**Advances in Linear Algebra & Matrix Theory**

Vol.05 No.01(2015), Article ID:53902,14 pages

10.4236/alamt.2015.51001

New Approach for the Inversion of Structured Matrices via Newton’s Iteration

Mohammad M. Tabanjeh

Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA, USA

Email: mtabanjeh@vsu.edu

Copyright © 2015 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

Received 14 January 2015; accepted 5 February 2015; published 10 February 2015

ABSTRACT

Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration rapidly refines a crude initial approximation to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this me- thod to structured matrices, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster compu- tations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the grow- th of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.

**Keywords:**

Newton Iteration, Structured Matrices, Superfast Algorithm, Displacement Operators, Matrix Inverse.

1. Introduction

Frequently, matrices encountered in practical computations that have some special structures which can be exploited to simplify the computations. In particular, computations with dense structured matrices are ubiquitous in sciences, communications and engineering. Exploitation of structure enables dramatic acceleration of the computations and a major decrease in memory space, but sometimes it also leads to numerical stability problems. The best-known classes of structured matrices are Toeplitz and Hankel matrices, but Cauchy and Vandermonde types are also quite popular. The computations with such matrices are widely applied in the areas of algebraic coding, control, signal processing, solution of partial differential equations and algebraic computing. For ex- ample, Toeplitz matrices arise in some major signal processing computations and Cauchy matrices appear in the study of integral equations and conformal mappings. The complexity of computations with dense structured matrices dramatically decreases in comparison with the general matrices, that is, from the order of words of storage space and arithmetic operations (ops) with in the best algorithms, to words of storage space and to ops (and sometimes to ops) (Table 1). The displacement rank for an Toeplitz and Hankel matrix is at most 2. A matrix is Toeplitz-like or Hankel-like if is small or bounded by a small constant independent of and. Those matrices can be represented by entries of its short displacement generators instead of entries which leads to more efficient storage of the entries in computer memory and much faster computations [1] . In this paper, we will focus our study on Newton’s iteration for computing the inverse of a structured matrix and we will analyze the resulting algorithms. This iteration is well-known for its numerically stability and faster convergence. Newton’s iteration

(1)

for matrix inversion was initially proposed by Schultz in 1933 and studied by Pan and others. The authors of [2] described quadratically convergent algorithms for the refinement of rough initial approximations to the inverse of Toeplitz and Toeplitz-like matrices and to the solutions of Toeplitz and Toeplitz linear systems of equations. The algorithms are based on the inversion formulas for Toeplitz matrices. The Cauchy-like case was studied in [3] . Since those matrices can be represented by their short generators, which allow faster com- putations based on the displacement operators tool, we can employ Newton’s iteration to compute the inverse of the input structured matrices. However, Newton’s iteration destroys the structure of the structured matrices when used. Therefore, in Section 5 we will study two compression techniques in details, namely, truncation of the smallest singular values of the displacement and the substitution of approximate inverses for the inverse matrix into its displacement expression to preserve the structure. Finally, each iteration step is reduced to two multiplications of structured matrices and each iteration is performed by using nearly linear time and optimal memory space which leads to superfast algorithm, especially if the input matrix is a well-conditioned structured matrix. Our main algorithm of Section 6 can be applied to more general classes of structured matrices. We will support our analysis with numerical experiments with Toeplitz matrices in Section 7.

2. Definition of Structured Matrices

Definition 1 A matrix is a Toeplitz matrix if for every pair of its entries and

. A matrix is a Hankel matrix if for every pair of its entries and

. For a given vector, the matrix of the form is called a Vander-

monde matrix. Given two vectors s and t such that for all i and, the matrix

is a Cauchy (generalized Hilbert) matrix where.

Example 1, , , and

are Teoplitz, Hankel, Vandermonde, and Cauchy matrices respectively that can be represented by the 5-dim vector, the 5-dim vector, the 3-dim vector, and the 3- dim vectors and respectively.

We refer the reader to [4] and [5] for more details on the basic properties and features of structured matrices.

3. Displacement Representation of Structured Matrices

The concept of displacement operators and displacement rank, initially introduced by Kailath, Kung, and Morf in 1979, and studied by other authors such as Bini and Pan, is a powerful tool for dealing with matrices with structure. The displacement rank approach was intended for more restricted use when it was initially introduced in [6] (also [7] ), namely, to measure how “close” to Toeplitz a given matrix is. Then the idea turned out to be even deeper and more powerful, thus it was developed, generalized and extended to other structured matrices. In this paper, we consider the most general and modern interpretation of the displacement of a matrix (see Definition 2). The main idea is, for a given structured matrix, we need to find an operator that transforms the matrix into a low rank matrix, such that one can easily recover from its image and operate with low rank matrices instead. Such operators that shift and scale the entries of the structured matrices turn out to be appropriate tools for introducing and defining the matrices of Toeplitz-like, Hankel-like, Vandermonde-like, and Cauchy-like types which we will introduce in Definition 4.

Definition 2 For any fixed field (say the complex field) and a fixed pair of operator matrices, we define the linear displacement operators of Sylvester type:

(2)

and Stein type:

. (3)

The image of the operator is called the displacement of the matrix. The operators of Sylvester and Stein types can be transformed easily into one another if at least one of the two associated operator matrices is nonsingular. This leads to the following theorem.

Theorem 1 if the operator matrix is nonsingular, and if the

operator matrix is nonsingular.

Proof., and

.

The operator matrices that we will use are the unit f-circulant matrix

, and the diagonal matrix. Note that

is unit lower triangular Toeplitz matrix.

is an f-circulant for a vector. For example,

is a lower triangular matrix.

Table 1. Parameters and flops (floating point arithmetic operations) count.

Example 2 (Toeplitz Matrices). Let. Then if we apply the displacement operator of Sylvester

type (2) to Toeplitz matrix using the shift operator matrices and, we will get:

.

This operator displaces the entries of the matrix so that the image can be represented by the first row and the last column. Furthermore, the image is a rank matrix

Notice that the operator matrices vary depending on the structure of the matrix. We use shift operator matrices (such as and for any scalar) for the structures of Toeplitz and Hankel type, scaling operator matrices (such as) for the Cauchy structure, and shift and scaling operator matrices for the structure of Vandermonde type. One can restrict the choices of to 0 and 1. However, other operator matrices may be used with other matrix structures (see Table 2).

Definition 3 For an structured matrix and an associated operator, or, the number is called the -rank or the displacement rank of the matrix.

The constant in Definition 3 usually remains relatively small as the size of grows large, that is, is

small relative to, say or as and grow large. In this case, we

call the matrix -like and having structure of type. Now let us recall the linear operators of (2) and (3) that map a matrix into its displacements:

(4)

and

(5)

where and are operator matrices of Table 2, and are ma- trices. Then the matrix pair is called a (nonunique) -generator (or displacement generator) of length for, (the -rank or the displacement rank of). If is an matrix and is small relative to and, then is said to have -structure or to be an -structured matrix. The displacement (as a matrix of small rank) can be represented with a short generator defined by only a small number of parameters. For instance, the singular value decomposition (SVD) of the matrix of size gives us its orthogonal generator that consists of two generator matrices and having orthogonal columns and of sizes and respectively, that is a total of entries. The matrix pair is called an -generator (or displacement generator) of length for. It is also a generator of length for. Note that the pair is nonunique for a fixed. In particular, for Toeplitz, Hankel, Vandermonde, and Cauchy matrices, the length of the associated generators (4) and (5) is as small as 1 or 2 for some appropriate choices of the operator matrices and of Table 2. The four classes of structured matrices are naturally extended to Toeplitz-like, Hankel-like, Vandermonde-like, and Cauchy- like matrices, for which is bounded by a fixed (not too large) constant.

Definition 4 An matrix is Toeplitz-like if is small relative to and if

Table 2. Operator Matrices for and.

is given with its -generator of length, where, , , and

for a fixed pair of scalars and. A matrix is Hankel-like if or is Toeplitz-

like where is the reflection matrix.

Therefore, the problems of solving Toeplitz-like and Hankel-like linear systems are reduced to each other.

Remark 1 In the case where the operator is associated with Toeplitz, Hankel, Vandermonde, or Cauchy matrices, we call an -like matrix Toeplitz-like, Hankel-like, Vandermonde-like, or Cauchy-like matrix, respectively, and we say that the matrix has the structure of Toeplitz, Hankel, Vandermonde, or Cauchy type respectively. To take advantage of the matrix structure, we are going to COMPRESS the structured input matrix via its short L-generators based on Theorem 2 below, then we will OPERATE with L-generators rather than with the matrix itself, and finally RECOVER the output from the computed short L-generators.

Definition 5 A linear operator is nonsingular if the matrix equation implies that.

Definition 6 We define the norm of a nonsingular linear operator and its inverse as follows:

and, for, or, and where

the supremum is taken over all matrices having a positive -rank of at most. In this case the condition

number of the operator is defined as

Theorem 2 [8] Let and be a pair of matrices, and, where and denote the -th columns of and respectively. Let. Then we have one of

the following Toeplitz type matrices:, where,.

, where,., where,., where,.

Theorem 3 Let M be a nonsingular matrix, then we have the following:

1).

2), if is a nonsingular matrix.

3), if is a nonsingular matrix.

Proof.

1)

2)

3)

Theorem 4 Let a pair of matrices G and form a -generator of length for a nonsingular matrix. Write and. Then.

Proof. By Theorem 3, , where and.

4. Newton’s Iteration for General Matrix Inversion

Given an nonsingular matrix and an initial approximation to its inverse that satisfies the bound for some positive and some fixed matrix norm, Newton’s iteration (1) can be written in the form

(6)

and rapidly improves the initial approximation to the inverse. can be supplied by either some computational methods such as preconditioning conjugate gradient method or by an exact solution algorithm that requires refinement due to rounding errors.

Lemma 1 Newton’s iteration (6) converges quadratically to the inverse of an nonsingular matrix.

Proof. From (6) , since, , then

. We define the error matrices

, and the residual matrices,. Since, then

and hence the norm of the residual matrices is bounded above by. This proves that

Newton’s iteration converges quadratically.

If is a given residual norm bound, the bound

(7)

is guaranteed in recursive steps (6), where stands for the smallest integer not exceeded by. (7) also implies that. Since, it follows that,

so that the matrix approximates with a relative error norm less than.

Remark 2 Each step of Newton’s iteration (6) involves the multiplication of by, the addition of 2 to all of the diagonal entries of the product, which takes additions, and the pre-multiplication of the resulting matrix by. The most costly steps are the two matrix multiplications; each step takes multiplications and additions for a pair of general input matrices and. Of course, this cost will be reduced if the input matrix is structured (see Section 5).

Example 3 (Newton’s iteration versus linear convergent schemes.) Let and. Then by quadratic convergence of Newton’s iteration, we only need 5 iterative steps of (6) to compute that satisfies the bound and hence. If we consider any linearly convergent scheme such as Jacobi’s or Gauss-Seidels’s iteration ([9] , p. 611) for the same input matrix, the norm of the error matrix of the computed approximation to decreases by factor 2 in each iteration step. The error norm decreases to below in iteration steps, so this will take 19 iteration steps to ensure the bound (7) for the same.

5. Newton’s Iteration for Structured Matrix Inversion

5.1. Newton-Structured Iteration

In Section 4, we showed that for a general input matrix, the computational cost of iteration (6) is quite high because matrix products must be computed at every step (see Remark 2). However, for structured matrices, matrix multiplication has a lower computational cost of flops for, provided that we operate with displacement generators of length and for the two input matrices and res- pectively. So if and are structured matrices, say Toeplitz or Toeplitz-like, the computations required for the Newton’s iteration can be carried out more efficiently. In general, for the four classes of structured matrices of Definition 1, we usually associate various linear displacement operators of Sylvester type and/or Stein type. In this case, Newton’s iteration can be efficiently applied and rapidly improves a rough initial approximation to the inverse, but also destroys the structure of the four classes of structured matrices. Therefore, Newton’s iteration has to be modified to preserve the initial displacement struc- ture of the input structure matrix during the iteration and maintain matrix structure throughout the computation. The main idea here is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. This can be done based on one of the following two techniques. The first technique is to periodically chop off the components corresponding to the smallest singular values in the SVDs of the displacement matrices defined by their generators. This tech- nique can be applied to a Toeplitz-like input matrix. For a Cauchy-like input matrix, there is no need to involve the SVD due to the availability of the inverse formula for [10] . The second technique is to control the growth of the length of the generators. This can be done by substituting the approximate inverses for the inverse matrix into its displacement. Here we assume that the matrices involved can be expressed via their displacement generators. Now, let us assume that the matrices and have short generators and respectively. So by operating with short -generators of the matrices, , and (or), we will have to modify Newton’s iteration and use one of the above two techniques to control the length of a -generator of, which seems to triple at every iterative step of (6). The same technique applies in the case when and -generators are used. In this paper, we restrict our presentation to -generators (see Theorem 1).

5.2. SVD-Based Compression of the Generators

For a fixed operator and a matrix, one can choose the orthogonal (SVD-based) -generator matrices to obtain better numerical stability [11] .

Definition 7 A matrix is said to be unitary iff. For real matrices, is orthogonal iff. If is orthogonal, then form an orthogonal basis for.

Definition 8 Every matrix has its SVD (singular value decomposition):

(8)

(9)

(10)

where, are singular values of the matrix W, and denote the Hermitian (conjugate) transpose of the orthogonal matrices and, respectively. That is, and.

Note that if and are real matrices, then and. Based on the SVD of a matrix W, its orthogonal generator of the minimum length is defined by

(11)

However, one can use an alternative diagonal scaling, in this case (11) can be written as

. (12)

Note that if for a matrix, then the pair is an orthogonal L-generator of minimum length for the matrix.

Example 4 Consider the following matrix.

Using Matlab command “” produces the above results. Clearly, is the largest singular value and is the smallest singular value., , and

(Frobenius norm). Furthermore, by (12), and

. Note that.

Remark 3 In numerical computation of the SVD with a fixed positive tolerance to the output errors caused by rounding (is the output error bound which is also an upper bound on all the output errors of the singular values and the entries of the computed matrices and and for), one truncates (sets to zero) all the smallest singular values up to. This will produce a numerical orthogonal generator of -length for a matrix, which is also a numerical orthogonal L-generator for. Here, is the -rank of the matrix, which is equal to the minimum number of the singular values of the displacement exceeding,.

Theorem 5 ([9] , p. 79) Given a matrix of rank and a nonnegative integer such that, it

holds that. That is, the error in the optimal choice of an approximate of whose

rank does not exceed is equal to the -th singular value of.

Theorem 6 If M is a nonsingular matrix, then.

Proof.. On the other hand,

. This

implies that.

Definition 9 Let, ,. We define

(13)

. (14)

6. Our Main Algorithm and Results

Outline of the Techniques

By Theorem 6, we assume that and. The ma-

trices that approximate for large have a nearby matrix of -displacement rank for large. This fact motivates the following approach to decrease the computational cost of the iteration in (6): We will replace in (6) with a nearby matrix having -displacement rank of at most and then restart the iteration with instead of. The advantage of this modification is the decrease of the computational cost at the -th Newton step and at all subsequent steps. However, the disadvantage is a possible deterioration of the approximation, since we do not know, and the transition from to may occur in a wrong direction. But since and are supposed to lie close to, hence they both lie close to each other. This observation enables us to bound the deterioration of the approximation relative to the current approximation error, so that the quadratic improvement in the error bound at the next step of Newton’s iteration will imme- diately compensate us for such a deterioration. Furthermore, the transition from to a nearby matrix of a small displacement rank can be done at a low computational cost. In the next main algorithm we describe this approach for Sylvester type operators only. The same results can be extended to Stein operators using Theorem 1 with slight modifications to the technique and the theory.

Algorithm 1 (Newton’s Iteration Using Sylverster Type Operators)

INPUT: An integer, two matrices and defining the operator, an nonsingular structured matrix having -rank and defined by its -generator of length at most, a sufficiently close initial approximation to the inverse given with its -generator of length at most, an upper bound on the number of Newton’s iteration steps, and a compression subalgorithm or for the transition from a -generator of length at most for an matrix approximating to -generator of length at most for a nearby matrix.

OUTPUT: A -generator of length at most for a matrix approximating.

COMPUTATION: For, recursively compute a -generators of length at most for the matrices

(15)

and then apply the compression subalgorithm or (transition from to) to the matrix to compute -generators of length at most for the matrices.

At the -th step of (15) of Algorithm 1, a -generator of length at most for the matrix can be computed using flops, for. As mentioned earlier, the main idea is to control the growth of the length of the short displacement generators. So the question is how can we control the length of the computed -generators? That is, we need to compress the length of the generators. Therefore, to complete Algorithm 1, we need to introduce two compression subalgorithms, namely, subalgorithm and. First,

we describe the compression subalgorithm based on the SVD of the displacement. To do this,

we compute the SVD based orthogonal generator of the displacement and decrease the length of the -generator to by zeroing (truncating) all singular values for. Then output the resulting orthogonal -generator of at most for a nearby matrix which is unique and lies near.

Remark 4 By Theorem 5 we can deduce that because

and this also implies that lies near if the operator is invertible.

Lemma 2 (1), (2), where as in (13).

Proof. 1) Since (largest singular value), then (1) follows from this fact. 2) Since

then. Now,

Lemma 3, where as in (13).

Proof. Using the estimate from ([9] , p. 442), that is, for any two matrices and,

for and conclude that

where are defined in (8) - (10) of

Definition 8. For all we have and then we obtain

. Now, we use (2) of Lemma 2 and deduce

, for. If we combine the latest inequality with equation (1) of Lemma 2

for, then we conclude that.

Now let us use Lemma 2 and Lemma 3 to prove the bound in the next theorem.

Theorem 7 If is a nonsingular linear operator and is a positive scalar defined in (13), then the

bound holds for of Definition 6.

Proof. By Definition 6 replacing L by, d = 2, and M by X_{i} − Y_{i} we have,

which implies that by linearity of. Now,

. Then replace and

to deduce

. Finally, use the inequality of Lemma 3 to

conclude that. This completes the proof.

Subalgorithm 1 (: Compression of Displacement by Truncation of their Smallest Singular Values.)

INPUT: An integer, two matrices and defining the operator for an nonsingular

structured matrices, , and a -generator of length

at most for a matrix such that,.

OUTPUT: A -displacement generator of length at most for a matrix such that

, where as in (13) and of Definition 6.

COMPUTATIONS: Compute the SVD of the displacement (the computation is not costly since it is performed with, where, and since is small, see [11] ). Set to zero the diagonal entries of, that is, turning into a diagonal matrix of rank at most. (are smallest singular values of the matrix.) Compute and output the matrices, obtained from and respectively, by deleting their last columns.

Example 5 For, consider the operator, where the

pair, is the orthogonal, SVD based generator for the displacement. Let

. Then Subalgorithm 1 is applied to the matrix and outputs the

matrix. The computational cost of performing Subalgorithm 1 is dominated

by the computation of the SVD of the displacement which requires flops for.

Next, we will present the second compression technique based on substitution which does not depend on the SVD. By Theorem 4 we conclude that, and by Theorem 3 we

also have if is nonsingular, and

if is nonsingular. Clearly this expresses the displacement of via the displacement of, the operator matrices and, and the product of the inverse with specified vectors, where for or. Now since

, let us substitute for on the right hand side and define a short -

generator for the matrix:

(16)

Since, we expect because.

Subalgorithm 2 (: Compression of the displacement by substitution.)

INPUT: An integer, a pair of operator matrices and defining the operator, a -generator of length r for a nonsingular matrix M where,

and a -generator of length at most for a matrix of Newton’s iteration (6).

OUTPUT: A -generator of length at most for a matrix satisfying the equation

with the bound

(17)

where, of Definition 6, of (14), and for of (13).

COMPUTATIONS: Compute the matrix products

(18)

and

. (19)

Now the pair is a -generator of length at most for a matrix satisfying the equation

. Here we assume the operator, then the matrix is uniquely defined by its L-

generators, and thus (18) and (19) completely define Newton’s process without involving the matrix

in. This subalgorithm achieves compression because,

and preserves approximation. Since, we deduce from Theorem 3 that

. The computation of the matrices of (16) is reduced to multi-

plication of the matrix by the matrix which requires flops for.

Now, we need to prove bound (17).

Theorem 8 For the matrices, , , and for, we

have.

Proof. Recall the matrix equations, and. Then we can write

, and

. Now, let us write, then

Now, take the norm of:

Simplifying the latest inequality yield the following bound:

(20)

Theorem 9 For and for in (13) and of

Definition 6. Then we have for.

Proof. Recall that, and since the operator is linear, then we

have. If we use (20) for, we get

. Therefore,.

7. Numerical Experiments

Algorithm 1 was tested numerically for Toeplitz input matrices by applying the compression Su- balgoritm 1. We use Matlab 8.3.0 on a Window 7 machine to run our tests. The Matlab SVD subroutine was used to obtain the singular value decomposition. In general, the complexity of the SVD on an square matrix is arithmetic operations. However, it is actually applied to structured matrices with smaller rank, so it is not really expensive. In addition, CLAPACK is one of the faster SVD algorithms that can be applied on real and complex matrices with less time complexity (see [11] ). The tests results are summarized in the follow- ing example.

Example 6 refers to number of Newton’s steps, is the matrix size, is number of times we ran the test, is a selected bound, and is the condition number of, where is the -th condition number of the matrix. We restricted our numerical experiments to the following classes of Toeplitz matrices:

1) Real symmetric tridiagonal Toeplitz matrices:, where, ,

(the first table of Table 3) or (the second table of Table 3) (see Figure 1).

2) The matrices (third table represents symmetric positive definite and the fourth table

represents symmetric positive indefinite of Table 3) (see Figure 1).

3) Randomly generated Toeplitz matrices, whose entries are chosen randomly from the interval and are uniformly distributed with mean 0 and standard deviation of 1 (see Table 4 and Figure 2).

For a symmetric positive definite matrix M, the initial matrix was used, whereas for unsym-

metric and symmetric indefinite Toeplitz matrices M, Newton’s iteration was initialized with.

Table 3. First top two tables represent Tridiagonal Matrices of Class 1 and the last two tables represent matrices of Class 2.

Table 4. Random Matrices of Class 3.

Figure 1. Error verses iterations.

Figure 2. Error verses iterations.

Acknowledgements

We thank the editor and the referees for their comments.

References

- Kailath, T. and Sayed, A. (1999) Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics, Philadelphia. http://dx.doi.org/10.1137/1.9781611971354
- Pan, V.Y., Branham, S., Rosholt, R. and Zheng, A. (1999) Newton’s Iteration for Structured Matrices and Linear Systems of Equations, SIAM Volume on Fast Reliable Algorithms for Matrices with Structure. Society for Industrial and Applied Mathematics, Philadelphia.
- Pan, V.Y., Zheng, A.L., Huang, X.H. and Dias, O. (1997) Newton’s Iteration for Inversion of Cauchy-Like and Other Structured Matrices. Journal of Complexity, 13, 108-124. http://dx.doi.org/10.1006/jcom.1997.0431
- Bini, D. and Pan, V.Y. (1994) Polynomial and Matrix Computations, Vol. 1 Fundamental Algorithms. Birkhäuser, Boston.
- Pan, V.Y. (2001) Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser, Boston.
- Kailath, T., Kung, S.-Y. and Morf, M. (1979) Displacement Ranks of Matrices and Linear Equations. Journal of Mathematical Analysis and Applications, 68, 395-407. http://dx.doi.org/10.1016/0022-247X(79)90124-0
- Kailath, T. and Sayed, A.H. (2002) Displacement Structure: Theory and Applications. SIAM Review, 37, 297-386. http://dx.doi.org/10.1137/1037082
- Pan, V.Y. and Rami, Y. (2001) Newton’s Iteration for the Inversion of Structured Matrices. In: Bini, D., Tyrtyshnikov, E. and Yalamov, P., Eds., Structured Matrices: Recent Developments in Theory and Computation, Nova Science Publishers, New York, 79-90.
- Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th Edition, John Hopkins University Press, Baltimore.
- Heinig, G. (1995) Inversion of Generalized Cauchy Matrices and the Other Classes of Structured Matrices. The IMA Volume in Mathematics and Its Applications, 69, 63-81.
- Pan, V.Y. (1993) Decreasing the Displacement Rank of a Matrix. SIAM Journal on Matrix Analysis and Application, 14, 118-121. http://dx.doi.org/10.1137/0614010