HQR(3)

SYNOPSIS
SUBROUTINE HQR (NM, N, LOW, IGH, H, WR, WI, IERR)

NAME
hqr - Compute the eigenvalues of a real upper Hessenberg matrix using the QR method.

LIBRARY
SLATEC (EISPACK)

CATEGORY
D4C2B

TYPE
SINGLE PRECISION (HQR-S, COMQR-C)

KEYWORDS
EIGENVALUES, EIGENVECTORS, EISPACK

AUTHOR
Smith, B. T., et al.

DESCRIPTION
This subroutine is a translation of the ALGOL procedure HQR, NUM. MATH. 14, 219-231(1970) by Martin, Peters, and Wilkinson. HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 359-371(1971). This subroutine finds the eigenvalues of a REAL UPPER Hessenberg matrix by the QR method. On INPUT NM must be set to the row dimension of the two-dimensional array parameter, H, as declared in the calling program dimension statement. NM is an INTEGER variable. N is the order of the matrix H. N is an INTEGER variable. N must be less than or equal to NM. LOW and IGH are two INTEGER variables determined by the balancing subroutine BALANC. If BALANC has not been used, set LOW=1 and IGH equal to the order of the matrix, N. H contains the upper Hessenberg matrix. Information about the transformations used in the reduction to Hessenberg form by ELMHES or ORTHES, if performed, is stored in the remaining triangle under the Hessenberg matrix. H is a two-dimensional REAL array, dimensioned H(NM,N). On OUTPUT H has been destroyed. Therefore, it must be saved before calling HQR if subsequent calculation and back transformation of eigenvectors is to be performed. WR and WI contain the real and imaginary parts, respectively, of the eigenvalues. The eigenvalues are unordered except that complex conjugate pairs of values appear consecutively with the eigenvalue having the positive imaginary part first. If an error exit is made, the eigenvalues should be correct for indices IERR+1, IERR+2, ..., N. WR and WI are one- dimensional REAL arrays, dimensioned WR(N) and WI(N). IERR is an INTEGER flag set to Zero for normal return, J if the J-th eigenvalue has not been determined after a total of 30*N iterations. The eigenvalues should be correct for indices IERR+1, IERR+2, ..., N. Questions and comments should be directed to B. S. Garbow, APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY ------------------------------------------------------------------

REFERENCES
B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen- system Routines - EISPACK Guide, Springer-Verlag, 1976.

ROUTINES CALLED
(NONE)

REVISION HISTORY (YYMMDD)
760101 DATE WRITTEN
890531 Changed all specific intrinsics to generic. (WRB)
890831 Modified array declarations. (WRB)
890831 REVISION DATE from Version 3.2
891214 Prologue converted to Version 4.0 format. (BAB)
920501 Reformatted the REFERENCES section. (WRB)