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Diffstat (limited to '2.3-1/src/fortran/lapack/dgeesx.f')
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diff --git a/2.3-1/src/fortran/lapack/dgeesx.f b/2.3-1/src/fortran/lapack/dgeesx.f new file mode 100644 index 00000000..deb30ab2 --- /dev/null +++ b/2.3-1/src/fortran/lapack/dgeesx.f @@ -0,0 +1,527 @@ + SUBROUTINE DGEESX( JOBVS, SORT, SELECT, SENSE, N, A, LDA, SDIM, + $ WR, WI, VS, LDVS, RCONDE, RCONDV, WORK, LWORK, + $ IWORK, LIWORK, BWORK, INFO ) +* +* -- LAPACK driver routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + CHARACTER JOBVS, SENSE, SORT + INTEGER INFO, LDA, LDVS, LIWORK, LWORK, N, SDIM + DOUBLE PRECISION RCONDE, RCONDV +* .. +* .. Array Arguments .. + LOGICAL BWORK( * ) + INTEGER IWORK( * ) + DOUBLE PRECISION A( LDA, * ), VS( LDVS, * ), WI( * ), WORK( * ), + $ WR( * ) +* .. +* .. Function Arguments .. + LOGICAL SELECT + EXTERNAL SELECT +* .. +* +* Purpose +* ======= +* +* DGEESX computes for an N-by-N real nonsymmetric matrix A, the +* eigenvalues, the real Schur form T, and, optionally, the matrix of +* Schur vectors Z. This gives the Schur factorization A = Z*T*(Z**T). +* +* Optionally, it also orders the eigenvalues on the diagonal of the +* real Schur form so that selected eigenvalues are at the top left; +* computes a reciprocal condition number for the average of the +* selected eigenvalues (RCONDE); and computes a reciprocal condition +* number for the right invariant subspace corresponding to the +* selected eigenvalues (RCONDV). The leading columns of Z form an +* orthonormal basis for this invariant subspace. +* +* For further explanation of the reciprocal condition numbers RCONDE +* and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where +* these quantities are called s and sep respectively). +* +* A real matrix is in real Schur form if it is upper quasi-triangular +* with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in +* the form +* [ a b ] +* [ c a ] +* +* where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc). +* +* Arguments +* ========= +* +* JOBVS (input) CHARACTER*1 +* = 'N': Schur vectors are not computed; +* = 'V': Schur vectors are computed. +* +* SORT (input) CHARACTER*1 +* Specifies whether or not to order the eigenvalues on the +* diagonal of the Schur form. +* = 'N': Eigenvalues are not ordered; +* = 'S': Eigenvalues are ordered (see SELECT). +* +* SELECT (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION arguments +* SELECT must be declared EXTERNAL in the calling subroutine. +* If SORT = 'S', SELECT is used to select eigenvalues to sort +* to the top left of the Schur form. +* If SORT = 'N', SELECT is not referenced. +* An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if +* SELECT(WR(j),WI(j)) is true; i.e., if either one of a +* complex conjugate pair of eigenvalues is selected, then both +* are. Note that a selected complex eigenvalue may no longer +* satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since +* ordering may change the value of complex eigenvalues +* (especially if the eigenvalue is ill-conditioned); in this +* case INFO may be set to N+3 (see INFO below). +* +* SENSE (input) CHARACTER*1 +* Determines which reciprocal condition numbers are computed. +* = 'N': None are computed; +* = 'E': Computed for average of selected eigenvalues only; +* = 'V': Computed for selected right invariant subspace only; +* = 'B': Computed for both. +* If SENSE = 'E', 'V' or 'B', SORT must equal 'S'. +* +* N (input) INTEGER +* The order of the matrix A. N >= 0. +* +* A (input/output) DOUBLE PRECISION array, dimension (LDA, N) +* On entry, the N-by-N matrix A. +* On exit, A is overwritten by its real Schur form T. +* +* LDA (input) INTEGER +* The leading dimension of the array A. LDA >= max(1,N). +* +* SDIM (output) INTEGER +* If SORT = 'N', SDIM = 0. +* If SORT = 'S', SDIM = number of eigenvalues (after sorting) +* for which SELECT is true. (Complex conjugate +* pairs for which SELECT is true for either +* eigenvalue count as 2.) +* +* WR (output) DOUBLE PRECISION array, dimension (N) +* WI (output) DOUBLE PRECISION array, dimension (N) +* WR and WI contain the real and imaginary parts, respectively, +* of the computed eigenvalues, in the same order that they +* appear on the diagonal of the output Schur form T. Complex +* conjugate pairs of eigenvalues appear consecutively with the +* eigenvalue having the positive imaginary part first. +* +* VS (output) DOUBLE PRECISION array, dimension (LDVS,N) +* If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur +* vectors. +* If JOBVS = 'N', VS is not referenced. +* +* LDVS (input) INTEGER +* The leading dimension of the array VS. LDVS >= 1, and if +* JOBVS = 'V', LDVS >= N. +* +* RCONDE (output) DOUBLE PRECISION +* If SENSE = 'E' or 'B', RCONDE contains the reciprocal +* condition number for the average of the selected eigenvalues. +* Not referenced if SENSE = 'N' or 'V'. +* +* RCONDV (output) DOUBLE PRECISION +* If SENSE = 'V' or 'B', RCONDV contains the reciprocal +* condition number for the selected right invariant subspace. +* Not referenced if SENSE = 'N' or 'E'. +* +* WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) +* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. +* +* LWORK (input) INTEGER +* The dimension of the array WORK. LWORK >= max(1,3*N). +* Also, if SENSE = 'E' or 'V' or 'B', +* LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of +* selected eigenvalues computed by this routine. Note that +* N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only +* returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or +* 'B' this may not be large enough. +* For good performance, LWORK must generally be larger. +* +* If LWORK = -1, then a workspace query is assumed; the routine +* only calculates upper bounds on the optimal sizes of the +* arrays WORK and IWORK, returns these values as the first +* entries of the WORK and IWORK arrays, and no error messages +* related to LWORK or LIWORK are issued by XERBLA. +* +* IWORK (workspace/output) INTEGER array, dimension (MAX(1,LIWORK)) +* On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK. +* +* LIWORK (input) INTEGER +* The dimension of the array IWORK. +* LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM). +* Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is +* only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this +* may not be large enough. +* +* If LIWORK = -1, then a workspace query is assumed; the +* routine only calculates upper bounds on the optimal sizes of +* the arrays WORK and IWORK, returns these values as the first +* entries of the WORK and IWORK arrays, and no error messages +* related to LWORK or LIWORK are issued by XERBLA. +* +* BWORK (workspace) LOGICAL array, dimension (N) +* Not referenced if SORT = 'N'. +* +* INFO (output) INTEGER +* = 0: successful exit +* < 0: if INFO = -i, the i-th argument had an illegal value. +* > 0: if INFO = i, and i is +* <= N: the QR algorithm failed to compute all the +* eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI +* contain those eigenvalues which have converged; if +* JOBVS = 'V', VS contains the transformation which +* reduces A to its partially converged Schur form. +* = N+1: the eigenvalues could not be reordered because some +* eigenvalues were too close to separate (the problem +* is very ill-conditioned); +* = N+2: after reordering, roundoff changed values of some +* complex eigenvalues so that leading eigenvalues in +* the Schur form no longer satisfy SELECT=.TRUE. This +* could also be caused by underflow due to scaling. +* +* ===================================================================== +* +* .. Parameters .. + DOUBLE PRECISION ZERO, ONE + PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 ) +* .. +* .. Local Scalars .. + LOGICAL CURSL, LASTSL, LQUERY, LST2SL, SCALEA, WANTSB, + $ WANTSE, WANTSN, WANTST, WANTSV, WANTVS + INTEGER HSWORK, I, I1, I2, IBAL, ICOND, IERR, IEVAL, + $ IHI, ILO, INXT, IP, ITAU, IWRK, LIWRK, LWRK, + $ MAXWRK, MINWRK + DOUBLE PRECISION ANRM, BIGNUM, CSCALE, EPS, SMLNUM +* .. +* .. Local Arrays .. + DOUBLE PRECISION DUM( 1 ) +* .. +* .. External Subroutines .. + EXTERNAL DCOPY, DGEBAK, DGEBAL, DGEHRD, DHSEQR, DLACPY, + $ DLASCL, DORGHR, DSWAP, DTRSEN, XERBLA +* .. +* .. External Functions .. + LOGICAL LSAME + INTEGER ILAENV + DOUBLE PRECISION DLAMCH, DLANGE + EXTERNAL LSAME, ILAENV, DLABAD, DLAMCH, DLANGE +* .. +* .. Intrinsic Functions .. + INTRINSIC MAX, SQRT +* .. +* .. Executable Statements .. +* +* Test the input arguments +* + INFO = 0 + WANTVS = LSAME( JOBVS, 'V' ) + WANTST = LSAME( SORT, 'S' ) + WANTSN = LSAME( SENSE, 'N' ) + WANTSE = LSAME( SENSE, 'E' ) + WANTSV = LSAME( SENSE, 'V' ) + WANTSB = LSAME( SENSE, 'B' ) + LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 ) + IF( ( .NOT.WANTVS ) .AND. ( .NOT.LSAME( JOBVS, 'N' ) ) ) THEN + INFO = -1 + ELSE IF( ( .NOT.WANTST ) .AND. ( .NOT.LSAME( SORT, 'N' ) ) ) THEN + INFO = -2 + ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSV .OR. WANTSB ) .OR. + $ ( .NOT.WANTST .AND. .NOT.WANTSN ) ) THEN + INFO = -4 + ELSE IF( N.LT.0 ) THEN + INFO = -5 + ELSE IF( LDA.LT.MAX( 1, N ) ) THEN + INFO = -7 + ELSE IF( LDVS.LT.1 .OR. ( WANTVS .AND. LDVS.LT.N ) ) THEN + INFO = -12 + END IF +* +* Compute workspace +* (Note: Comments in the code beginning "RWorkspace:" describe the +* minimal amount of real workspace needed at that point in the +* code, as well as the preferred amount for good performance. +* IWorkspace refers to integer workspace. +* NB refers to the optimal block size for the immediately +* following subroutine, as returned by ILAENV. +* HSWORK refers to the workspace preferred by DHSEQR, as +* calculated below. HSWORK is computed assuming ILO=1 and IHI=N, +* the worst case. +* If SENSE = 'E', 'V' or 'B', then the amount of workspace needed +* depends on SDIM, which is computed by the routine DTRSEN later +* in the code.) +* + IF( INFO.EQ.0 ) THEN + LIWRK = 1 + IF( N.EQ.0 ) THEN + MINWRK = 1 + LWRK = 1 + ELSE + MAXWRK = 2*N + N*ILAENV( 1, 'DGEHRD', ' ', N, 1, N, 0 ) + MINWRK = 3*N +* + CALL DHSEQR( 'S', JOBVS, N, 1, N, A, LDA, WR, WI, VS, LDVS, + $ WORK, -1, IEVAL ) + HSWORK = WORK( 1 ) +* + IF( .NOT.WANTVS ) THEN + MAXWRK = MAX( MAXWRK, N + HSWORK ) + ELSE + MAXWRK = MAX( MAXWRK, 2*N + ( N - 1 )*ILAENV( 1, + $ 'DORGHR', ' ', N, 1, N, -1 ) ) + MAXWRK = MAX( MAXWRK, N + HSWORK ) + END IF + LWRK = MAXWRK + IF( .NOT.WANTSN ) + $ LWRK = MAX( LWRK, N + ( N*N )/2 ) + IF( WANTSV .OR. WANTSB ) + $ LIWRK = ( N*N )/4 + END IF + IWORK( 1 ) = LIWRK + WORK( 1 ) = LWRK +* + IF( LWORK.LT.MINWRK .AND. .NOT.LQUERY ) THEN + INFO = -16 + ELSE IF( LIWORK.LT.1 .AND. .NOT.LQUERY ) THEN + INFO = -18 + END IF + END IF +* + IF( INFO.NE.0 ) THEN + CALL XERBLA( 'DGEESX', -INFO ) + RETURN + END IF +* +* Quick return if possible +* + IF( N.EQ.0 ) THEN + SDIM = 0 + RETURN + END IF +* +* Get machine constants +* + EPS = DLAMCH( 'P' ) + SMLNUM = DLAMCH( 'S' ) + BIGNUM = ONE / SMLNUM + CALL DLABAD( SMLNUM, BIGNUM ) + SMLNUM = SQRT( SMLNUM ) / EPS + BIGNUM = ONE / SMLNUM +* +* Scale A if max element outside range [SMLNUM,BIGNUM] +* + ANRM = DLANGE( 'M', N, N, A, LDA, DUM ) + SCALEA = .FALSE. + IF( ANRM.GT.ZERO .AND. ANRM.LT.SMLNUM ) THEN + SCALEA = .TRUE. + CSCALE = SMLNUM + ELSE IF( ANRM.GT.BIGNUM ) THEN + SCALEA = .TRUE. + CSCALE = BIGNUM + END IF + IF( SCALEA ) + $ CALL DLASCL( 'G', 0, 0, ANRM, CSCALE, N, N, A, LDA, IERR ) +* +* Permute the matrix to make it more nearly triangular +* (RWorkspace: need N) +* + IBAL = 1 + CALL DGEBAL( 'P', N, A, LDA, ILO, IHI, WORK( IBAL ), IERR ) +* +* Reduce to upper Hessenberg form +* (RWorkspace: need 3*N, prefer 2*N+N*NB) +* + ITAU = N + IBAL + IWRK = N + ITAU + CALL DGEHRD( N, ILO, IHI, A, LDA, WORK( ITAU ), WORK( IWRK ), + $ LWORK-IWRK+1, IERR ) +* + IF( WANTVS ) THEN +* +* Copy Householder vectors to VS +* + CALL DLACPY( 'L', N, N, A, LDA, VS, LDVS ) +* +* Generate orthogonal matrix in VS +* (RWorkspace: need 3*N-1, prefer 2*N+(N-1)*NB) +* + CALL DORGHR( N, ILO, IHI, VS, LDVS, WORK( ITAU ), WORK( IWRK ), + $ LWORK-IWRK+1, IERR ) + END IF +* + SDIM = 0 +* +* Perform QR iteration, accumulating Schur vectors in VS if desired +* (RWorkspace: need N+1, prefer N+HSWORK (see comments) ) +* + IWRK = ITAU + CALL DHSEQR( 'S', JOBVS, N, ILO, IHI, A, LDA, WR, WI, VS, LDVS, + $ WORK( IWRK ), LWORK-IWRK+1, IEVAL ) + IF( IEVAL.GT.0 ) + $ INFO = IEVAL +* +* Sort eigenvalues if desired +* + IF( WANTST .AND. INFO.EQ.0 ) THEN + IF( SCALEA ) THEN + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WR, N, IERR ) + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N, 1, WI, N, IERR ) + END IF + DO 10 I = 1, N + BWORK( I ) = SELECT( WR( I ), WI( I ) ) + 10 CONTINUE +* +* Reorder eigenvalues, transform Schur vectors, and compute +* reciprocal condition numbers +* (RWorkspace: if SENSE is not 'N', need N+2*SDIM*(N-SDIM) +* otherwise, need N ) +* (IWorkspace: if SENSE is 'V' or 'B', need SDIM*(N-SDIM) +* otherwise, need 0 ) +* + CALL DTRSEN( SENSE, JOBVS, BWORK, N, A, LDA, VS, LDVS, WR, WI, + $ SDIM, RCONDE, RCONDV, WORK( IWRK ), LWORK-IWRK+1, + $ IWORK, LIWORK, ICOND ) + IF( .NOT.WANTSN ) + $ MAXWRK = MAX( MAXWRK, N+2*SDIM*( N-SDIM ) ) + IF( ICOND.EQ.-15 ) THEN +* +* Not enough real workspace +* + INFO = -16 + ELSE IF( ICOND.EQ.-17 ) THEN +* +* Not enough integer workspace +* + INFO = -18 + ELSE IF( ICOND.GT.0 ) THEN +* +* DTRSEN failed to reorder or to restore standard Schur form +* + INFO = ICOND + N + END IF + END IF +* + IF( WANTVS ) THEN +* +* Undo balancing +* (RWorkspace: need N) +* + CALL DGEBAK( 'P', 'R', N, ILO, IHI, WORK( IBAL ), N, VS, LDVS, + $ IERR ) + END IF +* + IF( SCALEA ) THEN +* +* Undo scaling for the Schur form of A +* + CALL DLASCL( 'H', 0, 0, CSCALE, ANRM, N, N, A, LDA, IERR ) + CALL DCOPY( N, A, LDA+1, WR, 1 ) + IF( ( WANTSV .OR. WANTSB ) .AND. INFO.EQ.0 ) THEN + DUM( 1 ) = RCONDV + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, 1, 1, DUM, 1, IERR ) + RCONDV = DUM( 1 ) + END IF + IF( CSCALE.EQ.SMLNUM ) THEN +* +* If scaling back towards underflow, adjust WI if an +* offdiagonal element of a 2-by-2 block in the Schur form +* underflows. +* + IF( IEVAL.GT.0 ) THEN + I1 = IEVAL + 1 + I2 = IHI - 1 + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, ILO-1, 1, WI, N, + $ IERR ) + ELSE IF( WANTST ) THEN + I1 = 1 + I2 = N - 1 + ELSE + I1 = ILO + I2 = IHI - 1 + END IF + INXT = I1 - 1 + DO 20 I = I1, I2 + IF( I.LT.INXT ) + $ GO TO 20 + IF( WI( I ).EQ.ZERO ) THEN + INXT = I + 1 + ELSE + IF( A( I+1, I ).EQ.ZERO ) THEN + WI( I ) = ZERO + WI( I+1 ) = ZERO + ELSE IF( A( I+1, I ).NE.ZERO .AND. A( I, I+1 ).EQ. + $ ZERO ) THEN + WI( I ) = ZERO + WI( I+1 ) = ZERO + IF( I.GT.1 ) + $ CALL DSWAP( I-1, A( 1, I ), 1, A( 1, I+1 ), 1 ) + IF( N.GT.I+1 ) + $ CALL DSWAP( N-I-1, A( I, I+2 ), LDA, + $ A( I+1, I+2 ), LDA ) + CALL DSWAP( N, VS( 1, I ), 1, VS( 1, I+1 ), 1 ) + A( I, I+1 ) = A( I+1, I ) + A( I+1, I ) = ZERO + END IF + INXT = I + 2 + END IF + 20 CONTINUE + END IF + CALL DLASCL( 'G', 0, 0, CSCALE, ANRM, N-IEVAL, 1, + $ WI( IEVAL+1 ), MAX( N-IEVAL, 1 ), IERR ) + END IF +* + IF( WANTST .AND. INFO.EQ.0 ) THEN +* +* Check if reordering successful +* + LASTSL = .TRUE. + LST2SL = .TRUE. + SDIM = 0 + IP = 0 + DO 30 I = 1, N + CURSL = SELECT( WR( I ), WI( I ) ) + IF( WI( I ).EQ.ZERO ) THEN + IF( CURSL ) + $ SDIM = SDIM + 1 + IP = 0 + IF( CURSL .AND. .NOT.LASTSL ) + $ INFO = N + 2 + ELSE + IF( IP.EQ.1 ) THEN +* +* Last eigenvalue of conjugate pair +* + CURSL = CURSL .OR. LASTSL + LASTSL = CURSL + IF( CURSL ) + $ SDIM = SDIM + 2 + IP = -1 + IF( CURSL .AND. .NOT.LST2SL ) + $ INFO = N + 2 + ELSE +* +* First eigenvalue of conjugate pair +* + IP = 1 + END IF + END IF + LST2SL = LASTSL + LASTSL = CURSL + 30 CONTINUE + END IF +* + WORK( 1 ) = MAXWRK + IF( WANTSV .OR. WANTSB ) THEN + IWORK( 1 ) = MAX( 1, SDIM*( N-SDIM ) ) + ELSE + IWORK( 1 ) = 1 + END IF +* + RETURN +* +* End of DGEESX +* + END |