ctgsna-f(3)

ctgsna.f –

Synopsis

Functions/Subroutines

subroutine ctgsna (JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, IWORK, INFO)
CTGSNA

Function/Subroutine Documentation

subroutine ctgsna (characterJOB, characterHOWMNY, logical, dimension( * )SELECT, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldvl, * )VL, integerLDVL, complex, dimension( ldvr, * )VR, integerLDVR, real, dimension( * )S, real, dimension( * )DIF, integerMM, integerM, complex, dimension( * )WORK, integerLWORK, integer, dimension( * )IWORK, integerINFO)

CTGSNA

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.

Parameters:

JOB

 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).

HOWMNY

 HOWMNY is CHARACTER*1
 = 'A': compute condition numbers for all eigenpairs;
 = 'S': compute condition numbers for selected eigenpairs
 specified by the array SELECT.

SELECT

 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.

 N is INTEGER
 The order of the square matrix pair (A, B). N >= 0.

 A is COMPLEX array, dimension (LDA,N)
 The upper triangular matrix A in the pair (A,B).

LDA

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max(1,N).

 B is COMPLEX array, dimension (LDB,N)
 The upper triangular matrix B in the pair (A, B).

LDB

 LDB is INTEGER
 The leading dimension of the array B. LDB >= max(1,N).

 VL is COMPLEX 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 CTGEVC.
 If JOB = 'V', VL is not referenced.

LDVL

 LDVL is INTEGER
 The leading dimension of the array VL. LDVL >= 1; and
 If JOB = 'E' or 'B', LDVL >= N.

 VR is COMPLEX 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 CTGEVC.
 If JOB = 'V', VR is not referenced.

LDVR

 LDVR is INTEGER
 The leading dimension of the array VR. LDVR >= 1;
 If JOB = 'E' or 'B', LDVR >= N.

 S is REAL 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.

DIF

 DIF is REAL 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.

 MM is INTEGER
 The number of elements in the arrays S and DIF. MM >= M.

 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.

WORK

 WORK is COMPLEX array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

 LWORK is INTEGER
 The dimension of the array WORK. LWORK >= max(1,N).
 If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).

IWORK

 IWORK is INTEGER array, dimension (N+2)
 If JOB = 'E', IWORK is not referenced.

INFO

 INFO is INTEGER
 = 0: Successful exit
 < 0: If INFO = -i, the i-th argument had an illegal value

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Further Details:

 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 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.

Contributors:

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.

Definition at line 310 of file ctgsna.f.

Author

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