qzit.f
SUBROUTINE QZIT (NM, N, A, B, EPS1, MATZ, Z, IERR)
C***BEGIN PROLOGUE QZIT
C***PURPOSE The second step of the QZ algorithm for generalized
C eigenproblems. Accepts an upper Hessenberg and an upper
C triangular matrix and reduces the former to
C quasi-triangular form while preserving the form of the
C latter. Usually preceded by QZHES and followed by QZVAL
C and QZVEC.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C1B3
C***TYPE SINGLE PRECISION (QZIT-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is the second 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 as modified in technical note NASA TN D-7305(1973) by WARD.
C
C This subroutine accepts a pair of REAL matrices, one of them
C in upper Hessenberg form and the other in upper triangular form.
C It reduces the Hessenberg matrix to quasi-triangular form using
C orthogonal transformations while maintaining the triangular form
C of the other matrix. It is usually preceded by QZHES and
C followed by QZVAL and, possibly, QZVEC.
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 Hessenberg matrix. A is a two-
C dimensional REAL array, dimensioned A(NM,N).
C
C B contains a real upper triangular matrix. B is a two-
C dimensional REAL array, dimensioned B(NM,N).
C
C EPS1 is a tolerance used to determine negligible elements.
C EPS1 = 0.0 (or negative) may be input, in which case an
C element will be neglected only if it is less than roundoff
C error times the norm of its matrix. If the input EPS1 is
C positive, then an element will be considered negligible
C if it is less than EPS1 times the norm of its matrix. A
C positive value of EPS1 may result in faster execution,
C but less accurate results. EPS1 is a REAL variable.
C
C MATZ should be set to .TRUE. if the right hand transformations
C are to be accumulated for later use in computing
C eigenvectors, and to .FALSE. otherwise. MATZ is a LOGICAL
C variable.
C
C Z contains, if MATZ has been set to .TRUE., the transformation
C matrix produced in the reduction by QZHES, if performed, or
C else the identity matrix. If MATZ has been set to .FALSE.,
C Z is not referenced. Z is a two-dimensional REAL array,
C dimensioned Z(NM,N).
C
C On Output
C
C A has been reduced to quasi-triangular form. The elements
C below the first subdiagonal are still zero, and no two
C consecutive subdiagonal elements are nonzero.
C
C B is still in upper triangular form, although its elements
C have been altered. The location B(N,1) is used to store
C EPS1 times the norm of B for later use by QZVAL and QZVEC.
C
C Z contains the product of the right hand transformations
C (for both steps) if MATZ has been set to .TRUE.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C J if neither A(J,J-1) nor A(J-1,J-2) has become
C zero after a total of 30*N iterations.
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 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 QZIT