hqr2.f
SUBROUTINE HQR2 (NM, N, LOW, IGH, H, WR, WI, Z, IERR)
C***BEGIN PROLOGUE HQR2
C***PURPOSE Compute the eigenvalues and eigenvectors of a real upper
C Hessenberg matrix using QR method.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C2B
C***TYPE SINGLE PRECISION (HQR2-S, COMQR2-C)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure HQR2,
C NUM. MATH. 16, 181-204(1970) by Peters and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C
C This subroutine finds the eigenvalues and eigenvectors
C of a REAL UPPER Hessenberg matrix by the QR method. The
C eigenvectors of a REAL GENERAL matrix can also be found
C if ELMHES and ELTRAN or ORTHES and ORTRAN have
C been used to reduce this general matrix to Hessenberg form
C and to accumulate the similarity transformations.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, H and Z, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix H. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C LOW and IGH are two INTEGER variables determined by the
C balancing subroutine BALANC. If BALANC has not been
C used, set LOW=1 and IGH equal to the order of the matrix, N.
C
C H contains the upper Hessenberg matrix. H is a two-dimensional
C REAL array, dimensioned H(NM,N).
C
C Z contains the transformation matrix produced by ELTRAN
C after the reduction by ELMHES, or by ORTRAN after the
C reduction by ORTHES, if performed. If the eigenvectors
C of the Hessenberg matrix are desired, Z must contain the
C identity matrix. Z is a two-dimensional REAL array,
C dimensioned Z(NM,M).
C
C On OUTPUT
C
C H has been destroyed.
C
C WR and WI contain the real and imaginary parts, respectively,
C of the eigenvalues. The eigenvalues are unordered except
C that complex conjugate pairs of values appear consecutively
C with the eigenvalue having the positive imaginary part first.
C If an error exit is made, the eigenvalues should be correct
C for indices IERR+1, IERR+2, ..., N. WR and WI are one-
C dimensional REAL arrays, dimensioned WR(N) and WI(N).
C
C Z contains the real and imaginary parts of the eigenvectors.
C If the J-th eigenvalue is real, the J-th column of Z
C contains its eigenvector. If the J-th eigenvalue is complex
C with positive imaginary part, the J-th and (J+1)-th
C columns of Z contain the real and imaginary parts of its
C eigenvector. The eigenvectors are unnormalized. If an
C error exit is made, none of the eigenvectors has been found.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C J if the J-th eigenvalue has not been
C determined after a total of 30*N iterations.
C The eigenvalues should be correct for indices
C IERR+1, IERR+2, ..., N, but no eigenvectors are
C computed.
C
C Calls CDIV for complex division.
C
C Questions and comments should be directed to B. S. Garbow,
C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C ------------------------------------------------------------------
C
C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C system Routines - EISPACK Guide, Springer-Verlag,
C 1976.
C***ROUTINES CALLED CDIV
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE HQR2