a general admissible interpolation data set. For the case of simple interpolation conditions, the new feature contributed by the authors here compared to what is done in [1,2] is the procedure for constructing the particular minimal McMillan
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
347
degree interpolant: the present paper provides one with a direct approach which exactly generalizes the classical Hermite formula for the solution of a Lagrange–Sylvester interpolation, while the approach of [1,2] gives a more complicated formula arrived at through plugging 0 into a linear fractional parametrization for all solutions; its resolvent matrix in turn is constructed by solving a certain related but diJerent homogeneous interpolation problem. It should also be pointed out that the polynomial interpolants of the minimal McMillan degree, constructed carefully in [2], give rise to one and the same function as in this paper (although via a more complicated formula than that of the authors) in the complete data case. Throughout the paper, we use the following notation: c1 0 · · · 0 d1 0 · · · 0 0 c2 · · · 0 0 d2 · · · 0 Ac = . ; Ad = . ; .. . . .. .. .. .. .. . . . . . . . .
0
z1 z2 B= . ..
;
0
···
cN
0
0
···
dM
C = (y1 ; : : : ; yN );
zN
Vl; k = (B; Ac B; : : : ; Ack −1 B);
Vr; k
C
CAd = . ..
:
CAdk −1 We call Vl; k and Vr; k the generalized Vandermonde-type matrices for the pairs (Ac ; B) and (C; Ad ), respectively. In the case when p=q=1 and all zi =yi =1, they degenerate to a classical Vandermonde matrix and the transpose of a classical Vandermonde matrix (not necessarily square), respectively.
2. Basic lemmas In this section, we will lay out some known results about the LTLSI with N = np and the RTLSI with M = mq, which are basic in our analysis for a direct generalization of the classical Hermite formula. The reader can be referred to [1,6,7,10] for details. The construction of the minimal possible McMillan degree matrix polynomial for the homogeneous LTLSI/RTLSI for more general situation (in that, the integers N and M are not necessarily the multiples of the numbers of rows p and columns q of the unknown matrix functions F(z), respectively) and its connection with the Hermite formula given in Section 3 are considered simply in Section 4 (see also [2]).
348
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
Lemma 2.1 (Gohberg [6]). Suppose that N = nq and Vl; n is nonsingular, and Ul 0 . p ×p X1 = ); X2 = (e1 ; : : : ; eN ) (ei ∈ Cp×1 ) .. (Uls ∈ C U l ; n− 1 are the unique solutions of the matrix equations Vl; n X = Anc B and XVl; n = (0; : : : ; 0; Ip ), respectively. (1) There exists a monic p × p matrix polynomial Ul (z) of polynomial degree n satisfying zi Ul (ci ) = 0
(i = 1; 2; : : : ; N ):
(2.1)
Moreover, such a Ul (z) is unique and can be formulated as Ul (z) = z n Ip − z n−1 Ul; n−1 − · · · − zUl 1 − Ul 0 : (2) det Ul (z) = (z − c1 )(z − c2 ) · · · (z − cN ); rank Ul (ci ) = p − 1, and for each z = ci (1 6 i 6 N ), N ei zi Ul−1 (z) = X2 (zIN − Ac )−1 B = : (2.2) z − ci i=1 (3) A p × q matrix polynomial F(z) solves the homogeneous LTLSI : zi F(ci ) = 0 (1 6 i 6 N ) if and only if it is of the form: F(z) = Ul (z)L(z) for some p × q matrix polynomial L(z). In the terminology of [1], by this time the pair (Ac ; B) is a left null pair of the matrix polynomial Ul (z), and (Ac ; B; X2 ) is a left null triple of the same Ul (z). In the case, when p = 1 and all zi = 1, Ul (z) in Lemma 2.1 is identical with the annihilator polynomial ul (z), of degree N , of the interpolation nodes c1 ; : : : ; cN ul (z) = (z − c1 )(z − c2 ) · · · (z − cN ):
Lemma 2.2 (Hu and Zhang [10] and Heinig and Al-Musallam [7]). Let N = np and Vl; n be nonsingular, and Ul (z), ei (1 6 i 6 N ) be given as in Lemma 2.1. Then Ul (ci )ei = 0;
zi Ul (ci )ei = 1
(i = 1; 2; : : : ; N ):
(2.3)
Moreover, there is no other column vector ei in Cp×1 satisfying Eq. (2.3). By using Ul (z), ei (1 6 i 6 N ), we obtain in [10] a generalization of the classical Lagrange formula to the setting of the LTLSI with N = np N Ul (z)ei wi Fl (z) = : (2.4) z − ci i=1 The reason is that, in the case of p = q = 1 and all zi = 1, 1 (i = 1; 2; : : : ; N ); ei = ul (ci )
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
349
then Eq. (2.4) has the form Fl (z) =
N i=1
ul (z) wi ; ul (ci )(z − ci )
which coincides with the classical Lagrange formula. The following Lemmas 2.3 and 2.4 for the RTLSI with M = mq are counterparts of Lemmas 2.1 and 2.2 for the LTLSI with N = np, respectively. Lemma 2.3 (Gohberg et al. [6]). Suppose that M = mq and Vr; m is nonsingular, and f1 . 1× q X3 = (Ur0 ; : : : ; Ur; m−1 ) (Urt ∈ Cq×q ); X4 = .. (fj ∈ C ) fmq are the unique solutions to the matrix equations XVr; m = CAmd and Vr; m X = ( I0q ), respectively. (1) There exists a monic q × q matrix polynomial Ur (z) of polynomial degree m satisfying Ur (dj )yj = 0
(j = 1; 2; : : : ; M ):
(2.5)
Moreover, such a Ur (z) is unique and can be formulated as Ur (z) = z m Iq − z m−1 Ur; m−1 − · · · − zUr1 − Ur0 : (2) det Ur (z) = (z − d1 )(z − d2 ) · · · (z − dM ); rank Ur (dj ) = q − 1, and for each z = dj (1 6 j 6 M ), Ur−1 (z) = C(zIM − Ad )−1 X4 =
M yj fj : z − d j j=1
(2.6)
(3) A p × q matrix polynomial F(z) solves the homogeneous RTLSI: F(dj )yj = 0 (1 6 j 6 M ) if and only if it is of the form: F(z) = R(z)Ur (z) for some p × q matrix polynomial R(z). In the terminology of [1], by this time the pair (C; Ad ) is a right null pair of the matrix polynomial Ur (z), and (C; Ad ; X4 ) is a right null triple of the same Ur (z). In the case of q =1 and all yi =1, Ur (z) in Lemma 2.3 is identical with the annihilator polynomial ur (z), of degree M , of the interpolation nodes d1 ; : : : ; dM ur (z) = (z − d1 )(z − d2 ) · · · (z − dM ):
Lemma 2.4 (Hu and Zhang [10] and Heinig and Al-Musallam [7]). Let M = mq and Vr; m be nonsingular, and Ur (z), fj (1 6 j 6 M ) be given as in Lemma 2.3. Then fj Ur (dj ) = 0;
fj Ur (dj )yj = 1
(j = 1; 2; : : : ; M ):
Moreover, there is no other 1 × q row vector fj satisfying Eq. (2.7).
(2.7)
350
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
With Ur (z), fj (1 6 j 6 M ) as above, we obtain in [10] a generalized Lagrange formula for the RTLSI with M = mq M vj fj Ur (z)
Fr (z) =
j=1
z − dj
:
(2.8)
Similarly, in the case when p = q = 1 and all yj = 1, fj =
1 ur (dj )
(j = 1; 2; : : : ; M );
then Eq. (2.8) has the form Fr (z) =
M j=1
ur (z) vj ; ur (dj )(z − dj )
which coincides with the classical Lagrange formula. 3. Main results Let %ij = wi yj and &ij = zi vj (1 6 i 6 N; 1 6 j 6 M ). If F(z) is an arbitrary solution to the TSLSI, and ci = dj , then multiplication of Eq. (1.1) by yj and Eq. (1.2) by zi leads to %ij = &ij :
(3.1)
Thus that Eq. (3.1) holds whenever ci = dj is a necessary condition for the TSLSI to be solvable. Theorem 3.1. Let N = nq, M = mq, both Vl; n and Vr; m be nonsingular, and Ul (z), Ur (z), ei , fj be as in Section 2. If for all i; j such that ij = ci = dj Eq. (3.1) holds true, we have that (1) The system of linear equations x(Ur (ij ); Ur (ij )yj ) = (wi − %ij fj Ur (ij ); − 12 %ij fj Ur (ij )yj ) has a unique row-vector solution x. (2) The system of linear equations vj − %ij Ul (ij )ej Ul (ij ) x= zi Ul (ij ) − 12 %ij zi Ul (ij )ej
(3.2)
(3.3)
has a unique column-vector solution x. Proof. We show assertion (1) only. Assertion (2) can be proved in a similar way. According to the theory of linear equations, it suLces to verify that Ur (ij )yj Ur (ij ) rank (Ur (ij ); Ur (ij )yj ) = rank = q: (3.4) wi − %ij fj Ur (ij ) − 12 %ij fj Ur (ij )yj
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
351
Suppose that there exists a 1 × q row vector x0 such that x0 (Ur (ij ); Ur (ij )yj ) = 0; or equivalently, x0 Ur (ij ) = 0;
x0 Ur (ij )yj = 0:
From Lemmas 2.3(2) and 2.4, we conclude that the general solution to the system of linear equations xUr (ij ) = 0 is of the form: kfj , where k is an arbitrary complex number, and therefore x0 = kfj , and kfj Ur (ij )yj = k = 0. Thus, x0 = kfj = 0. This implies that rank(Ur (ij ); Ur (ij )yj ) = q:
(3.5)
On the other hand, observe that yj is nonzero p × 1 column vector because of the nonsingularity of Vr; m , and that Eqs. (2.5) and (2.7) force Ur (ij )yj Ur (ij ) yj 0 = : 1 0 0 wi − %ij fj Ur (ij ) − 2 %ij fj Ur (ij )yj Then we have by Eq. (3.5) Ur (ij ) rank wi − %ij fj Ur (ij )
Ur (ij )yj − 12 %ij fj Ur (ij )yj
= q:
(3.6)
Combining Eq. (3.5) with Eq. (3.6), we get immediately Eq. (3.4). Now we pull together the results of Lemmas 2.1–2.4 and Theorem 3.1 to present a generalized Hermite formula for the setting of the TSLSI with N = np and M = mq. Theorem 3.2. Let N = np, M = mq and both Vl; n and Vr; m be nonsingular. If Eq. (3.1) holds true for all i; j such that ij = ci = dj , and w˜ ij ∈ C1×q , v˜ij ∈ p × 1 are the unique solutions to the systems of linear equations (3.2) and (3.3), respectively, then the formula
ij ei fj + ei w˜ ij + v˜ij fj %ij ei fj Fts (z) = Ur (z) Ul (z) + z − ij (z − ij )2 (i; j):ij =ci =dj
+
Ul (z)
i:ci =dj
v˜j fj ei w˜ i Ur (z) + Ul (z) Ur (z); z − ci z − dj
(3.7)
j:dj =ci
where w˜ i = wi Ur−1 (ci ) =
M %ij fj ; ci − d j j=1
v˜j = Ul−1 (dj )vj =
N &ij ei ; dj − c i i=1
(3.8)
provides a matrix polynomial solution of polynomial degree at most m + n − 1 with the TSLSI. Proof. It follows from Lemmas 2.3 and 2.4 that fj Ur (z) Ul (z)ei (i = 1; : : : ; N ); rj (z) := li (z) := z − dj z − ci
(j = 1; : : : ; M )
352
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
are p ×1 and 1×q vector-valued polynomials with polynomial degrees n−1 and m−1, respectively. Then Fts (z) de=ned via Eq. (3.7) can be rewritten as Fts (z) = (ij li (z)rj (z)(z − ij ) + li (z)w˜ ij Ur (z) + Ul (z)v˜ij rj (z) (i; j):ij =ci =dj
+%ij li (z)rj (z)) +
li (z)w˜ i Ur (z) +
i:ci =dj
Ul (z)v˜j rj (z);
j:dj =ci
so that Fts (z) is a p × q matrix polynomial with polynomial degree at most m + n − 1. Now we verify that Fts (z) satis=es Eqs. (1.1)–(1.3). (1) If i is such that ci = dj = ij for some j, then we have by Eqs. (2.1), (2.3), (2.7) and (3.2) that zi Fts (ij ) = lim zi Fts (z) z →ij
ij ei fj + ei w˜ ij + v˜ij fj %ij ei fj = lim zi Ul (z) + z →ij z − ij (z − ij )2
Ur (z)
= zi Ul (ij )ei w˜ ij Ur (ij ) + %ij zi Ul (ij )ei fj Ur (ij ) = w˜ ij Ur (ij ) + %ij fj Ur (ij ) = wi : Similarly, we can obtain that Fts (ij )yj = vj for all i; j such that ij = ci = dj . Furthermore, for such i; j, we have Fts (z) − Fts (ij ) yj z →ij z − ij
ij ei fj + ei w˜ ij + v˜ij fj %ij ei fj %ij Ur (z)yj − = lim zi Ul (z) + z →ij (z − ij )2 (z − ij )3 z − ij zi Ul (z)(ij ei fj + ei w˜ ij + v˜ij fj )Ur (z)yj = lim z →ij (z − ij )2
%ij zi Ul (z)ei fj Ur (z)yj − %ij (z − ij )2 : + (z − ij )3
zi Fts (ij )yj = lim zi
By Lemmas 2.1–2.4 and the L’Hospital principal, we have that lim
z →ij
zi Ul (z)(ij ei fj + ei w˜ ij + v˜ij fj )Ur (z)yj (z − ij )2 = lim zi Ul (z)(ij ei fj + ei w˜ ij + v˜ij fj )Ur (z)yj z →ij
=ij + w˜ ij Ur (ij )yj + zi Ul (ij )v˜ij
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
353
and lim
z →ij
%ij zi Ul (z)ei fj Ur (z)yj − %ij (z − ij )2 (z − ij )3 = lim
z →ij
1 %ij (zi Ul (z)ei fj Ur (z)yj + zi Ul (ij )ei fj Ur (z)yj ) 2
1 = %ij (zi Ul (ij )ei + fj Ur (ij )yj ): 2 From the last three equations, we obtain by Eqs. (3.2) and (3.3) that zi Fts (ij )yj = ij + w˜ ij Ur (ij )yj + zi Ul (ij )v˜ij 1 + %ij (zi Ul (ij )ei + fj Ur (ij )yj ) 2 = ij : (2) If i is such that ci is not equal to any dj , we have by Eqs. (3.7) and (3.8) that ei w˜ i zi Fts (ci ) = lim zi Fts (z) = lim zi Ul (z) Ur (z) z →ci z →ij z − ci = zi Ul (ci )ei w˜ i Ur (ci ) = w˜ i Ur (ci ) = wi : (3) If j is such that dj is not equal to any ci , we have that v˜j fj Fts (dj )yj = lim Fts (z)yj = lim Ul (z) Ur (z)yj z →dj z →dj z − dj = Ul (dj )v˜j fj Ur (dj )fj = Ul (dj )v˜j = vj : Thus we have shown that Fts (z) satis=es Eqs. (1.1)–(1.3). The proof is completed. In the special case when p = q = 1, m = n and all ci = di , zi = yi = 1 (1 6 i 6 n), we check easily that Ul (z) = Ur (z) = ul (z) = ur (z), and both Eqs. (3.2) and (3.3) have the same solution w˜ ii = v˜ii = −
wi ul (ci ) 2ul (ci )
(i = 1; 2; : : : ; N );
so that Fts (z) de=ned by Eq. (3.7) admits of the form
N ul2 (z) ii − wi u (ci ) wi Fts (z) = + (ul (ci ))2 z − ci (z − ci )2 i=1 =
N
l2i (z)((ii − 2wi li (ci ))(z − ci ) + wi )
i=1
=
N i=1
ii (z −
ci )l2i (z)
+
N i=1
wi (1 − 2li (ci )(z − ci ))l2i (z);
(3.9)
354
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
in which li (z) =
ul (z) ul (ci )(z − ci )
(i = 1; 2; : : : ; N ):
In this case, Eq. (3.9) is identical with the classical Hermite formula for the scalar polynomial interpolation. Corollary 3.3. Let N = np, M = mq and both Vl; n and Vr; m be nonsingular. Then the TSLSI is solvable, if, and only if, Eq. (3.1) holds true whenever ci = dj . In particular, if the set of points {c1 ; c2 ; : : : ; cN } is disjoint from the set {d1 ; d2 ; : : : ; dM }, then the TSLSI is always solvable, and its Hermite formula (3.7) is identical with the Lagrange’s formula for the TSLSI, namely, formula (3.7) but with the =rst term on the right absent. Theorem 3.4. Let N = np, M = mq, and both Vl; n and Vr; m be nonsingular. There is at most one p × q matrix polynomial of polynomial degree at most m + n − 1, which solves the TSLSI. Proof. Suppose that there are two matrix polynomials F1 (z) and F2 (z), with polynomials degrees at most m + n − 1, which solve simultaneously the TSLSI. Let G(z) = F1 (z) − F2 (z): It remains now only to prove G(z) = 0. Note that G(z) is a matrix polynomial with polynomial degree at most m + n − 1, satisfying zi G(ci ) = 0 G(dj )yj = 0
(i = 1; 2; : : : ; N ); (j = 1; 2; : : : ; M );
(3.10) (3.11)
and for all i; j such that ij = ci = dj , zi G (ij )yj = 0:
(3.12)
From Eq. (3.11) and Lemma 2.3(3), we see that G(z) has to be of the form G(z) = R(z)Ur (z);
(3.13)
for some p × q matrix polynomial R(z). Inserting Eq. (3.13) into Eq. (3.10), we obtain zi R(ci )Ur (ci ) = 0
(i = 1; 2; : : : ; N ):
In the case of ci = dj (1 6 j 6 mq), Ur (ci ) is nonsingular, then zi R(ci )=0. In the case of ij =ci =dj for some j, zi R(ci )Ur (ij ) = 0. By Lemmas 2.3 and 2.4, we have that zi R(ci ) = kfj
(i = 1; 2; : : : ; N )
for some complex number k. By Eqs. (3.11), (3.13) and (3.14), we have 0 = zi G (ij )yj = zi (R (ij )Ur (ij ) + R(ij )Ur (ij ))yj = kfj Ur (ij )yj = k:
(3.14)
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
355
This implies that zi R(ci ) = kfj = 0. By Lemma 2.1(3), we have that R(z) = Ul (z)L(z) for some p × q matrix polynomial L(z). Then G(z) = Ul (z)L(z)Ur (z) for the same L(z). Note that the facts that Ul (z) and Ur (z) are monic matrix polynomials and deg Ul (z) = n, deg Ur (z) = m, then force L(z) = 0 and thus G(z) = 0, as required. Theorem 3.4 combined with Theorem 3.2 means that Fts (z) de=ned by Eq. (3.7) is the unique solution to the TSLSI for matrix polynomials with the polynomial degree at most m + n − 1. Making use of Ul (z), Ur (z) and Fts (z), we obtain in turn a description of the set of all matrix polynomial solutions to the TSLSI (than which for a more general result see [2, Theorem 6.1]). Theorem 3.5. Suppose that N = np, M = mq and both Vl; n and Vr; m are nonsingular. Then the general matrix polynomial solution to the solvable TSLSI can be formulated in the following divisor-remainder form: F(z) = Fts (z) + Ul (z)M (z)Ur (z);
(3.15)
where M (z) is an arbitrary p × q matrix polynomial. 4. Some extensions We devote this section to an extension of the main results to a more general situation, in that, the numbers of left and right interpolation nodes N and M of the TSLSI need not be the multiples of p and q, respectively. In that case, for convenience, we use the McMillan degree instead of the polynomial degree as measure of complexity for a given matrix polynomial. It is known that for a monic p × p matrix polynomial, the McMillan degree is p times of the polynomial degree. The pair (Ac ; B) is called a full-range pair if rank(Vl; i ) = N for some positive integer i, and (C; Ad ) is called a null-kernel pair if rank(Vr; j ) = M for some positive integer j. To simplify the consideration, we always assume in this section that ci ; dj = 0 for all i; j. This has the consequence that Ac and Ad are nonsingular matrices. According to [2], for the homogeneous LTLSI we have the following: Theorem 4.1 (Ball and Kang [2]). For a given full-range pair (Ac ; B), there is a comonic p × p matrix polynomial Ul (z), of the minimal possible McMillan degree, which satis=es zi Ul (ci ) = 0;
i = 1; 2; : : : ; N
(4.1)
and has the form 1 Ul (z) = Ip + zE(IN − zT∞ )−1 A− c B
(4.2)
with inverse given by Ul−1 (z) = Ip + zE(zIN − Ac )−1 B for an appropriate pair of matrices T∞ (nilpotent) and E.
(4.3)
356
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
We omit the procedure for constructing the matrices T∞ and E. The reader is referred to [2, Section 2.2] for details. In general, such a Ul (z) is far from unique, and cannot be taken to be monic. But for the complete data case: N = np and Vl; n is invertible, choosing 0 Ip .. . 0 −1 V ; E = −(Ip ; 0; : : : ; 0) V −1 ; T∞ = Vl; n l; n l; n .. . I p 0 we check easily that each Ul (z) in Theorem 4.1 degenerates exactly to one and the same Ul (z) given in Lemma 2.1, up to a right factor of nonsingular constant matrix. Now putting E = (e1 ; e2 ; : : : ; eN ) by Eq. (4.3) we have Ul−1 (z) = (Ip + EAc B) +
N ei zi : z − ci i=1
(4.4)
With notations above and using a Euclidean algorithm for matrix polynomials with respect to McMillan degree established in [2], we can prove that each Ul (z) in Eq. (4.2) enjoys the following properties. Corollary 4.2. Let (Ac ; B) be a full-range pair, and Ul (z), ei be as in Eqs. (4.2) and (4.4), respectively. (1) det Ul (z) (viewed as a scalar polynomial in z) has only the simple zeros at c1 ; c2 ; : : : ; cN . (2) For each i, rank Ul (ci ) = p − 1 and Ul (ci )ei = 0;
zi Ul (ci )ei = 1; i = 1; 2; : : : ; N:
(4.5)
Moreover, there is no other column vector ei in Cp×1 satisfying Eq. (4.5). (3) A p × q matrix polynomial F(z) satisfying the homogenous LTLSI: zi F(ci ) = 0 (1 6 i 6 N ) if and only if it is of the form: F(z) = Ul (z)L(z) for some p × q matrix polynomial L(z). The following Theorem 4.3 and Corollary 4.4 for the homogeneous RTLSI are parallel with Theorem 4.1 and Corollary 4.2 for the homogeneous LTLSI, respectively. Theorem 4.3 (Ball and Kang [2]). For a given null-kernel pair (C; Ad ), there is a comonic q × q matrix polynomial Ur (z), of the minimal possible McMillan degree, which satis=es Ur (dj )yj = 0;
j = 1; 2; : : : ; M
(4.6)
and has the form 1 −1 Ur (z) = Iq + zCA− d (IM − zS∞ ) F
(4.7)
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
357
with the inverse given by 1 −1 Ur−1 (z) = Iq + zCA− d (zIM − Ad ) F
(4.8)
for an appropriate pair of matrices S∞ (nilpotent) and F. As Ul (z) in Theorem 4.1, Ur (z) is generally far from unique and cannot be taken to be monic. But for the special case: M = mq and Vr; m is invertible, choosing 0 Iq Iq 0 0 1 −1 V S∞ = Vr− ; F = −V . ; r ; m ;m r ; m .. .. .. . . Iq
0
0
we can verify that each Ur (z) in Theorem 4.2 degenerates to one and the same Ur (z) given in Lemma 2.3, up to a left factor of nonsingular constant matrix. We now partition F in Eq. (4.7) into the following form: f1 f2 F = . ; fj ∈ C1×q : .. fM By Eq. (4.8) we have Ur−1 (z)
M fj yj = (Ip + CAd F) + : z − dj i=1
(4.9)
Corollary 4.4. Let (C; Ad ) be a null-kernel pair, and Ur (z) and fj be as in Eqs. (4.6) and (4.9), respectively. (1) det Ur (z) has only the simple zeros at d1 ; d2 ; : : : ; dM . (2) For each j, rank Ur (dj ) = p − 1 and fj Ur (dj ) = 0;
fj Ur (dj )yj = 1; j = 1; 2; : : : ; M:
(4.10)
1× q
satisfying Eq. (4.10). Moreover, there is no other row vector fj in C (3) A p×q matrix polynomial F(z) satisfying the homogeneous RTLSI: F(dj )yj =0 (1 6 j 6 M ) if and only if it is of the form: F(z) = R(z)Ur (z) for some p × q matrix polynomial R(z). Applying arguments similar to that given in Section 3, one can obtain some extensions of the main results. Theorem 4.5. Let (Ac ; B) be a full-range pair and (C; Ad ) be a null-kernel pair. Then the statement of Theorem 3.1 holds also if Ul (z); Ur (z); ei ; fj therein are placed with that de=ned via Eqs. (4.2), (4.4), (4.7) and (4.9), respectively.
358
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
By Theorem 4.5, we have in turn the following Theorem 4.6. Let (Ac ; B) be a full-range pair and (C; Ad ) be a null-kernel pair. Then formula (3.7) provides a matrix polynomial solution of McMillan degree at most M + N − 1 with the TSLSI, in which Ul (z), Ur (z); ei ; fj are as in Theorem 4.5. Moreover, the general matrix polynomial solution to the TSLSI can be formulated by Eq. (3.15), where M (z) is an arbitrary p×q matrix polynomial. It is worthwhile to point out that such matrix polynomial solution Fst of the minimal McMillan degree is far from unique in the generic case. Acknowledgements The authors acknowledge with many thanks helpful comments and suggestions of the referee, which encourage us to consider an extension of the main results given in Section 3 of the paper to the incomplete interpolation data case. References [1] J.A. Ball, I. Gohberg, L. Rodman, Interpolation of Rational Matrix Functions, Birkhauser, Basel, 1990. [2] J.A. Ball, J. Kang, Matrix polynomial solutions of tangential Lagrange–Sylvester interpolation conditions of low McMillan Degree, Linear Algebra Appl. 137/138 (1990) 699–746. [3] G.-N. Chen, Y.-J. Hu, The Nevanlinna–Pick interpolation problems and power moment problems for matrix-valued functions III: the in=nitely many data case, Linear Algebra Appl. 306 (2000) 59–87. [4] G.-N. Chen, Y.-J. Hu, Multiple Nevanlinna–Pick interpolation with both interior and boundary data and its connection with the power moment problem, Linear Algebra Appl. 323 (2001) 167–194. [5] G.-N. Chen, X.-Q. Li, The Nevanlinna–Pick interpolation problems and power moment problems for the matrix-valued functions, Linear Algebra Appl. 288 (1999) 123–248. [6] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. [7] G. Heinig, F. Al-Musallam, Lagrange formula for tangential interpolation with application to structured matrices, Integral Equations Operator Theory 30 (1998) 83–100. [8] G. Heinig, F. Al-Musallam, Hermite formula for vector polynomial interpolation with applications to structured matrices, Appl. Anal. 70 (1999) 331–346. [9] Y.-J. Hu, Z.-H. Yang, G.-N. Chen, Tangential Nevanlinna–Pick interpolation and its connection with Hamburger matrix moment problem, Integral Equations Operator Theory, to appear. [10] Y.-J. Hu, X.-N. Zhang, Lagrange’s formula for bi-tangential interpolation of matrix polynomials, J. Beijing Normal Univ. Natur. Sci. 38 (4) (2002) 427–431.
www.elsevier.com/locate/cam
Generalized Hermite formula for the two-sided Lagrange–Sylvester interpolation Yong-Jian Hua;∗ , Zheng-Hong Yangb , Gong-Ning Chena a
b
Department of Mathematics, Beijing Normal University, Beijing 100875, China Department of Mathematics, P.O. Box 71, East Campus, China Agriculture University, Beijing 100083, China Received 20 June 2003; received in revised form 28 February 2004
Abstract The classical Hermite formula of the scalar polynomial interpolation is generalized to the setting of two-sided Lagrange–Sylvester interpolation for matrix polynomials. The formula then is applied to a description of the set of all matrix polynomial solutions to that two-sided interpolation. c 2004 Elsevier B.V. All rights reserved. MSC: 65F05; 15A09 Keywords: Lagrange–Sylvester interpolation; Matrix polynomial; Lagrange formula; Hermite formula; Vandermonde-type matrix; McMillan degree
1. Introduction The two-sided Lagrange–Sylvester interpolation (TSLSI) under consideration is formulated in the following manner. It is required to =nd all rational p × q matrix-valued functions F(z) and/or p × q matrix polynomials F(z), satisfying simultaneously (1) For N distinct points c1 ; : : : ; cN in the complex plane C, nonzero row vectors z1 ; : : : ; zN in C1×p and row vectors w1 ; : : : ; wN in C1×q , zi F(ci ) = wi
(i = 1; : : : ; N ):
(1.1)
Project supported by National Natural Science Foundation of China (No. 10271018) and Tianyuan Youth Foundation of China (No. TY10126009). ∗ Corresponding author. E-mail address: [email protected] (Y.-J. Hu). c 2004 Elsevier B.V. All rights reserved. 0377-0427/$ - see front matter doi:10.1016/j.cam.2004.03.018
346
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
(2) For M distinct points d1 ; : : : ; dM in the complex plane C, nonzero column vectors y1 ; : : : ; yM in Cq×1 and column vectors v1 ; : : : ; vM in Cp×1 , F(dj )yj = vj
(j = 1; : : : ; M ):
(1.2)
(3) For all i; j such that ij = ci = dj , zi F (ij )yi = ij ;
(1.3)
where ij ’s are prescribed complex numbers. With the TSLSI, there are associated two tangential Lagrange–Sylvester interpolation problems: one is to =nd all p × q matrix-valued functions F(z) satisfying only Eq. (1.1), the other is to =nd all p × q matrix-valued functions F(z) satisfying only Eq. (1.2), and they will be called the left-sided Lagrange–Sylvester interpolation (LTLSI) and the right-sided Lagrange–Sylvester interpolation (RTLSI), respectively. For the sake of simplicity, we restrict ourselves here within the matrix polynomial case. Of particular importance are the questions of =nding a matrix polynomial solution with the minimal possible complexity measured by the polynomial degree or McMillan degree, and of describing all matrix polynomial solutions when the TSLSI is solvable. The present paper aims to give a direct generalization of the classical Hermite formula for polynomial interpolation to the case of the TSLSI with a complete interpolation data, and to give a parametrization of the set of all matrix polynomial solutions to the TSLSI on the basis of the generalized Hermite formula. This investigation is motivated by the recent papers [3–5,7,8] on the vector polynomial interpolation and the Nevanlinna–Pick interpolation. Heinig and Al-Musallam gave direct generalizations of classical Lagrange and Hermite formulas in [7,8], respectively, for vector polynomial interpolation, and their applications to the structured matrices. However, they did not oJer the generalized Lagrange or Hermite formula in the setting of two-sided (bi-tangential) interpolation problems. In [3–5], Chen and his collaborators presented the block-Hankel vector approach to deal with many kinds of nontangential Nevanlinna–Pick interpolations for matrix-valued functions, in which the classical Hermite formula plays an essential role. Also, applying the same approach to the tangential Nevanlinna–Pick interpolation, we need necessarily a generalization of the classical Lagrange or Hermite formula adapted to the indicated problem (see [9] for details) into the case of the TSLSI. We remark that the TSLSI, together with more intricate versions thereof, has been considered in the monograph [1] for the rational matrix-valued functions. The main strategy there is to reduce the solution of them to the inversion of structured matrices which is assumed to be a standard problem. From the corresponding formulas for the solutions of these interpolation problems even the classical Lagrange and Hermite formulas do not emerge as special cases (see the introduction part of [8] for further details). However, Ball and Kang [2] deal explicitly with the matrix polynomial solutions with the minimal possible McMillan degree via the methods of [1]. The problem considered in [2] is more general than that considered by the authors, in that, the interpolation conditions are allowed to be of possible incompleteness and to have higher multiplicity in terms of a general admissible interpolation data set. For the case of simple interpolation conditions, the new feature contributed by the authors here compared to what is done in [1,2] is the procedure for constructing the particular minimal McMillan
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
347
degree interpolant: the present paper provides one with a direct approach which exactly generalizes the classical Hermite formula for the solution of a Lagrange–Sylvester interpolation, while the approach of [1,2] gives a more complicated formula arrived at through plugging 0 into a linear fractional parametrization for all solutions; its resolvent matrix in turn is constructed by solving a certain related but diJerent homogeneous interpolation problem. It should also be pointed out that the polynomial interpolants of the minimal McMillan degree, constructed carefully in [2], give rise to one and the same function as in this paper (although via a more complicated formula than that of the authors) in the complete data case. Throughout the paper, we use the following notation: c1 0 · · · 0 d1 0 · · · 0 0 c2 · · · 0 0 d2 · · · 0 Ac = . ; Ad = . ; .. . . .. .. .. .. .. . . . . . . . .
0
z1 z2 B= . ..
;
0
···
cN
0
0
···
dM
C = (y1 ; : : : ; yN );
zN
Vl; k = (B; Ac B; : : : ; Ack −1 B);
Vr; k
C
CAd = . ..
:
CAdk −1 We call Vl; k and Vr; k the generalized Vandermonde-type matrices for the pairs (Ac ; B) and (C; Ad ), respectively. In the case when p=q=1 and all zi =yi =1, they degenerate to a classical Vandermonde matrix and the transpose of a classical Vandermonde matrix (not necessarily square), respectively.
2. Basic lemmas In this section, we will lay out some known results about the LTLSI with N = np and the RTLSI with M = mq, which are basic in our analysis for a direct generalization of the classical Hermite formula. The reader can be referred to [1,6,7,10] for details. The construction of the minimal possible McMillan degree matrix polynomial for the homogeneous LTLSI/RTLSI for more general situation (in that, the integers N and M are not necessarily the multiples of the numbers of rows p and columns q of the unknown matrix functions F(z), respectively) and its connection with the Hermite formula given in Section 3 are considered simply in Section 4 (see also [2]).
348
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
Lemma 2.1 (Gohberg [6]). Suppose that N = nq and Vl; n is nonsingular, and Ul 0 . p ×p X1 = ); X2 = (e1 ; : : : ; eN ) (ei ∈ Cp×1 ) .. (Uls ∈ C U l ; n− 1 are the unique solutions of the matrix equations Vl; n X = Anc B and XVl; n = (0; : : : ; 0; Ip ), respectively. (1) There exists a monic p × p matrix polynomial Ul (z) of polynomial degree n satisfying zi Ul (ci ) = 0
(i = 1; 2; : : : ; N ):
(2.1)
Moreover, such a Ul (z) is unique and can be formulated as Ul (z) = z n Ip − z n−1 Ul; n−1 − · · · − zUl 1 − Ul 0 : (2) det Ul (z) = (z − c1 )(z − c2 ) · · · (z − cN ); rank Ul (ci ) = p − 1, and for each z = ci (1 6 i 6 N ), N ei zi Ul−1 (z) = X2 (zIN − Ac )−1 B = : (2.2) z − ci i=1 (3) A p × q matrix polynomial F(z) solves the homogeneous LTLSI : zi F(ci ) = 0 (1 6 i 6 N ) if and only if it is of the form: F(z) = Ul (z)L(z) for some p × q matrix polynomial L(z). In the terminology of [1], by this time the pair (Ac ; B) is a left null pair of the matrix polynomial Ul (z), and (Ac ; B; X2 ) is a left null triple of the same Ul (z). In the case, when p = 1 and all zi = 1, Ul (z) in Lemma 2.1 is identical with the annihilator polynomial ul (z), of degree N , of the interpolation nodes c1 ; : : : ; cN ul (z) = (z − c1 )(z − c2 ) · · · (z − cN ):
Lemma 2.2 (Hu and Zhang [10] and Heinig and Al-Musallam [7]). Let N = np and Vl; n be nonsingular, and Ul (z), ei (1 6 i 6 N ) be given as in Lemma 2.1. Then Ul (ci )ei = 0;
zi Ul (ci )ei = 1
(i = 1; 2; : : : ; N ):
(2.3)
Moreover, there is no other column vector ei in Cp×1 satisfying Eq. (2.3). By using Ul (z), ei (1 6 i 6 N ), we obtain in [10] a generalization of the classical Lagrange formula to the setting of the LTLSI with N = np N Ul (z)ei wi Fl (z) = : (2.4) z − ci i=1 The reason is that, in the case of p = q = 1 and all zi = 1, 1 (i = 1; 2; : : : ; N ); ei = ul (ci )
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
349
then Eq. (2.4) has the form Fl (z) =
N i=1
ul (z) wi ; ul (ci )(z − ci )
which coincides with the classical Lagrange formula. The following Lemmas 2.3 and 2.4 for the RTLSI with M = mq are counterparts of Lemmas 2.1 and 2.2 for the LTLSI with N = np, respectively. Lemma 2.3 (Gohberg et al. [6]). Suppose that M = mq and Vr; m is nonsingular, and f1 . 1× q X3 = (Ur0 ; : : : ; Ur; m−1 ) (Urt ∈ Cq×q ); X4 = .. (fj ∈ C ) fmq are the unique solutions to the matrix equations XVr; m = CAmd and Vr; m X = ( I0q ), respectively. (1) There exists a monic q × q matrix polynomial Ur (z) of polynomial degree m satisfying Ur (dj )yj = 0
(j = 1; 2; : : : ; M ):
(2.5)
Moreover, such a Ur (z) is unique and can be formulated as Ur (z) = z m Iq − z m−1 Ur; m−1 − · · · − zUr1 − Ur0 : (2) det Ur (z) = (z − d1 )(z − d2 ) · · · (z − dM ); rank Ur (dj ) = q − 1, and for each z = dj (1 6 j 6 M ), Ur−1 (z) = C(zIM − Ad )−1 X4 =
M yj fj : z − d j j=1
(2.6)
(3) A p × q matrix polynomial F(z) solves the homogeneous RTLSI: F(dj )yj = 0 (1 6 j 6 M ) if and only if it is of the form: F(z) = R(z)Ur (z) for some p × q matrix polynomial R(z). In the terminology of [1], by this time the pair (C; Ad ) is a right null pair of the matrix polynomial Ur (z), and (C; Ad ; X4 ) is a right null triple of the same Ur (z). In the case of q =1 and all yi =1, Ur (z) in Lemma 2.3 is identical with the annihilator polynomial ur (z), of degree M , of the interpolation nodes d1 ; : : : ; dM ur (z) = (z − d1 )(z − d2 ) · · · (z − dM ):
Lemma 2.4 (Hu and Zhang [10] and Heinig and Al-Musallam [7]). Let M = mq and Vr; m be nonsingular, and Ur (z), fj (1 6 j 6 M ) be given as in Lemma 2.3. Then fj Ur (dj ) = 0;
fj Ur (dj )yj = 1
(j = 1; 2; : : : ; M ):
Moreover, there is no other 1 × q row vector fj satisfying Eq. (2.7).
(2.7)
350
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
With Ur (z), fj (1 6 j 6 M ) as above, we obtain in [10] a generalized Lagrange formula for the RTLSI with M = mq M vj fj Ur (z)
Fr (z) =
j=1
z − dj
:
(2.8)
Similarly, in the case when p = q = 1 and all yj = 1, fj =
1 ur (dj )
(j = 1; 2; : : : ; M );
then Eq. (2.8) has the form Fr (z) =
M j=1
ur (z) vj ; ur (dj )(z − dj )
which coincides with the classical Lagrange formula. 3. Main results Let %ij = wi yj and &ij = zi vj (1 6 i 6 N; 1 6 j 6 M ). If F(z) is an arbitrary solution to the TSLSI, and ci = dj , then multiplication of Eq. (1.1) by yj and Eq. (1.2) by zi leads to %ij = &ij :
(3.1)
Thus that Eq. (3.1) holds whenever ci = dj is a necessary condition for the TSLSI to be solvable. Theorem 3.1. Let N = nq, M = mq, both Vl; n and Vr; m be nonsingular, and Ul (z), Ur (z), ei , fj be as in Section 2. If for all i; j such that ij = ci = dj Eq. (3.1) holds true, we have that (1) The system of linear equations x(Ur (ij ); Ur (ij )yj ) = (wi − %ij fj Ur (ij ); − 12 %ij fj Ur (ij )yj ) has a unique row-vector solution x. (2) The system of linear equations vj − %ij Ul (ij )ej Ul (ij ) x= zi Ul (ij ) − 12 %ij zi Ul (ij )ej
(3.2)
(3.3)
has a unique column-vector solution x. Proof. We show assertion (1) only. Assertion (2) can be proved in a similar way. According to the theory of linear equations, it suLces to verify that Ur (ij )yj Ur (ij ) rank (Ur (ij ); Ur (ij )yj ) = rank = q: (3.4) wi − %ij fj Ur (ij ) − 12 %ij fj Ur (ij )yj
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
351
Suppose that there exists a 1 × q row vector x0 such that x0 (Ur (ij ); Ur (ij )yj ) = 0; or equivalently, x0 Ur (ij ) = 0;
x0 Ur (ij )yj = 0:
From Lemmas 2.3(2) and 2.4, we conclude that the general solution to the system of linear equations xUr (ij ) = 0 is of the form: kfj , where k is an arbitrary complex number, and therefore x0 = kfj , and kfj Ur (ij )yj = k = 0. Thus, x0 = kfj = 0. This implies that rank(Ur (ij ); Ur (ij )yj ) = q:
(3.5)
On the other hand, observe that yj is nonzero p × 1 column vector because of the nonsingularity of Vr; m , and that Eqs. (2.5) and (2.7) force Ur (ij )yj Ur (ij ) yj 0 = : 1 0 0 wi − %ij fj Ur (ij ) − 2 %ij fj Ur (ij )yj Then we have by Eq. (3.5) Ur (ij ) rank wi − %ij fj Ur (ij )
Ur (ij )yj − 12 %ij fj Ur (ij )yj
= q:
(3.6)
Combining Eq. (3.5) with Eq. (3.6), we get immediately Eq. (3.4). Now we pull together the results of Lemmas 2.1–2.4 and Theorem 3.1 to present a generalized Hermite formula for the setting of the TSLSI with N = np and M = mq. Theorem 3.2. Let N = np, M = mq and both Vl; n and Vr; m be nonsingular. If Eq. (3.1) holds true for all i; j such that ij = ci = dj , and w˜ ij ∈ C1×q , v˜ij ∈ p × 1 are the unique solutions to the systems of linear equations (3.2) and (3.3), respectively, then the formula
ij ei fj + ei w˜ ij + v˜ij fj %ij ei fj Fts (z) = Ur (z) Ul (z) + z − ij (z − ij )2 (i; j):ij =ci =dj
+
Ul (z)
i:ci =dj
v˜j fj ei w˜ i Ur (z) + Ul (z) Ur (z); z − ci z − dj
(3.7)
j:dj =ci
where w˜ i = wi Ur−1 (ci ) =
M %ij fj ; ci − d j j=1
v˜j = Ul−1 (dj )vj =
N &ij ei ; dj − c i i=1
(3.8)
provides a matrix polynomial solution of polynomial degree at most m + n − 1 with the TSLSI. Proof. It follows from Lemmas 2.3 and 2.4 that fj Ur (z) Ul (z)ei (i = 1; : : : ; N ); rj (z) := li (z) := z − dj z − ci
(j = 1; : : : ; M )
352
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
are p ×1 and 1×q vector-valued polynomials with polynomial degrees n−1 and m−1, respectively. Then Fts (z) de=ned via Eq. (3.7) can be rewritten as Fts (z) = (ij li (z)rj (z)(z − ij ) + li (z)w˜ ij Ur (z) + Ul (z)v˜ij rj (z) (i; j):ij =ci =dj
+%ij li (z)rj (z)) +
li (z)w˜ i Ur (z) +
i:ci =dj
Ul (z)v˜j rj (z);
j:dj =ci
so that Fts (z) is a p × q matrix polynomial with polynomial degree at most m + n − 1. Now we verify that Fts (z) satis=es Eqs. (1.1)–(1.3). (1) If i is such that ci = dj = ij for some j, then we have by Eqs. (2.1), (2.3), (2.7) and (3.2) that zi Fts (ij ) = lim zi Fts (z) z →ij
ij ei fj + ei w˜ ij + v˜ij fj %ij ei fj = lim zi Ul (z) + z →ij z − ij (z − ij )2
Ur (z)
= zi Ul (ij )ei w˜ ij Ur (ij ) + %ij zi Ul (ij )ei fj Ur (ij ) = w˜ ij Ur (ij ) + %ij fj Ur (ij ) = wi : Similarly, we can obtain that Fts (ij )yj = vj for all i; j such that ij = ci = dj . Furthermore, for such i; j, we have Fts (z) − Fts (ij ) yj z →ij z − ij
ij ei fj + ei w˜ ij + v˜ij fj %ij ei fj %ij Ur (z)yj − = lim zi Ul (z) + z →ij (z − ij )2 (z − ij )3 z − ij zi Ul (z)(ij ei fj + ei w˜ ij + v˜ij fj )Ur (z)yj = lim z →ij (z − ij )2
%ij zi Ul (z)ei fj Ur (z)yj − %ij (z − ij )2 : + (z − ij )3
zi Fts (ij )yj = lim zi
By Lemmas 2.1–2.4 and the L’Hospital principal, we have that lim
z →ij
zi Ul (z)(ij ei fj + ei w˜ ij + v˜ij fj )Ur (z)yj (z − ij )2 = lim zi Ul (z)(ij ei fj + ei w˜ ij + v˜ij fj )Ur (z)yj z →ij
=ij + w˜ ij Ur (ij )yj + zi Ul (ij )v˜ij
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
353
and lim
z →ij
%ij zi Ul (z)ei fj Ur (z)yj − %ij (z − ij )2 (z − ij )3 = lim
z →ij
1 %ij (zi Ul (z)ei fj Ur (z)yj + zi Ul (ij )ei fj Ur (z)yj ) 2
1 = %ij (zi Ul (ij )ei + fj Ur (ij )yj ): 2 From the last three equations, we obtain by Eqs. (3.2) and (3.3) that zi Fts (ij )yj = ij + w˜ ij Ur (ij )yj + zi Ul (ij )v˜ij 1 + %ij (zi Ul (ij )ei + fj Ur (ij )yj ) 2 = ij : (2) If i is such that ci is not equal to any dj , we have by Eqs. (3.7) and (3.8) that ei w˜ i zi Fts (ci ) = lim zi Fts (z) = lim zi Ul (z) Ur (z) z →ci z →ij z − ci = zi Ul (ci )ei w˜ i Ur (ci ) = w˜ i Ur (ci ) = wi : (3) If j is such that dj is not equal to any ci , we have that v˜j fj Fts (dj )yj = lim Fts (z)yj = lim Ul (z) Ur (z)yj z →dj z →dj z − dj = Ul (dj )v˜j fj Ur (dj )fj = Ul (dj )v˜j = vj : Thus we have shown that Fts (z) satis=es Eqs. (1.1)–(1.3). The proof is completed. In the special case when p = q = 1, m = n and all ci = di , zi = yi = 1 (1 6 i 6 n), we check easily that Ul (z) = Ur (z) = ul (z) = ur (z), and both Eqs. (3.2) and (3.3) have the same solution w˜ ii = v˜ii = −
wi ul (ci ) 2ul (ci )
(i = 1; 2; : : : ; N );
so that Fts (z) de=ned by Eq. (3.7) admits of the form
N ul2 (z) ii − wi u (ci ) wi Fts (z) = + (ul (ci ))2 z − ci (z − ci )2 i=1 =
N
l2i (z)((ii − 2wi li (ci ))(z − ci ) + wi )
i=1
=
N i=1
ii (z −
ci )l2i (z)
+
N i=1
wi (1 − 2li (ci )(z − ci ))l2i (z);
(3.9)
354
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
in which li (z) =
ul (z) ul (ci )(z − ci )
(i = 1; 2; : : : ; N ):
In this case, Eq. (3.9) is identical with the classical Hermite formula for the scalar polynomial interpolation. Corollary 3.3. Let N = np, M = mq and both Vl; n and Vr; m be nonsingular. Then the TSLSI is solvable, if, and only if, Eq. (3.1) holds true whenever ci = dj . In particular, if the set of points {c1 ; c2 ; : : : ; cN } is disjoint from the set {d1 ; d2 ; : : : ; dM }, then the TSLSI is always solvable, and its Hermite formula (3.7) is identical with the Lagrange’s formula for the TSLSI, namely, formula (3.7) but with the =rst term on the right absent. Theorem 3.4. Let N = np, M = mq, and both Vl; n and Vr; m be nonsingular. There is at most one p × q matrix polynomial of polynomial degree at most m + n − 1, which solves the TSLSI. Proof. Suppose that there are two matrix polynomials F1 (z) and F2 (z), with polynomials degrees at most m + n − 1, which solve simultaneously the TSLSI. Let G(z) = F1 (z) − F2 (z): It remains now only to prove G(z) = 0. Note that G(z) is a matrix polynomial with polynomial degree at most m + n − 1, satisfying zi G(ci ) = 0 G(dj )yj = 0
(i = 1; 2; : : : ; N ); (j = 1; 2; : : : ; M );
(3.10) (3.11)
and for all i; j such that ij = ci = dj , zi G (ij )yj = 0:
(3.12)
From Eq. (3.11) and Lemma 2.3(3), we see that G(z) has to be of the form G(z) = R(z)Ur (z);
(3.13)
for some p × q matrix polynomial R(z). Inserting Eq. (3.13) into Eq. (3.10), we obtain zi R(ci )Ur (ci ) = 0
(i = 1; 2; : : : ; N ):
In the case of ci = dj (1 6 j 6 mq), Ur (ci ) is nonsingular, then zi R(ci )=0. In the case of ij =ci =dj for some j, zi R(ci )Ur (ij ) = 0. By Lemmas 2.3 and 2.4, we have that zi R(ci ) = kfj
(i = 1; 2; : : : ; N )
for some complex number k. By Eqs. (3.11), (3.13) and (3.14), we have 0 = zi G (ij )yj = zi (R (ij )Ur (ij ) + R(ij )Ur (ij ))yj = kfj Ur (ij )yj = k:
(3.14)
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
355
This implies that zi R(ci ) = kfj = 0. By Lemma 2.1(3), we have that R(z) = Ul (z)L(z) for some p × q matrix polynomial L(z). Then G(z) = Ul (z)L(z)Ur (z) for the same L(z). Note that the facts that Ul (z) and Ur (z) are monic matrix polynomials and deg Ul (z) = n, deg Ur (z) = m, then force L(z) = 0 and thus G(z) = 0, as required. Theorem 3.4 combined with Theorem 3.2 means that Fts (z) de=ned by Eq. (3.7) is the unique solution to the TSLSI for matrix polynomials with the polynomial degree at most m + n − 1. Making use of Ul (z), Ur (z) and Fts (z), we obtain in turn a description of the set of all matrix polynomial solutions to the TSLSI (than which for a more general result see [2, Theorem 6.1]). Theorem 3.5. Suppose that N = np, M = mq and both Vl; n and Vr; m are nonsingular. Then the general matrix polynomial solution to the solvable TSLSI can be formulated in the following divisor-remainder form: F(z) = Fts (z) + Ul (z)M (z)Ur (z);
(3.15)
where M (z) is an arbitrary p × q matrix polynomial. 4. Some extensions We devote this section to an extension of the main results to a more general situation, in that, the numbers of left and right interpolation nodes N and M of the TSLSI need not be the multiples of p and q, respectively. In that case, for convenience, we use the McMillan degree instead of the polynomial degree as measure of complexity for a given matrix polynomial. It is known that for a monic p × p matrix polynomial, the McMillan degree is p times of the polynomial degree. The pair (Ac ; B) is called a full-range pair if rank(Vl; i ) = N for some positive integer i, and (C; Ad ) is called a null-kernel pair if rank(Vr; j ) = M for some positive integer j. To simplify the consideration, we always assume in this section that ci ; dj = 0 for all i; j. This has the consequence that Ac and Ad are nonsingular matrices. According to [2], for the homogeneous LTLSI we have the following: Theorem 4.1 (Ball and Kang [2]). For a given full-range pair (Ac ; B), there is a comonic p × p matrix polynomial Ul (z), of the minimal possible McMillan degree, which satis=es zi Ul (ci ) = 0;
i = 1; 2; : : : ; N
(4.1)
and has the form 1 Ul (z) = Ip + zE(IN − zT∞ )−1 A− c B
(4.2)
with inverse given by Ul−1 (z) = Ip + zE(zIN − Ac )−1 B for an appropriate pair of matrices T∞ (nilpotent) and E.
(4.3)
356
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
We omit the procedure for constructing the matrices T∞ and E. The reader is referred to [2, Section 2.2] for details. In general, such a Ul (z) is far from unique, and cannot be taken to be monic. But for the complete data case: N = np and Vl; n is invertible, choosing 0 Ip .. . 0 −1 V ; E = −(Ip ; 0; : : : ; 0) V −1 ; T∞ = Vl; n l; n l; n .. . I p 0 we check easily that each Ul (z) in Theorem 4.1 degenerates exactly to one and the same Ul (z) given in Lemma 2.1, up to a right factor of nonsingular constant matrix. Now putting E = (e1 ; e2 ; : : : ; eN ) by Eq. (4.3) we have Ul−1 (z) = (Ip + EAc B) +
N ei zi : z − ci i=1
(4.4)
With notations above and using a Euclidean algorithm for matrix polynomials with respect to McMillan degree established in [2], we can prove that each Ul (z) in Eq. (4.2) enjoys the following properties. Corollary 4.2. Let (Ac ; B) be a full-range pair, and Ul (z), ei be as in Eqs. (4.2) and (4.4), respectively. (1) det Ul (z) (viewed as a scalar polynomial in z) has only the simple zeros at c1 ; c2 ; : : : ; cN . (2) For each i, rank Ul (ci ) = p − 1 and Ul (ci )ei = 0;
zi Ul (ci )ei = 1; i = 1; 2; : : : ; N:
(4.5)
Moreover, there is no other column vector ei in Cp×1 satisfying Eq. (4.5). (3) A p × q matrix polynomial F(z) satisfying the homogenous LTLSI: zi F(ci ) = 0 (1 6 i 6 N ) if and only if it is of the form: F(z) = Ul (z)L(z) for some p × q matrix polynomial L(z). The following Theorem 4.3 and Corollary 4.4 for the homogeneous RTLSI are parallel with Theorem 4.1 and Corollary 4.2 for the homogeneous LTLSI, respectively. Theorem 4.3 (Ball and Kang [2]). For a given null-kernel pair (C; Ad ), there is a comonic q × q matrix polynomial Ur (z), of the minimal possible McMillan degree, which satis=es Ur (dj )yj = 0;
j = 1; 2; : : : ; M
(4.6)
and has the form 1 −1 Ur (z) = Iq + zCA− d (IM − zS∞ ) F
(4.7)
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
357
with the inverse given by 1 −1 Ur−1 (z) = Iq + zCA− d (zIM − Ad ) F
(4.8)
for an appropriate pair of matrices S∞ (nilpotent) and F. As Ul (z) in Theorem 4.1, Ur (z) is generally far from unique and cannot be taken to be monic. But for the special case: M = mq and Vr; m is invertible, choosing 0 Iq Iq 0 0 1 −1 V S∞ = Vr− ; F = −V . ; r ; m ;m r ; m .. .. .. . . Iq
0
0
we can verify that each Ur (z) in Theorem 4.2 degenerates to one and the same Ur (z) given in Lemma 2.3, up to a left factor of nonsingular constant matrix. We now partition F in Eq. (4.7) into the following form: f1 f2 F = . ; fj ∈ C1×q : .. fM By Eq. (4.8) we have Ur−1 (z)
M fj yj = (Ip + CAd F) + : z − dj i=1
(4.9)
Corollary 4.4. Let (C; Ad ) be a null-kernel pair, and Ur (z) and fj be as in Eqs. (4.6) and (4.9), respectively. (1) det Ur (z) has only the simple zeros at d1 ; d2 ; : : : ; dM . (2) For each j, rank Ur (dj ) = p − 1 and fj Ur (dj ) = 0;
fj Ur (dj )yj = 1; j = 1; 2; : : : ; M:
(4.10)
1× q
satisfying Eq. (4.10). Moreover, there is no other row vector fj in C (3) A p×q matrix polynomial F(z) satisfying the homogeneous RTLSI: F(dj )yj =0 (1 6 j 6 M ) if and only if it is of the form: F(z) = R(z)Ur (z) for some p × q matrix polynomial R(z). Applying arguments similar to that given in Section 3, one can obtain some extensions of the main results. Theorem 4.5. Let (Ac ; B) be a full-range pair and (C; Ad ) be a null-kernel pair. Then the statement of Theorem 3.1 holds also if Ul (z); Ur (z); ei ; fj therein are placed with that de=ned via Eqs. (4.2), (4.4), (4.7) and (4.9), respectively.
358
Y.-J. Hu et al. / Journal of Computational and Applied Mathematics 173 (2005) 345 – 358
By Theorem 4.5, we have in turn the following Theorem 4.6. Let (Ac ; B) be a full-range pair and (C; Ad ) be a null-kernel pair. Then formula (3.7) provides a matrix polynomial solution of McMillan degree at most M + N − 1 with the TSLSI, in which Ul (z), Ur (z); ei ; fj are as in Theorem 4.5. Moreover, the general matrix polynomial solution to the TSLSI can be formulated by Eq. (3.15), where M (z) is an arbitrary p×q matrix polynomial. It is worthwhile to point out that such matrix polynomial solution Fst of the minimal McMillan degree is far from unique in the generic case. Acknowledgements The authors acknowledge with many thanks helpful comments and suggestions of the referee, which encourage us to consider an extension of the main results given in Section 3 of the paper to the incomplete interpolation data case. References [1] J.A. Ball, I. Gohberg, L. Rodman, Interpolation of Rational Matrix Functions, Birkhauser, Basel, 1990. [2] J.A. Ball, J. Kang, Matrix polynomial solutions of tangential Lagrange–Sylvester interpolation conditions of low McMillan Degree, Linear Algebra Appl. 137/138 (1990) 699–746. [3] G.-N. Chen, Y.-J. Hu, The Nevanlinna–Pick interpolation problems and power moment problems for matrix-valued functions III: the in=nitely many data case, Linear Algebra Appl. 306 (2000) 59–87. [4] G.-N. Chen, Y.-J. Hu, Multiple Nevanlinna–Pick interpolation with both interior and boundary data and its connection with the power moment problem, Linear Algebra Appl. 323 (2001) 167–194. [5] G.-N. Chen, X.-Q. Li, The Nevanlinna–Pick interpolation problems and power moment problems for the matrix-valued functions, Linear Algebra Appl. 288 (1999) 123–248. [6] I. Gohberg, P. Lancaster, L. Rodman, Matrix Polynomials, Academic Press, New York, 1982. [7] G. Heinig, F. Al-Musallam, Lagrange formula for tangential interpolation with application to structured matrices, Integral Equations Operator Theory 30 (1998) 83–100. [8] G. Heinig, F. Al-Musallam, Hermite formula for vector polynomial interpolation with applications to structured matrices, Appl. Anal. 70 (1999) 331–346. [9] Y.-J. Hu, Z.-H. Yang, G.-N. Chen, Tangential Nevanlinna–Pick interpolation and its connection with Hamburger matrix moment problem, Integral Equations Operator Theory, to appear. [10] Y.-J. Hu, X.-N. Zhang, Lagrange’s formula for bi-tangential interpolation of matrix polynomials, J. Beijing Normal Univ. Natur. Sci. 38 (4) (2002) 427–431.