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