Approximate factoring of the inverse

Byckling, Mikko; Huhtanen, Marko

doi:10.1007/s00211-010-0341-4

Cited by 7 publications

(18 citation statements)

References 24 publications

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“…Second, in very large scale problems exact factoring is not even realistic. Then the dimensions of W and V 1 are very moderate compared with n 2 and thereby an arbitrary matrix can be factored only approximately [4]. The factors computed do satisfy…”

Section: Matrix Factorization Problemsmentioning

confidence: 99%

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Matrix intersection problems for conditioning

Huhtanen

Seiskari

2017

Banach Center Publ.

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Abstract. Conditioning of a nonsingular matrix subspace is addressed in terms of its best conditioned elements. Computationally the problem is seemingly challenging. By associating with the task an intersection problem with unitary matrices leads to a more accessible approach. A resulting matrix nearness problem can be viewed to generalize the so-called Löwdin problem in quantum chemistry. For critical points in the Frobenius norm, a differential equation on the manifold of unitary matrices is derived. Another resulting matrix nearness problem allows locating points of optimality more directly, once formulated as a problem in computational algebraic geometry.Key words. conditioning, matrix intersection problem, matrix nearness problem, Löwdin's problem, generalized eigenvalue problem AMS subject classifications. 15A12, 65F351. Introduction. This paper is concerned with the problem of conditioning of a nonsingular matrix subspace V of C n×n over C (or R). Matrix subspaces typically appear in large scale numerical linear algebra problems where assuming additional structure is quite unrealistic. Nonsingularity means that there exists invertible elements in V. The conditioning of V is then defined in terms of its best conditioned elements. In the applications that we have in mind, typically dim V ≪ n 2 . For example, in the generalized eigenvalue problem dim V = 2 only. In this paper the task of assessing conditioning is formulated as a matrix intersection problem for V and the set of unitary matrices.1 Since this can be done in many ways, the interpretation is amenable to computations through matrix nearness problems and versatile enough in view of addressing operator theoretic problems more generally.Denote by U (n) the set of unitary matrices in C n×n . The intersection problem for V and U (n), which are both smooth submanifolds of C n×n , can be formulated as a matrix nearness problem

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Section: Matrix Factorization Problemsmentioning

confidence: 99%

“…By regarding the set of unitary matrices as a smooth Riemannian submanifold of C n×n , a differential equation is derived for locating critical points. 4 Because of the structure of U (n), in what follows, we regard V as a vector space over R.…”

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confidence: 99%

Matrix intersection problems for conditioning

Huhtanen

Seiskari

2017

Banach Center Publ.

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“…Now the subspace V 1 is required to consist of matrices for which it is possible to solve linear systems fast, whenever V 1 is invertible [7].…”

Section: Corollary 216 Suppose There Exists M ∈V −1 Such That V * Mmentioning

confidence: 99%

Matrix subspaces and determinantal hypersurfaces

Huhtanen

2010

Ark. Mat.

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Nonsingular matrix subspaces can be separated into two categories: by being either invertible, or merely possessing invertible elements. The former class was introduced for factoring matrices into the product of two matrices. With the latter, the problem of characterizing the inverses and related nonlinear matrix geometries arises. For the singular elements there is a natural concept of spectrum defined in terms of determinantal hypersurfaces, linking matrix analysis with algebraic geometry. With this, matrix subspaces and the respective Grassmannians are split into equivalence classes. Conditioning of matrix subspaces is addressed.

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“…The set of Toeplitz and Hankel matrices are unitarily equivalent matrix subspaces often encountered in practice. 4 For the unitary equivalence, take X to be the permutation with ones on the antidiagonal and Y = I . The set of Toeplitz matrices contains the scalars.…”

Section: The Grassmannian Gr K (C N×n )mentioning

confidence: 99%

“…(See [24] for related historical remarks.) More recently, understanding the structure of Inv(V ) has turned out to be central in matrix factorization problems and large scale numerical linear algebra of preconditioning [13], [4]. It is noteworthy that in preconditioning n is very large whereas dim V n 2 , typically dim V = O(n).…”

Section: Introductionmentioning

confidence: 99%

Differential geometry of matrix inversion

Huhtanen¹

2010

MATH. SCAND.

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Essentially, there exists just the dimension segregating (square) matrix subspaces. In view of algebraic operations, this quantity is not particularly descriptive. For differential geometric information on matrix inversion, the second fundamental form is found for the set of inverses of the invertible elements of a matrix subspace. Several conditions for this form to vanish are given, such as being equivalent to a Jordan subalgebra. Global measures of curvature are introduced in terms of an analogy of the Nash fiber.

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Approximate factoring of the inverse

Cited by 7 publications

References 24 publications

Matrix intersection problems for conditioning

Matrix intersection problems for conditioning

Matrix subspaces and determinantal hypersurfaces

Differential geometry of matrix inversion

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