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Luca Gemignani

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SUNY Fredonia

Vijaykumar Chakka

Shiv Nadar University

University of Khartoum

University of Mysore

Prof.Dr Hamed Daei Kasmaei

Islamic Azad University of Tehran, Central branch

Southeast University (China)

Cairo University

Alfred Enahoro Owoloko

Covenant University Canaanland, Ota.

The University of Sheffield

Dimitrios Mavreas

Vrije Universiteit Brussel

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Papers by Luca Gemignani

Chebyshev rational interpolation

Computing the Inverse of a ϕ-Function by Rational Approximation

In this paper we introduce a family of rational approximations of the inverse of a ϕ function inv... more In this paper we introduce a family of rational approximations of the inverse of a ϕ function involved in the explicit solutions of certain linear differential equations as well as in integration schemes evolving on manifolds. For symmetric banded matrices these novel approximations provide a computable reconstruction of the associated matrix function which exhibits decaying properties comparable to the best existing theoretical bounds. Numerical examples show the benefits of the proposed rational approximations w.r.t. the classical Taylor polynomials.

From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks

In this paper we present a general strategy to deduce a family of interpolatory masks from a symm... more In this paper we present a general strategy to deduce a family of interpolatory masks from a symmetric Hurwitz non-interpolatory one. This brings back to a polynomial equation involving the symbol of the non-interpolatory scheme we start with. The solution of the polynomial equation here proposed, tai-lored for symmetric Hurwitz subdivision symbols, leads to an efficient pro-cedure for the computation of the coefficients of the corresponding family of interpolatory masks. Several examples of interpolatory masks associated with classical approximating masks are given. AMS classication: 65F05; 65D05

Zerofinding of analytic functions by structured matrix methods

Contemporary Mathematics, 2016

We propose a fast and numerically robust algorithm based on structured numerical linear algebra t... more We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV-like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples

Zerofinding of analytic functions by structured matrix methods

Contemporary Mathematics, 2016

We propose a fast and numerically robust algorithm based on structured numerical linear algebra t... more We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV-like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples

A fast implicit QR eigenvalue algorithm for companion matrices

Linear Algebra and its Applications, 2010

puting the eigenvalues of an n × n companion matrix using O(n 2) flops and O(n) memory storage. N... more

The Ehrlich-Aberth Method For The Nonsymmetric Tridiagonal Eigenvalue

An algorithm based on the Ehrlich-Aberth iteration is presented for the computation of the zeros ... more An algorithm based on the Ehrlich-Aberth iteration is presented for the computation of the zeros of p(#) = det(T - #I), where T is an irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of p(#)/p (#) = -1/trace(T - #I) , which is done by exploiting the QR factorization of T - #I and the semiseparable structure of (T - #I) . The choice of initial approximations relies on a divide-and-conquer strategy, some results motivating this strategy are given. Guaranteed a-posteriori error bounds based on a running error analysis are proved. A Fortran 95 module implementing the algorithm is provided and numerical experiments that confirm the e#ectiveness and the robustness of the approach are presented. In particular, comparisons with the LAPACK subroutine dhseqr show that our algorithm is faster for large dimensions.

4 Implicit QR for Companion-Like Pencils

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matric... more A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems. The modified QZ algorithm computes the generalized eigenvalues of certain N × N rank structured matrix pencils using O(N 2) flops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

Advances in Computational Mathematics

The algebraic characterization of dual univariate interpolating subdivision schemes is investigat... more The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes possible to identify conditions for the existence of the sought schemes.

On the Euclidean scheme for polynomials having interlaced real zeros

Proceedings of the second annual ACM symposium on Parallel algorithms and architectures - SPAA '90

Page 1. ON THE EUCLIDEAN SCHEME FOR POLYNOMIALS HAVING INTERLACED REAL ZEROS Dario Bini, Dipartim... more

Improved initialization of the accelerated and robust QR-like polynomial root-finding

We approximate polynomial roots numerically as the eigenvalues of a unitary diagonal plus rank-on... more We approximate polynomial roots numerically as the eigenvalues of a unitary diagonal plus rank-one matrix. We rely on our earlier adaptation of the algorithm, which exploits the semiseparable matrix structure to approximate the eigenvalues in a fast and robust way, but we substantially improve the performance of the resulting algorithm at the initial stage, as confirmed by our numerical tests.

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

ArXiv, 2020

The algebraic characterization of dual univariate interpolating subdivision schemes is investigat... more The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes possible to identify conditions for the existence of the sought schemes.

Efficient Reduction of Compressed Unitary Plus Low Rank Matrices to Hessenberg Form

SIAM J. Matrix Anal. Appl., 2020

We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-ra... more We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $G\in \mathbb C^{n\times n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$ are $n\times k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then we prove structural properties of LFR decompositions by giving conditions under which the LFR decomposition of $A$ implies its Hessenberg shape. Finally, we describe a bulge chasing scheme for converting the initial LFR decomposition of $A$ into the LFR decomposition of a Hessenberg matrix by means of unitary transformations. The reduction can be performed at the overall computational cost ...

Manipulating Polynomials in Generalized Form

We present new algorithms using structured matrix methods for manipulating polynomials expressed ... more We present new algorithms using structured matrix methods for manipulating polynomials expressed in generalized form, that is, relative to a polynomial basis $\{p_i(x)\}$ satisfying a three-term recurrence relation. This includes computing the Euclidean remainders as well as the greatest common divisor of two polynomials $u(x)$ and $v(x)$ of degrees less than $n$. The computational schemes involve solving linear systems with nested Hankel matrices represented in the given generalized basis and they reach the optimal sequential complexity of $O(n^2)$ arithmetical operations.

Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials

arXiv: Numerical Analysis, 2014

Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linea... more Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stat...

On the Shifted QR Iteration Applied to Companion Matrices

Electronic Transactions on Numerical Analysis, 2004

We show that the shifted QR iteration applied to a companion matrix F maintains the weakly semise... more We show that the shifted QR iteration applied to a companion matrix F maintains the weakly semiseparable structure of F. More precisely, if Ai iI = QiRi, Ai+1 := RiQi + iI, i = 0; 1; : : :, where A0 = F , then we prove that Qi, Ri and Ai are semiseparable matrices having semiseparability rank at most 1, 4 and 3, respectively. This structural property is used to design an algorithm for performing a single step of the QR iteration in just O(n) flops. The robustness and reliability of this algorithm is discussed. Applications to approximating polynomial roots are shown.

On the fast reduction of symmetric rationally generated Toeplitz matrices to tridiagonal form

Electronic Transactions on Numerical Analysis, 2009

In this paper two fast algorithms that use orthogonal simila rity transformations to convert a sy... more In this paper two fast algorithms that use orthogonal simila rity transformations to convert a symmetric rationally generated Toeplitz matrix to tridiagonal form a re developed, as a means of finding the eigenvalues of the matrix efficiently. The reduction algorithms achieve cost e fficiency by exploiting the rank structure of the input Toeplitz matrix. The proposed algorithms differ in the choi ce of the generator set for the rank structure of the input Toeplitz matrix.

Numerical Behavior of the DQR Method for Rank-Structured Matrices

In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalu... more In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalue problems associated with algebraic equations. The matrices under consideration have the form A = T +uv T , where u,v2 R n◊n and T = (ti,j)2 R n◊n is a tridiagonal matrix such that tj+1,j =±tj,j+1, 1 j n 1. We show that the DQR approach proposed in [Uhlig F., Numer. Math. 76 (1997), no. 4, 515‐553] can compute the eigenvalues of such matrices eciently. Our algorithm employs a fast Hessenberg reduction scheme for tridiagonal plus rank-one matrices combined with an ecient eigenvalue method for the resulting Hessenberg matrices. Exploiting the rank structure of the input matrix enables both steps to be carried out using quadratic time and linear storage. Numerical implementation of these techniques is presented and discussed for a number of test problems. AMS classification: 65F15

Comparison Theorems for Splittings of M-matrices in (block) Hessenberg Form

Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M -ma... more Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M -matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix ρ(PGS) ≥ ρ(PS) ≥ ρ(PAGS), where PGS , PS , PAGS are the iteration matrices of the Gauss–Seidel, staircase, and antiGauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation. AMS classification: 65F15

Accelerating the Sinkhorn-Knopp iteration by power methods

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can ... more It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.

Chebyshev rational interpolation

Computing the Inverse of a ϕ-Function by Rational Approximation

In this paper we introduce a family of rational approximations of the inverse of a ϕ function inv... more In this paper we introduce a family of rational approximations of the inverse of a ϕ function involved in the explicit solutions of certain linear differential equations as well as in integration schemes evolving on manifolds. For symmetric banded matrices these novel approximations provide a computable reconstruction of the associated matrix function which exhibits decaying properties comparable to the best existing theoretical bounds. Numerical examples show the benefits of the proposed rational approximations w.r.t. the classical Taylor polynomials.

From symmetric subdivision masks of Hurwitz type to interpolatory subdivision masks

In this paper we present a general strategy to deduce a family of interpolatory masks from a symm... more In this paper we present a general strategy to deduce a family of interpolatory masks from a symmetric Hurwitz non-interpolatory one. This brings back to a polynomial equation involving the symbol of the non-interpolatory scheme we start with. The solution of the polynomial equation here proposed, tai-lored for symmetric Hurwitz subdivision symbols, leads to an efficient pro-cedure for the computation of the coefficients of the corresponding family of interpolatory masks. Several examples of interpolatory masks associated with classical approximating masks are given. AMS classication: 65F05; 65D05

Zerofinding of analytic functions by structured matrix methods

Contemporary Mathematics, 2016

We propose a fast and numerically robust algorithm based on structured numerical linear algebra t... more We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV-like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples

Zerofinding of analytic functions by structured matrix methods

Contemporary Mathematics, 2016

We propose a fast and numerically robust algorithm based on structured numerical linear algebra t... more We propose a fast and numerically robust algorithm based on structured numerical linear algebra technology for the computation of the zeros of an analytic function inside the unit circle in the complex plane. At the core of our method there are two matrix algorithms: (a) a fast reduction of a certain linearization of the zerofinding problem to a matrix eigenvalue computation involving a perturbed CMV-like matrix and (b) a fast variant of the QR eigenvalue algorithm suited to exploit the structural properties of this latter matrix. We illustrate the reliability of the proposed method by several numerical examples

A fast implicit QR eigenvalue algorithm for companion matrices

Linear Algebra and its Applications, 2010

puting the eigenvalues of an n × n companion matrix using O(n 2) flops and O(n) memory storage. N... more

The Ehrlich-Aberth Method For The Nonsymmetric Tridiagonal Eigenvalue

An algorithm based on the Ehrlich-Aberth iteration is presented for the computation of the zeros ... more An algorithm based on the Ehrlich-Aberth iteration is presented for the computation of the zeros of p(#) = det(T - #I), where T is an irreducible nonsymmetric tridiagonal matrix. The algorithm requires the evaluation of p(#)/p (#) = -1/trace(T - #I) , which is done by exploiting the QR factorization of T - #I and the semiseparable structure of (T - #I) . The choice of initial approximations relies on a divide-and-conquer strategy, some results motivating this strategy are given. Guaranteed a-posteriori error bounds based on a running error analysis are proved. A Fortran 95 module implementing the algorithm is provided and numerical experiments that confirm the e#ectiveness and the robustness of the approach are presented. In particular, comparisons with the LAPACK subroutine dhseqr show that our algorithm is faster for large dimensions.

4 Implicit QR for Companion-Like Pencils

A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matric... more A fast implicit QR algorithm for eigenvalue computation of low rank corrections of unitary matrices is adjusted to work with matrix pencils arising from polynomial zerofinding problems. The modified QZ algorithm computes the generalized eigenvalues of certain N × N rank structured matrix pencils using O(N 2) flops and O(N) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

Advances in Computational Mathematics

The algebraic characterization of dual univariate interpolating subdivision schemes is investigat... more The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes possible to identify conditions for the existence of the sought schemes.

On the Euclidean scheme for polynomials having interlaced real zeros

Proceedings of the second annual ACM symposium on Parallel algorithms and architectures - SPAA '90

Page 1. ON THE EUCLIDEAN SCHEME FOR POLYNOMIALS HAVING INTERLACED REAL ZEROS Dario Bini, Dipartim... more

Improved initialization of the accelerated and robust QR-like polynomial root-finding

We approximate polynomial roots numerically as the eigenvalues of a unitary diagonal plus rank-on... more We approximate polynomial roots numerically as the eigenvalues of a unitary diagonal plus rank-one matrix. We rely on our earlier adaptation of the algorithm, which exploits the semiseparable matrix structure to approximate the eigenvalues in a fast and robust way, but we substantially improve the performance of the resulting algorithm at the initial stage, as confirmed by our numerical tests.

Bezout-like polynomial equations associated with dual univariate interpolating subdivision schemes

ArXiv, 2020

The algebraic characterization of dual univariate interpolating subdivision schemes is investigat... more The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes possible to identify conditions for the existence of the sought schemes.

Efficient Reduction of Compressed Unitary Plus Low Rank Matrices to Hessenberg Form

SIAM J. Matrix Anal. Appl., 2020

We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-ra... more We present fast numerical methods for computing the Hessenberg reduction of a unitary plus low-rank matrix $A=G+U V^H$, where $G\in \mathbb C^{n\times n}$ is a unitary matrix represented in some compressed format using $O(nk)$ parameters and $U$ and $V$ are $n\times k$ matrices with $k< n$. At the core of these methods is a certain structured decomposition, referred to as a LFR decomposition, of $A$ as product of three possibly perturbed unitary $k$ Hessenberg matrices of size $n$. It is shown that in most interesting cases an initial LFR decomposition of $A$ can be computed very cheaply. Then we prove structural properties of LFR decompositions by giving conditions under which the LFR decomposition of $A$ implies its Hessenberg shape. Finally, we describe a bulge chasing scheme for converting the initial LFR decomposition of $A$ into the LFR decomposition of a Hessenberg matrix by means of unitary transformations. The reduction can be performed at the overall computational cost ...

Manipulating Polynomials in Generalized Form

We present new algorithms using structured matrix methods for manipulating polynomials expressed ... more We present new algorithms using structured matrix methods for manipulating polynomials expressed in generalized form, that is, relative to a polynomial basis $\{p_i(x)\}$ satisfying a three-term recurrence relation. This includes computing the Euclidean remainders as well as the greatest common divisor of two polynomials $u(x)$ and $v(x)$ of degrees less than $n$. The computational schemes involve solving linear systems with nested Hankel matrices represented in the given generalized basis and they reach the optimal sequential complexity of $O(n^2)$ arithmetical operations.

Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials

arXiv: Numerical Analysis, 2014

Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linea... more Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stat...

On the Shifted QR Iteration Applied to Companion Matrices

Electronic Transactions on Numerical Analysis, 2004

We show that the shifted QR iteration applied to a companion matrix F maintains the weakly semise... more We show that the shifted QR iteration applied to a companion matrix F maintains the weakly semiseparable structure of F. More precisely, if Ai iI = QiRi, Ai+1 := RiQi + iI, i = 0; 1; : : :, where A0 = F , then we prove that Qi, Ri and Ai are semiseparable matrices having semiseparability rank at most 1, 4 and 3, respectively. This structural property is used to design an algorithm for performing a single step of the QR iteration in just O(n) flops. The robustness and reliability of this algorithm is discussed. Applications to approximating polynomial roots are shown.

On the fast reduction of symmetric rationally generated Toeplitz matrices to tridiagonal form

Electronic Transactions on Numerical Analysis, 2009

In this paper two fast algorithms that use orthogonal simila rity transformations to convert a sy... more In this paper two fast algorithms that use orthogonal simila rity transformations to convert a symmetric rationally generated Toeplitz matrix to tridiagonal form a re developed, as a means of finding the eigenvalues of the matrix efficiently. The reduction algorithms achieve cost e fficiency by exploiting the rank structure of the input Toeplitz matrix. The proposed algorithms differ in the choi ce of the generator set for the rank structure of the input Toeplitz matrix.

Numerical Behavior of the DQR Method for Rank-Structured Matrices

In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalu... more In this paper we consider fast numerical algorithms for solving certain modified matrix eigenvalue problems associated with algebraic equations. The matrices under consideration have the form A = T +uv T , where u,v2 R n◊n and T = (ti,j)2 R n◊n is a tridiagonal matrix such that tj+1,j =±tj,j+1, 1 j n 1. We show that the DQR approach proposed in [Uhlig F., Numer. Math. 76 (1997), no. 4, 515‐553] can compute the eigenvalues of such matrices eciently. Our algorithm employs a fast Hessenberg reduction scheme for tridiagonal plus rank-one matrices combined with an ecient eigenvalue method for the resulting Hessenberg matrices. Exploiting the rank structure of the input matrix enables both steps to be carried out using quadratic time and linear storage. Numerical implementation of these techniques is presented and discussed for a number of test problems. AMS classification: 65F15

Comparison Theorems for Splittings of M-matrices in (block) Hessenberg Form

Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M -ma... more Some variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M -matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix ρ(PGS) ≥ ρ(PS) ≥ ρ(PAGS), where PGS , PS , PAGS are the iteration matrices of the Gauss–Seidel, staircase, and antiGauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation. AMS classification: 65F15

Accelerating the Sinkhorn-Knopp iteration by power methods

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can ... more It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a constrained nonlinear multiparameter eigenvalue problem. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate that our novel methods accelerate significantly the convergence of the customary Sinkhorn-Knopp iteration for matrix balancing in the case of clustered dominant eigenvalues.