chgeqz - shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation det( A-w(i) B ) = 0 If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form (i.e., A and B are both upper triangular) by applying one unitary tranformation (usually called Q) on the left and another (usually called Z) on the right
SUBROUTINE CHGEQZ(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER*1 JOB, COMPQ, COMPZ COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*) INTEGER N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL RWORK(*) SUBROUTINE CHGEQZ_64(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER*1 JOB, COMPQ, COMPZ COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*) INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL RWORK(*) F95 INTERFACE SUBROUTINE HGEQZ(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, Q, Z INTEGER :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL, DIMENSION(:) :: RWORK SUBROUTINE HGEQZ_64(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, Q, Z INTEGER(8) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL, DIMENSION(:) :: RWORK C INTERFACE #include <sunperf.h> void chgeqz(char job, char compq, char compz, int n, int ilo, int ihi, complex *a, int lda, complex *b, int ldb, complex *alpha, complex *beta, complex *q, int ldq, complex *z, int ldz, int *info); void chgeqz_64(char job, char compq, char compz, long n, long ilo, long ihi, complex *a, long lda, complex *b, long ldb, complex *alpha, complex *beta, complex *q, long ldq, complex *z, long ldz, long *info);
Oracle Solaris Studio Performance Library chgeqz(3P) NAME chgeqz - implement a single-shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation det( A-w(i) B ) = 0 If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form (i.e., A and B are both upper triangular) by applying one unitary tranformation (usually called Q) on the left and another (usually called Z) on the right SYNOPSIS SUBROUTINE CHGEQZ(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER*1 JOB, COMPQ, COMPZ COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*) INTEGER N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL RWORK(*) SUBROUTINE CHGEQZ_64(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER*1 JOB, COMPQ, COMPZ COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*) INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL RWORK(*) F95 INTERFACE SUBROUTINE HGEQZ(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, Q, Z INTEGER :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL, DIMENSION(:) :: RWORK SUBROUTINE HGEQZ_64(JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO) CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ COMPLEX, DIMENSION(:) :: ALPHA, BETA, WORK COMPLEX, DIMENSION(:,:) :: A, B, Q, Z INTEGER(8) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO REAL, DIMENSION(:) :: RWORK C INTERFACE #include <sunperf.h> void chgeqz(char job, char compq, char compz, int n, int ilo, int ihi, complex *a, int lda, complex *b, int ldb, complex *alpha, complex *beta, complex *q, int ldq, complex *z, int ldz, int *info); void chgeqz_64(char job, char compq, char compz, long n, long ilo, long ihi, complex *a, long lda, complex *b, long ldb, complex *alpha, complex *beta, complex *q, long ldq, complex *z, long ldz, long *info); PURPOSE chgeqz implements a single-shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation A are then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N). If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary trans- formations used to reduce (A,B) are accumulated into the arrays Q and Z s.t.: (in) A(in) Z(in)* = Q(out) A(out) Z(out)* Ref: C.B. Moler & G.W. Stewart, "An Algorithm for Generalized Matrixi- genvalue Problems", SIAM J. Numer. Anal., 10(1973),p. 241--256. ARGUMENTS JOB (input) = 'E': compute only ALPHA and BETA. A and B will not neces- sarily be put into generalized Schur form. = 'S': put A and B into generalized Schur form, as well as computing ALPHA and BETA. COMPQ (input) = 'N': do not modify Q. = 'V': multiply the array Q on the right by the conjugate transpose of the unitary tranformation that is applied to the left side of A and B to reduce them to Schur form. = 'I': like COMPQ='V', except that Q will be initialized to the identity first. COMPZ (input) = 'N': do not modify Z. = 'V': multiply the array Z on the right by the unitary tran- formation that is applied to the right side of A and B to reduce them to Schur form. = 'I': like COMPZ='V', except that Z will be initialized to the identity first. N (input) The order of the matrices A, B, Q, and Z. N >= 0. ILO (input) It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. IHI (input) It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0. A (input) On entry, the N-by-N upper Hessenberg matrix A. Elements below the subdiagonal must be zero. If JOB='S', then on exit A and B will have been simultaneously reduced to upper trian- gular form. If JOB='E', then on exit A will have been destroyed. LDA (input) The leading dimension of the array A. LDA >= max( 1, N ). B (input) On entry, the N-by-N upper triangular matrix B. Elements below the diagonal must be zero. If JOB='S', then on exit A and B will have been simultaneously reduced to upper triangu- lar form. If JOB='E', then on exit B will have been destroyed. LDB (input) The leading dimension of the array B. LDB >= max( 1, N ). ALPHA (output) The diagonal elements of A when the pair (A,B) has been reduced to Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues. BETA (output) The diagonal elements of B when the pair (A,B) has been reduced to Schur form. ALPHA(i)/BETA(i) i=1,...,N are the generalized eigenvalues. A and B are normalized so that BETA(1),...,BETA(N) are non-negative real numbers. Q (input/output) If COMPQ='N', then Q will not be referenced. If COMPQ='V' or 'I', then the conjugate transpose of the unitary transforma- tions which are applied to A and B on the left will be applied to the array Q on the right. LDQ (input) The leading dimension of the array Q. LDQ >= 1. If COMPQ='V' or 'I', then LDQ >= N. Z (input/output) If COMPZ='N', then Z will not be referenced. If COMPZ='V' or 'I', then the unitary transformations which are applied to A and B on the right will be applied to the array Z on the right. LDZ (input) The leading dimension of the array Z. LDZ >= 1. If COMPZ='V' or 'I', then LDZ >= N. WORK (workspace) On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. LWORK (input) The dimension of the array WORK. LWORK >= max(1,N). If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA. RWORK (workspace) dimension(N) INFO (output) = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value = 1,...,N: the QZ iteration did not converge. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO+1,...,N should be correct. = N+1,...,2*N: the shift calculation failed. (A,B) is not in Schur form, but ALPHA(i) and BETA(i), i=INFO-N+1,...,N should be correct. > 2*N: var- ious "impossible" errors. FURTHER DETAILS We assume that complex ABS works as long as its value is less than overflow. 7 Nov 2015 chgeqz(3P)