.TH "SRC/ztgsna.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/ztgsna.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBztgsna\fP (job, howmny, select, n, a, lda, b, ldb, vl, ldvl, vr, ldvr, s, dif, mm, m, work, lwork, iwork, info)" .br .RI "\fBZTGSNA\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine ztgsna (character job, character howmny, logical, dimension( * ) select, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr, double precision, dimension( * ) s, double precision, dimension( * ) dif, integer mm, integer m, complex*16, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer info)" .PP \fBZTGSNA\fP .PP \fBPurpose:\fP .RS 4 .PP .nf ZTGSNA 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\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fIJOB\fP .PP .nf JOB is CHARACTER*1 Specifies whether condition numbers are required for eigenvalues (S) or eigenvectors (DIF): = 'E': for eigenvalues only (S); = 'V': for eigenvectors only (DIF); = 'B': for both eigenvalues and eigenvectors (S and DIF)\&. .fi .PP .br \fIHOWMNY\fP .PP .nf HOWMNY is CHARACTER*1 = 'A': compute condition numbers for all eigenpairs; = 'S': compute condition numbers for selected eigenpairs specified by the array SELECT\&. .fi .PP .br \fISELECT\fP .PP .nf SELECT is LOGICAL array, dimension (N) If HOWMNY = '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 HOWMNY = 'A', SELECT is not referenced\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the square matrix pair (A, B)\&. N >= 0\&. .fi .PP .br \fIA\fP .PP .nf A is COMPLEX*16 array, dimension (LDA,N) The upper triangular matrix A in the pair (A,B)\&. .fi .PP .br \fILDA\fP .PP .nf LDA is INTEGER The leading dimension of the array A\&. LDA >= max(1,N)\&. .fi .PP .br \fIB\fP .PP .nf B is COMPLEX*16 array, dimension (LDB,N) The upper triangular matrix B in the pair (A, B)\&. .fi .PP .br \fILDB\fP .PP .nf LDB is INTEGER The leading dimension of the array B\&. LDB >= max(1,N)\&. .fi .PP .br \fIVL\fP .PP .nf VL is COMPLEX*16 array, dimension (LDVL,M) IF JOB = 'E' or 'B', VL must contain left eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT\&. The eigenvectors must be stored in consecutive columns of VL, as returned by ZTGEVC\&. If JOB = 'V', VL is not referenced\&. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of the array VL\&. LDVL >= 1; and If JOB = 'E' or 'B', LDVL >= N\&. .fi .PP .br \fIVR\fP .PP .nf VR is COMPLEX*16 array, dimension (LDVR,M) IF JOB = 'E' or 'B', VR must contain right eigenvectors of (A, B), corresponding to the eigenpairs specified by HOWMNY and SELECT\&. The eigenvectors must be stored in consecutive columns of VR, as returned by ZTGEVC\&. If JOB = 'V', VR is not referenced\&. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR\&. LDVR >= 1; If JOB = 'E' or 'B', LDVR >= N\&. .fi .PP .br \fIS\fP .PP .nf S is DOUBLE PRECISION array, dimension (MM) 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\&. .fi .PP .br \fIDIF\fP .PP .nf DIF is DOUBLE PRECISION array, dimension (MM) If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the selected eigenvectors, stored in consecutive elements 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 eigenvalue/vector specified by SELECT, DIF stores a Frobenius norm-based estimate of Difl\&. If JOB = 'E', DIF is not referenced\&. .fi .PP .br \fIMM\fP .PP .nf MM is INTEGER The number of elements in the arrays S and DIF\&. MM >= M\&. .fi .PP .br \fIM\fP .PP .nf M is INTEGER 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 HOWMNY = 'A', M is set to N\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK\&. .fi .PP .br \fILWORK\fP .PP .nf LWORK is INTEGER The dimension of the array WORK\&. LWORK >= max(1,N)\&. If JOB = 'V' or 'B', LWORK >= max(1,2*N*N)\&. .fi .PP .br \fIIWORK\fP .PP .nf IWORK is INTEGER array, dimension (N+2) If JOB = 'E', IWORK is not referenced\&. .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: Successful exit < 0: If INFO = -i, the i-th argument had an illegal value .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf The reciprocal of the condition number of the i-th generalized eigenvalue w = (a, b) is defined as S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v)) where u and v are the right and left eigenvectors of (A, B) corresponding 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**HAu/v**HBu) 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 eigenvalue 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**H is the conjugate transpose of X\&. kron(X, Y) is the Kronecker product between the matrices X and Y\&. We approximate the smallest singular value of Zl with an upper bound\&. This is done by ZLATDF\&. 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\&. .fi .PP .RE .PP \fBContributors:\fP .RS 4 Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden\&. .RE .PP \fBReferences:\fP .RS 4 .PP .nf [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\&. .fi .PP .RE .PP .PP Definition at line \fB308\fP of file \fBztgsna\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.