numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
| .. | |||
| lapack/SRC/dtrevc3.f | 49690B | -rw-r--r-- |
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*> \brief \b DTREVC3 * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download DTREVC3 + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtrevc3.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtrevc3.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtrevc3.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, * VR, LDVR, MM, M, WORK, LWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER HOWMNY, SIDE * INTEGER INFO, LDT, LDVL, LDVR, LWORK, M, MM, N * .. * .. Array Arguments .. * LOGICAL SELECT( * ) * DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), * $ WORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> DTREVC3 computes some or all of the right and/or left eigenvectors of *> a real upper quasi-triangular matrix T. *> Matrices of this type are produced by the Schur factorization of *> a real general matrix: A = Q*T*Q**T, as computed by DHSEQR. *> *> The right eigenvector x and the left eigenvector y of T corresponding *> to an eigenvalue w are defined by: *> *> T*x = w*x, (y**T)*T = w*(y**T) *> *> where y**T denotes the transpose of the vector y. *> The eigenvalues are not input to this routine, but are read directly *> from the diagonal blocks of T. *> *> This routine returns the matrices X and/or Y of right and left *> eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an *> input matrix. If Q is the orthogonal factor that reduces a matrix *> A to Schur form T, then Q*X and Q*Y are the matrices of right and *> left eigenvectors of A. *> *> This uses a Level 3 BLAS version of the back transformation. *> \endverbatim * * Arguments: * ========== * *> \param[in] SIDE *> \verbatim *> SIDE is CHARACTER*1 *> = 'R': compute right eigenvectors only; *> = 'L': compute left eigenvectors only; *> = 'B': compute both right and left eigenvectors. *> \endverbatim *> *> \param[in] HOWMNY *> \verbatim *> HOWMNY is CHARACTER*1 *> = 'A': compute all right and/or left eigenvectors; *> = 'B': compute all right and/or left eigenvectors, *> backtransformed by the matrices in VR and/or VL; *> = 'S': compute selected right and/or left eigenvectors, *> as indicated by the logical array SELECT. *> \endverbatim *> *> \param[in,out] SELECT *> \verbatim *> SELECT is LOGICAL array, dimension (N) *> If HOWMNY = 'S', SELECT specifies the eigenvectors to be *> computed. *> If w(j) is a real eigenvalue, the corresponding real *> eigenvector is computed if SELECT(j) is .TRUE.. *> If w(j) and w(j+1) are the real and imaginary parts of a *> complex eigenvalue, the corresponding complex eigenvector is *> computed if either SELECT(j) or SELECT(j+1) is .TRUE., and *> on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to *> .FALSE.. *> Not referenced if HOWMNY = 'A' or 'B'. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix T. N >= 0. *> \endverbatim *> *> \param[in] T *> \verbatim *> T is DOUBLE PRECISION array, dimension (LDT,N) *> The upper quasi-triangular matrix T in Schur canonical form. *> \endverbatim *> *> \param[in] LDT *> \verbatim *> LDT is INTEGER *> The leading dimension of the array T. LDT >= max(1,N). *> \endverbatim *> *> \param[in,out] VL *> \verbatim *> VL is DOUBLE PRECISION array, dimension (LDVL,MM) *> On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must *> contain an N-by-N matrix Q (usually the orthogonal matrix Q *> of Schur vectors returned by DHSEQR). *> On exit, if SIDE = 'L' or 'B', VL contains: *> if HOWMNY = 'A', the matrix Y of left eigenvectors of T; *> if HOWMNY = 'B', the matrix Q*Y; *> if HOWMNY = 'S', the left eigenvectors of T specified by *> SELECT, stored consecutively in the columns *> of VL, in the same order as their *> eigenvalues. *> A complex eigenvector corresponding to a complex eigenvalue *> is stored in two consecutive columns, the first holding the *> real part, and the second the imaginary part. *> Not referenced if SIDE = 'R'. *> \endverbatim *> *> \param[in] LDVL *> \verbatim *> LDVL is INTEGER *> The leading dimension of the array VL. *> LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N. *> \endverbatim *> *> \param[in,out] VR *> \verbatim *> VR is DOUBLE PRECISION array, dimension (LDVR,MM) *> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must *> contain an N-by-N matrix Q (usually the orthogonal matrix Q *> of Schur vectors returned by DHSEQR). *> On exit, if SIDE = 'R' or 'B', VR contains: *> if HOWMNY = 'A', the matrix X of right eigenvectors of T; *> if HOWMNY = 'B', the matrix Q*X; *> if HOWMNY = 'S', the right eigenvectors of T specified by *> SELECT, stored consecutively in the columns *> of VR, in the same order as their *> eigenvalues. *> A complex eigenvector corresponding to a complex eigenvalue *> is stored in two consecutive columns, the first holding the *> real part and the second the imaginary part. *> Not referenced if SIDE = 'L'. *> \endverbatim *> *> \param[in] LDVR *> \verbatim *> LDVR is INTEGER *> The leading dimension of the array VR. *> LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N. *> \endverbatim *> *> \param[in] MM *> \verbatim *> MM is INTEGER *> The number of columns in the arrays VL and/or VR. MM >= M. *> \endverbatim *> *> \param[out] M *> \verbatim *> M is INTEGER *> The number of columns in the arrays VL and/or VR actually *> used to store the eigenvectors. *> If HOWMNY = 'A' or 'B', M is set to N. *> Each selected real eigenvector occupies one column and each *> selected complex eigenvector occupies two columns. *> \endverbatim *> *> \param[out] WORK *> \verbatim *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) *> \endverbatim *> *> \param[in] LWORK *> \verbatim *> LWORK is INTEGER *> The dimension of array WORK. LWORK >= max(1,3*N). *> For optimum performance, LWORK >= N + 2*N*NB, where NB is *> the optimal blocksize. *> *> 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 *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \ingroup trevc3 * *> \par Further Details: * ===================== *> *> \verbatim *> *> The algorithm used in this program is basically backward (forward) *> substitution, with scaling to make the the code robust against *> possible overflow. *> *> Each eigenvector is normalized so that the element of largest *> magnitude has magnitude 1; here the magnitude of a complex number *> (x,y) is taken to be |x| + |y|. *> \endverbatim *> * ===================================================================== SUBROUTINE DTREVC3( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, $ VR, LDVR, MM, M, WORK, LWORK, INFO ) IMPLICIT NONE * * -- LAPACK computational 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 HOWMNY, SIDE INTEGER INFO, LDT, LDVL, LDVR, LWORK, M, MM, N * .. * .. Array Arguments .. LOGICAL SELECT( * ) DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), $ WORK( * ) * .. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) INTEGER NBMIN, NBMAX PARAMETER ( NBMIN = 8, NBMAX = 128 ) * .. * .. Local Scalars .. LOGICAL ALLV, BOTHV, LEFTV, LQUERY, OVER, PAIR, $ RIGHTV, SOMEV INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, $ IV, MAXWRK, NB, KI2 DOUBLE PRECISION BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE, $ SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR, $ XNORM * .. * .. External Functions .. LOGICAL LSAME INTEGER IDAMAX, ILAENV DOUBLE PRECISION DDOT, DLAMCH EXTERNAL LSAME, IDAMAX, ILAENV, DDOT, DLAMCH * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMV, DLALN2, DSCAL, $ XERBLA, $ DGEMM, DLASET, DLACPY * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, SQRT * .. * .. Local Arrays .. DOUBLE PRECISION X( 2, 2 ) INTEGER ISCOMPLEX( NBMAX ) * .. * .. Executable Statements .. * * Decode and test the input parameters * BOTHV = LSAME( SIDE, 'B' ) RIGHTV = LSAME( SIDE, 'R' ) .OR. BOTHV LEFTV = LSAME( SIDE, 'L' ) .OR. BOTHV * ALLV = LSAME( HOWMNY, 'A' ) OVER = LSAME( HOWMNY, 'B' ) SOMEV = LSAME( HOWMNY, 'S' ) * INFO = 0 NB = ILAENV( 1, 'DTREVC', SIDE // HOWMNY, N, -1, -1, -1 ) MAXWRK = MAX( 1, N + 2*N*NB ) WORK(1) = MAXWRK LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.RIGHTV .AND. .NOT.LEFTV ) THEN INFO = -1 ELSE IF( .NOT.ALLV .AND. .NOT.OVER .AND. .NOT.SOMEV ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -4 ELSE IF( LDT.LT.MAX( 1, N ) ) THEN INFO = -6 ELSE IF( LDVL.LT.1 .OR. ( LEFTV .AND. LDVL.LT.N ) ) THEN INFO = -8 ELSE IF( LDVR.LT.1 .OR. ( RIGHTV .AND. LDVR.LT.N ) ) THEN INFO = -10 ELSE IF( LWORK.LT.MAX( 1, 3*N ) .AND. .NOT.LQUERY ) THEN INFO = -14 ELSE * * Set M to the number of columns required to store the selected * eigenvectors, standardize the array SELECT if necessary, and * test MM. * IF( SOMEV ) THEN M = 0 PAIR = .FALSE. DO 10 J = 1, N IF( PAIR ) THEN PAIR = .FALSE. SELECT( J ) = .FALSE. ELSE IF( J.LT.N ) THEN IF( T( J+1, J ).EQ.ZERO ) THEN IF( SELECT( J ) ) $ M = M + 1 ELSE PAIR = .TRUE. IF( SELECT( J ) .OR. SELECT( J+1 ) ) THEN SELECT( J ) = .TRUE. M = M + 2 END IF END IF ELSE IF( SELECT( N ) ) $ M = M + 1 END IF END IF 10 CONTINUE ELSE M = N END IF * IF( MM.LT.M ) THEN INFO = -11 END IF END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'DTREVC3', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible. * IF( N.EQ.0 ) $ RETURN * * Use blocked version of back-transformation if sufficient workspace. * Zero-out the workspace to avoid potential NaN propagation. * IF( OVER .AND. LWORK .GE. N + 2*N*NBMIN ) THEN NB = (LWORK - N) / (2*N) NB = MIN( NB, NBMAX ) CALL DLASET( 'F', N, 1+2*NB, ZERO, ZERO, WORK, N ) ELSE NB = 1 END IF * * Set the constants to control overflow. * UNFL = DLAMCH( 'Safe minimum' ) OVFL = ONE / UNFL ULP = DLAMCH( 'Precision' ) SMLNUM = UNFL*( N / ULP ) BIGNUM = ( ONE-ULP ) / SMLNUM * * Compute 1-norm of each column of strictly upper triangular * part of T to control overflow in triangular solver. * WORK( 1 ) = ZERO DO 30 J = 2, N WORK( J ) = ZERO DO 20 I = 1, J - 1 WORK( J ) = WORK( J ) + ABS( T( I, J ) ) 20 CONTINUE 30 CONTINUE * * Index IP is used to specify the real or complex eigenvalue: * IP = 0, real eigenvalue, * 1, first of conjugate complex pair: (wr,wi) * -1, second of conjugate complex pair: (wr,wi) * ISCOMPLEX array stores IP for each column in current block. * IF( RIGHTV ) THEN * * ============================================================ * Compute right eigenvectors. * * IV is index of column in current block. * For complex right vector, uses IV-1 for real part and IV for complex part. * Non-blocked version always uses IV=2; * blocked version starts with IV=NB, goes down to 1 or 2. * (Note the "0-th" column is used for 1-norms computed above.) IV = 2 IF( NB.GT.2 ) THEN IV = NB END IF IP = 0 IS = M DO 140 KI = N, 1, -1 IF( IP.EQ.-1 ) THEN * previous iteration (ki+1) was second of conjugate pair, * so this ki is first of conjugate pair; skip to end of loop IP = 1 GO TO 140 ELSE IF( KI.EQ.1 ) THEN * last column, so this ki must be real eigenvalue IP = 0 ELSE IF( T( KI, KI-1 ).EQ.ZERO ) THEN * zero on sub-diagonal, so this ki is real eigenvalue IP = 0 ELSE * non-zero on sub-diagonal, so this ki is second of conjugate pair IP = -1 END IF IF( SOMEV ) THEN IF( IP.EQ.0 ) THEN IF( .NOT.SELECT( KI ) ) $ GO TO 140 ELSE IF( .NOT.SELECT( KI-1 ) ) $ GO TO 140 END IF END IF * * Compute the KI-th eigenvalue (WR,WI). * WR = T( KI, KI ) WI = ZERO IF( IP.NE.0 ) $ WI = SQRT( ABS( T( KI, KI-1 ) ) )* $ SQRT( ABS( T( KI-1, KI ) ) ) SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM ) * IF( IP.EQ.0 ) THEN * * -------------------------------------------------------- * Real right eigenvector * WORK( KI + IV*N ) = ONE * * Form right-hand side. * DO 50 K = 1, KI - 1 WORK( K + IV*N ) = -T( K, KI ) 50 CONTINUE * * Solve upper quasi-triangular system: * [ T(1:KI-1,1:KI-1) - WR ]*X = SCALE*WORK. * JNXT = KI - 1 DO 60 J = KI - 1, 1, -1 IF( J.GT.JNXT ) $ GO TO 60 J1 = J J2 = J JNXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNXT = J - 2 END IF END IF * IF( J1.EQ.J2 ) THEN * * 1-by-1 diagonal block * CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, $ J ), $ LDT, ONE, ONE, WORK( J+IV*N ), N, WR, $ ZERO, X, 2, SCALE, XNORM, IERR ) * * Scale X(1,1) to avoid overflow when updating * the right-hand side. * IF( XNORM.GT.ONE ) THEN IF( WORK( J ).GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM SCALE = SCALE / XNORM END IF END IF * * Scale if necessary * IF( SCALE.NE.ONE ) $ CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 ) WORK( J+IV*N ) = X( 1, 1 ) * * Update right-hand side * CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1, $ WORK( 1+IV*N ), 1 ) * ELSE * * 2-by-2 diagonal block * CALL DLALN2( .FALSE., 2, 1, SMIN, ONE, $ T( J-1, J-1 ), LDT, ONE, ONE, $ WORK( J-1+IV*N ), N, WR, ZERO, X, 2, $ SCALE, XNORM, IERR ) * * Scale X(1,1) and X(2,1) to avoid overflow when * updating the right-hand side. * IF( XNORM.GT.ONE ) THEN BETA = MAX( WORK( J-1 ), WORK( J ) ) IF( BETA.GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM X( 2, 1 ) = X( 2, 1 ) / XNORM SCALE = SCALE / XNORM END IF END IF * * Scale if necessary * IF( SCALE.NE.ONE ) $ CALL DSCAL( KI, SCALE, WORK( 1+IV*N ), 1 ) WORK( J-1+IV*N ) = X( 1, 1 ) WORK( J +IV*N ) = X( 2, 1 ) * * Update right-hand side * CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1, $ WORK( 1+IV*N ), 1 ) CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1, $ WORK( 1+IV*N ), 1 ) END IF 60 CONTINUE * * Copy the vector x or Q*x to VR and normalize. * IF( .NOT.OVER ) THEN * ------------------------------ * no back-transform: copy x to VR and normalize. CALL DCOPY( KI, WORK( 1 + IV*N ), 1, VR( 1, IS ), $ 1 ) * II = IDAMAX( KI, VR( 1, IS ), 1 ) REMAX = ONE / ABS( VR( II, IS ) ) CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 ) * DO 70 K = KI + 1, N VR( K, IS ) = ZERO 70 CONTINUE * ELSE IF( NB.EQ.1 ) THEN * ------------------------------ * version 1: back-transform each vector with GEMV, Q*x. IF( KI.GT.1 ) $ CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR, $ WORK( 1 + IV*N ), 1, WORK( KI + IV*N ), $ VR( 1, KI ), 1 ) * II = IDAMAX( N, VR( 1, KI ), 1 ) REMAX = ONE / ABS( VR( II, KI ) ) CALL DSCAL( N, REMAX, VR( 1, KI ), 1 ) * ELSE * ------------------------------ * version 2: back-transform block of vectors with GEMM * zero out below vector DO K = KI + 1, N WORK( K + IV*N ) = ZERO END DO ISCOMPLEX( IV ) = IP * back-transform and normalization is done below END IF ELSE * * -------------------------------------------------------- * Complex right eigenvector. * * Initial solve * [ ( T(KI-1,KI-1) T(KI-1,KI) ) - (WR + I*WI) ]*X = 0. * [ ( T(KI, KI-1) T(KI, KI) ) ] * IF( ABS( T( KI-1, KI ) ).GE.ABS( T( KI, KI-1 ) ) ) THEN WORK( KI-1 + (IV-1)*N ) = ONE WORK( KI + (IV )*N ) = WI / T( KI-1, KI ) ELSE WORK( KI-1 + (IV-1)*N ) = -WI / T( KI, KI-1 ) WORK( KI + (IV )*N ) = ONE END IF WORK( KI + (IV-1)*N ) = ZERO WORK( KI-1 + (IV )*N ) = ZERO * * Form right-hand side. * DO 80 K = 1, KI - 2 WORK( K+(IV-1)*N ) = -WORK( KI-1+(IV-1)*N )*T(K,KI-1) WORK( K+(IV )*N ) = -WORK( KI +(IV )*N )*T(K,KI ) 80 CONTINUE * * Solve upper quasi-triangular system: * [ T(1:KI-2,1:KI-2) - (WR+i*WI) ]*X = SCALE*(WORK+i*WORK2) * JNXT = KI - 2 DO 90 J = KI - 2, 1, -1 IF( J.GT.JNXT ) $ GO TO 90 J1 = J J2 = J JNXT = J - 1 IF( J.GT.1 ) THEN IF( T( J, J-1 ).NE.ZERO ) THEN J1 = J - 1 JNXT = J - 2 END IF END IF * IF( J1.EQ.J2 ) THEN * * 1-by-1 diagonal block * CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, $ J ), $ LDT, ONE, ONE, WORK( J+(IV-1)*N ), N, $ WR, WI, X, 2, SCALE, XNORM, IERR ) * * Scale X(1,1) and X(1,2) to avoid overflow when * updating the right-hand side. * IF( XNORM.GT.ONE ) THEN IF( WORK( J ).GT.BIGNUM / XNORM ) THEN X( 1, 1 ) = X( 1, 1 ) / XNORM X( 1, 2 ) = X( 1, 2 ) / XNORM SCALE = SCALE / XNORM END IF END IF * * Scale if necessary * IF( SCALE.NE.ONE ) THEN CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ), $ 1 ) CALL DSCAL( KI, SCALE, WORK( 1+(IV )*N ), $ 1 ) END IF WORK( J+(IV-1)*N ) = X( 1, 1 ) WORK( J+(IV )*N ) = X( 1, 2 ) * * Update the right-hand side * CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1, $ WORK( 1+(IV-1)*N ), 1 ) CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1, $ WORK( 1+(IV )*N ), 1 ) * ELSE * * 2-by-2 diagonal block * CALL DLALN2( .FALSE., 2, 2, SMIN, ONE, $ T( J-1, J-1 ), LDT, ONE, ONE, $ WORK( J-1+(IV-1)*N ), N, WR, WI, X, 2, $ SCALE, XNORM, IERR ) * * Scale X to avoid overflow when updating * the right-hand side. * IF( XNORM.GT.ONE ) THEN BETA = MAX( WORK( J-1 ), WORK( J ) ) IF( BETA.GT.BIGNUM / XNORM ) THEN REC = ONE / XNORM X( 1, 1 ) = X( 1, 1 )*REC X( 1, 2 ) = X( 1, 2 )*REC X( 2, 1 ) = X( 2, 1 )*REC X( 2, 2 ) = X( 2, 2 )*REC SCALE = SCALE*REC END IF END IF * * Scale if necessary * IF( SCALE.NE.ONE ) THEN CALL DSCAL( KI, SCALE, WORK( 1+(IV-1)*N ), $ 1 ) CALL DSCAL( KI, SCALE, WORK( 1+(IV )*N ), $ 1 ) END IF WORK( J-1+(IV-1)*N ) = X( 1, 1 ) WORK( J +(IV-1)*N ) = X( 2, 1 ) WORK( J-1+(IV )*N ) = X( 1, 2 ) WORK( J +(IV )*N ) = X( 2, 2 ) * * Update the right-hand side * CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1, $ WORK( 1+(IV-1)*N ), 1 ) CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1, $ WORK( 1+(IV-1)*N ), 1 ) CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1, $ WORK( 1+(IV )*N ), 1 ) CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1, $ WORK( 1+(IV )*N ), 1 ) END IF 90 CONTINUE * * Copy the vector x or Q*x to VR and normalize. * IF( .NOT.OVER ) THEN * ------------------------------ * no back-transform: copy x to VR and normalize. CALL DCOPY( KI, WORK( 1+(IV-1)*N ), 1, VR(1,IS-1), $ 1 ) CALL DCOPY( KI, WORK( 1+(IV )*N ), 1, VR(1,IS ), $ 1 ) * EMAX = ZERO DO 100 K = 1, KI EMAX = MAX( EMAX, ABS( VR( K, IS-1 ) )+ $ ABS( VR( K, IS ) ) ) 100 CONTINUE REMAX = ONE / EMAX CALL DSCAL( KI, REMAX, VR( 1, IS-1 ), 1 ) CALL DSCAL( KI, REMAX, VR( 1, IS ), 1 ) * DO 110 K = KI + 1, N VR( K, IS-1 ) = ZERO VR( K, IS ) = ZERO 110 CONTINUE * ELSE IF( NB.EQ.1 ) THEN * ------------------------------ * version 1: back-transform each vector with GEMV, Q*x. IF( KI.GT.2 ) THEN CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR, $ WORK( 1 + (IV-1)*N ), 1, $ WORK( KI-1 + (IV-1)*N ), VR(1,KI-1), 1) CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR, $ WORK( 1 + (IV)*N ), 1, $ WORK( KI + (IV)*N ), VR( 1, KI ), 1 ) ELSE CALL DSCAL( N, WORK(KI-1+(IV-1)*N), VR(1,KI-1), $ 1) CALL DSCAL( N, WORK(KI +(IV )*N), VR(1,KI ), $ 1) END IF * EMAX = ZERO DO 120 K = 1, N EMAX = MAX( EMAX, ABS( VR( K, KI-1 ) )+ $ ABS( VR( K, KI ) ) ) 120 CONTINUE REMAX = ONE / EMAX CALL DSCAL( N, REMAX, VR( 1, KI-1 ), 1 ) CALL DSCAL( N, REMAX, VR( 1, KI ), 1 ) * ELSE * ------------------------------ * version 2: back-transform block of vectors with GEMM * zero out below vector DO K = KI + 1, N WORK( K + (IV-1)*N ) = ZERO WORK( K + (IV )*N ) = ZERO END DO ISCOMPLEX( IV-1 ) = -IP ISCOMPLEX( IV ) = IP IV = IV - 1 * back-transform and normalization is done below END IF END IF IF( NB.GT.1 ) THEN * -------------------------------------------------------- * Blocked version of back-transform * For complex case, KI2 includes both vectors (KI-1 and KI) IF( IP.EQ.0 ) THEN KI2 = KI ELSE KI2 = KI - 1 END IF * Columns IV:NB of work are valid vectors. * When the number of vectors stored reaches NB-1 or NB, * or if this was last vector, do the GEMM IF( (IV.LE.2) .OR. (KI2.EQ.1) ) THEN CALL DGEMM( 'N', 'N', N, NB-IV+1, KI2+NB-IV, ONE, $ VR, LDVR, $ WORK( 1 + (IV)*N ), N, $ ZERO, $ WORK( 1 + (NB+IV)*N ), N ) * normalize vectors DO K = IV, NB IF( ISCOMPLEX(K).EQ.0 ) THEN * real eigenvector II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 ) REMAX = ONE / ABS( WORK( II + (NB+K)*N ) ) ELSE IF( ISCOMPLEX(K).EQ.1 ) THEN * first eigenvector of conjugate pair EMAX = ZERO DO II = 1, N EMAX = MAX( EMAX, $ ABS( WORK( II + (NB+K )*N ) )+ $ ABS( WORK( II + (NB+K+1)*N ) ) ) END DO REMAX = ONE / EMAX * else if ISCOMPLEX(K).EQ.-1 * second eigenvector of conjugate pair * reuse same REMAX as previous K END IF CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 ) END DO CALL DLACPY( 'F', N, NB-IV+1, $ WORK( 1 + (NB+IV)*N ), N, $ VR( 1, KI2 ), LDVR ) IV = NB ELSE IV = IV - 1 END IF END IF ! blocked back-transform * IS = IS - 1 IF( IP.NE.0 ) $ IS = IS - 1 140 CONTINUE END IF IF( LEFTV ) THEN * * ============================================================ * Compute left eigenvectors. * * IV is index of column in current block. * For complex left vector, uses IV for real part and IV+1 for complex part. * Non-blocked version always uses IV=1; * blocked version starts with IV=1, goes up to NB-1 or NB. * (Note the "0-th" column is used for 1-norms computed above.) IV = 1 IP = 0 IS = 1 DO 260 KI = 1, N IF( IP.EQ.1 ) THEN * previous iteration (ki-1) was first of conjugate pair, * so this ki is second of conjugate pair; skip to end of loop IP = -1 GO TO 260 ELSE IF( KI.EQ.N ) THEN * last column, so this ki must be real eigenvalue IP = 0 ELSE IF( T( KI+1, KI ).EQ.ZERO ) THEN * zero on sub-diagonal, so this ki is real eigenvalue IP = 0 ELSE * non-zero on sub-diagonal, so this ki is first of conjugate pair IP = 1 END IF * IF( SOMEV ) THEN IF( .NOT.SELECT( KI ) ) $ GO TO 260 END IF * * Compute the KI-th eigenvalue (WR,WI). * WR = T( KI, KI ) WI = ZERO IF( IP.NE.0 ) $ WI = SQRT( ABS( T( KI, KI+1 ) ) )* $ SQRT( ABS( T( KI+1, KI ) ) ) SMIN = MAX( ULP*( ABS( WR )+ABS( WI ) ), SMLNUM ) * IF( IP.EQ.0 ) THEN * * -------------------------------------------------------- * Real left eigenvector * WORK( KI + IV*N ) = ONE * * Form right-hand side. * DO 160 K = KI + 1, N WORK( K + IV*N ) = -T( KI, K ) 160 CONTINUE * * Solve transposed quasi-triangular system: * [ T(KI+1:N,KI+1:N) - WR ]**T * X = SCALE*WORK * VMAX = ONE VCRIT = BIGNUM * JNXT = KI + 1 DO 170 J = KI + 1, N IF( J.LT.JNXT ) $ GO TO 170 J1 = J J2 = J JNXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNXT = J + 2 END IF END IF * IF( J1.EQ.J2 ) THEN * * 1-by-1 diagonal block * * Scale if necessary to avoid overflow when forming * the right-hand side. * IF( WORK( J ).GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF * WORK( J+IV*N ) = WORK( J+IV*N ) - $ DDOT( J-KI-1, T( KI+1, J ), 1, $ WORK( KI+1+IV*N ), 1 ) * * Solve [ T(J,J) - WR ]**T * X = WORK * CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, $ J ), $ LDT, ONE, ONE, WORK( J+IV*N ), N, WR, $ ZERO, X, 2, SCALE, XNORM, IERR ) * * Scale if necessary * IF( SCALE.NE.ONE ) $ CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ), $ 1 ) WORK( J+IV*N ) = X( 1, 1 ) VMAX = MAX( ABS( WORK( J+IV*N ) ), VMAX ) VCRIT = BIGNUM / VMAX * ELSE * * 2-by-2 diagonal block * * Scale if necessary to avoid overflow when forming * the right-hand side. * BETA = MAX( WORK( J ), WORK( J+1 ) ) IF( BETA.GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK( KI+IV*N ), 1 ) VMAX = ONE VCRIT = BIGNUM END IF * WORK( J+IV*N ) = WORK( J+IV*N ) - $ DDOT( J-KI-1, T( KI+1, J ), 1, $ WORK( KI+1+IV*N ), 1 ) * WORK( J+1+IV*N ) = WORK( J+1+IV*N ) - $ DDOT( J-KI-1, T( KI+1, J+1 ), $ 1, $ WORK( KI+1+IV*N ), 1 ) * * Solve * [ T(J,J)-WR T(J,J+1) ]**T * X = SCALE*( WORK1 ) * [ T(J+1,J) T(J+1,J+1)-WR ] ( WORK2 ) * CALL DLALN2( .TRUE., 2, 1, SMIN, ONE, T( J, J ), $ LDT, ONE, ONE, WORK( J+IV*N ), N, WR, $ ZERO, X, 2, SCALE, XNORM, IERR ) * * Scale if necessary * IF( SCALE.NE.ONE ) $ CALL DSCAL( N-KI+1, SCALE, WORK( KI+IV*N ), $ 1 ) WORK( J +IV*N ) = X( 1, 1 ) WORK( J+1+IV*N ) = X( 2, 1 ) * VMAX = MAX( ABS( WORK( J +IV*N ) ), $ ABS( WORK( J+1+IV*N ) ), VMAX ) VCRIT = BIGNUM / VMAX * END IF 170 CONTINUE * * Copy the vector x or Q*x to VL and normalize. * IF( .NOT.OVER ) THEN * ------------------------------ * no back-transform: copy x to VL and normalize. CALL DCOPY( N-KI+1, WORK( KI + IV*N ), 1, $ VL( KI, IS ), 1 ) * II = IDAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1 REMAX = ONE / ABS( VL( II, IS ) ) CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 ) * DO 180 K = 1, KI - 1 VL( K, IS ) = ZERO 180 CONTINUE * ELSE IF( NB.EQ.1 ) THEN * ------------------------------ * version 1: back-transform each vector with GEMV, Q*x. IF( KI.LT.N ) $ CALL DGEMV( 'N', N, N-KI, ONE, $ VL( 1, KI+1 ), LDVL, $ WORK( KI+1 + IV*N ), 1, $ WORK( KI + IV*N ), VL( 1, KI ), 1 ) * II = IDAMAX( N, VL( 1, KI ), 1 ) REMAX = ONE / ABS( VL( II, KI ) ) CALL DSCAL( N, REMAX, VL( 1, KI ), 1 ) * ELSE * ------------------------------ * version 2: back-transform block of vectors with GEMM * zero out above vector * could go from KI-NV+1 to KI-1 DO K = 1, KI - 1 WORK( K + IV*N ) = ZERO END DO ISCOMPLEX( IV ) = IP * back-transform and normalization is done below END IF ELSE * * -------------------------------------------------------- * Complex left eigenvector. * * Initial solve: * [ ( T(KI,KI) T(KI,KI+1) )**T - (WR - I* WI) ]*X = 0. * [ ( T(KI+1,KI) T(KI+1,KI+1) ) ] * IF( ABS( T( KI, KI+1 ) ).GE.ABS( T( KI+1, KI ) ) ) THEN WORK( KI + (IV )*N ) = WI / T( KI, KI+1 ) WORK( KI+1 + (IV+1)*N ) = ONE ELSE WORK( KI + (IV )*N ) = ONE WORK( KI+1 + (IV+1)*N ) = -WI / T( KI+1, KI ) END IF WORK( KI+1 + (IV )*N ) = ZERO WORK( KI + (IV+1)*N ) = ZERO * * Form right-hand side. * DO 190 K = KI + 2, N WORK( K+(IV )*N ) = -WORK( KI +(IV )*N )*T(KI, K) WORK( K+(IV+1)*N ) = -WORK( KI+1+(IV+1)*N )*T(KI+1,K) 190 CONTINUE * * Solve transposed quasi-triangular system: * [ T(KI+2:N,KI+2:N)**T - (WR-i*WI) ]*X = WORK1+i*WORK2 * VMAX = ONE VCRIT = BIGNUM * JNXT = KI + 2 DO 200 J = KI + 2, N IF( J.LT.JNXT ) $ GO TO 200 J1 = J J2 = J JNXT = J + 1 IF( J.LT.N ) THEN IF( T( J+1, J ).NE.ZERO ) THEN J2 = J + 1 JNXT = J + 2 END IF END IF * IF( J1.EQ.J2 ) THEN * * 1-by-1 diagonal block * * Scale if necessary to avoid overflow when * forming the right-hand side elements. * IF( WORK( J ).GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK(KI+(IV )*N), $ 1 ) CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N), $ 1 ) VMAX = ONE VCRIT = BIGNUM END IF * WORK( J+(IV )*N ) = WORK( J+(IV)*N ) - $ DDOT( J-KI-2, T( KI+2, J ), $ 1, $ WORK( KI+2+(IV)*N ), 1 ) WORK( J+(IV+1)*N ) = WORK( J+(IV+1)*N ) - $ DDOT( J-KI-2, T( KI+2, J ), $ 1, $ WORK( KI+2+(IV+1)*N ), 1 ) * * Solve [ T(J,J)-(WR-i*WI) ]*(X11+i*X12)= WK+I*WK2 * CALL DLALN2( .FALSE., 1, 2, SMIN, ONE, T( J, $ J ), $ LDT, ONE, ONE, WORK( J+IV*N ), N, WR, $ -WI, X, 2, SCALE, XNORM, IERR ) * * Scale if necessary * IF( SCALE.NE.ONE ) THEN CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV )*N), $ 1) CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N), $ 1) END IF WORK( J+(IV )*N ) = X( 1, 1 ) WORK( J+(IV+1)*N ) = X( 1, 2 ) VMAX = MAX( ABS( WORK( J+(IV )*N ) ), $ ABS( WORK( J+(IV+1)*N ) ), VMAX ) VCRIT = BIGNUM / VMAX * ELSE * * 2-by-2 diagonal block * * Scale if necessary to avoid overflow when forming * the right-hand side elements. * BETA = MAX( WORK( J ), WORK( J+1 ) ) IF( BETA.GT.VCRIT ) THEN REC = ONE / VMAX CALL DSCAL( N-KI+1, REC, WORK(KI+(IV )*N), $ 1 ) CALL DSCAL( N-KI+1, REC, WORK(KI+(IV+1)*N), $ 1 ) VMAX = ONE VCRIT = BIGNUM END IF * WORK( J +(IV )*N ) = WORK( J+(IV)*N ) - $ DDOT( J-KI-2, T( KI+2, J ), 1, $ WORK( KI+2+(IV)*N ), 1 ) * WORK( J +(IV+1)*N ) = WORK( J+(IV+1)*N ) - $ DDOT( J-KI-2, T( KI+2, J ), 1, $ WORK( KI+2+(IV+1)*N ), 1 ) * WORK( J+1+(IV )*N ) = WORK( J+1+(IV)*N ) - $ DDOT( J-KI-2, T( KI+2, J+1 ), $ 1, $ WORK( KI+2+(IV)*N ), 1 ) * WORK( J+1+(IV+1)*N ) = WORK( J+1+(IV+1)*N ) - $ DDOT( J-KI-2, T( KI+2, J+1 ), $ 1, $ WORK( KI+2+(IV+1)*N ), 1 ) * * Solve 2-by-2 complex linear equation * [ (T(j,j) T(j,j+1) )**T - (wr-i*wi)*I ]*X = SCALE*B * [ (T(j+1,j) T(j+1,j+1)) ] * CALL DLALN2( .TRUE., 2, 2, SMIN, ONE, T( J, J ), $ LDT, ONE, ONE, WORK( J+IV*N ), N, WR, $ -WI, X, 2, SCALE, XNORM, IERR ) * * Scale if necessary * IF( SCALE.NE.ONE ) THEN CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV )*N), $ 1) CALL DSCAL( N-KI+1, SCALE, WORK(KI+(IV+1)*N), $ 1) END IF WORK( J +(IV )*N ) = X( 1, 1 ) WORK( J +(IV+1)*N ) = X( 1, 2 ) WORK( J+1+(IV )*N ) = X( 2, 1 ) WORK( J+1+(IV+1)*N ) = X( 2, 2 ) VMAX = MAX( ABS( X( 1, 1 ) ), ABS( X( 1, 2 ) ), $ ABS( X( 2, 1 ) ), ABS( X( 2, 2 ) ), $ VMAX ) VCRIT = BIGNUM / VMAX * END IF 200 CONTINUE * * Copy the vector x or Q*x to VL and normalize. * IF( .NOT.OVER ) THEN * ------------------------------ * no back-transform: copy x to VL and normalize. CALL DCOPY( N-KI+1, WORK( KI + (IV )*N ), 1, $ VL( KI, IS ), 1 ) CALL DCOPY( N-KI+1, WORK( KI + (IV+1)*N ), 1, $ VL( KI, IS+1 ), 1 ) * EMAX = ZERO DO 220 K = KI, N EMAX = MAX( EMAX, ABS( VL( K, IS ) )+ $ ABS( VL( K, IS+1 ) ) ) 220 CONTINUE REMAX = ONE / EMAX CALL DSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 ) CALL DSCAL( N-KI+1, REMAX, VL( KI, IS+1 ), 1 ) * DO 230 K = 1, KI - 1 VL( K, IS ) = ZERO VL( K, IS+1 ) = ZERO 230 CONTINUE * ELSE IF( NB.EQ.1 ) THEN * ------------------------------ * version 1: back-transform each vector with GEMV, Q*x. IF( KI.LT.N-1 ) THEN CALL DGEMV( 'N', N, N-KI-1, ONE, $ VL( 1, KI+2 ), LDVL, $ WORK( KI+2 + (IV)*N ), 1, $ WORK( KI + (IV)*N ), $ VL( 1, KI ), 1 ) CALL DGEMV( 'N', N, N-KI-1, ONE, $ VL( 1, KI+2 ), LDVL, $ WORK( KI+2 + (IV+1)*N ), 1, $ WORK( KI+1 + (IV+1)*N ), $ VL( 1, KI+1 ), 1 ) ELSE CALL DSCAL( N, WORK(KI+ (IV )*N), VL(1, KI ), $ 1) CALL DSCAL( N, WORK(KI+1+(IV+1)*N), VL(1, KI+1), $ 1) END IF * EMAX = ZERO DO 240 K = 1, N EMAX = MAX( EMAX, ABS( VL( K, KI ) )+ $ ABS( VL( K, KI+1 ) ) ) 240 CONTINUE REMAX = ONE / EMAX CALL DSCAL( N, REMAX, VL( 1, KI ), 1 ) CALL DSCAL( N, REMAX, VL( 1, KI+1 ), 1 ) * ELSE * ------------------------------ * version 2: back-transform block of vectors with GEMM * zero out above vector * could go from KI-NV+1 to KI-1 DO K = 1, KI - 1 WORK( K + (IV )*N ) = ZERO WORK( K + (IV+1)*N ) = ZERO END DO ISCOMPLEX( IV ) = IP ISCOMPLEX( IV+1 ) = -IP IV = IV + 1 * back-transform and normalization is done below END IF END IF IF( NB.GT.1 ) THEN * -------------------------------------------------------- * Blocked version of back-transform * For complex case, KI2 includes both vectors (KI and KI+1) IF( IP.EQ.0 ) THEN KI2 = KI ELSE KI2 = KI + 1 END IF * Columns 1:IV of work are valid vectors. * When the number of vectors stored reaches NB-1 or NB, * or if this was last vector, do the GEMM IF( (IV.GE.NB-1) .OR. (KI2.EQ.N) ) THEN CALL DGEMM( 'N', 'N', N, IV, N-KI2+IV, ONE, $ VL( 1, KI2-IV+1 ), LDVL, $ WORK( KI2-IV+1 + (1)*N ), N, $ ZERO, $ WORK( 1 + (NB+1)*N ), N ) * normalize vectors DO K = 1, IV IF( ISCOMPLEX(K).EQ.0) THEN * real eigenvector II = IDAMAX( N, WORK( 1 + (NB+K)*N ), 1 ) REMAX = ONE / ABS( WORK( II + (NB+K)*N ) ) ELSE IF( ISCOMPLEX(K).EQ.1) THEN * first eigenvector of conjugate pair EMAX = ZERO DO II = 1, N EMAX = MAX( EMAX, $ ABS( WORK( II + (NB+K )*N ) )+ $ ABS( WORK( II + (NB+K+1)*N ) ) ) END DO REMAX = ONE / EMAX * else if ISCOMPLEX(K).EQ.-1 * second eigenvector of conjugate pair * reuse same REMAX as previous K END IF CALL DSCAL( N, REMAX, WORK( 1 + (NB+K)*N ), 1 ) END DO CALL DLACPY( 'F', N, IV, $ WORK( 1 + (NB+1)*N ), N, $ VL( 1, KI2-IV+1 ), LDVL ) IV = 1 ELSE IV = IV + 1 END IF END IF ! blocked back-transform * IS = IS + 1 IF( IP.NE.0 ) $ IS = IS + 1 260 CONTINUE END IF * RETURN * * End of DTREVC3 * END