398 lines
14 KiB
FortranFixed
398 lines
14 KiB
FortranFixed
SUBROUTINE ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
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$ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
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$ IWORK, INFO )
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*
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* -- LAPACK routine (version 3.1) --
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* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
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* November 2006
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*
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* .. Scalar Arguments ..
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CHARACTER HOWMNY, JOB
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INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
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* ..
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* .. Array Arguments ..
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LOGICAL SELECT( * )
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INTEGER IWORK( * )
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DOUBLE PRECISION DIF( * ), S( * )
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COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
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$ VR( LDVR, * ), WORK( * )
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* ..
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*
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* Purpose
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* =======
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*
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* ZTGSNA estimates reciprocal condition numbers for specified
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* eigenvalues and/or eigenvectors of a matrix pair (A, B).
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*
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* (A, B) must be in generalized Schur canonical form, that is, A and
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* B are both upper triangular.
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*
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* Arguments
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* =========
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*
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* JOB (input) CHARACTER*1
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* Specifies whether condition numbers are required for
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* eigenvalues (S) or eigenvectors (DIF):
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* = 'E': for eigenvalues only (S);
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* = 'V': for eigenvectors only (DIF);
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* = 'B': for both eigenvalues and eigenvectors (S and DIF).
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*
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* HOWMNY (input) CHARACTER*1
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* = 'A': compute condition numbers for all eigenpairs;
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* = 'S': compute condition numbers for selected eigenpairs
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* specified by the array SELECT.
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*
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* SELECT (input) LOGICAL array, dimension (N)
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* If HOWMNY = 'S', SELECT specifies the eigenpairs for which
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* condition numbers are required. To select condition numbers
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* for the corresponding j-th eigenvalue and/or eigenvector,
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* SELECT(j) must be set to .TRUE..
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* If HOWMNY = 'A', SELECT is not referenced.
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*
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* N (input) INTEGER
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* The order of the square matrix pair (A, B). N >= 0.
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*
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* A (input) COMPLEX*16 array, dimension (LDA,N)
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* The upper triangular matrix A in the pair (A,B).
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*
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* LDA (input) INTEGER
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* The leading dimension of the array A. LDA >= max(1,N).
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*
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* B (input) COMPLEX*16 array, dimension (LDB,N)
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* The upper triangular matrix B in the pair (A, B).
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*
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* LDB (input) INTEGER
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* The leading dimension of the array B. LDB >= max(1,N).
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*
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* VL (input) COMPLEX*16 array, dimension (LDVL,M)
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* IF JOB = 'E' or 'B', VL must contain left eigenvectors of
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* (A, B), corresponding to the eigenpairs specified by HOWMNY
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* and SELECT. The eigenvectors must be stored in consecutive
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* columns of VL, as returned by ZTGEVC.
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* If JOB = 'V', VL is not referenced.
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*
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* LDVL (input) INTEGER
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* The leading dimension of the array VL. LDVL >= 1; and
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* If JOB = 'E' or 'B', LDVL >= N.
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*
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* VR (input) COMPLEX*16 array, dimension (LDVR,M)
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* IF JOB = 'E' or 'B', VR must contain right eigenvectors of
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* (A, B), corresponding to the eigenpairs specified by HOWMNY
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* and SELECT. The eigenvectors must be stored in consecutive
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* columns of VR, as returned by ZTGEVC.
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* If JOB = 'V', VR is not referenced.
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*
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* LDVR (input) INTEGER
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* The leading dimension of the array VR. LDVR >= 1;
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* If JOB = 'E' or 'B', LDVR >= N.
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*
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* S (output) DOUBLE PRECISION array, dimension (MM)
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* If JOB = 'E' or 'B', the reciprocal condition numbers of the
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* selected eigenvalues, stored in consecutive elements of the
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* array.
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* If JOB = 'V', S is not referenced.
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*
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* DIF (output) DOUBLE PRECISION array, dimension (MM)
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* If JOB = 'V' or 'B', the estimated reciprocal condition
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* numbers of the selected eigenvectors, stored in consecutive
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* elements of the array.
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* If the eigenvalues cannot be reordered to compute DIF(j),
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* DIF(j) is set to 0; this can only occur when the true value
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* would be very small anyway.
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* For each eigenvalue/vector specified by SELECT, DIF stores
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* a Frobenius norm-based estimate of Difl.
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* If JOB = 'E', DIF is not referenced.
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*
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* MM (input) INTEGER
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* The number of elements in the arrays S and DIF. MM >= M.
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*
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* M (output) INTEGER
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* The number of elements of the arrays S and DIF used to store
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* the specified condition numbers; for each selected eigenvalue
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* one element is used. If HOWMNY = 'A', M is set to N.
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*
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* WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
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* On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
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*
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* LWORK (input) INTEGER
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* The dimension of the array WORK. LWORK >= max(1,N).
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* If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
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*
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* IWORK (workspace) INTEGER array, dimension (N+2)
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* If JOB = 'E', IWORK is not referenced.
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*
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* INFO (output) INTEGER
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* = 0: Successful exit
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* < 0: If INFO = -i, the i-th argument had an illegal value
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*
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* Further Details
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* ===============
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*
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* The reciprocal of the condition number of the i-th generalized
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* eigenvalue w = (a, b) is defined as
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*
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* S(I) = (|v'Au|**2 + |v'Bu|**2)**(1/2) / (norm(u)*norm(v))
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*
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* where u and v are the right and left eigenvectors of (A, B)
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* corresponding to w; |z| denotes the absolute value of the complex
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* number, and norm(u) denotes the 2-norm of the vector u. The pair
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* (a, b) corresponds to an eigenvalue w = a/b (= v'Au/v'Bu) of the
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* matrix pair (A, B). If both a and b equal zero, then (A,B) is
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* singular and S(I) = -1 is returned.
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*
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* An approximate error bound on the chordal distance between the i-th
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* computed generalized eigenvalue w and the corresponding exact
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* eigenvalue lambda is
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*
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* chord(w, lambda) <= EPS * norm(A, B) / S(I),
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*
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* where EPS is the machine precision.
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*
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* The reciprocal of the condition number of the right eigenvector u
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* and left eigenvector v corresponding to the generalized eigenvalue w
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* is defined as follows. Suppose
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*
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* (A, B) = ( a * ) ( b * ) 1
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* ( 0 A22 ),( 0 B22 ) n-1
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* 1 n-1 1 n-1
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*
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* Then the reciprocal condition number DIF(I) is
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*
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* Difl[(a, b), (A22, B22)] = sigma-min( Zl )
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*
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* where sigma-min(Zl) denotes the smallest singular value of
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*
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* Zl = [ kron(a, In-1) -kron(1, A22) ]
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* [ kron(b, In-1) -kron(1, B22) ].
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*
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* Here In-1 is the identity matrix of size n-1 and X' is the conjugate
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* transpose of X. kron(X, Y) is the Kronecker product between the
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* matrices X and Y.
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*
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* We approximate the smallest singular value of Zl with an upper
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* bound. This is done by ZLATDF.
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*
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* An approximate error bound for a computed eigenvector VL(i) or
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* VR(i) is given by
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*
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* EPS * norm(A, B) / DIF(i).
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*
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* See ref. [2-3] for more details and further references.
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*
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* Based on contributions by
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* Bo Kagstrom and Peter Poromaa, Department of Computing Science,
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* Umea University, S-901 87 Umea, Sweden.
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*
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* References
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* ==========
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*
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* [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
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* Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
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* M.S. Moonen et al (eds), Linear Algebra for Large Scale and
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* Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
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*
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* [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
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* Eigenvalues of a Regular Matrix Pair (A, B) and Condition
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* Estimation: Theory, Algorithms and Software, Report
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* UMINF - 94.04, Department of Computing Science, Umea University,
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* S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
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* To appear in Numerical Algorithms, 1996.
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*
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* [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
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* for Solving the Generalized Sylvester Equation and Estimating the
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* Separation between Regular Matrix Pairs, Report UMINF - 93.23,
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* Department of Computing Science, Umea University, S-901 87 Umea,
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* Sweden, December 1993, Revised April 1994, Also as LAPACK Working
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* Note 75.
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* To appear in ACM Trans. on Math. Software, Vol 22, No 1, 1996.
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*
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* =====================================================================
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*
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* .. Parameters ..
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DOUBLE PRECISION ZERO, ONE
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INTEGER IDIFJB
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PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 )
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* ..
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* .. Local Scalars ..
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LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
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INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
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DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
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COMPLEX*16 YHAX, YHBX
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* ..
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* .. Local Arrays ..
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COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 )
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2
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COMPLEX*16 ZDOTC
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EXTERNAL LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC
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* ..
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* .. External Subroutines ..
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EXTERNAL DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, DCMPLX, MAX
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* ..
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* .. Executable Statements ..
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*
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* Decode and test the input parameters
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*
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WANTBH = LSAME( JOB, 'B' )
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WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
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WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH
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*
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SOMCON = LSAME( HOWMNY, 'S' )
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*
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INFO = 0
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LQUERY = ( LWORK.EQ.-1 )
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*
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IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN
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INFO = -1
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ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -6
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( WANTS .AND. LDVL.LT.N ) THEN
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INFO = -10
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ELSE IF( WANTS .AND. LDVR.LT.N ) THEN
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INFO = -12
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ELSE
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*
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* Set M to the number of eigenpairs for which condition numbers
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* are required, and test MM.
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*
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IF( SOMCON ) THEN
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M = 0
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DO 10 K = 1, N
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IF( SELECT( K ) )
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$ M = M + 1
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10 CONTINUE
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ELSE
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M = N
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END IF
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*
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IF( N.EQ.0 ) THEN
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LWMIN = 1
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ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN
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LWMIN = 2*N*N
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ELSE
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LWMIN = N
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END IF
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WORK( 1 ) = LWMIN
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*
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IF( MM.LT.M ) THEN
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INFO = -15
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ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
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INFO = -18
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END IF
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZTGSNA', -INFO )
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RETURN
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ELSE IF( LQUERY ) THEN
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Get machine constants
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*
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EPS = DLAMCH( 'P' )
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SMLNUM = DLAMCH( 'S' ) / EPS
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BIGNUM = ONE / SMLNUM
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CALL DLABAD( SMLNUM, BIGNUM )
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KS = 0
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DO 20 K = 1, N
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*
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* Determine whether condition numbers are required for the k-th
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* eigenpair.
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*
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IF( SOMCON ) THEN
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IF( .NOT.SELECT( K ) )
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$ GO TO 20
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END IF
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*
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KS = KS + 1
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*
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IF( WANTS ) THEN
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*
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* Compute the reciprocal condition number of the k-th
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* eigenvalue.
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*
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RNRM = DZNRM2( N, VR( 1, KS ), 1 )
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LNRM = DZNRM2( N, VL( 1, KS ), 1 )
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CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA,
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$ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
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YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
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CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB,
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$ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
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YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
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COND = DLAPY2( ABS( YHAX ), ABS( YHBX ) )
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IF( COND.EQ.ZERO ) THEN
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S( KS ) = -ONE
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ELSE
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S( KS ) = COND / ( RNRM*LNRM )
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END IF
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END IF
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*
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IF( WANTDF ) THEN
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IF( N.EQ.1 ) THEN
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DIF( KS ) = DLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) )
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ELSE
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*
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* Estimate the reciprocal condition number of the k-th
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* eigenvectors.
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*
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* Copy the matrix (A, B) to the array WORK and move the
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* (k,k)th pair to the (1,1) position.
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*
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CALL ZLACPY( 'Full', N, N, A, LDA, WORK, N )
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CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
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IFST = K
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ILST = 1
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*
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CALL ZTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ),
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$ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR )
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*
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IF( IERR.GT.0 ) THEN
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*
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* Ill-conditioned problem - swap rejected.
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*
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DIF( KS ) = ZERO
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ELSE
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*
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* Reordering successful, solve generalized Sylvester
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* equation for R and L,
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* A22 * R - L * A11 = A12
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* B22 * R - L * B11 = B12,
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* and compute estimate of Difl[(A11,B11), (A22, B22)].
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*
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N1 = 1
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N2 = N - N1
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I = N*N + 1
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CALL ZTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ),
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$ N, WORK, N, WORK( N1+1 ), N,
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$ WORK( N*N1+N1+I ), N, WORK( I ), N,
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$ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY,
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$ 1, IWORK, IERR )
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END IF
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END IF
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END IF
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*
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20 CONTINUE
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WORK( 1 ) = LWMIN
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RETURN
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*
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* End of ZTGSNA
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*
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END
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