A binary powering Schur algorithm for computing primary matrix roots

Greco, Federico; Iannazzo, Bruno

doi:10.1007/s11075-009-9357-1

Cited by 16 publications

(12 citation statements)

References 16 publications

(30 reference statements)

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“…To show c k,i < 0 for k ≥ 2 and i ≥ 1, we only need to show that c 2,i < 0 for i ≥ 1. By (7) and (12)…”

Section: Sign Pattern Of Coefficients C Ki In Power Series Expansionsmentioning

confidence: 86%

A study of Schröder’s method for the matrix pth root using power series expansions

Guo

2019

Numer Algor

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When A is a matrix with all eigenvalues in the disk |z −1| < 1, the principal pth root of A can be computed by Schröder's method, among many other methods. In this paper we present a further study of Schröder's method for the matrix pth root, through an examination of power series expansions of some sequences of scalar functions. Specifically, we obtain a new and informative error estimate for the matrix sequence generated by the Schröder's method, a monotonic convergence result when A is a nonsingular M -matrix, and a structure preserving result when A is a nonsingular M -matrix or a real nonsingular H-matrix with positive diagonal entries.

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“…To show c k,i < 0 for k ≥ 2 and i ≥ 1, we only need to show that c 2,i < 0 for i ≥ 1. By (7) and (12)…”

Section: Sign Pattern Of Coefficients C Ki In Power Series Expansionsmentioning

confidence: 86%

A study of Schröder’s method for the matrix pth root using power series expansions

Guo

2019

Numer Algor

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“…In previous work [11,16], we have shown that, for the kth root, the computational cost of the straightforward algorithm [22] can be reduced by considering substitution algorithms that exploit more efficient matrix powering schemes. However, a fraction can be evaluated in several different ways, and some approaches require fewer matrix multiplications than applying Horner's method twice.…”

Section: Discussionmentioning

confidence: 99%

“…When T ∈ C N ×N is upper triangular, the blocks along the diagonal of T are of size 1 × 1 and ν = N . Equation (11) involves just scalars and can be written as ψ ij y ij = ϕ ij , where…”

Section: Complex Schur Formmentioning

confidence: 99%

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Computing primary solutions of equations involving primary matrix functions

Fasi

Iannazzo

2019

Linear Algebra and its Applications

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The matrix equation f (X) = A, where f is an analytic function and A is a square matrix, is considered. Some results on the classification of solutions are provided. When f is rational, a numerical algorithm is proposed to compute all solutions that can be written as a polynomial of A. For real data, the algorithm yields the real solutions using only real arithmetic. Numerical experiments show that the algorithm performs in a stable fashion when run in finite precision arithmetic.

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“…For example Smith [2003], Guo and Higham [2006], Greco and Iannazzo [2010], and Iannazzo and Manasse [2013] have all used the identity X p − A = 0 to assess algorithms for computing matrix pth roots. The identity e log A = A has been used by Davies and Higham [2003] and Dieci et al [1996] to test algorithms for the matrix logarithm.…”

Section: Introductionmentioning

confidence: 99%

Testing Matrix Function Algorithms Using Identities

Deadman

Higham

2016

ACM Trans. Math. Softw.

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Algorithms for computing matrix functions are typically tested by comparing the forward error with the product of the condition number and the unit roundoff. The forward error is computed with the aid of a reference solution, typically computed at high precision. An alternative approach is to use functional identities such as the "round trip tests" e log A = A and (A 1/p ) p = A, as are currently employed in a SciPy test module. We show how a linearized perturbation analysis for a functional identity allows the determination of a maximum residual consistent with backward stability of the constituent matrix function evaluations. Comparison of this maximum residual with a computed residual provides a necessary test for backward stability. We also show how the actual linearized backward error for these relations can be computed. Our approach makes use of Fréchet derivatives and estimates of their norms. Numerical experiments show that the proposed approaches are able both to detect instability and to confirm stability.

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A binary powering Schur algorithm for computing primary matrix roots

Cited by 16 publications

References 16 publications

A study of Schröder’s method for the matrix pth root using power series expansions

A study of Schröder’s method for the matrix pth root using power series expansions

Computing primary solutions of equations involving primary matrix functions

Testing Matrix Function Algorithms Using Identities

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