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authorSiddhesh Wani2015-05-25 14:46:31 +0530
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+ SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR,
+ $ LDVR, MM, M, WORK, 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 T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
+ $ WORK( * )
+* ..
+*
+* Purpose
+* =======
+*
+* DTREVC 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**H)*T = w*(y**H)
+*
+* where y**H denotes the conjugate transpose of 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.
+*
+* 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 by the matrices in VR and/or VL;
+* = 'S': compute selected right and/or left eigenvectors,
+* as indicated by the logical array SELECT.
+*
+* SELECT (input/output) 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'.
+*
+* N (input) INTEGER
+* The order of the matrix T. N >= 0.
+*
+* T (input) DOUBLE PRECISION array, dimension (LDT,N)
+* The upper quasi-triangular matrix T in Schur canonical form.
+*
+* LDT (input) INTEGER
+* The leading dimension of the array T. LDT >= max(1,N).
+*
+* VL (input/output) 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'.
+*
+* LDVL (input) INTEGER
+* The leading dimension of the array VL. LDVL >= 1, and if
+* SIDE = 'L' or 'B', LDVL >= N.
+*
+* VR (input/output) 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'.
+*
+* 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 real eigenvector occupies one column and each
+* selected complex eigenvector occupies two columns.
+*
+* WORK (workspace) DOUBLE PRECISION array, dimension (3*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 )
+* ..
+* .. Local Scalars ..
+ LOGICAL ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV
+ INTEGER I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2
+ DOUBLE PRECISION BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
+ $ SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
+ $ XNORM
+* ..
+* .. External Functions ..
+ LOGICAL LSAME
+ INTEGER IDAMAX
+ DOUBLE PRECISION DDOT, DLAMCH
+ EXTERNAL LSAME, IDAMAX, DDOT, DLAMCH
+* ..
+* .. External Subroutines ..
+ EXTERNAL DAXPY, DCOPY, DGEMV, DLALN2, DSCAL, XERBLA
+* ..
+* .. Intrinsic Functions ..
+ INTRINSIC ABS, MAX, SQRT
+* ..
+* .. Local Arrays ..
+ DOUBLE PRECISION X( 2, 2 )
+* ..
+* .. 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
+ 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
+*
+* 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( 'DTREVC', -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 )
+ 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)
+*
+ N2 = 2*N
+*
+ IF( RIGHTV ) THEN
+*
+* Compute right eigenvectors.
+*
+ IP = 0
+ IS = M
+ DO 140 KI = N, 1, -1
+*
+ IF( IP.EQ.1 )
+ $ GO TO 130
+ IF( KI.EQ.1 )
+ $ GO TO 40
+ IF( T( KI, KI-1 ).EQ.ZERO )
+ $ GO TO 40
+ IP = -1
+*
+ 40 CONTINUE
+ IF( SOMEV ) THEN
+ IF( IP.EQ.0 ) THEN
+ IF( .NOT.SELECT( KI ) )
+ $ GO TO 130
+ ELSE
+ IF( .NOT.SELECT( KI-1 ) )
+ $ GO TO 130
+ 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+N ) = ONE
+*
+* Form right-hand side
+*
+ DO 50 K = 1, KI - 1
+ WORK( K+N ) = -T( K, KI )
+ 50 CONTINUE
+*
+* Solve the 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+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+N ), 1 )
+ WORK( J+N ) = X( 1, 1 )
+*
+* Update right-hand side
+*
+ CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
+ $ WORK( 1+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+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+N ), 1 )
+ WORK( J-1+N ) = X( 1, 1 )
+ WORK( J+N ) = X( 2, 1 )
+*
+* Update right-hand side
+*
+ CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
+ $ WORK( 1+N ), 1 )
+ CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
+ $ WORK( 1+N ), 1 )
+ END IF
+ 60 CONTINUE
+*
+* Copy the vector x or Q*x to VR and normalize.
+*
+ IF( .NOT.OVER ) THEN
+ CALL DCOPY( KI, WORK( 1+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( KI.GT.1 )
+ $ CALL DGEMV( 'N', N, KI-1, ONE, VR, LDVR,
+ $ WORK( 1+N ), 1, WORK( KI+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 )
+ 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+N ) = ONE
+ WORK( KI+N2 ) = WI / T( KI-1, KI )
+ ELSE
+ WORK( KI-1+N ) = -WI / T( KI, KI-1 )
+ WORK( KI+N2 ) = ONE
+ END IF
+ WORK( KI+N ) = ZERO
+ WORK( KI-1+N2 ) = ZERO
+*
+* Form right-hand side
+*
+ DO 80 K = 1, KI - 2
+ WORK( K+N ) = -WORK( KI-1+N )*T( K, KI-1 )
+ WORK( K+N2 ) = -WORK( KI+N2 )*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+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+N ), 1 )
+ CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
+ END IF
+ WORK( J+N ) = X( 1, 1 )
+ WORK( J+N2 ) = X( 1, 2 )
+*
+* Update the right-hand side
+*
+ CALL DAXPY( J-1, -X( 1, 1 ), T( 1, J ), 1,
+ $ WORK( 1+N ), 1 )
+ CALL DAXPY( J-1, -X( 1, 2 ), T( 1, J ), 1,
+ $ WORK( 1+N2 ), 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+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+N ), 1 )
+ CALL DSCAL( KI, SCALE, WORK( 1+N2 ), 1 )
+ END IF
+ WORK( J-1+N ) = X( 1, 1 )
+ WORK( J+N ) = X( 2, 1 )
+ WORK( J-1+N2 ) = X( 1, 2 )
+ WORK( J+N2 ) = X( 2, 2 )
+*
+* Update the right-hand side
+*
+ CALL DAXPY( J-2, -X( 1, 1 ), T( 1, J-1 ), 1,
+ $ WORK( 1+N ), 1 )
+ CALL DAXPY( J-2, -X( 2, 1 ), T( 1, J ), 1,
+ $ WORK( 1+N ), 1 )
+ CALL DAXPY( J-2, -X( 1, 2 ), T( 1, J-1 ), 1,
+ $ WORK( 1+N2 ), 1 )
+ CALL DAXPY( J-2, -X( 2, 2 ), T( 1, J ), 1,
+ $ WORK( 1+N2 ), 1 )
+ END IF
+ 90 CONTINUE
+*
+* Copy the vector x or Q*x to VR and normalize.
+*
+ IF( .NOT.OVER ) THEN
+ CALL DCOPY( KI, WORK( 1+N ), 1, VR( 1, IS-1 ), 1 )
+ CALL DCOPY( KI, WORK( 1+N2 ), 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( KI.GT.2 ) THEN
+ CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
+ $ WORK( 1+N ), 1, WORK( KI-1+N ),
+ $ VR( 1, KI-1 ), 1 )
+ CALL DGEMV( 'N', N, KI-2, ONE, VR, LDVR,
+ $ WORK( 1+N2 ), 1, WORK( KI+N2 ),
+ $ VR( 1, KI ), 1 )
+ ELSE
+ CALL DSCAL( N, WORK( KI-1+N ), VR( 1, KI-1 ), 1 )
+ CALL DSCAL( N, WORK( KI+N2 ), 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 )
+ END IF
+ END IF
+*
+ IS = IS - 1
+ IF( IP.NE.0 )
+ $ IS = IS - 1
+ 130 CONTINUE
+ IF( IP.EQ.1 )
+ $ IP = 0
+ IF( IP.EQ.-1 )
+ $ IP = 1
+ 140 CONTINUE
+ END IF
+*
+ IF( LEFTV ) THEN
+*
+* Compute left eigenvectors.
+*
+ IP = 0
+ IS = 1
+ DO 260 KI = 1, N
+*
+ IF( IP.EQ.-1 )
+ $ GO TO 250
+ IF( KI.EQ.N )
+ $ GO TO 150
+ IF( T( KI+1, KI ).EQ.ZERO )
+ $ GO TO 150
+ IP = 1
+*
+ 150 CONTINUE
+ IF( SOMEV ) THEN
+ IF( .NOT.SELECT( KI ) )
+ $ GO TO 250
+ 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+N ) = ONE
+*
+* Form right-hand side
+*
+ DO 160 K = KI + 1, N
+ WORK( K+N ) = -T( KI, K )
+ 160 CONTINUE
+*
+* Solve the quasi-triangular system:
+* (T(KI+1:N,KI+1:N) - WR)'*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+N ), 1 )
+ VMAX = ONE
+ VCRIT = BIGNUM
+ END IF
+*
+ WORK( J+N ) = WORK( J+N ) -
+ $ DDOT( J-KI-1, T( KI+1, J ), 1,
+ $ WORK( KI+1+N ), 1 )
+*
+* Solve (T(J,J)-WR)'*X = WORK
+*
+ CALL DLALN2( .FALSE., 1, 1, SMIN, ONE, T( J, J ),
+ $ LDT, ONE, ONE, WORK( J+N ), N, WR,
+ $ ZERO, X, 2, SCALE, XNORM, IERR )
+*
+* Scale if necessary
+*
+ IF( SCALE.NE.ONE )
+ $ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
+ WORK( J+N ) = X( 1, 1 )
+ VMAX = MAX( ABS( WORK( J+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+N ), 1 )
+ VMAX = ONE
+ VCRIT = BIGNUM
+ END IF
+*
+ WORK( J+N ) = WORK( J+N ) -
+ $ DDOT( J-KI-1, T( KI+1, J ), 1,
+ $ WORK( KI+1+N ), 1 )
+*
+ WORK( J+1+N ) = WORK( J+1+N ) -
+ $ DDOT( J-KI-1, T( KI+1, J+1 ), 1,
+ $ WORK( KI+1+N ), 1 )
+*
+* Solve
+* [T(J,J)-WR T(J,J+1) ]'* 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+N ), N, WR,
+ $ ZERO, X, 2, SCALE, XNORM, IERR )
+*
+* Scale if necessary
+*
+ IF( SCALE.NE.ONE )
+ $ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N ), 1 )
+ WORK( J+N ) = X( 1, 1 )
+ WORK( J+1+N ) = X( 2, 1 )
+*
+ VMAX = MAX( ABS( WORK( J+N ) ),
+ $ ABS( WORK( J+1+N ) ), VMAX )
+ VCRIT = BIGNUM / VMAX
+*
+ END IF
+ 170 CONTINUE
+*
+* Copy the vector x or Q*x to VL and normalize.
+*
+ IF( .NOT.OVER ) THEN
+ CALL DCOPY( N-KI+1, WORK( KI+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( KI.LT.N )
+ $ CALL DGEMV( 'N', N, N-KI, ONE, VL( 1, KI+1 ), LDVL,
+ $ WORK( KI+1+N ), 1, WORK( KI+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 )
+*
+ END IF
+*
+ ELSE
+*
+* Complex left eigenvector.
+*
+* Initial solve:
+* ((T(KI,KI) T(KI,KI+1) )' - (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+N ) = WI / T( KI, KI+1 )
+ WORK( KI+1+N2 ) = ONE
+ ELSE
+ WORK( KI+N ) = ONE
+ WORK( KI+1+N2 ) = -WI / T( KI+1, KI )
+ END IF
+ WORK( KI+1+N ) = ZERO
+ WORK( KI+N2 ) = ZERO
+*
+* Form right-hand side
+*
+ DO 190 K = KI + 2, N
+ WORK( K+N ) = -WORK( KI+N )*T( KI, K )
+ WORK( K+N2 ) = -WORK( KI+1+N2 )*T( KI+1, K )
+ 190 CONTINUE
+*
+* Solve complex quasi-triangular system:
+* ( T(KI+2,N:KI+2,N) - (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+N ), 1 )
+ CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
+ VMAX = ONE
+ VCRIT = BIGNUM
+ END IF
+*
+ WORK( J+N ) = WORK( J+N ) -
+ $ DDOT( J-KI-2, T( KI+2, J ), 1,
+ $ WORK( KI+2+N ), 1 )
+ WORK( J+N2 ) = WORK( J+N2 ) -
+ $ DDOT( J-KI-2, T( KI+2, J ), 1,
+ $ WORK( KI+2+N2 ), 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+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+N ), 1 )
+ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
+ END IF
+ WORK( J+N ) = X( 1, 1 )
+ WORK( J+N2 ) = X( 1, 2 )
+ VMAX = MAX( ABS( WORK( J+N ) ),
+ $ ABS( WORK( J+N2 ) ), 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+N ), 1 )
+ CALL DSCAL( N-KI+1, REC, WORK( KI+N2 ), 1 )
+ VMAX = ONE
+ VCRIT = BIGNUM
+ END IF
+*
+ WORK( J+N ) = WORK( J+N ) -
+ $ DDOT( J-KI-2, T( KI+2, J ), 1,
+ $ WORK( KI+2+N ), 1 )
+*
+ WORK( J+N2 ) = WORK( J+N2 ) -
+ $ DDOT( J-KI-2, T( KI+2, J ), 1,
+ $ WORK( KI+2+N2 ), 1 )
+*
+ WORK( J+1+N ) = WORK( J+1+N ) -
+ $ DDOT( J-KI-2, T( KI+2, J+1 ), 1,
+ $ WORK( KI+2+N ), 1 )
+*
+ WORK( J+1+N2 ) = WORK( J+1+N2 ) -
+ $ DDOT( J-KI-2, T( KI+2, J+1 ), 1,
+ $ WORK( KI+2+N2 ), 1 )
+*
+* Solve 2-by-2 complex linear equation
+* ([T(j,j) T(j,j+1) ]'-(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+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+N ), 1 )
+ CALL DSCAL( N-KI+1, SCALE, WORK( KI+N2 ), 1 )
+ END IF
+ WORK( J+N ) = X( 1, 1 )
+ WORK( J+N2 ) = X( 1, 2 )
+ WORK( J+1+N ) = X( 2, 1 )
+ WORK( J+1+N2 ) = 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
+ CALL DCOPY( N-KI+1, WORK( KI+N ), 1, VL( KI, IS ), 1 )
+ CALL DCOPY( N-KI+1, WORK( KI+N2 ), 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( KI.LT.N-1 ) THEN
+ CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
+ $ LDVL, WORK( KI+2+N ), 1, WORK( KI+N ),
+ $ VL( 1, KI ), 1 )
+ CALL DGEMV( 'N', N, N-KI-1, ONE, VL( 1, KI+2 ),
+ $ LDVL, WORK( KI+2+N2 ), 1,
+ $ WORK( KI+1+N2 ), VL( 1, KI+1 ), 1 )
+ ELSE
+ CALL DSCAL( N, WORK( KI+N ), VL( 1, KI ), 1 )
+ CALL DSCAL( N, WORK( KI+1+N2 ), 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 )
+*
+ END IF
+*
+ END IF
+*
+ IS = IS + 1
+ IF( IP.NE.0 )
+ $ IS = IS + 1
+ 250 CONTINUE
+ IF( IP.EQ.-1 )
+ $ IP = 0
+ IF( IP.EQ.1 )
+ $ IP = -1
+*
+ 260 CONTINUE
+*
+ END IF
+*
+ RETURN
+*
+* End of DTREVC
+*
+ END