OSchip
/
OpenBLAS
Not watched
1
0
Fork 0
Code
Releases 66 Wiki evaluate Activity
Issues 0 Pull Requests 0 Datasets Model Cloudbrain HPC
Browse Source

Update brief description of function purpose (Reference-LAPACK 831)

pull/4058/head
Martin Kroeker GitHub 3 years ago
parent
e728744bd6
commit
3587fd0ea8
No known key found for this signature in database GPG Key ID: 4AEE18F83AFDEB23
8 changed files with 5070 additions and 0 deletions
Split View
Diff Options
Show Stats Download Patch File Download Diff File
  1. +528
    -0
      lapack-netlib/SRC/cgegs.f
  2. +703
    -0
      lapack-netlib/SRC/cgegv.f
  3. +538
    -0
      lapack-netlib/SRC/dgegs.f
  4. +766
    -0
      lapack-netlib/SRC/dgegv.f
  5. +538
    -0
      lapack-netlib/SRC/sgegs.f
  6. +766
    -0
      lapack-netlib/SRC/sgegv.f
  7. +528
    -0
      lapack-netlib/SRC/zgegs.f
  8. +703
    -0
      lapack-netlib/SRC/zgegv.f

+ 528
- 0
lapack-netlib/SRC/cgegs.f View File

@@ -0,0 +1,528 @@
*> \brief <b> CGEGS computes the eigenvalues, Schur form, and, optionally, the left and or/right Schur vectors of a complex matrix pair (A,B)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
* VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
* REAL RWORK( * )
* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine CGGES.
*>
*> CGEGS computes the eigenvalues, Schur form, and, optionally, the
*> left and or/right Schur vectors of a complex matrix pair (A,B).
*> Given two square matrices A and B, the generalized Schur
*> factorization has the form
*>
*> A = Q*S*Z**H, B = Q*T*Z**H
*>
*> where Q and Z are unitary matrices and S and T are upper triangular.
*> The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> CGEGV should be used instead. See CGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors (returned in VSR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, N)
*> On entry, the matrix A.
*> On exit, the upper triangular matrix S from the generalized
*> Schur factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB, N)
*> On entry, the matrix B.
*> On exit, the upper triangular matrix T from the generalized
*> Schur factorization.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is COMPLEX array, dimension (N)
*> The complex scalars alpha that define the eigenvalues of
*> GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur
*> form of A.
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is COMPLEX array, dimension (N)
*> The non-negative real scalars beta that define the
*> eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element
*> of the triangular factor T.
*>
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
*> represent the j-th eigenvalue of the matrix pair (A,B), in
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
*> Since either lambda or mu may overflow, they should not,
*> in general, be computed.
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is COMPLEX array, dimension (LDVSL,N)
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >= 1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is COMPLEX array, dimension (LDVSR,N)
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= 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(1,2*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
*> NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
*> the optimal LWORK is N*(NB+1).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> =1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHA(j) and BETA(j) should be correct for
*> j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from CGGBAL
*> =N+2: error return from CGEQRF
*> =N+3: error return from CUNMQR
*> =N+4: error return from CUNGQR
*> =N+5: error return from CGGHRD
*> =N+6: error return from CHGEQZ (other than failed
*> iteration)
*> =N+7: error return from CGGBAK (computing VSL)
*> =N+8: error return from CGGBAK (computing VSR)
*> =N+9: error return from CLASCL (various places)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexGEeigen
*
* =====================================================================
SUBROUTINE CGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
$ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
$ CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT,
$ ILO, IRIGHT, IROWS, IRWORK, ITAU, IWORK,
$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY,
$ CLASCL, CLASET, CUNGQR, CUNMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL CLANGE, SLAMCH
EXTERNAL ILAENV, LSAME, CLANGE, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
* Test the input arguments
*
LWKMIN = MAX( 2*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'CGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'CUNMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'CUNGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = N*(NB+1)
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGEGS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
SAFMIN = SLAMCH( 'S' )
SMLNUM = N*SAFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = CLANGE( 'M', N, N, A, LDA, RWORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
*
IF( ILASCL ) THEN
CALL CLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
*
IF( ILBSCL ) THEN
CALL CLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
*
ILEFT = 1
IRIGHT = N + 1
IRWORK = IRIGHT + N
IWORK = 1
CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWORK
IWORK = ITAU + IROWS
CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 10
END IF
*
CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 10
END IF
*
IF( ILVSL ) THEN
CALL CLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL )
CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL CUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 10
END IF
END IF
*
IF( ILVSR )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL CGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
*
IWORK = ITAU
CALL CHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWORK ),
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL CLASCL( 'U', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL CLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL CLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL CLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of CGEGS
*
END

+ 703
- 0
lapack-netlib/SRC/cgegv.f View File

@@ -0,0 +1,703 @@
*> \brief <b> CGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B).</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGEGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgegv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgegv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgegv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
* VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* REAL RWORK( * )
* COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine CGGEV.
*>
*> CGEGV computes the eigenvalues and, optionally, the left and/or right
*> eigenvectors of a complex matrix pair (A,B).
*> Given two square matrices A and B,
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
*> that
*> A*x = lambda*B*x.
*>
*> An alternate form is to find the eigenvalues mu and corresponding
*> eigenvectors y such that
*> mu*A*y = B*y.
*>
*> These two forms are equivalent with mu = 1/lambda and x = y if
*> neither lambda nor mu is zero. In order to deal with the case that
*> lambda or mu is zero or small, two values alpha and beta are returned
*> for each eigenvalue, such that lambda = alpha/beta and
*> mu = beta/alpha.
*>
*> The vectors x and y in the above equations are right eigenvectors of
*> the matrix pair (A,B). Vectors u and v satisfying
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
*> are left eigenvectors of (A,B).
*>
*> Note: this routine performs "full balancing" on A and B
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors (returned
*> in VL).
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors (returned
*> in VR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA, N)
*> On entry, the matrix A.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
*> contains the Schur form of A from the generalized Schur
*> factorization of the pair (A,B) after balancing. If no
*> eigenvectors were computed, then only the diagonal elements
*> of the Schur form will be correct. See CGGHRD and CHGEQZ
*> for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX array, dimension (LDB, N)
*> On entry, the matrix B.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
*> upper triangular matrix obtained from B in the generalized
*> Schur factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only the diagonal
*> elements of B will be correct. See CGGHRD and CHGEQZ for
*> details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is COMPLEX array, dimension (N)
*> The complex scalars alpha that define the eigenvalues of
*> GNEP.
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is COMPLEX array, dimension (N)
*> The complex scalars beta that define the eigenvalues of GNEP.
*>
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
*> represent the j-th eigenvalue of the matrix pair (A,B), in
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
*> Since either lambda or mu may overflow, they should not,
*> in general, be computed.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is COMPLEX array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
*> in the columns of VL, in the same order as their eigenvalues.
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is COMPLEX array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
*> in the columns of VR, in the same order as their eigenvalues.
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= 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(1,2*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for CGEQRF, CUNMQR, and CUNGQR.) Then compute:
*> NB -- MAX of the blocksizes for CGEQRF, CUNMQR, and CUNGQR;
*> The optimal LWORK is MAX( 2*N, N*(NB+1) ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> =1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHA(j) and BETA(j) should be
*> correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from CGGBAL
*> =N+2: error return from CGEQRF
*> =N+3: error return from CUNMQR
*> =N+4: error return from CUNGQR
*> =N+5: error return from CGGHRD
*> =N+6: error return from CHGEQZ (other than failed
*> iteration)
*> =N+7: error return from CTGEVC
*> =N+8: error return from CGGBAK (computing VL)
*> =N+9: error return from CGGBAK (computing VR)
*> =N+10: error return from CLASCL (various calls)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexGEeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing
*> ---------
*>
*> This driver calls CGGBAL to both permute and scale rows and columns
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
*> and PL*B*R will be upper triangular except for the diagonal blocks
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
*> possible. The diagonal scaling matrices DL and DR are chosen so
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
*> one (except for the elements that start out zero.)
*>
*> After the eigenvalues and eigenvectors of the balanced matrices
*> have been computed, CGGBAK transforms the eigenvectors back to what
*> they would have been (in perfect arithmetic) if they had not been
*> balanced.
*>
*> Contents of A and B on Exit
*> -------- -- - --- - -- ----
*>
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
*> both), then on exit the arrays A and B will contain the complex Schur
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
*> are computed, then only the diagonal blocks will be correct.
*>
*> [*] In other words, upper triangular form.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
REAL RWORK( * )
COMPLEX A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
COMPLEX CZERO, CONE
PARAMETER ( CZERO = ( 0.0E0, 0.0E0 ),
$ CONE = ( 1.0E0, 0.0E0 ) )
* ..
* .. Local Scalars ..
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR,
$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
REAL ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
$ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
$ SALFAR, SBETA, SCALE, TEMP
COMPLEX X
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL CGEQRF, CGGBAK, CGGBAL, CGGHRD, CHGEQZ, CLACPY,
$ CLASCL, CLASET, CTGEVC, CUNGQR, CUNMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL CLANGE, SLAMCH
EXTERNAL ILAENV, LSAME, CLANGE, SLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, AIMAG, CMPLX, INT, MAX, REAL
* ..
* .. Statement Functions ..
REAL ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( REAL( X ) ) + ABS( AIMAG( X ) )
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
LWKMIN = MAX( 2*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'CGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'CUNMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'CUNGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = MAX( 2*N, N*(NB+1) )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGEGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
SAFMIN = SLAMCH( 'S' )
SAFMIN = SAFMIN + SAFMIN
SAFMAX = ONE / SAFMIN
*
* Scale A
*
ANRM = CLANGE( 'M', N, N, A, LDA, RWORK )
ANRM1 = ANRM
ANRM2 = ONE
IF( ANRM.LT.ONE ) THEN
IF( SAFMAX*ANRM.LT.ONE ) THEN
ANRM1 = SAFMIN
ANRM2 = SAFMAX*ANRM
END IF
END IF
*
IF( ANRM.GT.ZERO ) THEN
CALL CLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Scale B
*
BNRM = CLANGE( 'M', N, N, B, LDB, RWORK )
BNRM1 = BNRM
BNRM2 = ONE
IF( BNRM.LT.ONE ) THEN
IF( SAFMAX*BNRM.LT.ONE ) THEN
BNRM1 = SAFMIN
BNRM2 = SAFMAX*BNRM
END IF
END IF
*
IF( BNRM.GT.ZERO ) THEN
CALL CLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Also "balance" the matrix.
*
ILEFT = 1
IRIGHT = N + 1
IRWORK = IRIGHT + N
CALL CGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 80
END IF
*
* Reduce B to triangular form, and initialize VL and/or VR
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = 1
IWORK = ITAU + IROWS
CALL CGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 80
END IF
*
CALL CUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 80
END IF
*
IF( ILVL ) THEN
CALL CLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
CALL CLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
CALL CUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 80
END IF
END IF
*
IF( ILVR )
$ CALL CLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL CGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IINFO )
ELSE
CALL CGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
END IF
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 80
END IF
*
* Perform QZ algorithm
*
IWORK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL CHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWORK ),
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 80
END IF
*
IF( ILV ) THEN
*
* Compute Eigenvectors
*
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL CTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWORK ), RWORK( IRWORK ),
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 80
END IF
*
* Undo balancing on VL and VR, rescale
*
IF( ILVL ) THEN
CALL CGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VL, LDVL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 80
END IF
DO 30 JC = 1, N
TEMP = ZERO
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
10 CONTINUE
IF( TEMP.LT.SAFMIN )
$ GO TO 30
TEMP = ONE / TEMP
DO 20 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
20 CONTINUE
30 CONTINUE
END IF
IF( ILVR ) THEN
CALL CGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VR, LDVR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
GO TO 80
END IF
DO 60 JC = 1, N
TEMP = ZERO
DO 40 JR = 1, N
TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
40 CONTINUE
IF( TEMP.LT.SAFMIN )
$ GO TO 60
TEMP = ONE / TEMP
DO 50 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
50 CONTINUE
60 CONTINUE
END IF
*
* End of eigenvector calculation
*
END IF
*
* Undo scaling in alpha, beta
*
* Note: this does not give the alpha and beta for the unscaled
* problem.
*
* Un-scaling is limited to avoid underflow in alpha and beta
* if they are significant.
*
DO 70 JC = 1, N
ABSAR = ABS( REAL( ALPHA( JC ) ) )
ABSAI = ABS( AIMAG( ALPHA( JC ) ) )
ABSB = ABS( REAL( BETA( JC ) ) )
SALFAR = ANRM*REAL( ALPHA( JC ) )
SALFAI = ANRM*AIMAG( ALPHA( JC ) )
SBETA = BNRM*REAL( BETA( JC ) )
ILIMIT = .FALSE.
SCALE = ONE
*
* Check for significant underflow in imaginary part of ALPHA
*
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI )
END IF
*
* Check for significant underflow in real part of ALPHA
*
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) /
$ MAX( SAFMIN, ANRM2*ABSAR ) )
END IF
*
* Check for significant underflow in BETA
*
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) /
$ MAX( SAFMIN, BNRM2*ABSB ) )
END IF
*
* Check for possible overflow when limiting scaling
*
IF( ILIMIT ) THEN
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
$ ABS( SBETA ) )
IF( TEMP.GT.ONE )
$ SCALE = SCALE / TEMP
IF( SCALE.LT.ONE )
$ ILIMIT = .FALSE.
END IF
*
* Recompute un-scaled ALPHA, BETA if necessary.
*
IF( ILIMIT ) THEN
SALFAR = ( SCALE*REAL( ALPHA( JC ) ) )*ANRM
SALFAI = ( SCALE*AIMAG( ALPHA( JC ) ) )*ANRM
SBETA = ( SCALE*BETA( JC ) )*BNRM
END IF
ALPHA( JC ) = CMPLX( SALFAR, SALFAI )
BETA( JC ) = SBETA
70 CONTINUE
*
80 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of CGEGV
*
END

+ 538
- 0
lapack-netlib/SRC/dgegs.f View File

@@ -0,0 +1,538 @@
*> \brief <b> DGEGS computes the eigenvalues, real Schur form, and, optionally, the left and/or right Schur vectors of a real matrix pair (A,B)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
* $ VSR( LDVSR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGES.
*>
*> DGEGS computes the eigenvalues, real Schur form, and, optionally,
*> left and or/right Schur vectors of a real matrix pair (A,B).
*> Given two square matrices A and B, the generalized real Schur
*> factorization has the form
*>
*> A = Q*S*Z**T, B = Q*T*Z**T
*>
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
*> of eigenvalues of (A,B). The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> DGEGV should be used instead. See DGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors (returned in VSR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the matrix A.
*> On exit, the upper quasi-triangular matrix S from the
*> generalized real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the matrix B.
*> On exit, the upper triangular matrix T from the generalized
*> real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue
*> of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
*> eigenvalue is real; if positive, then the j-th and (j+1)-st
*> eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >=1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION 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(1,4*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR
*> The optimal LWORK is 2*N + N*(NB+1).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*> be correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from DGGBAL
*> =N+2: error return from DGEQRF
*> =N+3: error return from DORMQR
*> =N+4: error return from DORGQR
*> =N+5: error return from DGGHRD
*> =N+6: error return from DHGEQZ (other than failed
*> iteration)
*> =N+7: error return from DGGBAK (computing VSL)
*> =N+8: error return from DGGBAK (computing VSR)
*> =N+9: error return from DLASCL (various places)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
* =====================================================================
SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
$ LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
* Test the input arguments
*
LWKMIN = MAX( 4*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N + N*( NB+1 )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEGS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SMLNUM = N*SAFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
*
IF( ILASCL ) THEN
CALL DLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (2*N words -- "work..." not actually used)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWORK
IWORK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 10
END IF
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 10
END IF
*
IF( ILVSL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 10
END IF
END IF
*
IF( ILVSR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
CALL DHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL DLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL DLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL DLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEGS
*
END

+ 766
- 0
lapack-netlib/SRC/dgegv.f View File

@@ -0,0 +1,766 @@
*> \brief <b> DGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGEGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgegv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgegv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgegv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine DGGEV.
*>
*> DGEGV computes the eigenvalues and, optionally, the left and/or right
*> eigenvectors of a real matrix pair (A,B).
*> Given two square matrices A and B,
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
*> that
*>
*> A*x = lambda*B*x.
*>
*> An alternate form is to find the eigenvalues mu and corresponding
*> eigenvectors y such that
*>
*> mu*A*y = B*y.
*>
*> These two forms are equivalent with mu = 1/lambda and x = y if
*> neither lambda nor mu is zero. In order to deal with the case that
*> lambda or mu is zero or small, two values alpha and beta are returned
*> for each eigenvalue, such that lambda = alpha/beta and
*> mu = beta/alpha.
*>
*> The vectors x and y in the above equations are right eigenvectors of
*> the matrix pair (A,B). Vectors u and v satisfying
*>
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
*>
*> are left eigenvectors of (A,B).
*>
*> Note: this routine performs "full balancing" on A and B
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors (returned
*> in VL).
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors (returned
*> in VR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (LDA, N)
*> On entry, the matrix A.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
*> contains the real Schur form of A from the generalized Schur
*> factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only the diagonal
*> blocks from the Schur form will be correct. See DGGHRD and
*> DHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (LDB, N)
*> On entry, the matrix B.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
*> upper triangular matrix obtained from B in the generalized
*> Schur factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only those elements of
*> B corresponding to the diagonal blocks from the Schur form of
*> A will be correct. See DGGHRD and DHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is DOUBLE PRECISION array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue of
*> GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is DOUBLE PRECISION array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
*> eigenvalue is real; if positive, then the j-th and
*> (j+1)-st eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is DOUBLE PRECISION array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*>
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is DOUBLE PRECISION array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
*> in the columns of VL, in the same order as their eigenvalues.
*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
*> pair, then
*> u(j) = VL(:,j) + i*VL(:,j+1)
*> and
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
*>
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is DOUBLE PRECISION array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
*> in the columns of VR, in the same order as their eigenvalues.
*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
*> pair, then
*> x(j) = VR(:,j) + i*VR(:,j+1)
*> and
*> x(j+1) = VR(:,j) - i*VR(:,j+1).
*>
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION 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(1,8*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for DGEQRF, DORMQR, and DORGQR.) Then compute:
*> NB -- MAX of the blocksizes for DGEQRF, DORMQR, and DORGQR;
*> The optimal LWORK is:
*> 2*N + MAX( 6*N, N*(NB+1) ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*> should be correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from DGGBAL
*> =N+2: error return from DGEQRF
*> =N+3: error return from DORMQR
*> =N+4: error return from DORGQR
*> =N+5: error return from DGGHRD
*> =N+6: error return from DHGEQZ (other than failed
*> iteration)
*> =N+7: error return from DTGEVC
*> =N+8: error return from DGGBAK (computing VL)
*> =N+9: error return from DGGBAK (computing VR)
*> =N+10: error return from DLASCL (various calls)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing
*> ---------
*>
*> This driver calls DGGBAL to both permute and scale rows and columns
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
*> and PL*B*R will be upper triangular except for the diagonal blocks
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
*> possible. The diagonal scaling matrices DL and DR are chosen so
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
*> one (except for the elements that start out zero.)
*>
*> After the eigenvalues and eigenvectors of the balanced matrices
*> have been computed, DGGBAK transforms the eigenvectors back to what
*> they would have been (in perfect arithmetic) if they had not been
*> balanced.
*>
*> Contents of A and B on Exit
*> -------- -- - --- - -- ----
*>
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
*> both), then on exit the arrays A and B will contain the real Schur
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
*> are computed, then only the diagonal blocks will be correct.
*>
*> [*] See DHGEQZ, DGEGS, or read the book "Matrix Computations",
*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
$ SALFAI, SALFAR, SBETA, SCALE, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ, DLACPY,
$ DLASCL, DLASET, DORGQR, DORMQR, DTGEVC, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, DLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
LWKMIN = MAX( 8*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'DGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'DORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'DORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N + MAX( 6*N, N*( NB+1 ) )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'DGEGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SAFMIN = SAFMIN + SAFMIN
SAFMAX = ONE / SAFMIN
ONEPLS = ONE + ( 4*EPS )
*
* Scale A
*
ANRM = DLANGE( 'M', N, N, A, LDA, WORK )
ANRM1 = ANRM
ANRM2 = ONE
IF( ANRM.LT.ONE ) THEN
IF( SAFMAX*ANRM.LT.ONE ) THEN
ANRM1 = SAFMIN
ANRM2 = SAFMAX*ANRM
END IF
END IF
*
IF( ANRM.GT.ZERO ) THEN
CALL DLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Scale B
*
BNRM = DLANGE( 'M', N, N, B, LDB, WORK )
BNRM1 = BNRM
BNRM2 = ONE
IF( BNRM.LT.ONE ) THEN
IF( SAFMAX*BNRM.LT.ONE ) THEN
BNRM1 = SAFMIN
BNRM2 = SAFMAX*BNRM
END IF
END IF
*
IF( BNRM.GT.ZERO ) THEN
CALL DLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (8*N words -- "work" requires 6*N words)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL DGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 120
END IF
*
* Reduce B to triangular form, and initialize VL and/or VR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = IWORK
IWORK = ITAU + IROWS
CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 120
END IF
*
CALL DORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 120
END IF
*
IF( ILVL ) THEN
CALL DLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
CALL DLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
CALL DORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 120
END IF
END IF
*
IF( ILVR )
$ CALL DLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL DGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IINFO )
ELSE
CALL DGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
END IF
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 120
END IF
*
* Perform QZ algorithm
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL DHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 120
END IF
*
IF( ILV ) THEN
*
* Compute Eigenvectors (DTGEVC requires 6*N words of workspace)
*
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL DTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 120
END IF
*
* Undo balancing on VL and VR, rescale
*
IF( ILVL ) THEN
CALL DGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 120
END IF
DO 50 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 50
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
10 CONTINUE
ELSE
DO 20 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
$ ABS( VL( JR, JC+1 ) ) )
20 CONTINUE
END IF
IF( TEMP.LT.SAFMIN )
$ GO TO 50
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 30 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
30 CONTINUE
ELSE
DO 40 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
40 CONTINUE
END IF
50 CONTINUE
END IF
IF( ILVR ) THEN
CALL DGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
GO TO 120
END IF
DO 100 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 100
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 60 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
60 CONTINUE
ELSE
DO 70 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
$ ABS( VR( JR, JC+1 ) ) )
70 CONTINUE
END IF
IF( TEMP.LT.SAFMIN )
$ GO TO 100
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 80 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
80 CONTINUE
ELSE
DO 90 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
90 CONTINUE
END IF
100 CONTINUE
END IF
*
* End of eigenvector calculation
*
END IF
*
* Undo scaling in alpha, beta
*
* Note: this does not give the alpha and beta for the unscaled
* problem.
*
* Un-scaling is limited to avoid underflow in alpha and beta
* if they are significant.
*
DO 110 JC = 1, N
ABSAR = ABS( ALPHAR( JC ) )
ABSAI = ABS( ALPHAI( JC ) )
ABSB = ABS( BETA( JC ) )
SALFAR = ANRM*ALPHAR( JC )
SALFAI = ANRM*ALPHAI( JC )
SBETA = BNRM*BETA( JC )
ILIMIT = .FALSE.
SCALE = ONE
*
* Check for significant underflow in ALPHAI
*
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
*
ELSE IF( SALFAI.EQ.ZERO ) THEN
*
* If insignificant underflow in ALPHAI, then make the
* conjugate eigenvalue real.
*
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
ALPHAI( JC-1 ) = ZERO
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
ALPHAI( JC+1 ) = ZERO
END IF
END IF
*
* Check for significant underflow in ALPHAR
*
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
END IF
*
* Check for significant underflow in BETA
*
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
END IF
*
* Check for possible overflow when limiting scaling
*
IF( ILIMIT ) THEN
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
$ ABS( SBETA ) )
IF( TEMP.GT.ONE )
$ SCALE = SCALE / TEMP
IF( SCALE.LT.ONE )
$ ILIMIT = .FALSE.
END IF
*
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
*
IF( ILIMIT ) THEN
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
SBETA = ( SCALE*BETA( JC ) )*BNRM
END IF
ALPHAR( JC ) = SALFAR
ALPHAI( JC ) = SALFAI
BETA( JC ) = SBETA
110 CONTINUE
*
120 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of DGEGV
*
END

+ 538
- 0
lapack-netlib/SRC/sgegs.f View File

@@ -0,0 +1,538 @@
*> \brief <b> SGEGS computes the eigenvalues, real Schur form, and, optionally, the left and/or right Schur vectors of a real matrix pair (A,B)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
* ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
* LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
* $ VSR( LDVSR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine SGGES.
*>
*> SGEGS computes the eigenvalues, real Schur form, and, optionally,
*> left and or/right Schur vectors of a real matrix pair (A,B).
*> Given two square matrices A and B, the generalized real Schur
*> factorization has the form
*>
*> A = Q*S*Z**T, B = Q*T*Z**T
*>
*> where Q and Z are orthogonal matrices, T is upper triangular, and S
*> is an upper quasi-triangular matrix with 1-by-1 and 2-by-2 diagonal
*> blocks, the 2-by-2 blocks corresponding to complex conjugate pairs
*> of eigenvalues of (A,B). The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> SGEGV should be used instead. See SGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors (returned in VSR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA, N)
*> On entry, the matrix A.
*> On exit, the upper quasi-triangular matrix S from the
*> generalized real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is REAL array, dimension (LDB, N)
*> On entry, the matrix B.
*> On exit, the upper triangular matrix T from the generalized
*> real Schur factorization.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is REAL array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue
*> of GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is REAL array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
*> eigenvalue is real; if positive, then the j-th and (j+1)-st
*> eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is REAL array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is REAL array, dimension (LDVSL,N)
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >=1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is REAL array, dimension (LDVSR,N)
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL 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(1,4*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute:
*> NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR
*> The optimal LWORK is 2*N + N*(NB+1).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*> be correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from SGGBAL
*> =N+2: error return from SGEQRF
*> =N+3: error return from SORMQR
*> =N+4: error return from SORGQR
*> =N+5: error return from SGGHRD
*> =N+6: error return from SHGEQZ (other than failed
*> iteration)
*> =N+7: error return from SGGBAK (computing VSL)
*> =N+8: error return from SGGBAK (computing VSR)
*> =N+9: error return from SLASCL (various places)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realGEeigen
*
* =====================================================================
SUBROUTINE SGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR,
$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VSL( LDVSL, * ),
$ VSR( LDVSR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT,
$ ILO, IRIGHT, IROWS, ITAU, IWORK, LOPT, LWKMIN,
$ LWKOPT, NB, NB1, NB2, NB3
REAL ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLACPY,
$ SLASCL, SLASET, SORGQR, SORMQR, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANGE
EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
* Test the input arguments
*
LWKMIN = MAX( 4*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'SGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'SORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'SORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N+N*(NB+1)
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEGS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
SAFMIN = SLAMCH( 'S' )
SMLNUM = N*SAFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = SLANGE( 'M', N, N, A, LDA, WORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
*
IF( ILASCL ) THEN
CALL SLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = SLANGE( 'M', N, N, B, LDB, WORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
*
IF( ILBSCL ) THEN
CALL SLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (2*N words -- "work..." not actually used)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWORK
IWORK = ITAU + IROWS
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 10
END IF
*
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 10
END IF
*
IF( ILVSL ) THEN
CALL SLASET( 'Full', N, N, ZERO, ONE, VSL, LDVSL )
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL SORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 10
END IF
END IF
*
IF( ILVSR )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL SGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
CALL SHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL SLASCL( 'H', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL SLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAR, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL SLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHAI, N,
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL SLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL SLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of SGEGS
*
END

+ 766
- 0
lapack-netlib/SRC/sgegv.f View File

@@ -0,0 +1,766 @@
*> \brief <b> SGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a real matrix pair (A,B).</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGEGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgegv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgegv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgegv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
* BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
* $ B( LDB, * ), BETA( * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine SGGEV.
*>
*> SGEGV computes the eigenvalues and, optionally, the left and/or right
*> eigenvectors of a real matrix pair (A,B).
*> Given two square matrices A and B,
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
*> that
*>
*> A*x = lambda*B*x.
*>
*> An alternate form is to find the eigenvalues mu and corresponding
*> eigenvectors y such that
*>
*> mu*A*y = B*y.
*>
*> These two forms are equivalent with mu = 1/lambda and x = y if
*> neither lambda nor mu is zero. In order to deal with the case that
*> lambda or mu is zero or small, two values alpha and beta are returned
*> for each eigenvalue, such that lambda = alpha/beta and
*> mu = beta/alpha.
*>
*> The vectors x and y in the above equations are right eigenvectors of
*> the matrix pair (A,B). Vectors u and v satisfying
*>
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
*>
*> are left eigenvectors of (A,B).
*>
*> Note: this routine performs "full balancing" on A and B
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors (returned
*> in VL).
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors (returned
*> in VR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is REAL array, dimension (LDA, N)
*> On entry, the matrix A.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
*> contains the real Schur form of A from the generalized Schur
*> factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only the diagonal
*> blocks from the Schur form will be correct. See SGGHRD and
*> SHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is REAL array, dimension (LDB, N)
*> On entry, the matrix B.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
*> upper triangular matrix obtained from B in the generalized
*> Schur factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only those elements of
*> B corresponding to the diagonal blocks from the Schur form of
*> A will be correct. See SGGHRD and SHGEQZ for details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*> ALPHAR is REAL array, dimension (N)
*> The real parts of each scalar alpha defining an eigenvalue of
*> GNEP.
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*> ALPHAI is REAL array, dimension (N)
*> The imaginary parts of each scalar alpha defining an
*> eigenvalue of GNEP. If ALPHAI(j) is zero, then the j-th
*> eigenvalue is real; if positive, then the j-th and
*> (j+1)-st eigenvalues are a complex conjugate pair, with
*> ALPHAI(j+1) = -ALPHAI(j).
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is REAL array, dimension (N)
*> The scalars beta that define the eigenvalues of GNEP.
*>
*> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and
*> beta = BETA(j) represent the j-th eigenvalue of the matrix
*> pair (A,B), in one of the forms lambda = alpha/beta or
*> mu = beta/alpha. Since either lambda or mu may overflow,
*> they should not, in general, be computed.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is REAL array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
*> in the columns of VL, in the same order as their eigenvalues.
*> If the j-th eigenvalue is real, then u(j) = VL(:,j).
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
*> pair, then
*> u(j) = VL(:,j) + i*VL(:,j+1)
*> and
*> u(j+1) = VL(:,j) - i*VL(:,j+1).
*>
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is REAL array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
*> in the columns of VR, in the same order as their eigenvalues.
*> If the j-th eigenvalue is real, then x(j) = VR(:,j).
*> If the j-th and (j+1)-st eigenvalues form a complex conjugate
*> pair, then
*> x(j) = VR(:,j) + i*VR(:,j+1)
*> and
*> x(j+1) = VR(:,j) - i*VR(:,j+1).
*>
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvalues
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL 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(1,8*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for SGEQRF, SORMQR, and SORGQR.) Then compute:
*> NB -- MAX of the blocksizes for SGEQRF, SORMQR, and SORGQR;
*> The optimal LWORK is:
*> 2*N + MAX( 6*N, N*(NB+1) ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> = 1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*> should be correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from SGGBAL
*> =N+2: error return from SGEQRF
*> =N+3: error return from SORMQR
*> =N+4: error return from SORGQR
*> =N+5: error return from SGGHRD
*> =N+6: error return from SHGEQZ (other than failed
*> iteration)
*> =N+7: error return from STGEVC
*> =N+8: error return from SGGBAK (computing VL)
*> =N+9: error return from SGGBAK (computing VR)
*> =N+10: error return from SLASCL (various calls)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realGEeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing
*> ---------
*>
*> This driver calls SGGBAL to both permute and scale rows and columns
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
*> and PL*B*R will be upper triangular except for the diagonal blocks
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
*> possible. The diagonal scaling matrices DL and DR are chosen so
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
*> one (except for the elements that start out zero.)
*>
*> After the eigenvalues and eigenvectors of the balanced matrices
*> have been computed, SGGBAK transforms the eigenvectors back to what
*> they would have been (in perfect arithmetic) if they had not been
*> balanced.
*>
*> Contents of A and B on Exit
*> -------- -- - --- - -- ----
*>
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
*> both), then on exit the arrays A and B will contain the real Schur
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
*> are computed, then only the diagonal blocks will be correct.
*>
*> [*] See SHGEQZ, SGEGS, or read the book "Matrix Computations",
*> by Golub & van Loan, pub. by Johns Hopkins U. Press.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHAR, ALPHAI,
$ BETA, VL, LDVL, VR, LDVR, WORK, LWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
$ B( LDB, * ), BETA( * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, ITAU, IWORK, JC, JR, LOPT,
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
REAL ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
$ BNRM1, BNRM2, EPS, ONEPLS, SAFMAX, SAFMIN,
$ SALFAI, SALFAR, SBETA, SCALE, TEMP
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLACPY,
$ SLASCL, SLASET, SORGQR, SORMQR, STGEVC, XERBLA
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
REAL SLAMCH, SLANGE
EXTERNAL ILAENV, LSAME, SLAMCH, SLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
LWKMIN = MAX( 8*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -12
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -14
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -16
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'SGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'SORMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'SORGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = 2*N + MAX( 6*N, N*(NB+1) )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SGEGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = SLAMCH( 'E' )*SLAMCH( 'B' )
SAFMIN = SLAMCH( 'S' )
SAFMIN = SAFMIN + SAFMIN
SAFMAX = ONE / SAFMIN
ONEPLS = ONE + ( 4*EPS )
*
* Scale A
*
ANRM = SLANGE( 'M', N, N, A, LDA, WORK )
ANRM1 = ANRM
ANRM2 = ONE
IF( ANRM.LT.ONE ) THEN
IF( SAFMAX*ANRM.LT.ONE ) THEN
ANRM1 = SAFMIN
ANRM2 = SAFMAX*ANRM
END IF
END IF
*
IF( ANRM.GT.ZERO ) THEN
CALL SLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Scale B
*
BNRM = SLANGE( 'M', N, N, B, LDB, WORK )
BNRM1 = BNRM
BNRM2 = ONE
IF( BNRM.LT.ONE ) THEN
IF( SAFMAX*BNRM.LT.ONE ) THEN
BNRM1 = SAFMIN
BNRM2 = SAFMAX*BNRM
END IF
END IF
*
IF( BNRM.GT.ZERO ) THEN
CALL SLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Workspace layout: (8*N words -- "work" requires 6*N words)
* left_permutation, right_permutation, work...
*
ILEFT = 1
IRIGHT = N + 1
IWORK = IRIGHT + N
CALL SGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 120
END IF
*
* Reduce B to triangular form, and initialize VL and/or VR
* Workspace layout: ("work..." must have at least N words)
* left_permutation, right_permutation, tau, work...
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = IWORK
IWORK = ITAU + IROWS
CALL SGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 120
END IF
*
CALL SORMQR( 'L', 'T', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 120
END IF
*
IF( ILVL ) THEN
CALL SLASET( 'Full', N, N, ZERO, ONE, VL, LDVL )
CALL SLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
CALL SORGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 120
END IF
END IF
*
IF( ILVR )
$ CALL SLASET( 'Full', N, N, ZERO, ONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL SGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IINFO )
ELSE
CALL SGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
END IF
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 120
END IF
*
* Perform QZ algorithm
* Workspace layout: ("work..." must have at least 1 word)
* left_permutation, right_permutation, work...
*
IWORK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL SHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR,
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 120
END IF
*
IF( ILV ) THEN
*
* Compute Eigenvectors (STGEVC requires 6*N words of workspace)
*
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL STGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 120
END IF
*
* Undo balancing on VL and VR, rescale
*
IF( ILVL ) THEN
CALL SGGBAK( 'P', 'L', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VL, LDVL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 120
END IF
DO 50 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 50
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) ) )
10 CONTINUE
ELSE
DO 20 JR = 1, N
TEMP = MAX( TEMP, ABS( VL( JR, JC ) )+
$ ABS( VL( JR, JC+1 ) ) )
20 CONTINUE
END IF
IF( TEMP.LT.SAFMIN )
$ GO TO 50
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 30 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
30 CONTINUE
ELSE
DO 40 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
VL( JR, JC+1 ) = VL( JR, JC+1 )*TEMP
40 CONTINUE
END IF
50 CONTINUE
END IF
IF( ILVR ) THEN
CALL SGGBAK( 'P', 'R', N, ILO, IHI, WORK( ILEFT ),
$ WORK( IRIGHT ), N, VR, LDVR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
GO TO 120
END IF
DO 100 JC = 1, N
IF( ALPHAI( JC ).LT.ZERO )
$ GO TO 100
TEMP = ZERO
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 60 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) ) )
60 CONTINUE
ELSE
DO 70 JR = 1, N
TEMP = MAX( TEMP, ABS( VR( JR, JC ) )+
$ ABS( VR( JR, JC+1 ) ) )
70 CONTINUE
END IF
IF( TEMP.LT.SAFMIN )
$ GO TO 100
TEMP = ONE / TEMP
IF( ALPHAI( JC ).EQ.ZERO ) THEN
DO 80 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
80 CONTINUE
ELSE
DO 90 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
VR( JR, JC+1 ) = VR( JR, JC+1 )*TEMP
90 CONTINUE
END IF
100 CONTINUE
END IF
*
* End of eigenvector calculation
*
END IF
*
* Undo scaling in alpha, beta
*
* Note: this does not give the alpha and beta for the unscaled
* problem.
*
* Un-scaling is limited to avoid underflow in alpha and beta
* if they are significant.
*
DO 110 JC = 1, N
ABSAR = ABS( ALPHAR( JC ) )
ABSAI = ABS( ALPHAI( JC ) )
ABSB = ABS( BETA( JC ) )
SALFAR = ANRM*ALPHAR( JC )
SALFAI = ANRM*ALPHAI( JC )
SBETA = BNRM*BETA( JC )
ILIMIT = .FALSE.
SCALE = ONE
*
* Check for significant underflow in ALPHAI
*
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = ( ONEPLS*SAFMIN / ANRM1 ) /
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAI )
*
ELSE IF( SALFAI.EQ.ZERO ) THEN
*
* If insignificant underflow in ALPHAI, then make the
* conjugate eigenvalue real.
*
IF( ALPHAI( JC ).LT.ZERO .AND. JC.GT.1 ) THEN
ALPHAI( JC-1 ) = ZERO
ELSE IF( ALPHAI( JC ).GT.ZERO .AND. JC.LT.N ) THEN
ALPHAI( JC+1 ) = ZERO
END IF
END IF
*
* Check for significant underflow in ALPHAR
*
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / ANRM1 ) /
$ MAX( ONEPLS*SAFMIN, ANRM2*ABSAR ) )
END IF
*
* Check for significant underflow in BETA
*
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( ONEPLS*SAFMIN / BNRM1 ) /
$ MAX( ONEPLS*SAFMIN, BNRM2*ABSB ) )
END IF
*
* Check for possible overflow when limiting scaling
*
IF( ILIMIT ) THEN
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
$ ABS( SBETA ) )
IF( TEMP.GT.ONE )
$ SCALE = SCALE / TEMP
IF( SCALE.LT.ONE )
$ ILIMIT = .FALSE.
END IF
*
* Recompute un-scaled ALPHAR, ALPHAI, BETA if necessary.
*
IF( ILIMIT ) THEN
SALFAR = ( SCALE*ALPHAR( JC ) )*ANRM
SALFAI = ( SCALE*ALPHAI( JC ) )*ANRM
SBETA = ( SCALE*BETA( JC ) )*BNRM
END IF
ALPHAR( JC ) = SALFAR
ALPHAI( JC ) = SALFAI
BETA( JC ) = SBETA
110 CONTINUE
*
120 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of SGEGV
*
END

+ 528
- 0
lapack-netlib/SRC/zgegs.f View File

@@ -0,0 +1,528 @@
*> \brief <b> ZGEGS computes the eigenvalues, Schur form, and, optionally, the left and or/right Schur vectors of a complex matrix pair (A,B)</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZGEGS + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgegs.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgegs.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgegs.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
* VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVSL, JOBVSR
* INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine ZGGES.
*>
*> ZGEGS computes the eigenvalues, Schur form, and, optionally, the
*> left and or/right Schur vectors of a complex matrix pair (A,B).
*> Given two square matrices A and B, the generalized Schur
*> factorization has the form
*>
*> A = Q*S*Z**H, B = Q*T*Z**H
*>
*> where Q and Z are unitary matrices and S and T are upper triangular.
*> The columns of Q are the left Schur vectors
*> and the columns of Z are the right Schur vectors.
*>
*> If only the eigenvalues of (A,B) are needed, the driver routine
*> ZGEGV should be used instead. See ZGEGV for a description of the
*> eigenvalues of the generalized nonsymmetric eigenvalue problem
*> (GNEP).
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVSL
*> \verbatim
*> JOBVSL is CHARACTER*1
*> = 'N': do not compute the left Schur vectors;
*> = 'V': compute the left Schur vectors (returned in VSL).
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*> JOBVSR is CHARACTER*1
*> = 'N': do not compute the right Schur vectors;
*> = 'V': compute the right Schur vectors (returned in VSR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VSL, and VSR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA, N)
*> On entry, the matrix A.
*> On exit, the upper triangular matrix S from the generalized
*> Schur factorization.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 array, dimension (LDB, N)
*> On entry, the matrix B.
*> On exit, the upper triangular matrix T from the generalized
*> Schur factorization.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is COMPLEX*16 array, dimension (N)
*> The complex scalars alpha that define the eigenvalues of
*> GNEP. ALPHA(j) = S(j,j), the diagonal element of the Schur
*> form of A.
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is COMPLEX*16 array, dimension (N)
*> The non-negative real scalars beta that define the
*> eigenvalues of GNEP. BETA(j) = T(j,j), the diagonal element
*> of the triangular factor T.
*>
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
*> represent the j-th eigenvalue of the matrix pair (A,B), in
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
*> Since either lambda or mu may overflow, they should not,
*> in general, be computed.
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*> VSL is COMPLEX*16 array, dimension (LDVSL,N)
*> If JOBVSL = 'V', the matrix of left Schur vectors Q.
*> Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*> LDVSL is INTEGER
*> The leading dimension of the matrix VSL. LDVSL >= 1, and
*> if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*> VSR is COMPLEX*16 array, dimension (LDVSR,N)
*> If JOBVSR = 'V', the matrix of right Schur vectors Z.
*> Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*> LDVSR is INTEGER
*> The leading dimension of the matrix VSR. LDVSR >= 1, and
*> if JOBVSR = 'V', LDVSR >= 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(1,2*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for ZGEQRF, ZUNMQR, and CUNGQR.) Then compute:
*> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and CUNGQR;
*> the optimal LWORK is N*(NB+1).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (3*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> =1,...,N:
*> The QZ iteration failed. (A,B) are not in Schur
*> form, but ALPHA(j) and BETA(j) should be correct for
*> j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from ZGGBAL
*> =N+2: error return from ZGEQRF
*> =N+3: error return from ZUNMQR
*> =N+4: error return from ZUNGQR
*> =N+5: error return from ZGGHRD
*> =N+6: error return from ZHGEQZ (other than failed
*> iteration)
*> =N+7: error return from ZGGBAK (computing VSL)
*> =N+8: error return from ZGGBAK (computing VSR)
*> =N+9: error return from ZLASCL (various places)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16GEeigen
*
* =====================================================================
SUBROUTINE ZGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHA, BETA,
$ VSL, LDVSL, VSR, LDVSR, WORK, LWORK, RWORK,
$ INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVSL, JOBVSR
INTEGER INFO, LDA, LDB, LDVSL, LDVSR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VSL( LDVSL, * ), VSR( LDVSR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
$ CONE = ( 1.0D0, 0.0D0 ) )
* ..
* .. Local Scalars ..
LOGICAL ILASCL, ILBSCL, ILVSL, ILVSR, LQUERY
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IRIGHT, IROWS, IRWORK, ITAU, IWORK, LOPT,
$ LWKMIN, LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
$ SAFMIN, SMLNUM
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ,
$ ZLACPY, ZLASCL, ZLASET, ZUNGQR, ZUNMQR
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC INT, MAX
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVSL, 'N' ) ) THEN
IJOBVL = 1
ILVSL = .FALSE.
ELSE IF( LSAME( JOBVSL, 'V' ) ) THEN
IJOBVL = 2
ILVSL = .TRUE.
ELSE
IJOBVL = -1
ILVSL = .FALSE.
END IF
*
IF( LSAME( JOBVSR, 'N' ) ) THEN
IJOBVR = 1
ILVSR = .FALSE.
ELSE IF( LSAME( JOBVSR, 'V' ) ) THEN
IJOBVR = 2
ILVSR = .TRUE.
ELSE
IJOBVR = -1
ILVSR = .FALSE.
END IF
*
* Test the input arguments
*
LWKMIN = MAX( 2*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVSL.LT.1 .OR. ( ILVSL .AND. LDVSL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVSR.LT.1 .OR. ( ILVSR .AND. LDVSR.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = N*( NB+1 )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGEGS ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SMLNUM = N*SAFMIN / EPS
BIGNUM = ONE / SMLNUM
*
* Scale A if max element outside range [SMLNUM,BIGNUM]
*
ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
ILASCL = .FALSE.
IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN
ANRMTO = SMLNUM
ILASCL = .TRUE.
ELSE IF( ANRM.GT.BIGNUM ) THEN
ANRMTO = BIGNUM
ILASCL = .TRUE.
END IF
*
IF( ILASCL ) THEN
CALL ZLASCL( 'G', -1, -1, ANRM, ANRMTO, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Scale B if max element outside range [SMLNUM,BIGNUM]
*
BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
ILBSCL = .FALSE.
IF( BNRM.GT.ZERO .AND. BNRM.LT.SMLNUM ) THEN
BNRMTO = SMLNUM
ILBSCL = .TRUE.
ELSE IF( BNRM.GT.BIGNUM ) THEN
BNRMTO = BIGNUM
ILBSCL = .TRUE.
END IF
*
IF( ILBSCL ) THEN
CALL ZLASCL( 'G', -1, -1, BNRM, BNRMTO, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
*
ILEFT = 1
IRIGHT = N + 1
IRWORK = IRIGHT + N
IWORK = 1
CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 10
END IF
*
* Reduce B to triangular form, and initialize VSL and/or VSR
*
IROWS = IHI + 1 - ILO
ICOLS = N + 1 - ILO
ITAU = IWORK
IWORK = ITAU + IROWS
CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 10
END IF
*
CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 10
END IF
*
IF( ILVSL ) THEN
CALL ZLASET( 'Full', N, N, CZERO, CONE, VSL, LDVSL )
CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VSL( ILO+1, ILO ), LDVSL )
CALL ZUNGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 10
END IF
END IF
*
IF( ILVSR )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VSR, LDVSR )
*
* Reduce to generalized Hessenberg form
*
CALL ZGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
$ LDVSL, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 10
END IF
*
* Perform QZ algorithm, computing Schur vectors if desired
*
IWORK = ITAU
CALL ZHGEQZ( 'S', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, WORK( IWORK ),
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 10
END IF
*
* Apply permutation to VSL and VSR
*
IF( ILVSL ) THEN
CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VSL, LDVSL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 10
END IF
END IF
IF( ILVSR ) THEN
CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VSR, LDVSR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 10
END IF
END IF
*
* Undo scaling
*
IF( ILASCL ) THEN
CALL ZLASCL( 'U', -1, -1, ANRMTO, ANRM, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL ZLASCL( 'G', -1, -1, ANRMTO, ANRM, N, 1, ALPHA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
IF( ILBSCL ) THEN
CALL ZLASCL( 'U', -1, -1, BNRMTO, BNRM, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
CALL ZLASCL( 'G', -1, -1, BNRMTO, BNRM, N, 1, BETA, N, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
RETURN
END IF
END IF
*
10 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of ZGEGS
*
END

+ 703
- 0
lapack-netlib/SRC/zgegv.f View File

@@ -0,0 +1,703 @@
*> \brief <b> ZGEGV computes the eigenvalues and, optionally, the left and/or right eigenvectors of a complex matrix pair (A,B).</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZGEGV + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgegv.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgegv.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgegv.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
* VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBVL, JOBVR
* INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION RWORK( * )
* COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
* $ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
* $ WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine ZGGEV.
*>
*> ZGEGV computes the eigenvalues and, optionally, the left and/or right
*> eigenvectors of a complex matrix pair (A,B).
*> Given two square matrices A and B,
*> the generalized nonsymmetric eigenvalue problem (GNEP) is to find the
*> eigenvalues lambda and corresponding (non-zero) eigenvectors x such
*> that
*> A*x = lambda*B*x.
*>
*> An alternate form is to find the eigenvalues mu and corresponding
*> eigenvectors y such that
*> mu*A*y = B*y.
*>
*> These two forms are equivalent with mu = 1/lambda and x = y if
*> neither lambda nor mu is zero. In order to deal with the case that
*> lambda or mu is zero or small, two values alpha and beta are returned
*> for each eigenvalue, such that lambda = alpha/beta and
*> mu = beta/alpha.
*>
*> The vectors x and y in the above equations are right eigenvectors of
*> the matrix pair (A,B). Vectors u and v satisfying
*> u**H*A = lambda*u**H*B or mu*v**H*A = v**H*B
*> are left eigenvectors of (A,B).
*>
*> Note: this routine performs "full balancing" on A and B
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBVL
*> \verbatim
*> JOBVL is CHARACTER*1
*> = 'N': do not compute the left generalized eigenvectors;
*> = 'V': compute the left generalized eigenvectors (returned
*> in VL).
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*> JOBVR is CHARACTER*1
*> = 'N': do not compute the right generalized eigenvectors;
*> = 'V': compute the right generalized eigenvectors (returned
*> in VR).
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrices A, B, VL, and VR. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA, N)
*> On entry, the matrix A.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit A
*> contains the Schur form of A from the generalized Schur
*> factorization of the pair (A,B) after balancing. If no
*> eigenvectors were computed, then only the diagonal elements
*> of the Schur form will be correct. See ZGGHRD and ZHGEQZ
*> for details.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 array, dimension (LDB, N)
*> On entry, the matrix B.
*> If JOBVL = 'V' or JOBVR = 'V', then on exit B contains the
*> upper triangular matrix obtained from B in the generalized
*> Schur factorization of the pair (A,B) after balancing.
*> If no eigenvectors were computed, then only the diagonal
*> elements of B will be correct. See ZGGHRD and ZHGEQZ for
*> details.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*> ALPHA is COMPLEX*16 array, dimension (N)
*> The complex scalars alpha that define the eigenvalues of
*> GNEP.
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*> BETA is COMPLEX*16 array, dimension (N)
*> The complex scalars beta that define the eigenvalues of GNEP.
*>
*> Together, the quantities alpha = ALPHA(j) and beta = BETA(j)
*> represent the j-th eigenvalue of the matrix pair (A,B), in
*> one of the forms lambda = alpha/beta or mu = beta/alpha.
*> Since either lambda or mu may overflow, they should not,
*> in general, be computed.
*> \endverbatim
*>
*> \param[out] VL
*> \verbatim
*> VL is COMPLEX*16 array, dimension (LDVL,N)
*> If JOBVL = 'V', the left eigenvectors u(j) are stored
*> in the columns of VL, in the same order as their eigenvalues.
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*> LDVL is INTEGER
*> The leading dimension of the matrix VL. LDVL >= 1, and
*> if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*> VR is COMPLEX*16 array, dimension (LDVR,N)
*> If JOBVR = 'V', the right eigenvectors x(j) are stored
*> in the columns of VR, in the same order as their eigenvalues.
*> Each eigenvector is scaled so that its largest component has
*> abs(real part) + abs(imag. part) = 1, except for eigenvectors
*> corresponding to an eigenvalue with alpha = beta = 0, which
*> are set to zero.
*> Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*> LDVR is INTEGER
*> The leading dimension of the matrix VR. LDVR >= 1, and
*> if JOBVR = 'V', LDVR >= 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(1,2*N).
*> For good performance, LWORK must generally be larger.
*> To compute the optimal value of LWORK, call ILAENV to get
*> blocksizes (for ZGEQRF, ZUNMQR, and ZUNGQR.) Then compute:
*> NB -- MAX of the blocksizes for ZGEQRF, ZUNMQR, and ZUNGQR;
*> The optimal LWORK is MAX( 2*N, N*(NB+1) ).
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, returns
*> this value as the first entry of the WORK array, and no error
*> message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (8*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> =1,...,N:
*> The QZ iteration failed. No eigenvectors have been
*> calculated, but ALPHA(j) and BETA(j) should be
*> correct for j=INFO+1,...,N.
*> > N: errors that usually indicate LAPACK problems:
*> =N+1: error return from ZGGBAL
*> =N+2: error return from ZGEQRF
*> =N+3: error return from ZUNMQR
*> =N+4: error return from ZUNGQR
*> =N+5: error return from ZGGHRD
*> =N+6: error return from ZHGEQZ (other than failed
*> iteration)
*> =N+7: error return from ZTGEVC
*> =N+8: error return from ZGGBAK (computing VL)
*> =N+9: error return from ZGGBAK (computing VR)
*> =N+10: error return from ZLASCL (various calls)
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16GEeigen
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Balancing
*> ---------
*>
*> This driver calls ZGGBAL to both permute and scale rows and columns
*> of A and B. The permutations PL and PR are chosen so that PL*A*PR
*> and PL*B*R will be upper triangular except for the diagonal blocks
*> A(i:j,i:j) and B(i:j,i:j), with i and j as close together as
*> possible. The diagonal scaling matrices DL and DR are chosen so
*> that the pair DL*PL*A*PR*DR, DL*PL*B*PR*DR have elements close to
*> one (except for the elements that start out zero.)
*>
*> After the eigenvalues and eigenvectors of the balanced matrices
*> have been computed, ZGGBAK transforms the eigenvectors back to what
*> they would have been (in perfect arithmetic) if they had not been
*> balanced.
*>
*> Contents of A and B on Exit
*> -------- -- - --- - -- ----
*>
*> If any eigenvectors are computed (either JOBVL='V' or JOBVR='V' or
*> both), then on exit the arrays A and B will contain the complex Schur
*> form[*] of the "balanced" versions of A and B. If no eigenvectors
*> are computed, then only the diagonal blocks will be correct.
*>
*> [*] In other words, upper triangular form.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZGEGV( JOBVL, JOBVR, N, A, LDA, B, LDB, ALPHA, BETA,
$ VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO )
*
* -- LAPACK driver routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER JOBVL, JOBVR
INTEGER INFO, LDA, LDB, LDVL, LDVR, LWORK, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION RWORK( * )
COMPLEX*16 A( LDA, * ), ALPHA( * ), B( LDB, * ),
$ BETA( * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ),
$ CONE = ( 1.0D0, 0.0D0 ) )
* ..
* .. Local Scalars ..
LOGICAL ILIMIT, ILV, ILVL, ILVR, LQUERY
CHARACTER CHTEMP
INTEGER ICOLS, IHI, IINFO, IJOBVL, IJOBVR, ILEFT, ILO,
$ IN, IRIGHT, IROWS, IRWORK, ITAU, IWORK, JC, JR,
$ LOPT, LWKMIN, LWKOPT, NB, NB1, NB2, NB3
DOUBLE PRECISION ABSAI, ABSAR, ABSB, ANRM, ANRM1, ANRM2, BNRM,
$ BNRM1, BNRM2, EPS, SAFMAX, SAFMIN, SALFAI,
$ SALFAR, SBETA, SCALE, TEMP
COMPLEX*16 X
* ..
* .. Local Arrays ..
LOGICAL LDUMMA( 1 )
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZGEQRF, ZGGBAK, ZGGBAL, ZGGHRD, ZHGEQZ,
$ ZLACPY, ZLASCL, ZLASET, ZTGEVC, ZUNGQR, ZUNMQR
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANGE
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANGE
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DIMAG, INT, MAX
* ..
* .. Statement Functions ..
DOUBLE PRECISION ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
* ..
* .. Executable Statements ..
*
* Decode the input arguments
*
IF( LSAME( JOBVL, 'N' ) ) THEN
IJOBVL = 1
ILVL = .FALSE.
ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
IJOBVL = 2
ILVL = .TRUE.
ELSE
IJOBVL = -1
ILVL = .FALSE.
END IF
*
IF( LSAME( JOBVR, 'N' ) ) THEN
IJOBVR = 1
ILVR = .FALSE.
ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
IJOBVR = 2
ILVR = .TRUE.
ELSE
IJOBVR = -1
ILVR = .FALSE.
END IF
ILV = ILVL .OR. ILVR
*
* Test the input arguments
*
LWKMIN = MAX( 2*N, 1 )
LWKOPT = LWKMIN
WORK( 1 ) = LWKOPT
LQUERY = ( LWORK.EQ.-1 )
INFO = 0
IF( IJOBVL.LE.0 ) THEN
INFO = -1
ELSE IF( IJOBVR.LE.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -5
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
INFO = -11
ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.LWKMIN .AND. .NOT.LQUERY ) THEN
INFO = -15
END IF
*
IF( INFO.EQ.0 ) THEN
NB1 = ILAENV( 1, 'ZGEQRF', ' ', N, N, -1, -1 )
NB2 = ILAENV( 1, 'ZUNMQR', ' ', N, N, N, -1 )
NB3 = ILAENV( 1, 'ZUNGQR', ' ', N, N, N, -1 )
NB = MAX( NB1, NB2, NB3 )
LOPT = MAX( 2*N, N*( NB+1 ) )
WORK( 1 ) = LOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZGEGV ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
* Get machine constants
*
EPS = DLAMCH( 'E' )*DLAMCH( 'B' )
SAFMIN = DLAMCH( 'S' )
SAFMIN = SAFMIN + SAFMIN
SAFMAX = ONE / SAFMIN
*
* Scale A
*
ANRM = ZLANGE( 'M', N, N, A, LDA, RWORK )
ANRM1 = ANRM
ANRM2 = ONE
IF( ANRM.LT.ONE ) THEN
IF( SAFMAX*ANRM.LT.ONE ) THEN
ANRM1 = SAFMIN
ANRM2 = SAFMAX*ANRM
END IF
END IF
*
IF( ANRM.GT.ZERO ) THEN
CALL ZLASCL( 'G', -1, -1, ANRM, ONE, N, N, A, LDA, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Scale B
*
BNRM = ZLANGE( 'M', N, N, B, LDB, RWORK )
BNRM1 = BNRM
BNRM2 = ONE
IF( BNRM.LT.ONE ) THEN
IF( SAFMAX*BNRM.LT.ONE ) THEN
BNRM1 = SAFMIN
BNRM2 = SAFMAX*BNRM
END IF
END IF
*
IF( BNRM.GT.ZERO ) THEN
CALL ZLASCL( 'G', -1, -1, BNRM, ONE, N, N, B, LDB, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 10
RETURN
END IF
END IF
*
* Permute the matrix to make it more nearly triangular
* Also "balance" the matrix.
*
ILEFT = 1
IRIGHT = N + 1
IRWORK = IRIGHT + N
CALL ZGGBAL( 'P', N, A, LDA, B, LDB, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), RWORK( IRWORK ), IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 1
GO TO 80
END IF
*
* Reduce B to triangular form, and initialize VL and/or VR
*
IROWS = IHI + 1 - ILO
IF( ILV ) THEN
ICOLS = N + 1 - ILO
ELSE
ICOLS = IROWS
END IF
ITAU = 1
IWORK = ITAU + IROWS
CALL ZGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
$ WORK( IWORK ), LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 2
GO TO 80
END IF
*
CALL ZUNMQR( 'L', 'C', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
$ WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWORK ),
$ LWORK+1-IWORK, IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 3
GO TO 80
END IF
*
IF( ILVL ) THEN
CALL ZLASET( 'Full', N, N, CZERO, CONE, VL, LDVL )
CALL ZLACPY( 'L', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
$ VL( ILO+1, ILO ), LDVL )
CALL ZUNGQR( IROWS, IROWS, IROWS, VL( ILO, ILO ), LDVL,
$ WORK( ITAU ), WORK( IWORK ), LWORK+1-IWORK,
$ IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
INFO = N + 4
GO TO 80
END IF
END IF
*
IF( ILVR )
$ CALL ZLASET( 'Full', N, N, CZERO, CONE, VR, LDVR )
*
* Reduce to generalized Hessenberg form
*
IF( ILV ) THEN
*
* Eigenvectors requested -- work on whole matrix.
*
CALL ZGGHRD( JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB, VL,
$ LDVL, VR, LDVR, IINFO )
ELSE
CALL ZGGHRD( 'N', 'N', IROWS, 1, IROWS, A( ILO, ILO ), LDA,
$ B( ILO, ILO ), LDB, VL, LDVL, VR, LDVR, IINFO )
END IF
IF( IINFO.NE.0 ) THEN
INFO = N + 5
GO TO 80
END IF
*
* Perform QZ algorithm
*
IWORK = ITAU
IF( ILV ) THEN
CHTEMP = 'S'
ELSE
CHTEMP = 'E'
END IF
CALL ZHGEQZ( CHTEMP, JOBVL, JOBVR, N, ILO, IHI, A, LDA, B, LDB,
$ ALPHA, BETA, VL, LDVL, VR, LDVR, WORK( IWORK ),
$ LWORK+1-IWORK, RWORK( IRWORK ), IINFO )
IF( IINFO.GE.0 )
$ LWKOPT = MAX( LWKOPT, INT( WORK( IWORK ) )+IWORK-1 )
IF( IINFO.NE.0 ) THEN
IF( IINFO.GT.0 .AND. IINFO.LE.N ) THEN
INFO = IINFO
ELSE IF( IINFO.GT.N .AND. IINFO.LE.2*N ) THEN
INFO = IINFO - N
ELSE
INFO = N + 6
END IF
GO TO 80
END IF
*
IF( ILV ) THEN
*
* Compute Eigenvectors
*
IF( ILVL ) THEN
IF( ILVR ) THEN
CHTEMP = 'B'
ELSE
CHTEMP = 'L'
END IF
ELSE
CHTEMP = 'R'
END IF
*
CALL ZTGEVC( CHTEMP, 'B', LDUMMA, N, A, LDA, B, LDB, VL, LDVL,
$ VR, LDVR, N, IN, WORK( IWORK ), RWORK( IRWORK ),
$ IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 7
GO TO 80
END IF
*
* Undo balancing on VL and VR, rescale
*
IF( ILVL ) THEN
CALL ZGGBAK( 'P', 'L', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VL, LDVL, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 8
GO TO 80
END IF
DO 30 JC = 1, N
TEMP = ZERO
DO 10 JR = 1, N
TEMP = MAX( TEMP, ABS1( VL( JR, JC ) ) )
10 CONTINUE
IF( TEMP.LT.SAFMIN )
$ GO TO 30
TEMP = ONE / TEMP
DO 20 JR = 1, N
VL( JR, JC ) = VL( JR, JC )*TEMP
20 CONTINUE
30 CONTINUE
END IF
IF( ILVR ) THEN
CALL ZGGBAK( 'P', 'R', N, ILO, IHI, RWORK( ILEFT ),
$ RWORK( IRIGHT ), N, VR, LDVR, IINFO )
IF( IINFO.NE.0 ) THEN
INFO = N + 9
GO TO 80
END IF
DO 60 JC = 1, N
TEMP = ZERO
DO 40 JR = 1, N
TEMP = MAX( TEMP, ABS1( VR( JR, JC ) ) )
40 CONTINUE
IF( TEMP.LT.SAFMIN )
$ GO TO 60
TEMP = ONE / TEMP
DO 50 JR = 1, N
VR( JR, JC ) = VR( JR, JC )*TEMP
50 CONTINUE
60 CONTINUE
END IF
*
* End of eigenvector calculation
*
END IF
*
* Undo scaling in alpha, beta
*
* Note: this does not give the alpha and beta for the unscaled
* problem.
*
* Un-scaling is limited to avoid underflow in alpha and beta
* if they are significant.
*
DO 70 JC = 1, N
ABSAR = ABS( DBLE( ALPHA( JC ) ) )
ABSAI = ABS( DIMAG( ALPHA( JC ) ) )
ABSB = ABS( DBLE( BETA( JC ) ) )
SALFAR = ANRM*DBLE( ALPHA( JC ) )
SALFAI = ANRM*DIMAG( ALPHA( JC ) )
SBETA = BNRM*DBLE( BETA( JC ) )
ILIMIT = .FALSE.
SCALE = ONE
*
* Check for significant underflow in imaginary part of ALPHA
*
IF( ABS( SALFAI ).LT.SAFMIN .AND. ABSAI.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = ( SAFMIN / ANRM1 ) / MAX( SAFMIN, ANRM2*ABSAI )
END IF
*
* Check for significant underflow in real part of ALPHA
*
IF( ABS( SALFAR ).LT.SAFMIN .AND. ABSAR.GE.
$ MAX( SAFMIN, EPS*ABSAI, EPS*ABSB ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( SAFMIN / ANRM1 ) /
$ MAX( SAFMIN, ANRM2*ABSAR ) )
END IF
*
* Check for significant underflow in BETA
*
IF( ABS( SBETA ).LT.SAFMIN .AND. ABSB.GE.
$ MAX( SAFMIN, EPS*ABSAR, EPS*ABSAI ) ) THEN
ILIMIT = .TRUE.
SCALE = MAX( SCALE, ( SAFMIN / BNRM1 ) /
$ MAX( SAFMIN, BNRM2*ABSB ) )
END IF
*
* Check for possible overflow when limiting scaling
*
IF( ILIMIT ) THEN
TEMP = ( SCALE*SAFMIN )*MAX( ABS( SALFAR ), ABS( SALFAI ),
$ ABS( SBETA ) )
IF( TEMP.GT.ONE )
$ SCALE = SCALE / TEMP
IF( SCALE.LT.ONE )
$ ILIMIT = .FALSE.
END IF
*
* Recompute un-scaled ALPHA, BETA if necessary.
*
IF( ILIMIT ) THEN
SALFAR = ( SCALE*DBLE( ALPHA( JC ) ) )*ANRM
SALFAI = ( SCALE*DIMAG( ALPHA( JC ) ) )*ANRM
SBETA = ( SCALE*BETA( JC ) )*BNRM
END IF
ALPHA( JC ) = DCMPLX( SALFAR, SALFAI )
BETA( JC ) = SBETA
70 CONTINUE
*
80 CONTINUE
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of ZGEGV
*
END

Write Preview
Loading…
Cancel
Save