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OpenBLAS/lapack-netlib/SRC/ztgsna.f

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*> \brief \b ZTGSNA
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZTGSNA + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsna.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztgsna.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztgsna.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZTGSNA( 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( * )
* DOUBLE PRECISION DIF( * ), S( * )
* COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \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*16 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*16 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*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.
*> \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*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.
*> \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 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.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*> 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.
*> \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*16 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 complex16OTHERcomputational
*
*> \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 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.
*> \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 ZTGSNA( 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( * )
DOUBLE PRECISION DIF( * ), S( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
INTEGER IDIFJB
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, IDIFJB = 3 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, SOMCON, WANTBH, WANTDF, WANTS
INTEGER I, IERR, IFST, ILST, K, KS, LWMIN, N1, N2
DOUBLE PRECISION BIGNUM, COND, EPS, LNRM, RNRM, SCALE, SMLNUM
COMPLEX*16 YHAX, YHBX
* ..
* .. Local Arrays ..
COMPLEX*16 DUMMY( 1 ), DUMMY1( 1 )
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLAPY2, DZNRM2
COMPLEX*16 ZDOTC
EXTERNAL LSAME, DLAMCH, DLAPY2, DZNRM2, ZDOTC
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, XERBLA, ZGEMV, ZLACPY, ZTGEXC, ZTGSYL
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DCMPLX, 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( 'ZTGSNA', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'P' )
SMLNUM = DLAMCH( 'S' ) / EPS
BIGNUM = ONE / SMLNUM
CALL DLABAD( 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 = DZNRM2( N, VR( 1, KS ), 1 )
LNRM = DZNRM2( N, VL( 1, KS ), 1 )
CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), A, LDA,
$ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
YHAX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
CALL ZGEMV( 'N', N, N, DCMPLX( ONE, ZERO ), B, LDB,
$ VR( 1, KS ), 1, DCMPLX( ZERO, ZERO ), WORK, 1 )
YHBX = ZDOTC( N, WORK, 1, VL( 1, KS ), 1 )
COND = DLAPY2( 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 ) = DLAPY2( 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 ZLACPY( 'Full', N, N, A, LDA, WORK, N )
CALL ZLACPY( 'Full', N, N, B, LDB, WORK( N*N+1 ), N )
IFST = K
ILST = 1
*
CALL ZTGEXC( .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 ZTGSYL( '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 ZTGSNA
*
END
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