A parallel divide and conquer algorithm for the generalized real symmetric definite tridiagonal eigenproblem

Borges, Carlos F.; Gragg, William B.

doi:10.1515/9783110857658.11

Cited by 22 publications

(51 citation statements)

References 9 publications

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“…The convergence of Euler's method for this particular case, or Gragg's zero finder, was also proven in [1]. Our proof was presented from a more general viewpoint, with the additional advantage of avoiding the relatively messy computation of the coefficients in the approximation.…”

Section: Theorem 42 Euler's and Halley's Methods Converge From Any mentioning

confidence: 75%

“…Let us have another look at Gragg's zero finder ( [1,13]). We recall that we are solving the equation f (t) = 0 on the interval (0, δ), where f (t) is given by (2) and where we have defined δ = δ i+1 .…”

Section: Gragg's Zero Findermentioning

confidence: 99%

“…We recall that we are solving the equation f (t) = 0 on the interval (0, δ), where f (t) is given by (2) and where we have defined δ = δ i+1 . The method proposed in [1,13] is an iterative method for the solution of this equation, based on a third-order approximation of the form…”

Section: Gragg's Zero Findermentioning

confidence: 99%

“…This and similar problems appear in several applications in numerical linear algebra, such as the singular value decomposition of a matrix and eigenvalue problems, to name but a few. The following list is a small sample of the many related references : [1,4,5,6,8,9,10,11,12,13,14]. The secular function g(s) we will consider is a real function of the form…”

Section: Introductionmentioning

confidence: 99%

“…In fact only one method, Gragg's zero finder ( [1,13]), seems to be practically used. We start off with this method and show that hiding behind it, is the classical Euler 1 approximation (second-order Taylor polynomial) for a different function, but one having the same root.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of third-order methods for secular equations

Melman

1998

Math. Comp.

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Abstract. Third-order numerical methods are analyzed for secular equations. These equations arise in several matrix problems and numerical linear algebra applications. A closer look at an existing method shows that it can be considered as a classical method for an equivalent problem. This not only leads to other third-order methods, it also provides the means for a unifying convergence analysis of these methods and for their comparisons. Finally, we consider approximated versions of the aforementioned methods.

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Section: Theorem 42 Euler's and Halley's Methods Converge From Any mentioning

confidence: 75%

Section: Gragg's Zero Findermentioning

confidence: 99%

Section: Gragg's Zero Findermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of third-order methods for secular equations

Melman

1998

Math. Comp.

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An implicit QR algorithm for symmetric semiseparable matrices

Vandebril

Barel

Mastronardi

2005

Numerical Linear Algebra App

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The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n x n matrix, this algorithm requires O(n 3 ) operations per iteration step. To reduce this complexity for a sytmmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. Article information AbstractThe QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n 3 ) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is £rst reduced to tridiagonal form using orthogonal similarity transformations.In the report ] a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones.The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices.A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has * The research of the £rst and second author was supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scienti£c Research-Flanders (Belgium)

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An adaptive Richardson iteration method for indefinite linear systems

Calvetti

Reichel

1996

Numer Algor

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A parallel divide and conquer algorithm for the generalized real symmetric definite tridiagonal eigenproblem

Cited by 22 publications

References 9 publications

Analysis of third-order methods for secular equations

Analysis of third-order methods for secular equations

An implicit QR algorithm for symmetric semiseparable matrices

An adaptive Richardson iteration method for indefinite linear systems

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