qzvec.f
SUBROUTINE QZVEC (NM, N, A, B, ALFR, ALFI, BETA, Z)
C***BEGIN PROLOGUE QZVEC
C***PURPOSE The optional fourth step of the QZ algorithm for
C generalized eigenproblems. Accepts a matrix in
C quasi-triangular form and another in upper triangular
C and computes the eigenvectors of the triangular problem
C and transforms them back to the original coordinates
C Usually preceded by QZHES, QZIT, and QZVAL.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C3
C***TYPE SINGLE PRECISION (QZVEC-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is the optional fourth step of the QZ algorithm
C for solving generalized matrix eigenvalue problems,
C SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART.
C
C This subroutine accepts a pair of REAL matrices, one of them in
C quasi-triangular form (in which each 2-by-2 block corresponds to
C a pair of complex eigenvalues) and the other in upper triangular
C form. It computes the eigenvectors of the triangular problem and
C transforms the results back to the original coordinate system.
C It is usually preceded by QZHES, QZIT, and QZVAL.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, A, B, and Z, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrices A and B. N is an INTEGER
C variable. N must be less than or equal to NM.
C
C A contains a real upper quasi-triangular matrix. A is a two-
C dimensional REAL array, dimensioned A(NM,N).
C
C B contains a real upper triangular matrix. In addition,
C location B(N,1) contains the tolerance quantity (EPSB)
C computed and saved in QZIT. B is a two-dimensional REAL
C array, dimensioned B(NM,N).
C
C ALFR, ALFI, and BETA are one-dimensional REAL arrays with
C components whose ratios ((ALFR+I*ALFI)/BETA) are the
C generalized eigenvalues. They are usually obtained from
C QZVAL. They are dimensioned ALFR(N), ALFI(N), and BETA(N).
C
C Z contains the transformation matrix produced in the reductions
C by QZHES, QZIT, and QZVAL, if performed. If the
C eigenvectors of the triangular problem are desired, Z must
C contain the identity matrix. Z is a two-dimensional REAL
C array, dimensioned Z(NM,N).
C
C On Output
C
C A is unaltered. Its subdiagonal elements provide information
C about the storage of the complex eigenvectors.
C
C B has been destroyed.
C
C ALFR, ALFI, and BETA are unaltered.
C
C Z contains the real and imaginary parts of the eigenvectors.
C If ALFI(J) .EQ. 0.0, the J-th eigenvalue is real and
C the J-th column of Z contains its eigenvector.
C If ALFI(J) .NE. 0.0, the J-th eigenvalue is complex.
C If ALFI(J) .GT. 0.0, the eigenvalue is the first of
C a complex pair and the J-th and (J+1)-th columns
C of Z contain its eigenvector.
C If ALFI(J) .LT. 0.0, the eigenvalue is the second of
C a complex pair and the (J-1)-th and J-th columns
C of Z contain the conjugate of its eigenvector.
C Each eigenvector is normalized so that the modulus
C of its largest component is 1.0 .
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 (NONE)
C***REVISION HISTORY (YYMMDD)
C 760101 DATE WRITTEN
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 QZVEC