Calculation of the Eigenvalues of a Symmetric Tridiagonal Matrix by the Method of Bisection

Barth, Wilhelm; Martin, R. S.; Wilkinson, J. H.

doi:10.1007/978-3-662-39778-7_16

Cited by 16 publications

(38 citation statements)

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“…[Barth, Martin, Wilkinson 1971] give bounds for all the eigenvalues. Godunov [Godunov et al 1993], page 315, gives a similar result but uses a different sequence and guarantees no overflow.…”

Section: Setting It To Work In Floating Point Arithmeticmentioning

confidence: 99%

“…The general structure of the algorithm is the same as in [Barth, Martin, Wilkinson 1971], except the final test: for each eigenvalue the bisection process is continued until the interval [x, y] is such that x and y are two consecutive machinenumbers (denormalized numbers excluded), or until a fixed maximum number of steps is reached. The machine precision used is eps = 2 −64 ≈ 5,5 × 10 −20 .…”

Section: Algorithm and Examplementioning

confidence: 99%

“…The results have been obtained using the software [Pavec 1994]. Consider the matrix of order n = 30 from [Barth, Martin, Wilkinson 1971]:…”

Section: Algorithm and Examplementioning

confidence: 99%

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Some Algorithms Providing Rigourous Bounds for the Eigenvalues of a Matrix

Pavec

1996

J.UCS the Journal of Universal Computer Science

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Three algorithms providing rigourous bounds for the eigenvalues of a real matrix are presented. The first is an implementation of the bisection algorithm for a symmetric tridiagonal matrix using IEEE floating-point arithmetic. The two others use interval arithmetic with directed rounding and are deduced from the Jacobi method for a symmetric matrix and the Jacobi-like method of Eberlein for an unsymmetric matrix.

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Section: Setting It To Work In Floating Point Arithmeticmentioning

confidence: 99%

Section: Algorithm and Examplementioning

confidence: 99%

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Some Algorithms Providing Rigourous Bounds for the Eigenvalues of a Matrix

Pavec

1996

J.UCS the Journal of Universal Computer Science

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“…It should be mentioned that the g, are the well-known quotients of consecutive principal minors of the shifted matrix A, which are also used in the bisection process [1]. This can be used to advantage to assign ordinals to computed eigenvalues.…”

Section: Introduction In 1961 Francismentioning

confidence: 99%

A Stable, Rational QR Algorithm for the Computation of the Eigenvalues of an Hermitian, Tridiagonal Matrix

Reinsch¹

1971

Mathematics of Computation

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Abstract. The most efficient program for finding all the eigenvalues of a symmetric matrix is a combination of the Householder tridiagonalization and the QR algorithm. The latter, if carried out in a natural way, requires An additions, 10« multiplications, In divisions, and n square roots per iteration (n the order of the matrix). In 1963, Ortega and Kaiser showed that the process can be carried out using no square roots (and saving In multiplications). However, their algorithm is unstable and several modifications were suggested to increase its accuracy. We, too, want to give such a modification together with some examples demonstrating the achieved accuracy.

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“…., p n (X) is terminated. For example, if eigenvalues from m, to m 2 are being sought and the sequence in (1) has been evaluated as far as p,(X) and this in turn has so far produced k negative signs, then if (m 2 -1 -k) is less than zero or if (m t -k -n + i) is greater than zero, no further computations of the elements of the sequence are necessary since they would not provide any information related to the eigenvalues from m, to m-,. Size= 1000 x 1000.…”

Section: A Modification To the Algorithmsmentioning

confidence: 99%

A Modified Bisection Algorithm for the Determination of the Eigenvalues of a Symmetric Quindiagonal Matrix

Shanehchi

Evans

1982

The Computer Journal

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In this note, we consider the implementation of the incomplete Sturm sequence evaluation, 2 for the determination of the eigenvalues of a symmetric quindiagonal matrix A based on the bisection method. The application of the bisection method for quindiagonal matrices is described in Ref. 3 and therefore, we discuss the changes to the proposed method in order to improve efficiency when dealing with only a few of the lowest or highest eigenvalues of quindiagonal matrices. IntroductionFollowing the recognition of the bisection algorithm as being one of the most efficient methods for evaluating the eigenvalues of symmetric tridiagonal matrices,' efforts have been made to implement a similar procedure to matrices with wider diagonal band than 3 (e.g. quindiagonal matrices). 23 The paper of Sentance and Cliff 3 employs a Gauss-type elimination algorithm called triangularization by elementary stabilized matrices, 4 where the necessary interchanges are made within the rows of individual minors and hence maintaining the correct signs for their determinantal values. However, in the above report the determination of the principal minors is carried out completely and therefore, producing an inefficient algorithm when only several of the end eigenvalues are required. A Modification to the AlgorithmsIt is shown in Ref. 2 that a great improvement can be achieved in the bisection method for the determination of eigenvalues of symmetric tridiagonal method by calculating the incomplete Sturm sequence. The idea can also be implemented in this case where the principal minors for a sample point are evaluated as long as they provide information on the next and all the following eigenvalues required. That is, if we denote the values of the principal minors at point X as: then, if after calculating/;, W),/> 2 (A),... ,p<(X), there is sufficient information to decide where to bisect next, then the determination of the remaining />, + , (X), p i+2 (X),. . ., p n (X) is terminated. For example, if eigenvalues from m, to m 2 are being sought and the sequence in (1) has been evaluated as far as p,(X) and this in turn has so far produced k negative signs, then if (m 2 -1 -k) is less than zero or if (m t -k -n + i) is greater than zero, no further computations of the elements of the sequence are necessary since they would not provide any information related to the eigenvalues from m, to m-,. Size= 1000 x 1000. Eigenvalues 478-500. Numerical Experiments and ConclusionsThe above strategy was incorporated into a FORTRAN version of the program given in Ref.3. The purpose of the investigation was to obtain the percentage gains for different cases when the incomplete evaluation of the sequence (I) is performed. The matrix chosen for the experiments is the symmetric quindiagonal matrix A defined as: except for

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Calculation of the Eigenvalues of a Symmetric Tridiagonal Matrix by the Method of Bisection

Cited by 16 publications

References 5 publications

Some Algorithms Providing Rigourous Bounds for the Eigenvalues of a Matrix

Some Algorithms Providing Rigourous Bounds for the Eigenvalues of a Matrix

A Stable, Rational QR Algorithm for the Computation of the Eigenvalues of an Hermitian, Tridiagonal Matrix

A Modified Bisection Algorithm for the Determination of the Eigenvalues of a Symmetric Quindiagonal Matrix

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