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SUBROUTINE ZTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
$ LDVR, MM, M, WORK, RWORK, INFO )
*
* -- LAPACK routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
CHARACTER HOWMNY, SIDE
INTEGER INFO, LDT, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
$ WORK( * )
* ..
*
* Purpose
* =======
*
* ZTREVC computes some or all of the right and/or left eigenvectors of
* a complex upper triangular matrix T.
* Matrices of this type are produced by the Schur factorization of
* a complex general matrix: A = Q*T*Q**H, as computed by ZHSEQR.
*
* The right eigenvector x and the left eigenvector y of T corresponding
* to an eigenvalue w are defined by:
*
* T*x = w*x, (y**H)*T = w*(y**H)
*
* where y**H denotes the conjugate transpose of the vector y.
* The eigenvalues are not input to this routine, but are read directly
* from the diagonal 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 unitary 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.
*
* Arguments
* =========
*
* SIDE (input) CHARACTER*1
* = 'R': compute right eigenvectors only;
* = 'L': compute left eigenvectors only;
* = 'B': compute both right and left eigenvectors.
*
* HOWMNY (input) CHARACTER*1
* = 'A': compute all right and/or left eigenvectors;
* = 'B': compute all right and/or left eigenvectors,
* backtransformed using the matrices supplied in
* VR and/or VL;
* = 'S': compute selected right and/or left eigenvectors,
* as indicated by the logical array SELECT.
*
* SELECT (input) LOGICAL array, dimension (N)
* If HOWMNY = 'S', SELECT specifies the eigenvectors to be
* computed.
* The eigenvector corresponding to the j-th eigenvalue is
* computed if SELECT(j) = .TRUE..
* Not referenced if HOWMNY = 'A' or 'B'.
*
* N (input) INTEGER
* The order of the matrix T. N >= 0.
*
* T (input/output) COMPLEX*16 array, dimension (LDT,N)
* The upper triangular matrix T. T is modified, but restored
* on exit.
*
* LDT (input) INTEGER
* The leading dimension of the array T. LDT >= max(1,N).
*
* VL (input/output) COMPLEX*16 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 unitary matrix Q of
* Schur vectors returned by ZHSEQR).
* 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.
* Not referenced if SIDE = 'R'.
*
* LDVL (input) INTEGER
* The leading dimension of the array VL. LDVL >= 1, and if
* SIDE = 'L' or 'B', LDVL >= N.
*
* VR (input/output) COMPLEX*16 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 unitary matrix Q of
* Schur vectors returned by ZHSEQR).
* 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.
* Not referenced if SIDE = 'L'.
*
* LDVR (input) INTEGER
* The leading dimension of the array VR. LDVR >= 1, and if
* SIDE = 'R' or 'B'; LDVR >= N.
*
* MM (input) INTEGER
* The number of columns in the arrays VL and/or VR. MM >= M.
*
* M (output) 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 eigenvector occupies one
* column.
*
* WORK (workspace) COMPLEX*16 array, dimension (2*N)
*
* RWORK (workspace) DOUBLE PRECISION array, dimension (N)
*
* INFO (output) INTEGER
* = 0: successful exit
* < 0: if INFO = -i, the i-th argument had an illegal value
*
* Further Details
* ===============
*
* 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|.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CMZERO, CMONE
PARAMETER ( CMZERO = ( 0.0D+0, 0.0D+0 ),
$ CMONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL ALLV, BOTHV, LEFTV, OVER, RIGHTV, SOMEV
INTEGER I, II, IS, J, K, KI
DOUBLE PRECISION OVFL, REMAX, SCALE, SMIN, SMLNUM, ULP, UNFL
COMPLEX*16 CDUM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER IZAMAX
DOUBLE PRECISION DLAMCH, DZASUM
EXTERNAL LSAME, IZAMAX, DLAMCH, DZASUM
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZCOPY, ZDSCAL, ZGEMV, ZLATRS
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX
* ..
* .. Statement Functions ..
DOUBLE PRECISION CABS1
* ..
* .. Statement Function definitions ..
CABS1( CDUM ) = ABS( DBLE( CDUM ) ) + ABS( DIMAG( CDUM ) )
* ..
* .. 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' )
*
* Set M to the number of columns required to store the selected
* eigenvectors.
*
IF( SOMEV ) THEN
M = 0
DO 10 J = 1, N
IF( SELECT( J ) )
$ M = M + 1
10 CONTINUE
ELSE
M = N
END IF
*
INFO = 0
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( MM.LT.M ) THEN
INFO = -11
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTREVC', -INFO )
RETURN
END IF
*
* Quick return if possible.
*
IF( N.EQ.0 )
$ RETURN
*
* Set the constants to control overflow.
*
UNFL = DLAMCH( 'Safe minimum' )
OVFL = ONE / UNFL
CALL DLABAD( UNFL, OVFL )
ULP = DLAMCH( 'Precision' )
SMLNUM = UNFL*( N / ULP )
*
* Store the diagonal elements of T in working array WORK.
*
DO 20 I = 1, N
WORK( I+N ) = T( I, I )
20 CONTINUE
*
* Compute 1-norm of each column of strictly upper triangular
* part of T to control overflow in triangular solver.
*
RWORK( 1 ) = ZERO
DO 30 J = 2, N
RWORK( J ) = DZASUM( J-1, T( 1, J ), 1 )
30 CONTINUE
*
IF( RIGHTV ) THEN
*
* Compute right eigenvectors.
*
IS = M
DO 80 KI = N, 1, -1
*
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 80
END IF
SMIN = MAX( ULP*( CABS1( T( KI, KI ) ) ), SMLNUM )
*
WORK( 1 ) = CMONE
*
* Form right-hand side.
*
DO 40 K = 1, KI - 1
WORK( K ) = -T( K, KI )
40 CONTINUE
*
* Solve the triangular system:
* (T(1:KI-1,1:KI-1) - T(KI,KI))*X = SCALE*WORK.
*
DO 50 K = 1, KI - 1
T( K, K ) = T( K, K ) - T( KI, KI )
IF( CABS1( T( K, K ) ).LT.SMIN )
$ T( K, K ) = SMIN
50 CONTINUE
*
IF( KI.GT.1 ) THEN
CALL ZLATRS( 'Upper', 'No transpose', 'Non-unit', 'Y',
$ KI-1, T, LDT, WORK( 1 ), SCALE, RWORK,
$ INFO )
WORK( KI ) = SCALE
END IF
*
* Copy the vector x or Q*x to VR and normalize.
*
IF( .NOT.OVER ) THEN
CALL ZCOPY( KI, WORK( 1 ), 1, VR( 1, IS ), 1 )
*
II = IZAMAX( KI, VR( 1, IS ), 1 )
REMAX = ONE / CABS1( VR( II, IS ) )
CALL ZDSCAL( KI, REMAX, VR( 1, IS ), 1 )
*
DO 60 K = KI + 1, N
VR( K, IS ) = CMZERO
60 CONTINUE
ELSE
IF( KI.GT.1 )
$ CALL ZGEMV( 'N', N, KI-1, CMONE, VR, LDVR, WORK( 1 ),
$ 1, DCMPLX( SCALE ), VR( 1, KI ), 1 )
*
II = IZAMAX( N, VR( 1, KI ), 1 )
REMAX = ONE / CABS1( VR( II, KI ) )
CALL ZDSCAL( N, REMAX, VR( 1, KI ), 1 )
END IF
*
* Set back the original diagonal elements of T.
*
DO 70 K = 1, KI - 1
T( K, K ) = WORK( K+N )
70 CONTINUE
*
IS = IS - 1
80 CONTINUE
END IF
*
IF( LEFTV ) THEN
*
* Compute left eigenvectors.
*
IS = 1
DO 130 KI = 1, N
*
IF( SOMEV ) THEN
IF( .NOT.SELECT( KI ) )
$ GO TO 130
END IF
SMIN = MAX( ULP*( CABS1( T( KI, KI ) ) ), SMLNUM )
*
WORK( N ) = CMONE
*
* Form right-hand side.
*
DO 90 K = KI + 1, N
WORK( K ) = -DCONJG( T( KI, K ) )
90 CONTINUE
*
* Solve the triangular system:
* (T(KI+1:N,KI+1:N) - T(KI,KI))'*X = SCALE*WORK.
*
DO 100 K = KI + 1, N
T( K, K ) = T( K, K ) - T( KI, KI )
IF( CABS1( T( K, K ) ).LT.SMIN )
$ T( K, K ) = SMIN
100 CONTINUE
*
IF( KI.LT.N ) THEN
CALL ZLATRS( 'Upper', 'Conjugate transpose', 'Non-unit',
$ 'Y', N-KI, T( KI+1, KI+1 ), LDT,
$ WORK( KI+1 ), SCALE, RWORK, INFO )
WORK( KI ) = SCALE
END IF
*
* Copy the vector x or Q*x to VL and normalize.
*
IF( .NOT.OVER ) THEN
CALL ZCOPY( N-KI+1, WORK( KI ), 1, VL( KI, IS ), 1 )
*
II = IZAMAX( N-KI+1, VL( KI, IS ), 1 ) + KI - 1
REMAX = ONE / CABS1( VL( II, IS ) )
CALL ZDSCAL( N-KI+1, REMAX, VL( KI, IS ), 1 )
*
DO 110 K = 1, KI - 1
VL( K, IS ) = CMZERO
110 CONTINUE
ELSE
IF( KI.LT.N )
$ CALL ZGEMV( 'N', N, N-KI, CMONE, VL( 1, KI+1 ), LDVL,
$ WORK( KI+1 ), 1, DCMPLX( SCALE ),
$ VL( 1, KI ), 1 )
*
II = IZAMAX( N, VL( 1, KI ), 1 )
REMAX = ONE / CABS1( VL( II, KI ) )
CALL ZDSCAL( N, REMAX, VL( 1, KI ), 1 )
END IF
*
* Set back the original diagonal elements of T.
*
DO 120 K = KI + 1, N
T( K, K ) = WORK( K+N )
120 CONTINUE
*
IS = IS + 1
130 CONTINUE
END IF
*
RETURN
*
* End of ZTREVC
*
END
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