CC BY
17
MSC 47A68 DOI: 10.14529/mmp230104
NOTE ON EXACT FACTORIZATION ALGORITHM FOR MATRIX POLYNOMIALS
V.M. Adukov1, N. V. Adukova1'2, G. Mishuris2
1 South Ural State University, Chelyabinsk, Russian Federation 2Aberystwyth University, Aberystwyth, United Kingdom
E-mail: [email protected], [email protected], [email protected], [email protected]
There are two major obstacles for a wide utilisation of the Wiener-Hopf factorization technique for matrix functions used to solve vectorial Riemann boundary problems. The first one reflects the absence of a general explicit factorization method in the matrix case, even though there are some explicit (constructive) factorizations available for specific classes of matrix functions. The second obstacle follows from the fact that the factorization of a matrix function is, generally speaking, not stable operation with respect to a small perturbation of the original function. As a result of the latter, a realisation of any constructive algorithm, even if it exists for the given matrix function, cannot be performed in practice. Moreover, developing explicit methods, authors do not often analyze its numerical implementation, implicitly assuming that all steps of the proposed constructive algorithm can be carried out exactly. In the proposed work, we continue studying a relation between the explicit and exact solutions of the factorization problem in the class of matrix polynomials. The main goal is to obtain an algorithm for the exact evaluation of the so-called indices and essential polynomials of a finite sequence of matrices. This is the cornerstone of the problem of exact factorization of matrix polynomials.
Keywords: Wiener-Hopf factorization; toeplitz matrices; essential polynomials of sequence.
Introduction
Let A(t) be a matrix function from the matrix Wiener algebra Wpxp(T) that is invertible on the unit circle T. The representation
A(t) = A+ (t)D(t)A- (t), t G T, (1)
is called a left Wiener-Hopf factorization of A(t). Here A± (t) belong to the group GW±xp(T) of invertible elements of the subalgebra W±xp(T), the middle factor D(t) is the diagonal matrix D(t) = diag [tAl,..., tAp], where integers Ai > ... > Ap are the left partial indices of A(t). The relation A1 + ... + Ap = k = indT det A(z) is valid. A similar representation in which the factors A± are rearranged is called the right Wiener-Hopf factorization.
Mathematical modelling of wave diffraction, problems of dynamic elasticity and fracture mechanics, and geophysical problems are often reduced to the Wiener-Hopf factorization problem for matrix functions [1-4]. The factorization of matrix functions is also a powerful tool itself used in various areas of mathematics [5-7,9].
Unfortunately, for the matrix case, there is no constrictive solution of the factorization problem in a general setting and it is very important to find cases when the problem can be solved effectively or explicitly. By the explicit (or constructive) solution of the factorization problem we understand a clearly defined algorithmic procedure that should definitely terminate after a finite number of steps. There are not that many classes of matrix functions for which an explicit solution to factorization problem has been found.
The most important of them are classes of matrix polynomials [10,11] and meromorphic matrix functions [12]. A detailed review of constructive methods for the factorization problem is presented in the works [13-15].
In addition to the aforementioned lack of availability of explicit solution to the factorization problem, in the general case, there is another obstacle to use the technique. This is possible instability of the factorization problem. Even if an explicit method for solving the particular factorization problem exists, each step of the respective algorithm can be executed exactly or approximately (numerically).
We say that the problem can be solved exactly if (i) the input data belonging to the Gaussian field Q(i) of complex rational numbers, and (ii) all steps of the explicit algorithm can be perform in the exact arithmetic. The instability of the problem leads to the fact that the explicit algorithm cannot be implemented numerically. As a rule, researchers developing a particular explicit factorization method usually ignore this issue. In fact, they implicitly assume that all steps of the proposed explicit algorithm can be carried out exactly that, unfortunately, is not always possible.
For the first time, the need to accurate study the way of numerical implementation of the explicit algorithm was highlighted in [16]. This has been done for matrix polynomials, where existence of an explicit solution of the factorization problem was proved in [11]. In [16], based on this work, a criterion for the exact factorizability of a matrix polynomial was obtained, and an exact algorithm for a solution of the factorization problem was developed. This algorithm was also implemented as the package ExactMPF in Maple. Thus, if the condition satisfies, the problem of an instability does not arise.
The package makes it easy to carry out numerical experiments with the Wiener-Hopf factorization for matrix polynomials. It can be used to construct an approximate canonical factorization with quaranteed accuracy for strictly nonsingular 2 x 2 matrix functions and to the integration of a discrete analog of the nonlinear Schrodinger equation by Inverse Scattering Transform. We hope that the application of the package will not be exhausted by these examples.
This paper is complimentary to [16], where the length was limited by the publisher rules. As a result, some crucial technical results have been omitted there. In particular, the algorithm for constructing essential polynomials was not described. In this work we fill this gap.
1. Explicit Solution of the Factorization Problems for Matrix Polynomials
In this section, we present an explicit algorithm for the factorization of an arbitrary matrix polynomial. Our presentation is based on the results from [11,12,16].
N
Supposed that the matrix polynomial a(z) = Y1 akzk, ak £ Cpxp, is invertible on the
k=0
unit circle T. We will write its left and right Wiener-Hopf factorizations in the form
a(t) = l+(t)dL(t)l-(t), a(t) = r-(t)dR(t)r+ (t), t £ T. (2)
Here dL(t) = diag[tXl ,...,tx* ], and X1 > ••• > Xp; dR(t) = diag[tp1 ,...,tp* ], where p1 < • • • < Pp. Note that left Xj and right pj indices are usually different sets of integers and constructions of the right and left factorizations are usually considered as two separate problems. For explicit construction of these factorizations we will use the method proposed in the work [11]. The method requires simultaneous considerations of the both factorizations.
Let A(z) = deta(z) and A(z) = A-(z)zKA+ (z), A-(œ) = 1, be the Wiener-Hopf factorization of A(z). The factorization is unique with the additional condition at infinity for the polynomial A-(z). In the sequel, we use, in fact, only one of the factors, namely, A-(z) = 1 + A-z-1 +-----+ A-z-K, k = indT det a(z).
We expand the rational matrix function A-1 (z)a(z) in the Laurent series at infinity: A-1(z)a(z) = cjzj• The coefficients Cj G Cpxp are computed recurrently in terms
of matrix coefficients aj of the original matrix polynomial, a(z), and the coefficients A-of the scalar polynomial, A-(z), (see [16, Eq.(2.5)]).
To construct the factorizations of a(z) we only need a finite number of the coefficients, ck, for k = — k, ..., 0, • • •, k. Denote c-k := {c-k,• • •, co,•• •, ck}. The main tools for computations of the partial indices and factors in the factorizations of matrix functions are the so-called indices and essential polynomials of the sequence c-k (see [11,17]). Let us define these notions.
Form a finite family of the block Toeplitz matrices of finite sizes:
Tk = ||ci-j||i=k,k+i,..,K , —k < k < k, (3)
j=0,1,...,K+k
and study the structure of the right kerfTk = {R G C( k+K+1)x1 |TkR = 0} and left kerL Tk = {L G C1x(K-k+1)|LTk = 0} kernels of Tk. Further it is more convenient to deal not with vectors R = (r0, r1, • • •, rk+K)T G kerf Tk, rj G Cpx1, but with their generating column-valued polynomials R(z) = r0 + r1 z + ■ ■ ■ + r k+K zk+K. We will use the spaces Nk of the generating polynomials instead of the spaces ker Tk.
By Nf, —k < k < k, we denote the space of generating vector polynomials for vectors in kerf Tk• Put NfK-1 := {0} and let Nf+1 be (2k + 2)p-dimensional space of all column-valued polynomials whose degrees are not greater than 2k + 1.
Repeating the same line of reasoning, we can define the space Nf, — k < k < k, of the row-valued generating polynomials in z-1 for the rows from kerL Tk.
By df, we denote a dimension of the right kernel Nf and introduce the following integers: Af = df — df-1 for —k < k < k + 1. A sequence c-k is called regular if A-k = 0 and Af+1 = 2p.
For a regular sequence, we have (see [11,17])
0 = A-K < ARK+1 < ■ ■ ■ < Af < Af+1 =
Since a monotone integer sequence is piecewise constant, then there are 2p integers < ■ ■ ■ < ^ 2p such that
AR = ■■■ = Af =0,
AS
AR
2p+1
ARi+i = (4)
= AR+1 = 2p.
The absence of the j-th row here means that +1 = .
Definition 1. The integers ... defined by the relations (4) are the indices of the sequence .
Similarly, we can consider the sequence of the left kernel Nf that will lead, however, to the same indices.
Furthermore, we define the right essential polynomials of the sequence . Note that Nf and zNR are subspaces of NR1 as it follows from the definition of the spaces Nf. The dimension, hf+1, of the complement H f+1 of Nf + zNf in NAf_1 is equal to Af+1 — Af.
Then, Eqs. (4) imply that hR+1 = 0 if and only if k = ^j, j = 1,..., 2p. Moreover, in this case, hR+1 is equal to the multiplicity, Kj, of the index ^j. Therefore, for k = ^j we have
NR+1 = NR + zNR,
and for k = ^j
NR+1 = (NR + zNR) ®H R+1.
Definition 2. Any polynomials Rj(z),..., Rj+Kj-1(z) forming a basis for a complement
H-j+1 are called right essential polynomials of the sequence c-K corresponding to the index; ■
As a result, we have defined 2p indices ^1,... ,^2p and 2p right essential polynomials R1(z),..., R2p(z) for any regular sequence c-K. Similarly, we can define the left essential polynomials L1 (z),..., L2p(z) of the sequence c-K.
In factorization problems, there are natural candidates for the role of indices and essential polynomials. To check that this is indeed the case, the following essentiality criterion can be used ( [17, Theorem 4.1], see also [11, p. 258],):
Theorem 1. The integers are the indices and R1(z),..., R2p (z) are right
essential polynomials of the regular sequence c-K if and only if the matrix
A
R
dR{z-K-1R1(z)} ••• d- {z-K-1 R2p(z)}
R1 (0) ••• R2p (0)
v-K-1 o z^M _ + 1 Jj)
is invertible. Here aR{z K 1Rj(z)} = cK+1-mrm'
By Theorem 3.1 from [11], the sequence c—K is regular and there exist respective essential polynomials R1 (z),..., R2p(z); L1 (z),..., L2p(z) such that
(i) the constant terms of the polynomials R1 (z),..., Rp(z) are equal to zero,
(ii) the leading terms of the polynomials Lp+1(z),..., L2p (z) are equal to zero.
Definition 3. The essential polynomials R1(z),..., R2p(z); L1 (z),..., L2p(z) satisfying the conditions (i), (ii) are called the factorization essential polynomials of the sequence.
Now we can formulate a final result on the explicit Wiener-Hopf factorization of a matrix polynomial a(z).
Theorem 2. [11, Theorem 3.2] Let be the indices and R1 (z),..., R2p(z)
(L1(z),..., L2p(z)) are the right (left) factorization essential polynomials of the regular sequence c-K. Let us introduce the p x p matrix functions
¡Lp+1(zY
R1(z) = (R1(z) ... Rp(z)), L2(z)= I .
V L2p (z)
and dL(z) = diag[z-^1,..., z-^p], dR(z) = diag[z^p+1,..., z^2p].
Then the left (X1 > • • • > Xp) and right (p1 < • • • < pp) partial indices and the factors (l±(z), r±(z)) of the respective factorizations of the matrix polynomial a(z) are defined by the formulas
= = P1 = №p+1,. — ,Pp = №2p, (5)
l-(z) = zK+1A-(z)d-1 (z)R-1 (z), l+ (t) = z-K-1 A-1 (z)a(z)R (z), (6)
r- (z) = A-(z)L-1 (z), r+(z) = A-1(z)d-R1 (z)L2(z)a(z). (7)
In the statement of this theorem, we have corrected the misprints appeared in the formulas for the factors l+ (z), r+ (z) in [11, Theorem 3.2].
Let us list the basic steps of the presented factorization algorithm.
1. Calculation of the Laurent coefficients cj, — k < j < k, for the rational matrix functions A-1(z)a(z).
Here k = indT det a(z) is a number of zeros of det a(z) in open disc |z| < 1.
Finding k and constructing the factorization of scalar polynomial A(z) can be considered as explicit procedures. Now calculation of the Laurent coefficient cj using recurrence relations requires a finite number of operations.
2. Calculation of the indices for the sequence c-K.
To calculate the indices ... , ^2p it is needed to find ranks of the matrices , — k < j < k. We can do it by means of linear algebra in a finite number of steps.
3. Calculation of the essential polynomials for the sequence c-K.
For this it is necessary to find bases of the kerf,L , — k < j < k. We can do it by means of linear algebra in a finite number of steps.
4. Constructing the factorizations in accordance with Th.2. Now this step can be done in an explicit form.
Thus, in accordance with our understanding of the explicit solution of the factorization problem given above, the presented algorithm indeed belongs to this class.
2. Exact Solution of the Factorization Problems for Matrix Polynomials
In this section, we find a condition when the proposed explicit algorithm can be implemented numerically. Due to the instability of the factorization problem, there are two obstacles for doing this.
1. The factorization of scalar polynomial A(z), in general case, can only be constructed approximately.
2. Finding the indices and essential polynomials of the sequence c-K requires calculating ranks and constructing bases of kernels for matrices . Unfortunately, those operations can be unstable.
Thus, in general, the proposed explicit factorization algorithm can not be implemented numerically.
Remark 1. A numerical implementation of the algorithm proposed in [10] meets into the same difficulties.
However, there is still a possibility to implement the algorithm exactly by utilising calculations in rational arithmetic. Obviously, we must demand that the coefficients aj of the original matrix polynomial a(z) must belong the Gaussian field Q(i) and the factorization of A(z) should be performed exactly. In this case the calculations of the Laurent coefficients Cj and finding the indices ... , ^2p can also be made exactly.
Now we have to make sure that finding the factorization essential polynomials can also be performed exactly. This was not done in [16] and it is the main goal of this work.
In the following theorem we describe the algorithm of finding these essential polynomials and prove that this algorithm can be implemented in the exact arithmetic if entries of the matrices Cj, — k < j < k, belong to the field Q(i).
Theorem 3. Let c—K := {c-K,..., co,..., cK} be a regular sequence of complex p x p matrices with entries from the field Q(i). Suppose that the indices ^1,...,^2p of the
sequence satisfy the condition ^j = —K,J2j=p+i ^j = K and the sequence has the factorization essential polynomials. Then these polynomials can be found by calculations in the exact arithmetic.
Proof. Let us restrict ourselves to considering only the right essential polynomials R1(z),..., Rjp(z). Algorithm of finding the factorization essential polynomials is based on the criterion of essentiality (Th. 1).
By Definition 3, these polynomials have zero constant terms: R1(0) = ■ ■ ■ = Rp(0) = 0. Hence, to construct the factorization essential polynomials R1(z),..., Rp(z) we must select first p vector polynomials Rj(z) G NR.+1, j = 1,... ,p, such that the 2p x 2p matrix
A
R
Or{Z-*-1rx{Z)} ■ • aR{z-*~lRp(z)} oR{z-*~lRp+l(z)} ■ ■ aR{z~*-lR2p(z)}
0 0 Rp+l{0) R2P(0)
will be invertible or, in other words, the p x p submatrices
All = (Rr {z-K-1Ri (z)}---;Rr {z-K-1 Rp (z)}) , Ajj = (Rp+i(0) ■■■ Rjp(0)) will be invertible.
Now it will be convenient to introduce the distinct indices and to assign them the respective multiplicities. Moreover, it is necessary to highlight the border index ^p. Let v1 < ■ ■ ■ < vs be the distinct indices of the sequence c-K and K1,..., Ks their multiplicities. Let the index ^p coincides with vt.
We will use induction by a number of indices v1 ,...,vt. First we select the factorization essential polynomials R1 (z),..., RKl (z) corresponding to the first index v1 of ^-multiplicity. They are the generating polynomials of vectors forming a basis of ker TVl+1. Since there are the factorization essential polynomials, there exist a basis of NVl+1 = ker TVl+1 such that R1(z) = zR1 (z),..., RK1 (z) = zRKl (z) for some polynonials R1(z),..., RKl(z) from the space JRVl+1 = ker TVl+1, where the matrix TVl+1 is obtained from TVl+1 by deleting of the first p columns, i.e. by deleting the first block column of the matrix. It is easily seen that corresponding vectors R1,..., RKl form a basis ker TVl+1. Thus, by virtue the essentiality criterion (Th.1) there exists a basis of ker TVl+1 such that the 2p x k1 submatrix
&r {z-K-1R1 (z)} ■■■ aR {z-k-1Rki (z)} 0 ■■■ 0
of Ar has the rank is equal to k1 . In fact, it is easy to show that this condition is fulfilled for any choice of a basis R1,..., RKl. Since the entries of the matrices Cj belong to Q(i), the construction of this basis and calculation of the rank can be done in the exact arithmetic.
Thus, we can exactly construct the first K1 polynomials R1 (z),..., RKl(z) such that R1(0) = 0,..., RKl (0) = 0, entries of the p x k1 matrix
(0Rr{z-k-1R1(z)}^^ Rr{z-k-1Rki (z)})
belong to Q(i), and this matrix has the rank equal to K1.
Now we repeat these considerations for the other indices v2,... ,vt. Assume first that
< ^p+1. Recall that vt = ^p and has the multiplicity Kt. In this case K1 + ■ ■ ■ + Kt coincides with the number of the indices ..., p,p, that is K1 + ■ ■ ■ + Kt = p.
Suppose that we construct the polynomials
R1 (z)^ . . , RKl (z); RKl+1(z), . . . , RKl+K2 (z); ... ; RKl+•••+кj-l+1(z), . . . , RKl+-+Kj (z)
corresponding to the indices vb...,vj, 2 < j < vt-1, such that Ri(0) = 0,..., RK1+^+Kj (0) = 0, entries of the p x (к1 + ■ ■ ■ + Kj) matrix
(ад{z-K-1 Ri(z)} ■ ■ ■ {z-K-1RK1+^+Kj(z)})
belong to Q(i), and this matrix has the rank equal to K1 + ■ ■ ■ + Kj.
Let us define the polynomials RK1+_+Kj+1(z),..., R^+^+k^ (z) corresponding to the index Vj+1 of the multiplicity Kj+1. These polynomials belong to the space NVj+1+1 =
ker TVj.+1+1. Let nj+1 is the dimension of the space NVVj+1+1 = ker TVj+1+1 and the polynomials Q1(z),..., Qnj+1 (z) be a basis of this space. Here TVj+1+1 is obtained from Tv.+1+1 by deleting the first block column of the matrix. Hence, zQ1(z),..., zQnj+1 (z) is a basis of the space NVj+1+1.
From this basis , we consequently select the required polynomials. Such a selection we will do consequently. First we choose among zQ1(z),..., zQnj+1 (z) a polynomial RK1+^+Kj+1(z) such that the matrix
(ад{z-K-1 R1(z)} ■ ■ ■ aR{z-K-1RK1+^^^+Kj (z) ад{z-K-1RK1+^+Kj+1(z)})
has the rank equal to K1 + ■ ■ ■ + Kj + 1. This selection is always possible since the sequence c-K has the factorization essential polynomials. In a similar way, we select the other polynomials RK1+_+Kj+2(z),..., (z) for which the matrix
(ад {z-K-1R1(z)}--- aR{z-K-1RK1+^^^+Kj+1 (z)})
has the rank equal to K1 + ■ ■ ■ + Kj+1.
Hence, in the case of к1 + ■ ■ ■ + Kt = p , we obtain, by induction, the polynomials R1(z),..., Rp(z), for which the matrix
Л11 = (ад{z-K-1R1(z)} ■ ■ ■ CTr{z-k-1Rp(z)})
over Q(i) has the rank equal to p. Thus, in this case, the first p factorization essential polynomials R1(z),..., Rp(z) are exactly constructed.
Now, we build the polynomials Rp+1 (z),..., R2p(z) such that the matrix Л22 is invertible. These polynomials must be sequentially chosen from the spaces Nj+1 = ker Tj+1, j = vt+1,..., vs .It is clear that we can always choose polynomials Rp+1(z),..., Rp+Kt+1 (z) from the basis NVt+1+1, such that vectors Rp+1(0),..., Rp+Kt+1 (0) are linear independent. Otherwise, the sequence c-K would not have factorization essential polynomials. Repeating these arguments for the indices vt+2 ..., vs we arrive to polynomials Rp+1(z),..., R2p(z) for which the matrix Л22 is invertible. Therefore, in the case of < ^p+1, the right factorization essential polynomials can always be found by the exact computation.
Let us consider now the case when the border index satisfies the equality = ^p+1, or more precisely, when
< ... < ^p-l = Vt-1 < ^p-l+1 = ■ ■ ■ = ^p = ■ ■ ■ = ^p+m = Vt < ^p+m+1 < ... < ^2p
for some I > 0, m > 0. Then k1 + ■ ■ ■ + Kt-1 = p — /, Kt = I + m, Kt+1 + ■ ■ ■ + Ks = p — m and K1 + ■ ■ ■ + Kt = p + m > p. The right factorization polynomials R1 (z),..., RK1+_Kt-1 (z) corresponding to the indices v1,..., vt-1 we can construct as above. Recall that Kt is the number of the right essential polynomials RK1+_Kt_1+1(z),..., RK1+_Kt(z) corresponding to the index vt. They belong to the space NVt+1 = ker TVt+1. These polynomials are divided into two type. For the first I polynomials RK1+^^^Kt_1+1(z),..., Rp(z), the conditions
RKl+„ret_l+1(0) = 0,..., Rp(0) = 0 and the invertibility of the matrix An must be fulfilled.
The remaining m polynomials Rp+1(z),..., RKl+_____+Kt (z) G NVt+1 must be chosen so that
the vectors Rp+1(0),..., RKl+...+Kt (0) are linearly independent.
The first type polynomials we can construct as above by choosing successively l polynomials from a basis Q1(z),... ,Qnt+1 of the space JRVt+1 — kerTVt+1. The existence of the factorization essential polynomials guarantees that this process can be carried out.
The remaining m polynomials Rp+1(z),..., RKl+...+Kt (z) must be chosen from the elements of a basis of the space = ker TVt+1 in a way that the rank of the matrix
(Rp+1(0) ■ ■ ■ RKl+...+Kt (0)) is equal to m = k1 + ■ ■ ■ + Kt — p. It is again possible since the sequence c—K possesses the factorization essential polynomials.
By repeating this choice for the spaces NVj+1 — ker TVj+1, j = vt+1,... ,vs, we obtain the polynomials Rp+1(z),..., Rjp(z) for which the matrix Aj2 is invertible. Then, for the polynomials R1 (z),...,Rp (z), Rp+1(z),..., Rjp(z), the matrix AR is invertible and these polynomials are the right factorization essential polynomials. To evaluate these polynomials, we have solved block Toeplitz systems with the coefficients belonging to Q(i) and have found the ranks of matrices with entries from this field. All such operations can be performed exactly.
To obtain the left factorization essential polynomials, we can carry out similar construction with the sequence of left kernels of matrices Tk, —k < k < k, or can apply a conformance procedure (see [17], Def. 5.3). This procedure can be also fulfilled exactly. The conformance procedure that is used to construct the left factorization essential polynomials, is described in [17].
□
After finding the indices and factorization essential polynomials, we can exactly construct the Wiener-Hopf factorizations using the formulas (5) - (7).
3. Pseudo-Code for an Exact Constructing the Right Factorization Essential Polynomials
The full variant of the pseudo-code for the algorithm of simultaneous construction of the left and right factorizations is given in [18]. However, if only one type of the factorization is needed (for instance, the left factorization), using the full algorithm leads to a significant increasing in execution time. For this reason, in this section we give the pseudo-code for construction of the left factorization only. For simplicity, here we restrict ourselves to the case when ^p < ^p+1.
Algorithm. Indices and right factorization essential polynomials of a sequence
Input. The sequence c—x := {c-K,... ,c0 ..., cK}, Cj G <Qpxp(i).
Output. The indices ^1 ,...,^jp and the matrix of the right factorization essential polynomials, TZ\ := (Ri(z) ■ ■ ■ Rp(z)),_
1. find the distinct indices v1,...,vs, their multiplicities K1,...,Ks, form the indices
..., ^ jp, and the number t such that ^p = vt
2. find the polynomials R1(z),..., RKl (z) forming a basis of the space JJiVl+1 = ker TVl+1, define the matrix R1 := (zR1 (z) ■ ■ ■ zR,Kl (z))
3. form the matrix an := (aR{z-KT1(z)} ■ ■ ■ aR{z-KTKl(z)})
4. for j = 2,...,t do
10
5. find a basis Qi(z),..., Qn (z) of the space NVVj+1 — ker TVj+1
6. for k = 1,..., n do
7. form the matrix a2 := (a11 aR{z-K<Tk (z)})
8. if rank a2 = rank a11 + 1 then
9. an := a2 R := (r zQfc (z))
11. end if
12. end do
13. end do
14. if rank o11 = p then
15. print "The factorization essential polynomials were not constructed. The factorization process is interrupted"
16. stop
17. end if
/ zM ... 0 \
18. form D(z) = . ... . 1 .
V 0 ... /
19. return R1 (z)
Now by formulas (5), (6) we can construct the left factorization of a matrix polynomial.
4. Numerical Example
Based on the proposed algorithm, a procedure ExactFEP was developed, which is the main part of the ExactMPF package in Maple. The package is designed for the exact solution of the factorization problem for matrix polynomials. To access ExactMPF use the commands
> read("ExactMPF.txt");
> with(ExactMPF);
> with(LinearAlgebra);
To obtain the factorizations of a(z) we run the module SolverExactMPF with the argument a(z):
> lplus, dl, lminus, rminus, dr, rplus := SolverExactMPF(a):
The module SolverExactMPF returns the factors lplus, dl, lminus of the left factorization and the factors rminus, dr, rplus of the right factorization. Let us give an example of using this package.
Example 1. Consider
'36z2 + 17z - 14 z4 - z2 + 3z - 1 z + 10^
a(z) :=
0 0
z2 + 13z +15 0
1
The module SolverExactMPF gives in this case the following expression for the factors of a(z):
> lplus; dl; lminus;
2
z
36 z + 10 z4 - z2 + 3z - 1
1
z2 + 13z + 15 0
0 1 0 0 0 1
1 + JL__1— n
1 + 36z 18z2 u
0 0
0 1
The executing time is 0,500 seconds when computations were performed on a home desktop computer HP with Intel(R) Core(TM)i3-415T CPU, 3.00 GHz, 4G RAM, operating system Windows 10.
Acknowledgments. V.M. Adukov and N. V. Adukova were supported by funding from, RFBR grant no. 20-41-740024. G. Mishuris was supported by funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement EffectFact no. 101008140.
G. Mishuris is thankful to the Royal Society for the Wolfson Research Merit Award and to the Welsh Government for the Future Generation Industrial Fellowship.
z2 0 0
2
z
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Received December 8, 2022
УДК 517.544.8 БЭТ: 10.14529/mmp230104
ЗАМЕЧАНИЕ ОБ АЛГОРИТМЕ ТОЧНОЙ ФАКТОРИЗАЦИИ ДЛЯ МАТРИЧНЫХ МНОГОЧЛЕНОВ
В.М. Адуков1, Н.В. Адукова1'2, Г. Мишурис2
1Южно-Уральский государственный университет, г. Челябинск, Российская Федерация
2 Аберистуит университет, г. Аберистуит, Великобритания
Существуют два основных препятствия для широкого использования метода факторизации Винера - Хопфа для матриц-функций, используемых для решения векторных краевых задач Римана. Первое препятствие связано с отсутствием общего явного
метода факторизации в матричном случае, хотя для конкретных классов матричных функций могут существовать явные (конструктивные) методы факторизации. Второе препятствие является следствием того, что факторизация матриц-функций, вообще говоря, является неустойчивой по отношению к малому возмущению исходной функции. В результате последнего, реализация любого конструктивного алгоритма, даже если он существует для данной матрицы-функции, на практике не может быть осуществлена. Более того, разрабатывая явные методы, авторы часто не анализируют его численную реализацию, неявно предполагая, что все шаги предложенного конструктивного алгоритма могут быть выполнены точно. В предлагаемой работе мы продолжаем изучение связи между явным и точным решениями задачи факторизации в классе матричных многочленов. Основная цель - получить алгоритм точного вычисления так называемых индексов и существенных многочленов конечной последовательности матриц. Это краеугольный камень проблемы точной факторизации матричных многочленов.
Ключевые слова: факторизация Винера - Хопфа; теплицевы матрицы; существенные многочлены последовательности.
Виктор Михайлович Адуков, доктор физико-математических наук, ведущий научный сотрудник, кафедра математического анализа и методики преподавания математики, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Наталия Викторовна Адукова, кафедра математического анализа и методики преподавания математики, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected]; аспирант, кафедра математики, Аберистуит университет (г. Аберистуит, Великобритания), [email protected].
Геннадий Мишурис, профессор, кафедра математики, Аберистуит университет (г. Аберистуит, Великобритания), [email protected].
Поступила в редакцию 8 декабря 2022 г.
