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dtrevc3.f
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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 doubleOTHERcomputational
*
*> \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 = 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