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In this paper, we investigate Cramer’s rule for the general solution to the system of quaternion matrix equations A 1 X B 1 = C 1 , A 2 X B 2 = C 2 , and Cramer’s rule for the general solution to the generalized Sylvester quaternion matrix equation A X B + C Y D = E , respectively. As applications, we derive the determinantal expressions for the Hermitian solutions to some quaternion matrix equations. The findings of this paper extend some known results in the literature. [ABSTRACT FROM AUTHOR]

In previous papers, the authors introduced and characterized a class of matrices called { K , s + 1 } -potent. Also, they established a method to construct these matrices. The purpose of this paper is to solve the associated inverse problem. Several algorithms are developed in order to find all involutory matrices K satisfying K A s + 1 K = A for a given matrix A ∈ C n × n and a given natural number s . The cases s = 0 and s ≥ 1 are separately studied since they produce different situations. In addition, some examples are presented showing the numerical performance of the methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new decomposition (called the core-EP decomposition in the present paper) for square matrices and give some of its applications. By applying the decomposition, we derive several characterizations of the core-EP inverse, introduce two orders – the core-EP order and the core-minus partial order, and characterize their properties. We also give some properties of the decomposition. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations A 1 X = C 1 , Y B 1 = D 1 , A 2 W = C 2 , Z B 2 = D 2 , A 3 V = C 3 , V B 3 = C 4 , A 4 V B 4 = C 5 , A 5 X + Y B 5 + C 6 W + Z D 6 + E 6 V F 6 = G 6 , and constitute an expression of the general solution to the system when it is solvable. The outcomes of this paper encompass some recognized results in the collected works. In addition, we establish an algorithm and a numerical example to illustrate the theory constructed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the bi-periodic Lucas matrix sequence and present some fundamental properties of this generalized matrix sequence. Moreover, we investigate the important relationships between the bi-periodic Fibonacci and Lucas matrix sequences. We express that some behaviors of bi-periodic Lucas numbers also can be obtained by considering properties of this new matrix sequence. Finally, we say that the matrix sequences as Lucas, k -Lucas and Pell–Lucas are special cases of this generalized matrix sequence. [ABSTRACT FROM AUTHOR]

The T-congruence Sylvester equation is the matrix equation A X + X T B = C , where A ∈ R m × n , B ∈ R n × m and C ∈ R m × m are given, and matrix X ∈ R n × m is to be determined. The T-congruence Sylvester equation has recently attracted attention because of a relationship with palindromic eigenvalue problems. For example, necessary and sufficient conditions for the existence and uniqueness of solutions, and numerical solvers have been intensively studied. In this paper, we will show that, under a certain condition and n = m , the T-congruence Sylvester equation can be transformed into the Lyapunov equation. This may lead to further properties and efficient numerical solvers by utilizing the rich literature on the Lyapunov equation. [ABSTRACT FROM AUTHOR]

The well-known linear matrix equation A X = B is the simplest representative of all linear matrix equations. In this paper, we study quadratic properties of weighted least-squares solutions of this matrix equation. We first establish two groups of closed-form formulas for calculating the global maximum and minimum ranks and inertias of matrices in the two quadratical matrix-valued functions Q 1 − X P 1 X ′ and Q 2 − X ′ P 2 X subject to the restriction trace [ ( A X − B ) ′ W ( A X − B ) ] = min , where both P i and Q i are real symmetric matrices, i = 1 , 2 , W is a positive semi-definite matrix, and X ′ is the transpose of X . We then use the rank and inertia formulas to characterize quadratic properties of weighted least-squares solutions of A X = B , including necessary and sufficient conditions for weighted least-squares solutions of A X = B to satisfy the quadratic symmetric matrix equalities X P 1 X ′ = Q 1 an X ′ P 2 X = Q 2 , respectively, and necessary and sufficient conditions for the quadratic matrix inequalities XP 1 X ′≻ Q 1  (≽ Q 1 , ≺ Q 1 , ≼ Q 1 ) and X ′ P 2 X ≻ Q 2  (≽ Q 2 , ≺ Q 2 , ≼ Q 2 ) in the Löwner partial ordering to hold, respectively. In addition, we give closed-form solutions to four Löwner partial ordering optimization problems on Q 1 − X P 1 X ′ and Q 2 − X ′ P 2 X subject to weighted least-squares solutions of A X = B . [ABSTRACT FROM AUTHOR]

In this paper, a new iteration method is proposed for solving the complex symmetric linear systems. In theory, we show that the convergence factor or the upper bound of the spectral radius of the iteration matrix of the new method are smaller than that of the PMHSS method proposed by Bai et al. (2011). Moreover, we analyze and compare the parameter-free versions and the spectrum distributions of the preconditioned matrix of the new method and the PMHSS method. Finally, we present some numerical experiments on a few model problems to illustrate the theoretical results and show the effectiveness of our new method. [ABSTRACT FROM AUTHOR]

This paper is the second part of [15] . Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similar transformation introduced by [18] to reduce H to a Hamiltonian Jordan canonical form J . The asymptotic analysis of the structure-preserving flows and RDEs is studied by using e J t . The convergence of the SDA as well as its rate can thus result from the study of the structure-preserving flows. A complete asymptotic dynamics of the SDA is investigated, including the linear and quadratic convergence studied in the literature [3,12,13] . [ABSTRACT FROM AUTHOR]

In this paper we propose an iteration method to solve the multiple constrained least squares matrix problem. We first transform the multiple constrained least squares matrix problems into the multiple constrained matrix optimal approximation problem, and then we use the idea of Dykstra’s algorithm to derive the basic iterative pattern. We observe that we only need to solve multiple single constrained least squares matrix problems at each iteration step of the proposed algorithm. We give a numerical example to illustrate the effectiveness of the proposed method to solve the original problems. Also, we give an example to illustrate that the method proposed by Escalante and Li to solve the single constrained least squares matrix problem is not correct. [ABSTRACT FROM AUTHOR]

In this paper we solve nonlinear matrix equations of the form X δ = Q + ∑ i = 1 p ( A i ⁎ F i ( X ) A i ) r i and X δ = Q + ∑ i = 1 p ( A i ⁎ F i ( X ) A i ) r i + ∑ j = 1 q ( B j ⁎ G j ( X ) B j ) q j , where δ ∈ ( − ∞ , − 1 ] ∪ [ 1 , ∞ ) , r i , q j ∈ [ − 1 , 1 ] , Q ∈ P ( n ) , the collection of all n × n Hermitian positive definite matrices and A i , B j 's are n × n matrices, also F i , G j 's are monotone mappings from P ( n ) into P ( n ) . Examples are given to illustrate that the equations can not be solved by previously known theorems. [ABSTRACT FROM AUTHOR]

An explicit solution to the partial eigenvalue assignment problem of high order control system is presented by the method of receptance. Conventional methods, e.g. finite elements, are known to contain inaccuracies and assumptions that may hinder the calculations. An alternative approach was given by Ram and Mottershead [1] in the form of receptances, typically available from a modal test. This paper generalizes the earlier work on partial assignment that is applicable to multi-input delayed system without use of the Sherman–Morrison formula. The results of our numerical experiments support the validity of our proposed method. [ABSTRACT FROM AUTHOR]

Let R and H m × n stand, respectively, for the real number field and the set of all m × n matrices over the real quaternion algebra H = { a 0 + a 1 i + a 2 j + a 3 k | i 2 = j 2 = k 2 = ijk = − 1 , a 0 , a 1 , a 2 , a 3 ∈ R } . For η ∈ { i, j, k }, a real quaternion matrix A ∈ H n × n is said to be η -Hermitian if A η * = A where A η * = − η A * η , and A * stands for the conjugate transpose of A , arising in widely linear modeling. We present a simultaneous decomposition for a set of nine real quaternion matrices involving η -Hermicity with compatible sizes: A i ∈ H p i × t i , B i ∈ H p i × t i + 1 , and C i ∈ H p i × p i , where C i are η -Hermitian matrices, ( i = 1 , 2 , 3 ) . As applications of the simultaneous decomposition, we give necessary and sufficient conditions for the existence of an η -Hermitian solution to the system of coupled real quaternion matrix equations A i X i A i η * + B i X i + 1 B i η * = C i , ( i = 1 , 2 , 3 ) , and provide an expression of the general η -Hermitian solutions to this system. Moreover, we derive the rank bounds of the general η -Hermitian solutions to the above-mentioned system using ranks of the given matrices A i , B i , and C i as well as the block matrices formed by them. Finally some numerical examples are given to illustrate the results of this paper. [ABSTRACT FROM AUTHOR]

In this paper we introduce an isospectral matrix flow (Lax flow) that preserves some structures of an initial matrix. This flow is given by d A d t = [ A u − A l , A ] , A ( 0 ) = A 0 , where A is a real n × n matrix (not necessarily symmetric), [ A , B ] = A B − B A is the matrix commutator (also known as the Lie bracket), A u is the strictly upper triangular part of A and A l is the strictly lower triangular part of A . We prove that if the initial matrix A 0 is staircase, so is A ( t ) . Moreover, we prove that this flow preserves the certain positivity properties of A 0 . Also we prove that if the initial matrix A 0 is totally positive or totally nonnegative with non-zero codiagonal elements and distinct eigenvalues, then the solution A ( t ) converges to a diagonal matrix while preserving the spectrum of A 0 . Some simulations are provided to confirm the convergence properties. [ABSTRACT FROM AUTHOR]

Between 1970 and 1982 Hans Schneider and co-authors produced a number of results regarding matrix scalings. They demonstrated that a matrix has a diagonal similarity scaling to any matrix with entries in the subgroup generated by the cycle weights of the associated digraph, and that a matrix has an equivalent scaling to any matrix with entries related to the weights of cycles in an associated bipartite graph. Further, given matrices A and B , they produced a description of all diagonal X such that X − 1 A X = B . In 2005 Butkovič and Schneider used max-algebra to give a simple and efficient description of this set of scalings. In this paper we focus on the additive group of integers, and work in the max-plus algebra to give a full description of all scalings of a real matrix A to any integer matrix. We do this for four types of scalings; beginning with the familiar X − 1 A X , XAY and XAX scalings and finishing with a new scaling which we call a signed similarity scaling. This is a scaling of the form XAY where we specify for each row i , either x i = y i or x i = − y i . In all of our results we use necessary and sufficient conditions for existence which are based on integer weighted cycles in the associated digraph, or associated bipartite graph, of the matrix. [ABSTRACT FROM AUTHOR]

Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max ⁡ n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]

Let H m × n be the set of all m × n matrices over the real quaternion algebra. We call that A ∈ H n × n is η -Hermitian if A = − η A * η , η ∈ { i, j, k }, where i, j, k are the quaternion units. Denote A η * = − η A * η . In this paper, we derive some necessary and sufficient conditions for the solvability to the system of generalized Sylvester real quaternion matrix equations A i X i + Y i B i + C i Z D i = E i , ( i = 1 , 2 ) , and give an expression of the general solution to the above mentioned system. As applications, we give some solvability conditions and general solution for the generalized Sylvester real quaternion matrix equation A 1 X + Y B 1 + C 1 Z D 1 = E 1 , where Z is required to be η -Hermitian. We also present some solvability conditions and general solution for the system of real quaternion matrix equations involving η -Hermicity A i X i + ( A i X i ) η * + B i Y B i η * = C i , ( i = 1 , 2 ) , where Y is required to be η -Hermitian. Our results include some well-known results as special cases. [ABSTRACT FROM AUTHOR]

Given a Laurent polynomial with matrix coefficients that is positive semi-definite over the unit circle in the complex plane, the Fejér–Riesz theorem asserts that it can always be factorized as the product of a polynomial with matrix coefficients and its adjoint. This paper exploits such a factorization in its simplest form of degree one and its relationship with the nonlinear matrix equation X + A ⁎ X − 1 A = Q . In particular, the nonlinear equation can be recast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation is readily obtainable from hermitian eigenvalue computation. The unitary constraint can be enforced by a hybrid of a straightforward alternating projection for low precision estimation and a coordinate-free Newton iteration for high precision calculation. This approach offers a complete parametrization of all solutions and, in contrast to most existent algorithms, makes it possible to find all solutions if so desired. [ABSTRACT FROM AUTHOR]

This article discusses an approach to solving large-scale algebraic Riccati equations (AREs) by computing a low-dimensional stable invariant subspace of the associated Hamiltonian matrix. We give conditions on AREs to admit solutions of low numerical rank and show that these can be approximated via Hamiltonian eigenspaces. We discuss strategies on choosing the proper eigenspace that yields a good approximation, and different formulas for building the approximation itself. Similarities of our approach with several other methods for solving AREs are shown: closely related are the projection-type methods that use various Krylov subspaces and the qADI algorithm. The aim of this paper is merely to analyze the possibilities of computing approximate Riccati solutions from low-dimensional subspaces related to the corresponding Hamiltonian matrix and to explain commonalities among existing methods rather than providing a new algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we study partial orders in terms of the core-nilpotent decomposition. We derive some characterizations of the C-N partial ordering, create a new partial ordering (the G-Drazin partial ordering) which is a generalization of C-N partial ordering, and give some properties and characterizations of the G-Drazin partial ordering. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the nonlinear matrix equations X + A * X q A = Q ( 0 < q ≤ 1 ) and X − A * X q A = Q ( q > 1 ) . Necessary and sufficient conditions for the (unique) existence of Hermitian positive definite solutions of these equations are derived. Effective iterative algorithms are also provided to obtain the unique solution of X + A * X q A = Q ( 0 < q ≤ 1 ) and the minimal and maximal Hermitian positive definite solutions of X − A * X q A = Q ( q > 1 ) . Numerical examples are included to illustrate the efficiency of the presented iteration algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the following two minimum rank matrix approximation problems in the spectral norm: (i) For given A = A H ∈ C m × m , B ∈ C m × n , determining X ∈ S 1 , such that rank ( X ) = min Y ∈ S 1 rank ( Y ) , S 1 = { Y = Y H ∈ C n × n : ‖ A − B Y B H ‖ 2 = min } . (ii) For given A = − A H ∈ C m × m , B ∈ C m × n , determining X ∈ S 2 , such that rank ( X ) = min Y ∈ S 2 rank ( Y ) , S 2 = { Y = − Y H ∈ C n × n : ‖ A − B Y B H ‖ 2 = min } . By applying the norm-preserving dilation theorem, the Hermitian-type (skew-Hermitian-type) generalized singular value decomposition (HGSVD, SHGSVD), we characterize the expressions of the minimum rank and derive a general form of minimum rank (skew) Hermitian solutions to the matrix approximation problem. [ABSTRACT FROM AUTHOR]

For the large-scale nonlinear matrix equations with low rank structure, the well-developed doubling algorithm in low rank form (DA-LR) is known as an efficient method to compute the stabilizing solution. By further analyzing the global efficiency index constructed in this paper, we propose a tripling algorithm in low rank form (TA-LR) from two points of view, the cyclic reduction and the symplectic structure preservation. The new presented algorithm shares the same pre-processing complexity with that of DA-LR, but can attain the prescribed normalized residual level within less iterations by only consuming some negligible iteration time as an offset. Under the solvability condition, the proposed algorithm is demonstrated to inherit a cubic convergence and is capable of delivering errors from the current iteration to the next with the same order. Numerical experiments including some from nano research show that the TA-LR is highly efficient to compute the stabilizing solution of large-scale nonlinear matrix equations with low rank structure. [ABSTRACT FROM AUTHOR]

In this paper, we first establish some necessary and sufficient conditions for the existence of Hermitian solution of A X A * = B subject to CXC * ≥ D , where B and D are Hermitian matrices. Furthermore, a general expression for this Hermitian constrained solution is derived, several special cases are also considered. [ABSTRACT FROM AUTHOR]

This paper deals with a characterization of the solution set of algebraic Riccati equation (ARE) (over reals) for both controllable and uncontrollable systems. We characterize all solutions using simple linear algebraic arguments. It turns out that solutions of ARE of maximal rank have lower rank solutions encoded within it. We demonstrate how these lower rank solutions are encoded within the full rank solution and how one can retrieve the lower rank solutions from the maximal rank solution. We characterize situations where there are no full rank solutions to the ARE. We also characterize situations when the number of solutions to the ARE is finite, when they are infinite and when they are bounded. We also explore the poset structure on the solution set of ARE, which in some specific cases turns out to be a lattice which is isomorphic to lattice of invariant subspaces of a certain matrix. We provide several examples that bring out the essence of these results. [ABSTRACT FROM AUTHOR]

In this paper, we extend the generalized product-type bi-conjugate gradient (GPBiCG) method for solving the generalized Sylvester-conjugate matrix equations A 1 X B 1 + C 1 Y ¯ D 1 = S 1 , A 2 X ¯ B 2 + C 2 Y D 2 = S 2 by the real representation of the complex matrix and the properties of Kronecker product and vectorization operator. Some numerical experiments demonstrate that the introduced iteration approach is efficient. [ABSTRACT FROM AUTHOR]

The generalized Sylvester matrix equation ∑ i = 1 s A i X B i + ∑ j = 1 t C j Y D j = E with unknown matrices X and Y is encountered in many system and control applications. In this paper, a direct method is established to solve the least-squares symmetric and skew-symmetric solutions of the equation by using the Kronecker product and the generalized inverses and, the expression of the solution set S are provided. Moreover, an optimal approximation between a given matrix pair and the affine subspace S is discussed, and an explicit formula for the unique optimal approximation solution is presented. Finally, two numerical examples are given which demonstrate that the introduced algorithm is quite efficient. [ABSTRACT FROM AUTHOR]

Let H m × n be the set of all m × n matrices over the real quaternion algebra H = { c 0 + c 1 i + c 2 j + c 3 k ∣ i 2 = j 2 = k 2 = i j k = − 1 , c 0 , c 1 , c 2 , c 3 ∈ R } . A ∈ H n × n is known to be η -Hermitian if A = A η * = − η A * η , η ∈ { i , j , k } and A * means the conjugate transpose of A . We mention some necessary and sufficient conditions for the existence of the solution to the system of real quaternion matrix equations including η -Hermicity A 1 X = C 1 , A 2 Y = C 2 , Y B 2 = D 2 , Y = Y η * , A 3 Z = C 3 , Z B 3 = D 3 , Z = Z η * , A 4 X + ( A 4 X ) η * + B 4 Y B 4 η * + C 4 Z C 4 η * = D 4 , and also construct the general solution to the system when it is consistent. The outcome of this paper diversifies some particular results in the literature. Furthermore, we constitute an algorithm and a numerical example to comprehend the approach established in this treatise. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce the construction of the Bézout matrix of two univariate polynomials given by values in the Hermite interpolation basis, namely the confluent Bézout matrix. Moreover, if such polynomials have exactly one common simple zero, we describe how to compute it from the null space of the confluent Bézout matrix. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of factorizing the n × n matrix J n of all ones into the n × n binary matrices. We show that under some conditions on the factors, these are isomorphic to a row permutation of a De Bruijn matrix. Moreover, we consider in particular the binary roots of J n , i.e. the binary solutions to A m = J n . On the one hand, we prove that any binary root with minimum rank is isomorphic to a row permutation of a De Bruijn matrix whose row permutation is represented by a block diagonal matrix. On the other hand, we partially solve Hoffman's open problem of characterizing the binary solutions to A 2 = J n by providing a characterization of the binary solutions to A 2 = J n with minimum rank. Finally, we provide a class of roots which are isomorphic to a De Bruijn matrix. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to solve the problem of finding a geodesic that best fits a given set of time-labelled points on the Grassmann manifold. To achieve this goal, we first derive a very useful simplified formula for the geodesic arc joining two points on the Grassmannian depending explicitly only on the given points. This allows to simplify the expression for the geodesic distance, which is crucial to generalize the fitting problem, and is also used to obtain a simpler characterization of the geometric mean of a finite set of points lying on the Grassmannian, where the given points enter explicitly. [ABSTRACT FROM AUTHOR]

In this paper, we derive some equivalent conditions for block two-by-two matrices to be nonsingular in an elementary way. [ABSTRACT FROM AUTHOR]

The semigroup of Hamiltonians acting on the cone of positive definite matrices via linear fractional transformations satisfies the Birkhoff contraction formula for the Thompson metric. In this paper we describe the action of the Hamiltonians lying in the boundary of the semigroup. This involves in particular a construction of linear transformations leaving invariant the cone of positive definite matrices (strictly positive linear mappings) parameterized over all square matrices. Its invertibility and relation to the Lyapunov and Stein operators are investigated in detail. In particular, it is shown that each of these linear transformations commutes with the corresponding Lyapunov operator and contracts the Thompson metric. [ABSTRACT FROM AUTHOR]

Abstract: The main aim of this paper is to provide a higher-order convergent iterative method in order to calculate the generalized inverse of a given matrix. We extend the iterative method proposed in Li et al. [W.G. Li, Z. Li, A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Applied Mathematics and Computation 215 (2010) 3433-3442] to compute the -inverse, generalized inverse -inverse and -inverse. Moreover, we modify the iterative method to compute the generalized inverse of Toeplitz matrices by using the displacement theory. [Copyright &y& Elsevier]

Abstract: This paper studies the solvability, existence of unique solution, closed-form solution and numerical solution of matrix equation with and where is the unknown. It is proven that the solvability of these equations is equivalent to the solvability of some auxiliary standard Stein equations in the form of where the dimensions of the coefficient matrices and are the same as those of the original equation. Closed-form solutions of equation can then be obtained by utilizing standard results on the standard Stein equation. On the other hand, some generalized Stein iterations and accelerated Stein iterations are proposed to obtain numerical solutions of equation . Necessary and sufficient conditions are established to guarantee the convergence of the iterations. [Copyright &y& Elsevier]

Abstract: In this paper, we obtain the additive and multiplicative perturbation bounds for the Moore-Penrose inverse under the unitarily invariant norm and the Q - norm, which improve the corresponding ones in [P.Å. Wedin, Perturbation theory for pseudo-inverses, BIT 13(1973)217–232]. [ABSTRACT FROM AUTHOR]

Abstract: The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation , as well as the extremal inertias of a partial block Hermitian matrix. [Copyright &y& Elsevier]

Abstract: In this paper, the nonlinear matrix equation is investigated. Based on the fixed-point theory, the existence and the uniqueness of the positive definite solution are studied. An effective iterative method to obtain the unique positive definite solution is established given . In addition, some computable estimates of the unique positive definite solution are derived. Finally, numerical examples are given to illustrate the effectiveness of the algorithm and the perturbation estimates. [Copyright &y& Elsevier]