SLATEC Routines --- INVIT ---


*DECK INVIT
      SUBROUTINE INVIT (NM, N, A, WR, WI, SELECT, MM, M, Z, IERR, RM1,
     +   RV1, RV2)
C***BEGIN PROLOGUE  INVIT
C***PURPOSE  Compute the eigenvectors of a real upper Hessenberg
C            matrix associated with specified eigenvalues by inverse
C            iteration.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4C2B
C***TYPE      SINGLE PRECISION (INVIT-S, CINVIT-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 INVIT
C     by Peters and Wilkinson.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 418-439(1971).
C
C     This subroutine finds those eigenvectors of a REAL UPPER
C     Hessenberg matrix corresponding to specified eigenvalues,
C     using inverse iteration.
C
C     On INPUT
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, A 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 A.  N is an INTEGER variable.
C          N must be less than or equal to NM.
C
C        A contains the upper Hessenberg matrix.  A is a two-dimensional
C          REAL array, dimensioned A(NM,N).
C
C        WR and WI contain the real and imaginary parts, respectively,
C          of the eigenvalues of the Hessenberg matrix.  The eigenvalues
C          must be stored in a manner identical to that output by
C          subroutine  HQR,  which recognizes possible splitting of the
C          matrix.  WR and WI are one-dimensional REAL arrays,
C          dimensioned WR(N) and WI(N).
C
C        SELECT specifies the eigenvectors to be found. The
C          eigenvector corresponding to the J-th eigenvalue is
C          specified by setting SELECT(J) to .TRUE.  SELECT is a
C          one-dimensional LOGICAL array, dimensioned SELECT(N).
C
C        MM should be set to an upper bound for the number of
C          columns required to store the eigenvectors to be found.
C          NOTE that two columns are required to store the
C          eigenvector corresponding to a complex eigenvalue.  One
C          column is required to store the eigenvector corresponding
C          to a real eigenvalue.  MM is an INTEGER variable.
C
C     On OUTPUT
C
C        A and WI are unaltered.
C
C        WR may have been altered since close eigenvalues are perturbed
C          slightly in searching for independent eigenvectors.
C
C        SELECT may have been altered.  If the elements corresponding
C          to a pair of conjugate complex eigenvalues were each
C          initially set to .TRUE., the program resets the second of
C          the two elements to .FALSE.
C
C        M is the number of columns actually used to store the
C          eigenvectors.  M is an INTEGER variable.
C
C        Z contains the real and imaginary parts of the eigenvectors.
C          The eigenvectors are packed into the columns of Z starting
C          at the first column.  If the next selected eigenvalue is
C          real, the next column of Z contains its eigenvector.  If the
C          eigenvalue is complex, the next two columns of Z contain the
C          real and imaginary parts of its eigenvector, with the real
C          part first.  The eigenvectors are normalized so that the
C          component of largest magnitude is 1. Any vector which fails
C          the acceptance test is set to zero.  Z is a two-dimensional
C          REAL array, dimensioned Z(NM,MM).
C
C        IERR is an INTEGER flag set to
C          Zero       for normal return,
C          -(2*N+1)   if more than MM columns of Z are necessary
C                     to store the eigenvectors corresponding to
C                     the specified eigenvalues (in this case, M is
C                     equal to the number of columns of Z containing
C                     eigenvectors already computed),
C          -K         if the iteration corresponding to the K-th
C                     value fails (if this occurs more than once, K
C                     is the index of the last occurrence); the
C                     corresponding columns of Z are set to zero
C                     vectors,
C          -(N+K)     if both error situations occur.
C
C        RM1 is a two-dimensional REAL array used for temporary storage.
C          This array holds the triangularized form of the upper
C          Hessenberg matrix used in the inverse iteration process.
C          RM1 is dimensioned RM1(N,N).
C
C        RV1 and RV2 are one-dimensional REAL arrays used for temporary
C          storage.  They hold the approximate eigenvectors during the
C          inverse iteration process.  RV1 and RV2 are dimensioned
C          RV1(N) and RV2(N).
C
C     The ALGOL procedure GUESSVEC appears in INVIT in-line.
C
C     Calls PYTHAG(A,B) for sqrt(A**2 + B**2).
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, PYTHAG
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  INVIT