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18
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 322 (25 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
StructurePreserving Methods for Computing Eigenpairs of Large Sparse SkewHamiltonian/Hamiltonian Pencils
 SIAM J. Sci. Comput
, 2000
"... We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linearquadr ..."
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Cited by 66 (17 self)
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We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linearquadratic control problem for partial differential equations. We develop a structurepreserving skewHamiltonian, isotropic, implicitlyrestarted shiftandinvert Arnoldi algorithm (SHIRA). Several numerical examples demonstrate the superiority of SHIRA over a competing unstructured method.
Structured factorizations in scalar product spaces
 SIAM J. Matrix Anal. Appl
"... Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. Fo ..."
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Cited by 21 (7 self)
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Abstract. Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decomposition, and we relate it to other such decompositions in the literature. Finally, we study eigendecompositions and structured singular value decompositions, considering in particular the structure in eigenvalues, eigenvectors and singular values that persists across a wide range of scalar products. A key feature of our analysis is the identification of two particular classes of scalar products, termed unitary and orthosymmetric, which serve to unify assumptions for the existence of structured factorizations. A variety of different characterizations of these scalar product classes is given.
Canonical Forms for Doubly Structured Matrices and Pencils
, 2000
"... In this paper we derive canonical forms under structure preserving equivalence transformations for matrices and matrix pencils that have a multiple structure, which is either an Hselfadjoint or Hskewadjoint structure, where the matrix H is a complex nonsingular Hermitian or skewHermitian matrix. ..."
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Cited by 17 (0 self)
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In this paper we derive canonical forms under structure preserving equivalence transformations for matrices and matrix pencils that have a multiple structure, which is either an Hselfadjoint or Hskewadjoint structure, where the matrix H is a complex nonsingular Hermitian or skewHermitian matrix. Matrices and pencils of such multiple structures arise for example in quantum chemistry in HartreeFock models or random phase approximation.
Structured tools for structured matrices
 Electron. J. Linear Algebra
"... Abstract. An extensive and unified collection of structurepreserving transformations is presented and organized for easy reference. The structures involved arise in the context of a nondegenerate bilinear or sesquilinear form on R n or C n. A variety of transformations belonging to the automorphism ..."
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Cited by 14 (2 self)
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Abstract. An extensive and unified collection of structurepreserving transformations is presented and organized for easy reference. The structures involved arise in the context of a nondegenerate bilinear or sesquilinear form on R n or C n. A variety of transformations belonging to the automorphism groups of these forms, that imitate the action of Givens rotations, Householder reflectors, and Gauss transformations are constructed. Transformations for performing structured scaling actions are also described. The matrix groups considered in this paper are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudoorthogonal, and pseudounitary groups. In addition to deriving new transformations, this paper collects and unifies existing structurepreserving tools.
STRUCTURED JORDAN CANONICAL FORMS FOR STRUCTURED MATRICES THAT ARE HERMITIAN, SKEW HERMITIAN OR UNITARY WITH RESPECT TO INDEFINITE INNER PRODUCTS
, 1999
"... For inner products defined by a symmetric indefinite matrix p;q, canonical forms for real or complex p;qHermitian matrices, p;qskew Hermitian matrices and p;qunitary matrices are studied under equivalence transformations which keep the class invariant. ..."
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Cited by 14 (4 self)
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For inner products defined by a symmetric indefinite matrix p;q, canonical forms for real or complex p;qHermitian matrices, p;qskew Hermitian matrices and p;qunitary matrices are studied under equivalence transformations which keep the class invariant.
Structure Preserving Algorithms for Perplectic Eigenproblems
, 2003
"... Structured real canonical forms for matrices in R that are symmetric or skewsymmetric about the antidiagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solu ..."
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Cited by 8 (1 self)
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Structured real canonical forms for matrices in R that are symmetric or skewsymmetric about the antidiagonal as well as the main diagonal are presented, and Jacobi algorithms for solving the complete eigenproblem for three of these four classes of matrices are developed. Based on the direct solution of 4 × 4 subproblems constructed via quaternions, the algorithms calculate structured orthogonal bases for the invariant subspaces of the associated matrix. In addition to preserving structure, these methods are inherently parallelizable, numerically stable, and show asymptotic quadratic convergence.
On classification of normal matrices in indefinite inner product spaces
, 2006
"... Canonical forms are developed for several sets of matrices that are normal with respect to an indefinite inner product induced bya nonsingular Hermitian, symmetric, or skewsymmetric matrix. The most general result covers the case of polynomially normal matrices, i.e., matrices whose adjoint with res ..."
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Cited by 6 (3 self)
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Canonical forms are developed for several sets of matrices that are normal with respect to an indefinite inner product induced bya nonsingular Hermitian, symmetric, or skewsymmetric matrix. The most general result covers the case of polynomially normal matrices, i.e., matrices whose adjoint with respect to the indefinite inner product is a polynomial of the original matrix. From this result, canonical forms for complex matrices that are selfadjoint, skewadjoint, or unitarywith respect to the given indefinite inner product are derived. Most of the canonical forms for the latter three special types of normal matrices are known in the literature, but it is the aim of this paper to present a general theorythat allows the unified treatment of all different cases and to collect known results and new results such that all canonical forms for the complex case can be found in a single source.
Singularvaluelike decompositions for complex matrix triples
 DFG Research Center Matheon, Mathematics for
, 2007
"... Dedicated to William B. Gragg on the occasion of his 70th birthday The classical singular value decomposition for a matrix A ∈ Cm×n is a canonical form In for A that also displays the eigenvalues of the Hermitian matrices AA ∗ and A∗A. this paper, we develop a corresponding decomposition for A that ..."
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Cited by 2 (1 self)
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Dedicated to William B. Gragg on the occasion of his 70th birthday The classical singular value decomposition for a matrix A ∈ Cm×n is a canonical form In for A that also displays the eigenvalues of the Hermitian matrices AA ∗ and A∗A. this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices AAT and AT A. More generally, we consider the matrix triple (A, G, ˆ G), where G ∈ Cm×m, ˆ G ∈ Cn×n are invertible and either complex symmetric or complex skewsymmetric, and we provide a canonical form under transformations of the form (A, G, ˆ G) ↦ → (XT AY, XT GX, Y T GY ˆ), where X, Y are nonsingular.