ctgsna - ues and/or eigenvectors of a matrix pair (A, B)
SUBROUTINE CTGSNA(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER*1 JOB, HOWMNT COMPLEX A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(*) INTEGER N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER IWORK(*) LOGICAL SELECT(*) REAL S(*), DIF(*) SUBROUTINE CTGSNA_64(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER*1 JOB, HOWMNT COMPLEX A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(*) INTEGER*8 N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER*8 IWORK(*) LOGICAL*8 SELECT(*) REAL S(*), DIF(*) F95 INTERFACE SUBROUTINE TGSNA(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: JOB, HOWMNT COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, VL, VR INTEGER :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER, DIMENSION(:) :: IWORK LOGICAL, DIMENSION(:) :: SELECT REAL, DIMENSION(:) :: S, DIF SUBROUTINE TGSNA_64(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: JOB, HOWMNT COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, VL, VR INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER(8), DIMENSION(:) :: IWORK LOGICAL(8), DIMENSION(:) :: SELECT REAL, DIMENSION(:) :: S, DIF C INTERFACE #include <sunperf.h> void ctgsna(char job, char howmnt, int *select, int n, complex *a, int lda, complex *b, int ldb, complex *vl, int ldvl, complex *vr, int ldvr, float *s, float *dif, int mm, int *m, int *info); void ctgsna_64(char job, char howmnt, long *select, long n, complex *a, long lda, complex *b, long ldb, complex *vl, long ldvl, com- plex *vr, long ldvr, float *s, float *dif, long mm, long *m, long *info);
Oracle Solaris Studio Performance Library ctgsna(3P) NAME ctgsna - estimate reciprocal condition numbers for specified eigenval- ues and/or eigenvectors of a matrix pair (A, B) SYNOPSIS SUBROUTINE CTGSNA(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER*1 JOB, HOWMNT COMPLEX A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(*) INTEGER N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER IWORK(*) LOGICAL SELECT(*) REAL S(*), DIF(*) SUBROUTINE CTGSNA_64(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER*1 JOB, HOWMNT COMPLEX A(LDA,*), B(LDB,*), VL(LDVL,*), VR(LDVR,*), WORK(*) INTEGER*8 N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER*8 IWORK(*) LOGICAL*8 SELECT(*) REAL S(*), DIF(*) F95 INTERFACE SUBROUTINE TGSNA(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: JOB, HOWMNT COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, VL, VR INTEGER :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER, DIMENSION(:) :: IWORK LOGICAL, DIMENSION(:) :: SELECT REAL, DIMENSION(:) :: S, DIF SUBROUTINE TGSNA_64(JOB, HOWMNT, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO) CHARACTER(LEN=1) :: JOB, HOWMNT COMPLEX, DIMENSION(:) :: WORK COMPLEX, DIMENSION(:,:) :: A, B, VL, VR INTEGER(8) :: N, LDA, LDB, LDVL, LDVR, MM, M, LWORK, INFO INTEGER(8), DIMENSION(:) :: IWORK LOGICAL(8), DIMENSION(:) :: SELECT REAL, DIMENSION(:) :: S, DIF C INTERFACE #include <sunperf.h> void ctgsna(char job, char howmnt, int *select, int n, complex *a, int lda, complex *b, int ldb, complex *vl, int ldvl, complex *vr, int ldvr, float *s, float *dif, int mm, int *m, int *info); void ctgsna_64(char job, char howmnt, long *select, long n, complex *a, long lda, complex *b, long ldb, complex *vl, long ldvl, com- plex *vr, long ldvr, float *s, float *dif, long mm, long *m, long *info); PURPOSE ctgsna estimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a matrix pair (A, B). (A, B) must be in generalized Schur canonical form, that is, A and B are both upper triangular. ARGUMENTS JOB (input) Specifies whether condition numbers are required for eigen- values (S) or eigenvectors (DIF): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (DIF); = 'B': for both eigenvalues and eigenvectors (S and DIF). HOWMNT (input) = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT. SELECT (input) If HOWMNT = 'S', SELECT specifies the eigenpairs for which condition numbers are required. To select condition numbers for the corresponding j-th eigenvalue and/or eigenvector, SELECT(j) must be set to .TRUE.. If HOWMNT = 'A', SELECT is not referenced. N (input) The order of the square matrix pair (A, B). N >= 0. A (input) The upper triangular matrix A in the pair (A,B). LDA (input) The leading dimension of the array A. LDA >= max(1,N). B (input) The upper triangular matrix B in the pair (A, B). LDB (input) The leading dimension of the array B. LDB >= max(1,N). VL (input) If JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNT and SELECT. The eigenvectors must be stored in consecutive col- umns of VL, as returned by CTGEVC. If JOB = 'V', VL is not referenced. LDVL (input) The leading dimension of the array VL. LDVL >= 1; and If JOB = 'E' or 'B', LDVL >= N. VR (input) If JOB = 'E' or 'B', VR must contain right eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNT and SELECT. The eigenvectors must be stored in consecutive columns of VR, as returned by CTGEVC. If JOB = 'V', VR is not referenced. LDVR (input) The leading dimension of the array VR. LDVR >= 1; If JOB = 'E' or 'B', LDVR >= N. S (output) If JOB = 'E' or 'B', the reciprocal condition numbers of the selected eigenvalues, stored in consecutive elements of the array. If JOB = 'V', S is not referenced. DIF (output) If JOB = 'V' or 'B', the estimated reciprocal condition num- bers of the selected eigenvectors, stored in consecutive ele- ments of the array. If the eigenvalues cannot be reordered to compute DIF(j), DIF(j) is set to 0; this can only occur when the true value would be very small anyway. For each ei- genvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl. If JOB = 'E', DIF is not refer- enced. MM (input) The number of elements in the arrays S and DIF. MM >= M. M (output) The number of elements of the arrays S and DIF used to store the specified condition numbers; for each selected eigenvalue one element is used. If HOWMNT = 'A', M is set to N. WORK (workspace) If JOB = 'E', WORK is not referenced. Otherwise, on exit, if INFO = 0, WORK(1) returns the optimal LWORK. LWORK (input) The dimension of the array WORK. LWORK >= max(1, N). If JOB = 'V' or 'B', LWORK >= max(1, 2*N*N). IWORK (workspace) dimension(N+2) If JOB = 'E', IWORK is not referenced. INFO (output) = 0: Successful exit < 0: If INFO = -i, the i-th argument had an illegal value FURTHER DETAILS The reciprocal of the condition number of the i-th generalized eigen- value w = (a, b) is defined as S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of (A, B) correspond- ing to w; |z| denotes the absolute value of the complex number, and norm(u) denotes the 2-norm of the vector u. The pair (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the matrix pair (A, B). If both a and b equal zero, then (A,B) is singular and S(I) = -1 is returned. An approximate error bound on the chordal distance between the i-th computed generalized eigenvalue w and the corresponding exact eigenval- ue lambda is chord(w, lambda) <= EPS * norm(A, B) / S(I), where EPS is the machine precision. The reciprocal of the condition number of the right eigenvector u and left eigenvector v corresponding to the generalized eigenvalue w is defined as follows. Suppose (A, B) = ( a * ) ( b * ) 1 ( 0 A22 ),( 0 B22 ) n-1 1 n-1 1 n-1 Then the reciprocal condition number DIF(I) is Difl[(a, b), (A22, B22)] = sigma-min( Zl ) where sigma-min(Zl) denotes the smallest singular value of Zl = [ kron(a, In-1) -kron(1, A22) ] [ kron(b, In-1) -kron(1, B22) ]. Here In-1 is the identity matrix of size n-1 and X' is the conjugate transpose of X. kron(X, Y) is the Kronecker product between the matri- ces X and Y. We approximate the smallest singular value of Zl with an upper bound. This is done by CLATDF. An approximate error bound for a computed eigenvector VL(i) or VR(i) is given by EPS * norm(A, B) / DIF(i). See ref. [2-3] for more details and further references. Based on contributions by Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden. References ========== [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the Generalized Real Schur Form of a Regular Matrix Pair (A, B), in M.S. Moonen et al (eds), Linear Algebra for Large Scale and Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218. [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified Eigenvalues of a Regular Matrix Pair (A, B) and Condition Estimation: Theory, Algorithms and Software, Report UMINF - 94.04, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87. To appear in Numerical Algorithms, 1996. [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software for Solving the Generalized Sylvester Equation and Estimating the Separation between Regular Matrix Pairs, Report UMINF - 93.23, Department of Computing Science, Umea University, S-901 87 Umea, Sweden, December 1993, Revised April 1994, Also as LAPACK Working Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996. 7 Nov 2015 ctgsna(3P)