SUBROUTINE ZTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, 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, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
DOUBLE PRECISION RWORK( * )
COMPLEX*16 P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
*
* Purpose
* =======
*
* ZTGEVC computes some or all of the right and/or left eigenvectors of
* a pair of complex matrices (S,P), where S and P are upper triangular.
* Matrix pairs of this type are produced by the generalized Schur
* factorization of a complex matrix pair (A,B):
*
* A = Q*S*Z**H, B = Q*P*Z**H
*
* as computed by ZGGHRD + ZHGEQZ.
*
* The right eigenvector x and the left eigenvector y of (S,P)
* corresponding to an eigenvalue w are defined by:
*
* S*x = w*P*x, (y**H)*S = w*(y**H)*P,
*
* where y**H denotes the conjugate tranpose of y.
* The eigenvalues are not input to this routine, but are computed
* directly from the diagonal elements of S and P.
*
* This routine returns the matrices X and/or Y of right and left
* eigenvectors of (S,P), or the products Z*X and/or Q*Y,
* where Z and Q are input matrices.
* If Q and Z are the unitary factors from the generalized Schur
* factorization of a matrix pair (A,B), then Z*X and Q*Y
* are the matrices of right and left eigenvectors of (A,B).
*
* 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,
* specified 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 matrices S and P. N >= 0.
*
* S (input) COMPLEX*16 array, dimension (LDS,N)
* The upper triangular matrix S from a generalized Schur
* factorization, as computed by ZHGEQZ.
*
* LDS (input) INTEGER
* The leading dimension of array S. LDS >= max(1,N).
*
* P (input) COMPLEX*16 array, dimension (LDP,N)
* The upper triangular matrix P from a generalized Schur
* factorization, as computed by ZHGEQZ. P must have real
* diagonal elements.
*
* LDP (input) INTEGER
* The leading dimension of array P. LDP >= 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 left Schur vectors returned by ZHGEQZ).
* On exit, if SIDE = 'L' or 'B', VL contains:
* if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
* if HOWMNY = 'B', the matrix Q*Y;
* if HOWMNY = 'S', the left eigenvectors of (S,P) 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 array VL. LDVL >= 1, and if
* SIDE = 'L' or 'l' or 'B' 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 Z
* of right Schur vectors returned by ZHGEQZ).
* On exit, if SIDE = 'R' or 'B', VR contains:
* if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
* if HOWMNY = 'B', the matrix Z*X;
* if HOWMNY = 'S', the right eigenvectors of (S,P) 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 (2*N)
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
COMPLEX*16 CZERO, CONE
PARAMETER ( CZERO = ( 0.0D+0, 0.0D+0 ),
$ CONE = ( 1.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
LOGICAL COMPL, COMPR, ILALL, ILBACK, ILBBAD, ILCOMP,
$ LSA, LSB
INTEGER I, IBEG, IEIG, IEND, IHWMNY, IM, ISIDE, ISRC,
$ J, JE, JR
DOUBLE PRECISION ACOEFA, ACOEFF, ANORM, ASCALE, BCOEFA, BIG,
$ BIGNUM, BNORM, BSCALE, DMIN, SAFMIN, SBETA,
$ SCALE, SMALL, TEMP, ULP, XMAX
COMPLEX*16 BCOEFF, CA, CB, D, SALPHA, SUM, SUMA, SUMB, X
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
COMPLEX*16 ZLADIV
EXTERNAL LSAME, DLAMCH, ZLADIV
* ..
* .. External Subroutines ..
EXTERNAL DLABAD, XERBLA, ZGEMV
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN
* ..
* .. Statement Functions ..
DOUBLE PRECISION ABS1
* ..
* .. Statement Function definitions ..
ABS1( X ) = ABS( DBLE( X ) ) + ABS( DIMAG( X ) )
* ..
* .. Executable Statements ..
*
* Decode and Test the input parameters
*
IF( LSAME( HOWMNY, 'A' ) ) THEN
IHWMNY = 1
ILALL = .TRUE.
ILBACK = .FALSE.
ELSE IF( LSAME( HOWMNY, 'S' ) ) THEN
IHWMNY = 2
ILALL = .FALSE.
ILBACK = .FALSE.
ELSE IF( LSAME( HOWMNY, 'B' ) ) THEN
IHWMNY = 3
ILALL = .TRUE.
ILBACK = .TRUE.
ELSE
IHWMNY = -1
END IF
*
IF( LSAME( SIDE, 'R' ) ) THEN
ISIDE = 1
COMPL = .FALSE.
COMPR = .TRUE.
ELSE IF( LSAME( SIDE, 'L' ) ) THEN
ISIDE = 2
COMPL = .TRUE.
COMPR = .FALSE.
ELSE IF( LSAME( SIDE, 'B' ) ) THEN
ISIDE = 3
COMPL = .TRUE.
COMPR = .TRUE.
ELSE
ISIDE = -1
END IF
*
INFO = 0
IF( ISIDE.LT.0 ) THEN
INFO = -1
ELSE IF( IHWMNY.LT.0 ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -4
ELSE IF( LDS.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDP.LT.MAX( 1, N ) ) THEN
INFO = -8
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTGEVC', -INFO )
RETURN
END IF
*
* Count the number of eigenvectors
*
IF( .NOT.ILALL ) THEN
IM = 0
DO 10 J = 1, N
IF( SELECT( J ) )
$ IM = IM + 1
10 CONTINUE
ELSE
IM = N
END IF
*
* Check diagonal of B
*
ILBBAD = .FALSE.
DO 20 J = 1, N
IF( DIMAG( P( J, J ) ).NE.ZERO )
$ ILBBAD = .TRUE.
20 CONTINUE
*
IF( ILBBAD ) THEN
INFO = -7
ELSE IF( COMPL .AND. LDVL.LT.N .OR. LDVL.LT.1 ) THEN
INFO = -10
ELSE IF( COMPR .AND. LDVR.LT.N .OR. LDVR.LT.1 ) THEN
INFO = -12
ELSE IF( MM.LT.IM ) THEN
INFO = -13
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZTGEVC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = IM
IF( N.EQ.0 )
$ RETURN
*
* Machine Constants
*
SAFMIN = DLAMCH( 'Safe minimum' )
BIG = ONE / SAFMIN
CALL DLABAD( SAFMIN, BIG )
ULP = DLAMCH( 'Epsilon' )*DLAMCH( 'Base' )
SMALL = SAFMIN*N / ULP
BIG = ONE / SMALL
BIGNUM = ONE / ( SAFMIN*N )
*
* Compute the 1-norm of each column of the strictly upper triangular
* part of A and B to check for possible overflow in the triangular
* solver.
*
ANORM = ABS1( S( 1, 1 ) )
BNORM = ABS1( P( 1, 1 ) )
RWORK( 1 ) = ZERO
RWORK( N+1 ) = ZERO
DO 40 J = 2, N
RWORK( J ) = ZERO
RWORK( N+J ) = ZERO
DO 30 I = 1, J - 1
RWORK( J ) = RWORK( J ) + ABS1( S( I, J ) )
RWORK( N+J ) = RWORK( N+J ) + ABS1( P( I, J ) )
30 CONTINUE
ANORM = MAX( ANORM, RWORK( J )+ABS1( S( J, J ) ) )
BNORM = MAX( BNORM, RWORK( N+J )+ABS1( P( J, J ) ) )
40 CONTINUE
*
ASCALE = ONE / MAX( ANORM, SAFMIN )
BSCALE = ONE / MAX( BNORM, SAFMIN )
*
* Left eigenvectors
*
IF( COMPL ) THEN
IEIG = 0
*
* Main loop over eigenvalues
*
DO 140 JE = 1, N
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE
ILCOMP = SELECT( JE )
END IF
IF( ILCOMP ) THEN
IEIG = IEIG + 1
*
IF( ABS1( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( DBLE( P( JE, JE ) ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- return unit eigenvector
*
DO 50 JR = 1, N
VL( JR, IEIG ) = CZERO
50 CONTINUE
VL( IEIG, IEIG ) = CONE
GO TO 140
END IF
*
* Non-singular eigenvalue:
* Compute coefficients a and b in
* H
* y ( a A - b B ) = 0
*
TEMP = ONE / MAX( ABS1( S( JE, JE ) )*ASCALE,
$ ABS( DBLE( P( JE, JE ) ) )*BSCALE, SAFMIN )
SALPHA = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*DBLE( P( JE, JE ) ) )*BSCALE
ACOEFF = SBETA*ASCALE
BCOEFF = SALPHA*BSCALE
*
* Scale to avoid underflow
*
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEFF ).LT.SMALL
LSB = ABS1( SALPHA ).GE.SAFMIN .AND. ABS1( BCOEFF ).LT.
$ SMALL
*
SCALE = ONE
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS1( SALPHA ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEFF ),
$ ABS1( BCOEFF ) ) ) )
IF( LSA ) THEN
ACOEFF = ASCALE*( SCALE*SBETA )
ELSE
ACOEFF = SCALE*ACOEFF
END IF
IF( LSB ) THEN
BCOEFF = BSCALE*( SCALE*SALPHA )
ELSE
BCOEFF = SCALE*BCOEFF
END IF
END IF
*
ACOEFA = ABS( ACOEFF )
BCOEFA = ABS1( BCOEFF )
XMAX = ONE
DO 60 JR = 1, N
WORK( JR ) = CZERO
60 CONTINUE
WORK( JE ) = CONE
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* H
* Triangular solve of (a A - b B) y = 0
*
* H
* (rowwise in (a A - b B) , or columnwise in a A - b B)
*
DO 100 J = JE + 1, N
*
* Compute
* j-1
* SUM = sum conjg( a*S(k,j) - b*P(k,j) )*x(k)
* k=je
* (Scale if necessary)
*
TEMP = ONE / XMAX
IF( ACOEFA*RWORK( J )+BCOEFA*RWORK( N+J ).GT.BIGNUM*
$ TEMP ) THEN
DO 70 JR = JE, J - 1
WORK( JR ) = TEMP*WORK( JR )
70 CONTINUE
XMAX = ONE
END IF
SUMA = CZERO
SUMB = CZERO
*
DO 80 JR = JE, J - 1
SUMA = SUMA + DCONJG( S( JR, J ) )*WORK( JR )
SUMB = SUMB + DCONJG( P( JR, J ) )*WORK( JR )
80 CONTINUE
SUM = ACOEFF*SUMA - DCONJG( BCOEFF )*SUMB
*
* Form x(j) = - SUM / conjg( a*S(j,j) - b*P(j,j) )
*
* with scaling and perturbation of the denominator
*
D = DCONJG( ACOEFF*S( J, J )-BCOEFF*P( J, J ) )
IF( ABS1( D ).LE.DMIN )
$ D = DCMPLX( DMIN )
*
IF( ABS1( D ).LT.ONE ) THEN
IF( ABS1( SUM ).GE.BIGNUM*ABS1( D ) ) THEN
TEMP = ONE / ABS1( SUM )
DO 90 JR = JE, J - 1
WORK( JR ) = TEMP*WORK( JR )
90 CONTINUE
XMAX = TEMP*XMAX
SUM = TEMP*SUM
END IF
END IF
WORK( J ) = ZLADIV( -SUM, D )
XMAX = MAX( XMAX, ABS1( WORK( J ) ) )
100 CONTINUE
*
* Back transform eigenvector if HOWMNY='B'.
*
IF( ILBACK ) THEN
CALL ZGEMV( 'N', N, N+1-JE, CONE, VL( 1, JE ), LDVL,
$ WORK( JE ), 1, CZERO, WORK( N+1 ), 1 )
ISRC = 2
IBEG = 1
ELSE
ISRC = 1
IBEG = JE
END IF
*
* Copy and scale eigenvector into column of VL
*
XMAX = ZERO
DO 110 JR = IBEG, N
XMAX = MAX( XMAX, ABS1( WORK( ( ISRC-1 )*N+JR ) ) )
110 CONTINUE
*
IF( XMAX.GT.SAFMIN ) THEN
TEMP = ONE / XMAX
DO 120 JR = IBEG, N
VL( JR, IEIG ) = TEMP*WORK( ( ISRC-1 )*N+JR )
120 CONTINUE
ELSE
IBEG = N + 1
END IF
*
DO 130 JR = 1, IBEG - 1
VL( JR, IEIG ) = CZERO
130 CONTINUE
*
END IF
140 CONTINUE
END IF
*
* Right eigenvectors
*
IF( COMPR ) THEN
IEIG = IM + 1
*
* Main loop over eigenvalues
*
DO 250 JE = N, 1, -1
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE
ILCOMP = SELECT( JE )
END IF
IF( ILCOMP ) THEN
IEIG = IEIG - 1
*
IF( ABS1( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( DBLE( P( JE, JE ) ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- return unit eigenvector
*
DO 150 JR = 1, N
VR( JR, IEIG ) = CZERO
150 CONTINUE
VR( IEIG, IEIG ) = CONE
GO TO 250
END IF
*
* Non-singular eigenvalue:
* Compute coefficients a and b in
*
* ( a A - b B ) x = 0
*
TEMP = ONE / MAX( ABS1( S( JE, JE ) )*ASCALE,
$ ABS( DBLE( P( JE, JE ) ) )*BSCALE, SAFMIN )
SALPHA = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*DBLE( P( JE, JE ) ) )*BSCALE
ACOEFF = SBETA*ASCALE
BCOEFF = SALPHA*BSCALE
*
* Scale to avoid underflow
*
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEFF ).LT.SMALL
LSB = ABS1( SALPHA ).GE.SAFMIN .AND. ABS1( BCOEFF ).LT.
$ SMALL
*
SCALE = ONE
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS1( SALPHA ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEFF ),
$ ABS1( BCOEFF ) ) ) )
IF( LSA ) THEN
ACOEFF = ASCALE*( SCALE*SBETA )
ELSE
ACOEFF = SCALE*ACOEFF
END IF
IF( LSB ) THEN
BCOEFF = BSCALE*( SCALE*SALPHA )
ELSE
BCOEFF = SCALE*BCOEFF
END IF
END IF
*
ACOEFA = ABS( ACOEFF )
BCOEFA = ABS1( BCOEFF )
XMAX = ONE
DO 160 JR = 1, N
WORK( JR ) = CZERO
160 CONTINUE
WORK( JE ) = CONE
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* Triangular solve of (a A - b B) x = 0 (columnwise)
*
* WORK(1:j-1) contains sums w,
* WORK(j+1:JE) contains x
*
DO 170 JR = 1, JE - 1
WORK( JR ) = ACOEFF*S( JR, JE ) - BCOEFF*P( JR, JE )
170 CONTINUE
WORK( JE ) = CONE
*
DO 210 J = JE - 1, 1, -1
*
* Form x(j) := - w(j) / d
* with scaling and perturbation of the denominator
*
D = ACOEFF*S( J, J ) - BCOEFF*P( J, J )
IF( ABS1( D ).LE.DMIN )
$ D = DCMPLX( DMIN )
*
IF( ABS1( D ).LT.ONE ) THEN
IF( ABS1( WORK( J ) ).GE.BIGNUM*ABS1( D ) ) THEN
TEMP = ONE / ABS1( WORK( J ) )
DO 180 JR = 1, JE
WORK( JR ) = TEMP*WORK( JR )
180 CONTINUE
END IF
END IF
*
WORK( J ) = ZLADIV( -WORK( J ), D )
*
IF( J.GT.1 ) THEN
*
* w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling
*
IF( ABS1( WORK( J ) ).GT.ONE ) THEN
TEMP = ONE / ABS1( WORK( J ) )
IF( ACOEFA*RWORK( J )+BCOEFA*RWORK( N+J ).GE.
$ BIGNUM*TEMP ) THEN
DO 190 JR = 1, JE
WORK( JR ) = TEMP*WORK( JR )
190 CONTINUE
END IF
END IF
*
CA = ACOEFF*WORK( J )
CB = BCOEFF*WORK( J )
DO 200 JR = 1, J - 1
WORK( JR ) = WORK( JR ) + CA*S( JR, J ) -
$ CB*P( JR, J )
200 CONTINUE
END IF
210 CONTINUE
*
* Back transform eigenvector if HOWMNY='B'.
*
IF( ILBACK ) THEN
CALL ZGEMV( 'N', N, JE, CONE, VR, LDVR, WORK, 1,
$ CZERO, WORK( N+1 ), 1 )
ISRC = 2
IEND = N
ELSE
ISRC = 1
IEND = JE
END IF
*
* Copy and scale eigenvector into column of VR
*
XMAX = ZERO
DO 220 JR = 1, IEND
XMAX = MAX( XMAX, ABS1( WORK( ( ISRC-1 )*N+JR ) ) )
220 CONTINUE
*
IF( XMAX.GT.SAFMIN ) THEN
TEMP = ONE / XMAX
DO 230 JR = 1, IEND
VR( JR, IEIG ) = TEMP*WORK( ( ISRC-1 )*N+JR )
230 CONTINUE
ELSE
IEND = 0
END IF
*
DO 240 JR = IEND + 1, N
VR( JR, IEIG ) = CZERO
240 CONTINUE
*
END IF
250 CONTINUE
END IF
*
RETURN
*
* End of ZTGEVC
*
END