# 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