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