*> \brief \b CTGSNA * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CTGSNA + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, * LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, * IWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER HOWMNY, JOB * INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N * .. * .. Array Arguments .. * LOGICAL SELECT( * ) * INTEGER IWORK( * ) * REAL DIF( * ), S( * ) * COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ), * $ VR( LDVR, * ), WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] JOB *> \verbatim *> 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). *> \endverbatim *> *> \param[in] HOWMNY *> \verbatim *> HOWMNY is CHARACTER*1 *> = 'A': compute condition numbers for all eigenpairs; *> = 'S': compute condition numbers for selected eigenpairs *> specified by the array SELECT. *> \endverbatim *> *> \param[in] SELECT *> \verbatim *> 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. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the square matrix pair (A, B). N >= 0. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is COMPLEX array, dimension (LDA,N) *> The upper triangular matrix A in the pair (A,B). *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] B *> \verbatim *> B is COMPLEX array, dimension (LDB,N) *> The upper triangular matrix B in the pair (A, B). *> \endverbatim *> *> \param[in] LDB *> \verbatim *> LDB is INTEGER *> The leading dimension of the array B. LDB >= max(1,N). *> \endverbatim *> *> \param[in] VL *> \verbatim *> 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. *> \endverbatim *> *> \param[in] LDVL *> \verbatim *> LDVL is INTEGER *> The leading dimension of the array VL. LDVL >= 1; and *> If JOB = 'E' or 'B', LDVL >= N. *> \endverbatim *> *> \param[in] VR *> \verbatim *> 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. *> \endverbatim *> *> \param[in] LDVR *> \verbatim *> LDVR is INTEGER *> The leading dimension of the array VR. LDVR >= 1; *> If JOB = 'E' or 'B', LDVR >= N. *> \endverbatim *> *> \param[out] S *> \verbatim *> 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. *> \endverbatim *> *> \param[out] DIF *> \verbatim *> 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. *> \endverbatim *> *> \param[in] MM *> \verbatim *> MM is INTEGER *> The number of elements in the arrays S and DIF. MM >= M. *> \endverbatim *> *> \param[out] M *> \verbatim *> 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. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is COMPLEX array, dimension (MAX(1,LWORK)) *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of the array WORK. LWORK >= max(1,N). *> If JOB = 'V' or 'B', LWORK >= max(1,2*N*N). *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (N+2) *> If JOB = 'E', IWORK is not referenced. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: Successful exit *> < 0: If INFO = -i, the i-th argument had an illegal value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date November 2011 * *> \ingroup complexOTHERcomputational * *> \par Further Details: * ===================== *> *> \verbatim *> *> 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. *> \endverbatim * *> \par Contributors: * ================== *> *> Bo Kagstrom and Peter Poromaa, Department of Computing Science, *> Umea University, S-901 87 Umea, Sweden. * *> \par References: * ================ *> *> \verbatim *> *> [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. *> \endverbatim *> * ===================================================================== SUBROUTINE CTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL, $ LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK, $ IWORK, INFO ) * * -- LAPACK computational routine (version 3.4.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2011 * * .. Scalar Arguments .. CHARACTER HOWMNY, JOB INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N * .. * .. Array Arguments .. LOGICAL SELECT( * ) INTEGER IWORK( * ) REAL DIF( * ), S( * ) COMPLEX A( LDA, * ), B( LDB, * ), VL( LDVL, * ), $ VR( LDVR, * ), WORK( * ) * .. * * ===================================================================== * * .. Parameters .. REAL ZERO, ONE INTEGER IDIFJB PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, IDIFJB = 3 ) * .. * .. Local Scalars .. LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2 REAL BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM COMPLEX YHAX, YHBX * .. * .. Local Arrays .. COMPLEX DUMMY( 1 ), DUMMY1( 1 ) * .. * .. External Functions .. LOGICAL LSAME REAL SCNRM2, SLAMCH, SLAPY2 COMPLEX CDOTC EXTERNAL LSAME, SCNRM2, SLAMCH, SLAPY2, CDOTC * .. * .. External Subroutines .. EXTERNAL CGEMV, CLACPY, CTGEXC, CTGSYL, SLABAD, XERBLA * .. * .. Intrinsic Functions .. INTRINSIC ABS, CMPLX, MAX * .. * .. Executable Statements .. * * Decode and test the input parameters * WANTBH = LSAME( JOB, 'B' ) WANTS = LSAME( JOB, 'E' ) .OR. WANTBH WANTDF = LSAME( JOB, 'V' ) .OR. WANTBH * SOMCON = LSAME( HOWMNY, 'S' ) * INFO = 0 LQUERY = ( LWORK.EQ.-1 ) * IF( .NOT.WANTS .AND. .NOT.WANTDF ) THEN INFO = -1 ELSE IF( .NOT.LSAME( HOWMNY, 'A' ) .AND. .NOT.SOMCON ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -8 ELSE IF( WANTS .AND. LDVL.LT.N ) THEN INFO = -10 ELSE IF( WANTS .AND. LDVR.LT.N ) THEN INFO = -12 ELSE * * Set M to the number of eigenpairs for which condition numbers * are required, and test MM. * IF( SOMCON ) THEN M = 0 DO 10 K = 1, N IF( SELECT( K ) ) $ M = M + 1 10 CONTINUE ELSE M = N END IF * IF( N.EQ.0 ) THEN LWMIN = 1 ELSE IF( LSAME( JOB, 'V' ) .OR. LSAME( JOB, 'B' ) ) THEN LWMIN = 2*N*N ELSE LWMIN = N END IF WORK( 1 ) = LWMIN * IF( MM.LT.M ) THEN INFO = -15 ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN INFO = -18 END IF END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'CTGSNA', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) $ RETURN * * Get machine constants * EPS = SLAMCH( 'P' ) SMLNUM = SLAMCH( 'S' ) / EPS BIGNUM = ONE / SMLNUM CALL SLABAD( SMLNUM, BIGNUM ) KS = 0 DO 20 K = 1, N * * Determine whether condition numbers are required for the k-th * eigenpair. * IF( SOMCON ) THEN IF( .NOT.SELECT( K ) ) $ GO TO 20 END IF * KS = KS + 1 * IF( WANTS ) THEN * * Compute the reciprocal condition number of the k-th * eigenvalue. * RNRM = SCNRM2( N, VR( 1, KS ), 1 ) LNRM = SCNRM2( N, VL( 1, KS ), 1 ) CALL CGEMV( 'N', N, N, CMPLX( ONE, ZERO ), A, LDA, $ VR( 1, KS ), 1, CMPLX( ZERO, ZERO ), WORK, 1 ) YHAX = CDOTC( N, WORK, 1, VL( 1, KS ), 1 ) CALL CGEMV( 'N', N, N, CMPLX( ONE, ZERO ), B, LDB, $ VR( 1, KS ), 1, CMPLX( ZERO, ZERO ), WORK, 1 ) YHBX = CDOTC( N, WORK, 1, VL( 1, KS ), 1 ) COND = SLAPY2( ABS( YHAX ), ABS( YHBX ) ) IF( COND.EQ.ZERO ) THEN S( KS ) = -ONE ELSE S( KS ) = COND / ( RNRM*LNRM ) END IF END IF * IF( WANTDF ) THEN IF( N.EQ.1 ) THEN DIF( KS ) = SLAPY2( ABS( A( 1, 1 ) ), ABS( B( 1, 1 ) ) ) ELSE * * Estimate the reciprocal condition number of the k-th * eigenvectors. * * Copy the matrix (A, B) to the array WORK and move the * (k,k)th pair to the (1,1) position. * CALL CLACPY( 'Full', N, N, A, LDA, WORK, N ) CALL CLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N ) IFST = K ILST = 1 * CALL CTGEXC( .FALSE., .FALSE., N, WORK, N, WORK( N*N+1 ), $ N, DUMMY, 1, DUMMY1, 1, IFST, ILST, IERR ) * IF( IERR.GT.0 ) THEN * * Ill-conditioned problem - swap rejected. * DIF( KS ) = ZERO ELSE * * Reordering successful, solve generalized Sylvester * equation for R and L, * A22 * R - L * A11 = A12 * B22 * R - L * B11 = B12, * and compute estimate of Difl[(A11,B11), (A22, B22)]. * N1 = 1 N2 = N - N1 I = N*N + 1 CALL CTGSYL( 'N', IDIFJB, N2, N1, WORK( N*N1+N1+1 ), $ N, WORK, N, WORK( N1+1 ), N, $ WORK( N*N1+N1+I ), N, WORK( I ), N, $ WORK( N1+I ), N, SCALE, DIF( KS ), DUMMY, $ 1, IWORK, IERR ) END IF END IF END IF * 20 CONTINUE WORK( 1 ) = LWMIN RETURN * * End of CTGSNA * END