SUBROUTINE DGEGS( JOBVSL, JOBVSR, N, A, LDA, B, LDB, ALPHAR, \$ ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, \$ LWORK, INFO ) * * -- LAPACK driver routine (version 3.2) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * November 2006 * * .. 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( * ) * .. * * Purpose * ======= * * This routine is deprecated and has been replaced by routine DGGES.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). * * Arguments * ========= * * JOBVSL (input) CHARACTER*1 * = 'N': do not compute the left Schur vectors; * = 'V': compute the left Schur vectors (returned in VSL). * * JOBVSR (input) CHARACTER*1 * = 'N': do not compute the right Schur vectors; * = 'V': compute the right Schur vectors (returned in VSR). * * N (input) INTEGER * The order of the matrices A, B, VSL, and VSR. N >= 0. * * A (input/output) 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. * * LDA (input) INTEGER * The leading dimension of A. LDA >= max(1,N). * * B (input/output) DOUBLE PRECISION array, dimension (LDB, N) * On entry, the matrix B. * On exit, the upper triangular matrix T from the generalized * real Schur factorization. * * LDB (input) INTEGER * The leading dimension of B. LDB >= max(1,N). * * ALPHAR (output) DOUBLE PRECISION array, dimension (N) * The real parts of each scalar alpha defining an eigenvalue * of GNEP. * * ALPHAI (output) 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). * * BETA (output) 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. * * VSL (output) DOUBLE PRECISION array, dimension (LDVSL,N) * If JOBVSL = 'V', the matrix of left Schur vectors Q. * Not referenced if JOBVSL = 'N'. * * LDVSL (input) INTEGER * The leading dimension of the matrix VSL. LDVSL >=1, and * if JOBVSL = 'V', LDVSL >= N. * * VSR (output) DOUBLE PRECISION array, dimension (LDVSR,N) * If JOBVSR = 'V', the matrix of right Schur vectors Z. * Not referenced if JOBVSR = 'N'. * * LDVSR (input) INTEGER * The leading dimension of the matrix VSR. LDVSR >= 1, and * if JOBVSR = 'V', LDVSR >= N. * * 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(1,4*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. * * INFO (output) 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) * * ===================================================================== * * .. 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