Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

Al-Mohy, Awad H.; Higham, Nicholas J.; Relton, Samuel D.

doi:10.1137/120885991

Cited by 56 publications

(95 citation statements)

References 29 publications

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“…Because of the importance of the (quasi-) triangular square root, which arises in algorithms for computing the matrix logarithm [2], [3], matrix pth roots [5], [8], and arbitrary matrix powers [13], this computational kernel is a strong contender for inclusion in any future extensions of the BLAS.…”

Section: Discussionmentioning

confidence: 99%

Blocked Schur Algorithms for Computing the Matrix Square Root

Deadman

Higham

Ralha

2013

Lecture Notes in Computer Science

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Abstract. The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using either standard blocking or recursive blocking the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speedups over the point algorithm, both in the square root phase and in the algorithm as a whole. In parallel implementations, recursive blocking is found to provide better performance than standard blocking when the parallelism comes only from threaded BLAS, but the reverse is true when parallelism is explicitly expressed using OpenMP. The excellent numerical stability of the point algorithm is shown to be preserved by blocking. These results are extended to the real Schur method. Blocking is also shown to be effective for multiplying triangular matrices.

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Section: Discussionmentioning

confidence: 99%

Blocked Schur Algorithms for Computing the Matrix Square Root

Deadman

Higham

Ralha

2013

Lecture Notes in Computer Science

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“…(In contrast, for the Schur-Parlett algorithm usually only the complex Schur form can be used [4], [14]). The logarithm evaluations are now done using the algorithm of Al-Mohy, Higham, and Relton [3], which is designed for real matrices and works entirely in real arithmetic.…”

Section: The Algorithmmentioning

confidence: 99%

An Algorithm for the Matrix Lambert $W$ Function

Fasi

Higham

Iannazzo³

2015

SIAM J. Matrix Anal. & Appl.

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Abstract. An algorithm is proposed for computing primary matrix Lambert W functions of a square matrix A, which are solutions of the matrix equation W e W = A. The algorithm employs the Schur decomposition and blocks the triangular form in such a way that Newton's method can be used on each diagonal block, with a starting matrix depending on the block. A natural simplification of Newton's method for the Lambert W function is shown to be numerically unstable. By reorganizing the iteration a new Newton variant is constructed that is proved to be numerically stable. Numerical experiments demonstrate that the algorithm is able to compute the branches of the matrix Lambert W function in a numerically reliable way.

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“…and σ = 2 G · dev 3 (V − 11) + K · tr[V − 11] · 11 = 2 G · (V − 11) + Λ · tr[V − 11] · 11 (2) respectively, where U = √ F T F is the right Biot stretch tensor, V = √ F F T is the left Biot stretch tensor, σ is the Cauchy stress tensor, T Biot is the Biot stress tensor, dev 3 X = X − 1 3 tr(X) · 11 denotes the deviatoric part of X ∈ R 3×3 , K is the bulk modulus and G, Λ are the Lamé constants. According to Becker, it was already "universally acknowledged that either law [(1), (2)] is applicable only to strains so small that their squares are negligible" (3, p. 337).…”

Section: A Modern Interpretation Of Becker's Developmentmentioning

confidence: 99%

“…According to Becker, it was already "universally acknowledged that either law [(1), (2)] is applicable only to strains so small that their squares are negligible" (3, p. 337). He gives a number of reasons for this rejection of Hooke's law as a model for finite deformations, including the fact that it allows for infinite distortions (det F = 0) under finite stresses (4, p. 337).…”

Section: A Modern Interpretation Of Becker's Developmentmentioning

confidence: 99%

Rediscovering GF Becker’s early axiomatic deduction of a multiaxial nonlinear stress–strain relation based on logarithmic strain

Neff

Münch

Martín

2014

Mathematics and Mechanics of Solids

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We discuss a completely forgotten work of the geologist G.F. Becker on the ideal isotropic nonlinear stress-strain function [8]. In doing this we provide the original paper from 1893 newly typeset in L A T E X and with corrections of typographical errors as well as an updated notation. Due to the fact that the mathematical modelling of elastic deformations has evolved greatly since the original publication we give a modern reinterpretation of Becker's work, combining his approach with the current framework of the theory of nonlinear elasticity.Interestingly, Becker introduces a multiaxial constitutive law incorporating the logarithmic strain tensor, more than 35 years before the quadratic Hencky strain energy was introduced by Heinrich Hencky in 1929. Becker's deduction is purely axiomatic in nature. He considers the finite strain response to applied shear stresses and spherical stresses, formulated in terms of the principal strains and stresses, and postulates a principle of superposition for principal forces which leads, in a straightforward way, to a unique invertible constitutive relation, which in today's notation can be written aswhere T Biot is the Biot stress tensor, log(U ) is the principal matrix logarithm of the right Biot stretch tensor U = √ F T F , tr X = 3 i=1 X i,i denotes the trace and dev 3 X = X − 1 3 tr(X) · 11 denotes the deviatoric part of a matrix X ∈ R 3×3 . Here, G is the shear modulus and K is the bulk modulus. For Poisson's number ν = 0 the formulation is hyperelastic and the corresponding strain energyBecker (U ) = 2 G [ < U, log(U ) − 11 > +3 ] has the form of the maximum entropy function. B Notation 70 References 71The Finite Elastic Stress-Strain Function by G.F. Becker 76 3 1 Introduction Some reflections on constitutive assumptions in nonlinear elasticityThe question of proper constitutive assumptions in nonlinear elasticity has puzzled many generations of researchers. The problem of finding simple enough constitutive assumptions which are sufficient to characterize a physically plausible behaviour of "completely elastic" materials was even called "das ungelöste Hauptproblem der endlichen Elastizitätstheorie" (the unsolved main problem of finite elasticity theory) by C. Truesdell [79]. While such assumptions can only lead to an idealized material behaviour, the merits of such an ideal model were already described by H. Hencky in his 1928 article On the form of the law of elasticity for ideally elastic materials [31,55]: Like so many mathematical and geometric concepts, it is a useful ideal, because once its deducible properties are known it can be used as a comparative rule for assessing the actual elastic behaviour of physical bodies. [. . . ] While it is certainly a matter of empirical observation to determine how actual materials compare to the ideally elastic body, the law itself acts as a measuring instrument which is extended into the realm of the intellect, making it possible for the experimental researcher to make systematic observations.The range of applicabili...

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Computing the Fréchet Derivative of the Matrix Logarithm and Estimating the Condition Number

Cited by 56 publications

References 29 publications

Blocked Schur Algorithms for Computing the Matrix Square Root

Blocked Schur Algorithms for Computing the Matrix Square Root

An Algorithm for the Matrix Lambert $W$ Function

Rediscovering GF Becker’s early axiomatic deduction of a multiaxial nonlinear stress–strain relation based on logarithmic strain

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