*> \brief \b STGEVC
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download STGEVC + dependencies
*>
*> [TGZ]
*>
*> [ZIP]
*>
*> [TXT]
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE STGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
* LDVL, VR, LDVR, MM, M, WORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER HOWMNY, SIDE
* INTEGER INFO, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
* LOGICAL SELECT( * )
* REAL P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
* $ VR( LDVR, * ), WORK( * )
* ..
*
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> STGEVC computes some or all of the right and/or left eigenvectors of
*> a pair of real matrices (S,P), where S is a quasi-triangular matrix
*> and P is upper triangular. Matrix pairs of this type are produced by
*> the generalized Schur factorization of a matrix pair (A,B):
*>
*> A = Q*S*Z**T, B = Q*P*Z**T
*>
*> as computed by SGGHRD + SHGEQZ.
*>
*> 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 blocks 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 orthogonal 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).
*>
*> \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,
*> specified by the logical array SELECT.
*> \endverbatim
*>
*> \param[in] 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 matrices S and P. N >= 0.
*> \endverbatim
*>
*> \param[in] S
*> \verbatim
*> S is REAL array, dimension (LDS,N)
*> The upper quasi-triangular matrix S from a generalized Schur
*> factorization, as computed by SHGEQZ.
*> \endverbatim
*>
*> \param[in] LDS
*> \verbatim
*> LDS is INTEGER
*> The leading dimension of array S. LDS >= max(1,N).
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is REAL array, dimension (LDP,N)
*> The upper triangular matrix P from a generalized Schur
*> factorization, as computed by SHGEQZ.
*> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
*> of S must be in positive diagonal form.
*> \endverbatim
*>
*> \param[in] LDP
*> \verbatim
*> LDP is INTEGER
*> The leading dimension of array P. LDP >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] VL
*> \verbatim
*> VL is REAL 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 left Schur vectors returned by SHGEQZ).
*> 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.
*>
*> 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 array VL. LDVL >= 1, and if
*> SIDE = 'L' or 'B', LDVL >= N.
*> \endverbatim
*>
*> \param[in,out] VR
*> \verbatim
*> VR is REAL array, dimension (LDVR,MM)
*> On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
*> contain an N-by-N matrix Z (usually the orthogonal matrix Z
*> of right Schur vectors returned by SHGEQZ).
*>
*> On exit, if SIDE = 'R' or 'B', VR contains:
*> if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
*> if HOWMNY = 'B' or 'b', the matrix Z*X;
*> if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
*> 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 REAL array, dimension (6*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex
*> eigenvalue.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> Allocation of workspace:
*> ---------- -- ---------
*>
*> WORK( j ) = 1-norm of j-th column of A, above the diagonal
*> WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
*> WORK( 2*N+1:3*N ) = real part of eigenvector
*> WORK( 3*N+1:4*N ) = imaginary part of eigenvector
*> WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
*> WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
*>
*> Rowwise vs. columnwise solution methods:
*> ------- -- ---------- -------- -------
*>
*> Finding a generalized eigenvector consists basically of solving the
*> singular triangular system
*>
*> (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)
*>
*> Consider finding the i-th right eigenvector (assume all eigenvalues
*> are real). The equation to be solved is:
*> n i
*> 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1
*> k=j k=j
*>
*> where C = (A - w B) (The components v(i+1:n) are 0.)
*>
*> The "rowwise" method is:
*>
*> (1) v(i) := 1
*> for j = i-1,. . .,1:
*> i
*> (2) compute s = - sum C(j,k) v(k) and
*> k=j+1
*>
*> (3) v(j) := s / C(j,j)
*>
*> Step 2 is sometimes called the "dot product" step, since it is an
*> inner product between the j-th row and the portion of the eigenvector
*> that has been computed so far.
*>
*> The "columnwise" method consists basically in doing the sums
*> for all the rows in parallel. As each v(j) is computed, the
*> contribution of v(j) times the j-th column of C is added to the
*> partial sums. Since FORTRAN arrays are stored columnwise, this has
*> the advantage that at each step, the elements of C that are accessed
*> are adjacent to one another, whereas with the rowwise method, the
*> elements accessed at a step are spaced LDS (and LDP) words apart.
*>
*> When finding left eigenvectors, the matrix in question is the
*> transpose of the one in storage, so the rowwise method then
*> actually accesses columns of A and B at each step, and so is the
*> preferred method.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE STGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL,
$ LDVL, VR, LDVR, MM, M, WORK, INFO )
*
* -- 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, LDP, LDS, LDVL, LDVR, M, MM, N
* ..
* .. Array Arguments ..
LOGICAL SELECT( * )
REAL P( LDP, * ), S( LDS, * ), VL( LDVL, * ),
$ VR( LDVR, * ), WORK( * )
* ..
*
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, SAFETY
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0,
$ SAFETY = 1.0E+2 )
* ..
* .. Local Scalars ..
LOGICAL COMPL, COMPR, IL2BY2, ILABAD, ILALL, ILBACK,
$ ILBBAD, ILCOMP, ILCPLX, LSA, LSB
INTEGER I, IBEG, IEIG, IEND, IHWMNY, IINFO, IM, ISIDE,
$ J, JA, JC, JE, JR, JW, NA, NW
REAL ACOEF, ACOEFA, ANORM, ASCALE, BCOEFA, BCOEFI,
$ BCOEFR, BIG, BIGNUM, BNORM, BSCALE, CIM2A,
$ CIM2B, CIMAGA, CIMAGB, CRE2A, CRE2B, CREALA,
$ CREALB, DMIN, SAFMIN, SALFAR, SBETA, SCALE,
$ SMALL, TEMP, TEMP2, TEMP2I, TEMP2R, ULP, XMAX,
$ XSCALE
* ..
* .. Local Arrays ..
REAL BDIAG( 2 ), SUM( 2, 2 ), SUMS( 2, 2 ),
$ SUMP( 2, 2 )
* ..
* .. External Functions ..
LOGICAL LSAME
REAL SLAMCH
EXTERNAL LSAME, SLAMCH
* ..
* .. External Subroutines ..
EXTERNAL SGEMV, SLABAD, SLACPY, SLAG2, SLALN2, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN
* ..
* .. 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
ILALL = .TRUE.
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( 'STGEVC', -INFO )
RETURN
END IF
*
* Count the number of eigenvectors to be computed
*
IF( .NOT.ILALL ) THEN
IM = 0
ILCPLX = .FALSE.
DO 10 J = 1, N
IF( ILCPLX ) THEN
ILCPLX = .FALSE.
GO TO 10
END IF
IF( J.LT.N ) THEN
IF( S( J+1, J ).NE.ZERO )
$ ILCPLX = .TRUE.
END IF
IF( ILCPLX ) THEN
IF( SELECT( J ) .OR. SELECT( J+1 ) )
$ IM = IM + 2
ELSE
IF( SELECT( J ) )
$ IM = IM + 1
END IF
10 CONTINUE
ELSE
IM = N
END IF
*
* Check 2-by-2 diagonal blocks of A, B
*
ILABAD = .FALSE.
ILBBAD = .FALSE.
DO 20 J = 1, N - 1
IF( S( J+1, J ).NE.ZERO ) THEN
IF( P( J, J ).EQ.ZERO .OR. P( J+1, J+1 ).EQ.ZERO .OR.
$ P( J, J+1 ).NE.ZERO )ILBBAD = .TRUE.
IF( J.LT.N-1 ) THEN
IF( S( J+2, J+1 ).NE.ZERO )
$ ILABAD = .TRUE.
END IF
END IF
20 CONTINUE
*
IF( ILABAD ) THEN
INFO = -5
ELSE 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( 'STGEVC', -INFO )
RETURN
END IF
*
* Quick return if possible
*
M = IM
IF( N.EQ.0 )
$ RETURN
*
* Machine Constants
*
SAFMIN = SLAMCH( 'Safe minimum' )
BIG = ONE / SAFMIN
CALL SLABAD( SAFMIN, BIG )
ULP = SLAMCH( 'Epsilon' )*SLAMCH( '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 (i.e., excluding all elements belonging to the diagonal
* blocks) of A and B to check for possible overflow in the
* triangular solver.
*
ANORM = ABS( S( 1, 1 ) )
IF( N.GT.1 )
$ ANORM = ANORM + ABS( S( 2, 1 ) )
BNORM = ABS( P( 1, 1 ) )
WORK( 1 ) = ZERO
WORK( N+1 ) = ZERO
*
DO 50 J = 2, N
TEMP = ZERO
TEMP2 = ZERO
IF( S( J, J-1 ).EQ.ZERO ) THEN
IEND = J - 1
ELSE
IEND = J - 2
END IF
DO 30 I = 1, IEND
TEMP = TEMP + ABS( S( I, J ) )
TEMP2 = TEMP2 + ABS( P( I, J ) )
30 CONTINUE
WORK( J ) = TEMP
WORK( N+J ) = TEMP2
DO 40 I = IEND + 1, MIN( J+1, N )
TEMP = TEMP + ABS( S( I, J ) )
TEMP2 = TEMP2 + ABS( P( I, J ) )
40 CONTINUE
ANORM = MAX( ANORM, TEMP )
BNORM = MAX( BNORM, TEMP2 )
50 CONTINUE
*
ASCALE = ONE / MAX( ANORM, SAFMIN )
BSCALE = ONE / MAX( BNORM, SAFMIN )
*
* Left eigenvectors
*
IF( COMPL ) THEN
IEIG = 0
*
* Main loop over eigenvalues
*
ILCPLX = .FALSE.
DO 220 JE = 1, N
*
* Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
* (b) this would be the second of a complex pair.
* Check for complex eigenvalue, so as to be sure of which
* entry(-ies) of SELECT to look at.
*
IF( ILCPLX ) THEN
ILCPLX = .FALSE.
GO TO 220
END IF
NW = 1
IF( JE.LT.N ) THEN
IF( S( JE+1, JE ).NE.ZERO ) THEN
ILCPLX = .TRUE.
NW = 2
END IF
END IF
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE IF( ILCPLX ) THEN
ILCOMP = SELECT( JE ) .OR. SELECT( JE+1 )
ELSE
ILCOMP = SELECT( JE )
END IF
IF( .NOT.ILCOMP )
$ GO TO 220
*
* Decide if (a) singular pencil, (b) real eigenvalue, or
* (c) complex eigenvalue.
*
IF( .NOT.ILCPLX ) THEN
IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- return unit eigenvector
*
IEIG = IEIG + 1
DO 60 JR = 1, N
VL( JR, IEIG ) = ZERO
60 CONTINUE
VL( IEIG, IEIG ) = ONE
GO TO 220
END IF
END IF
*
* Clear vector
*
DO 70 JR = 1, NW*N
WORK( 2*N+JR ) = ZERO
70 CONTINUE
* T
* Compute coefficients in ( a A - b B ) y = 0
* a is ACOEF
* b is BCOEFR + i*BCOEFI
*
IF( .NOT.ILCPLX ) THEN
*
* Real eigenvalue
*
TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
$ ABS( P( JE, JE ) )*BSCALE, SAFMIN )
SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*P( JE, JE ) )*BSCALE
ACOEF = SBETA*ASCALE
BCOEFR = SALFAR*BSCALE
BCOEFI = ZERO
*
* Scale to avoid underflow
*
SCALE = ONE
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
$ SMALL
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEF ),
$ ABS( BCOEFR ) ) ) )
IF( LSA ) THEN
ACOEF = ASCALE*( SCALE*SBETA )
ELSE
ACOEF = SCALE*ACOEF
END IF
IF( LSB ) THEN
BCOEFR = BSCALE*( SCALE*SALFAR )
ELSE
BCOEFR = SCALE*BCOEFR
END IF
END IF
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR )
*
* First component is 1
*
WORK( 2*N+JE ) = ONE
XMAX = ONE
ELSE
*
* Complex eigenvalue
*
CALL SLAG2( S( JE, JE ), LDS, P( JE, JE ), LDP,
$ SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
$ BCOEFI )
BCOEFI = -BCOEFI
IF( BCOEFI.EQ.ZERO ) THEN
INFO = JE
RETURN
END IF
*
* Scale to avoid over/underflow
*
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
SCALE = ONE
IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
$ SCALE = ( SAFMIN / ULP ) / ACOEFA
IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
$ SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
IF( SAFMIN*ACOEFA.GT.ASCALE )
$ SCALE = ASCALE / ( SAFMIN*ACOEFA )
IF( SAFMIN*BCOEFA.GT.BSCALE )
$ SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
IF( SCALE.NE.ONE ) THEN
ACOEF = SCALE*ACOEF
ACOEFA = ABS( ACOEF )
BCOEFR = SCALE*BCOEFR
BCOEFI = SCALE*BCOEFI
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
END IF
*
* Compute first two components of eigenvector
*
TEMP = ACOEF*S( JE+1, JE )
TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
TEMP2I = -BCOEFI*P( JE, JE )
IF( ABS( TEMP ).GT.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
WORK( 2*N+JE ) = ONE
WORK( 3*N+JE ) = ZERO
WORK( 2*N+JE+1 ) = -TEMP2R / TEMP
WORK( 3*N+JE+1 ) = -TEMP2I / TEMP
ELSE
WORK( 2*N+JE+1 ) = ONE
WORK( 3*N+JE+1 ) = ZERO
TEMP = ACOEF*S( JE, JE+1 )
WORK( 2*N+JE ) = ( BCOEFR*P( JE+1, JE+1 )-ACOEF*
$ S( JE+1, JE+1 ) ) / TEMP
WORK( 3*N+JE ) = BCOEFI*P( JE+1, JE+1 ) / TEMP
END IF
XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
$ ABS( WORK( 2*N+JE+1 ) )+ABS( WORK( 3*N+JE+1 ) ) )
END IF
*
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* T
* Triangular solve of (a A - b B) y = 0
*
* T
* (rowwise in (a A - b B) , or columnwise in (a A - b B) )
*
IL2BY2 = .FALSE.
*
DO 160 J = JE + NW, N
IF( IL2BY2 ) THEN
IL2BY2 = .FALSE.
GO TO 160
END IF
*
NA = 1
BDIAG( 1 ) = P( J, J )
IF( J.LT.N ) THEN
IF( S( J+1, J ).NE.ZERO ) THEN
IL2BY2 = .TRUE.
BDIAG( 2 ) = P( J+1, J+1 )
NA = 2
END IF
END IF
*
* Check whether scaling is necessary for dot products
*
XSCALE = ONE / MAX( ONE, XMAX )
TEMP = MAX( WORK( J ), WORK( N+J ),
$ ACOEFA*WORK( J )+BCOEFA*WORK( N+J ) )
IF( IL2BY2 )
$ TEMP = MAX( TEMP, WORK( J+1 ), WORK( N+J+1 ),
$ ACOEFA*WORK( J+1 )+BCOEFA*WORK( N+J+1 ) )
IF( TEMP.GT.BIGNUM*XSCALE ) THEN
DO 90 JW = 0, NW - 1
DO 80 JR = JE, J - 1
WORK( ( JW+2 )*N+JR ) = XSCALE*
$ WORK( ( JW+2 )*N+JR )
80 CONTINUE
90 CONTINUE
XMAX = XMAX*XSCALE
END IF
*
* Compute dot products
*
* j-1
* SUM = sum conjg( a*S(k,j) - b*P(k,j) )*x(k)
* k=je
*
* To reduce the op count, this is done as
*
* _ j-1 _ j-1
* a*conjg( sum S(k,j)*x(k) ) - b*conjg( sum P(k,j)*x(k) )
* k=je k=je
*
* which may cause underflow problems if A or B are close
* to underflow. (E.g., less than SMALL.)
*
*
DO 120 JW = 1, NW
DO 110 JA = 1, NA
SUMS( JA, JW ) = ZERO
SUMP( JA, JW ) = ZERO
*
DO 100 JR = JE, J - 1
SUMS( JA, JW ) = SUMS( JA, JW ) +
$ S( JR, J+JA-1 )*
$ WORK( ( JW+1 )*N+JR )
SUMP( JA, JW ) = SUMP( JA, JW ) +
$ P( JR, J+JA-1 )*
$ WORK( ( JW+1 )*N+JR )
100 CONTINUE
110 CONTINUE
120 CONTINUE
*
DO 130 JA = 1, NA
IF( ILCPLX ) THEN
SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
$ BCOEFR*SUMP( JA, 1 ) -
$ BCOEFI*SUMP( JA, 2 )
SUM( JA, 2 ) = -ACOEF*SUMS( JA, 2 ) +
$ BCOEFR*SUMP( JA, 2 ) +
$ BCOEFI*SUMP( JA, 1 )
ELSE
SUM( JA, 1 ) = -ACOEF*SUMS( JA, 1 ) +
$ BCOEFR*SUMP( JA, 1 )
END IF
130 CONTINUE
*
* T
* Solve ( a A - b B ) y = SUM(,)
* with scaling and perturbation of the denominator
*
CALL SLALN2( .TRUE., NA, NW, DMIN, ACOEF, S( J, J ), LDS,
$ BDIAG( 1 ), BDIAG( 2 ), SUM, 2, BCOEFR,
$ BCOEFI, WORK( 2*N+J ), N, SCALE, TEMP,
$ IINFO )
IF( SCALE.LT.ONE ) THEN
DO 150 JW = 0, NW - 1
DO 140 JR = JE, J - 1
WORK( ( JW+2 )*N+JR ) = SCALE*
$ WORK( ( JW+2 )*N+JR )
140 CONTINUE
150 CONTINUE
XMAX = SCALE*XMAX
END IF
XMAX = MAX( XMAX, TEMP )
160 CONTINUE
*
* Copy eigenvector to VL, back transforming if
* HOWMNY='B'.
*
IEIG = IEIG + 1
IF( ILBACK ) THEN
DO 170 JW = 0, NW - 1
CALL SGEMV( 'N', N, N+1-JE, ONE, VL( 1, JE ), LDVL,
$ WORK( ( JW+2 )*N+JE ), 1, ZERO,
$ WORK( ( JW+4 )*N+1 ), 1 )
170 CONTINUE
CALL SLACPY( ' ', N, NW, WORK( 4*N+1 ), N, VL( 1, JE ),
$ LDVL )
IBEG = 1
ELSE
CALL SLACPY( ' ', N, NW, WORK( 2*N+1 ), N, VL( 1, IEIG ),
$ LDVL )
IBEG = JE
END IF
*
* Scale eigenvector
*
XMAX = ZERO
IF( ILCPLX ) THEN
DO 180 J = IBEG, N
XMAX = MAX( XMAX, ABS( VL( J, IEIG ) )+
$ ABS( VL( J, IEIG+1 ) ) )
180 CONTINUE
ELSE
DO 190 J = IBEG, N
XMAX = MAX( XMAX, ABS( VL( J, IEIG ) ) )
190 CONTINUE
END IF
*
IF( XMAX.GT.SAFMIN ) THEN
XSCALE = ONE / XMAX
*
DO 210 JW = 0, NW - 1
DO 200 JR = IBEG, N
VL( JR, IEIG+JW ) = XSCALE*VL( JR, IEIG+JW )
200 CONTINUE
210 CONTINUE
END IF
IEIG = IEIG + NW - 1
*
220 CONTINUE
END IF
*
* Right eigenvectors
*
IF( COMPR ) THEN
IEIG = IM + 1
*
* Main loop over eigenvalues
*
ILCPLX = .FALSE.
DO 500 JE = N, 1, -1
*
* Skip this iteration if (a) HOWMNY='S' and SELECT=.FALSE., or
* (b) this would be the second of a complex pair.
* Check for complex eigenvalue, so as to be sure of which
* entry(-ies) of SELECT to look at -- if complex, SELECT(JE)
* or SELECT(JE-1).
* If this is a complex pair, the 2-by-2 diagonal block
* corresponding to the eigenvalue is in rows/columns JE-1:JE
*
IF( ILCPLX ) THEN
ILCPLX = .FALSE.
GO TO 500
END IF
NW = 1
IF( JE.GT.1 ) THEN
IF( S( JE, JE-1 ).NE.ZERO ) THEN
ILCPLX = .TRUE.
NW = 2
END IF
END IF
IF( ILALL ) THEN
ILCOMP = .TRUE.
ELSE IF( ILCPLX ) THEN
ILCOMP = SELECT( JE ) .OR. SELECT( JE-1 )
ELSE
ILCOMP = SELECT( JE )
END IF
IF( .NOT.ILCOMP )
$ GO TO 500
*
* Decide if (a) singular pencil, (b) real eigenvalue, or
* (c) complex eigenvalue.
*
IF( .NOT.ILCPLX ) THEN
IF( ABS( S( JE, JE ) ).LE.SAFMIN .AND.
$ ABS( P( JE, JE ) ).LE.SAFMIN ) THEN
*
* Singular matrix pencil -- unit eigenvector
*
IEIG = IEIG - 1
DO 230 JR = 1, N
VR( JR, IEIG ) = ZERO
230 CONTINUE
VR( IEIG, IEIG ) = ONE
GO TO 500
END IF
END IF
*
* Clear vector
*
DO 250 JW = 0, NW - 1
DO 240 JR = 1, N
WORK( ( JW+2 )*N+JR ) = ZERO
240 CONTINUE
250 CONTINUE
*
* Compute coefficients in ( a A - b B ) x = 0
* a is ACOEF
* b is BCOEFR + i*BCOEFI
*
IF( .NOT.ILCPLX ) THEN
*
* Real eigenvalue
*
TEMP = ONE / MAX( ABS( S( JE, JE ) )*ASCALE,
$ ABS( P( JE, JE ) )*BSCALE, SAFMIN )
SALFAR = ( TEMP*S( JE, JE ) )*ASCALE
SBETA = ( TEMP*P( JE, JE ) )*BSCALE
ACOEF = SBETA*ASCALE
BCOEFR = SALFAR*BSCALE
BCOEFI = ZERO
*
* Scale to avoid underflow
*
SCALE = ONE
LSA = ABS( SBETA ).GE.SAFMIN .AND. ABS( ACOEF ).LT.SMALL
LSB = ABS( SALFAR ).GE.SAFMIN .AND. ABS( BCOEFR ).LT.
$ SMALL
IF( LSA )
$ SCALE = ( SMALL / ABS( SBETA ) )*MIN( ANORM, BIG )
IF( LSB )
$ SCALE = MAX( SCALE, ( SMALL / ABS( SALFAR ) )*
$ MIN( BNORM, BIG ) )
IF( LSA .OR. LSB ) THEN
SCALE = MIN( SCALE, ONE /
$ ( SAFMIN*MAX( ONE, ABS( ACOEF ),
$ ABS( BCOEFR ) ) ) )
IF( LSA ) THEN
ACOEF = ASCALE*( SCALE*SBETA )
ELSE
ACOEF = SCALE*ACOEF
END IF
IF( LSB ) THEN
BCOEFR = BSCALE*( SCALE*SALFAR )
ELSE
BCOEFR = SCALE*BCOEFR
END IF
END IF
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR )
*
* First component is 1
*
WORK( 2*N+JE ) = ONE
XMAX = ONE
*
* Compute contribution from column JE of A and B to sum
* (See "Further Details", above.)
*
DO 260 JR = 1, JE - 1
WORK( 2*N+JR ) = BCOEFR*P( JR, JE ) -
$ ACOEF*S( JR, JE )
260 CONTINUE
ELSE
*
* Complex eigenvalue
*
CALL SLAG2( S( JE-1, JE-1 ), LDS, P( JE-1, JE-1 ), LDP,
$ SAFMIN*SAFETY, ACOEF, TEMP, BCOEFR, TEMP2,
$ BCOEFI )
IF( BCOEFI.EQ.ZERO ) THEN
INFO = JE - 1
RETURN
END IF
*
* Scale to avoid over/underflow
*
ACOEFA = ABS( ACOEF )
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
SCALE = ONE
IF( ACOEFA*ULP.LT.SAFMIN .AND. ACOEFA.GE.SAFMIN )
$ SCALE = ( SAFMIN / ULP ) / ACOEFA
IF( BCOEFA*ULP.LT.SAFMIN .AND. BCOEFA.GE.SAFMIN )
$ SCALE = MAX( SCALE, ( SAFMIN / ULP ) / BCOEFA )
IF( SAFMIN*ACOEFA.GT.ASCALE )
$ SCALE = ASCALE / ( SAFMIN*ACOEFA )
IF( SAFMIN*BCOEFA.GT.BSCALE )
$ SCALE = MIN( SCALE, BSCALE / ( SAFMIN*BCOEFA ) )
IF( SCALE.NE.ONE ) THEN
ACOEF = SCALE*ACOEF
ACOEFA = ABS( ACOEF )
BCOEFR = SCALE*BCOEFR
BCOEFI = SCALE*BCOEFI
BCOEFA = ABS( BCOEFR ) + ABS( BCOEFI )
END IF
*
* Compute first two components of eigenvector
* and contribution to sums
*
TEMP = ACOEF*S( JE, JE-1 )
TEMP2R = ACOEF*S( JE, JE ) - BCOEFR*P( JE, JE )
TEMP2I = -BCOEFI*P( JE, JE )
IF( ABS( TEMP ).GE.ABS( TEMP2R )+ABS( TEMP2I ) ) THEN
WORK( 2*N+JE ) = ONE
WORK( 3*N+JE ) = ZERO
WORK( 2*N+JE-1 ) = -TEMP2R / TEMP
WORK( 3*N+JE-1 ) = -TEMP2I / TEMP
ELSE
WORK( 2*N+JE-1 ) = ONE
WORK( 3*N+JE-1 ) = ZERO
TEMP = ACOEF*S( JE-1, JE )
WORK( 2*N+JE ) = ( BCOEFR*P( JE-1, JE-1 )-ACOEF*
$ S( JE-1, JE-1 ) ) / TEMP
WORK( 3*N+JE ) = BCOEFI*P( JE-1, JE-1 ) / TEMP
END IF
*
XMAX = MAX( ABS( WORK( 2*N+JE ) )+ABS( WORK( 3*N+JE ) ),
$ ABS( WORK( 2*N+JE-1 ) )+ABS( WORK( 3*N+JE-1 ) ) )
*
* Compute contribution from columns JE and JE-1
* of A and B to the sums.
*
CREALA = ACOEF*WORK( 2*N+JE-1 )
CIMAGA = ACOEF*WORK( 3*N+JE-1 )
CREALB = BCOEFR*WORK( 2*N+JE-1 ) -
$ BCOEFI*WORK( 3*N+JE-1 )
CIMAGB = BCOEFI*WORK( 2*N+JE-1 ) +
$ BCOEFR*WORK( 3*N+JE-1 )
CRE2A = ACOEF*WORK( 2*N+JE )
CIM2A = ACOEF*WORK( 3*N+JE )
CRE2B = BCOEFR*WORK( 2*N+JE ) - BCOEFI*WORK( 3*N+JE )
CIM2B = BCOEFI*WORK( 2*N+JE ) + BCOEFR*WORK( 3*N+JE )
DO 270 JR = 1, JE - 2
WORK( 2*N+JR ) = -CREALA*S( JR, JE-1 ) +
$ CREALB*P( JR, JE-1 ) -
$ CRE2A*S( JR, JE ) + CRE2B*P( JR, JE )
WORK( 3*N+JR ) = -CIMAGA*S( JR, JE-1 ) +
$ CIMAGB*P( JR, JE-1 ) -
$ CIM2A*S( JR, JE ) + CIM2B*P( JR, JE )
270 CONTINUE
END IF
*
DMIN = MAX( ULP*ACOEFA*ANORM, ULP*BCOEFA*BNORM, SAFMIN )
*
* Columnwise triangular solve of (a A - b B) x = 0
*
IL2BY2 = .FALSE.
DO 370 J = JE - NW, 1, -1
*
* If a 2-by-2 block, is in position j-1:j, wait until
* next iteration to process it (when it will be j:j+1)
*
IF( .NOT.IL2BY2 .AND. J.GT.1 ) THEN
IF( S( J, J-1 ).NE.ZERO ) THEN
IL2BY2 = .TRUE.
GO TO 370
END IF
END IF
BDIAG( 1 ) = P( J, J )
IF( IL2BY2 ) THEN
NA = 2
BDIAG( 2 ) = P( J+1, J+1 )
ELSE
NA = 1
END IF
*
* Compute x(j) (and x(j+1), if 2-by-2 block)
*
CALL SLALN2( .FALSE., NA, NW, DMIN, ACOEF, S( J, J ),
$ LDS, BDIAG( 1 ), BDIAG( 2 ), WORK( 2*N+J ),
$ N, BCOEFR, BCOEFI, SUM, 2, SCALE, TEMP,
$ IINFO )
IF( SCALE.LT.ONE ) THEN
*
DO 290 JW = 0, NW - 1
DO 280 JR = 1, JE
WORK( ( JW+2 )*N+JR ) = SCALE*
$ WORK( ( JW+2 )*N+JR )
280 CONTINUE
290 CONTINUE
END IF
XMAX = MAX( SCALE*XMAX, TEMP )
*
DO 310 JW = 1, NW
DO 300 JA = 1, NA
WORK( ( JW+1 )*N+J+JA-1 ) = SUM( JA, JW )
300 CONTINUE
310 CONTINUE
*
* w = w + x(j)*(a S(*,j) - b P(*,j) ) with scaling
*
IF( J.GT.1 ) THEN
*
* Check whether scaling is necessary for sum.
*
XSCALE = ONE / MAX( ONE, XMAX )
TEMP = ACOEFA*WORK( J ) + BCOEFA*WORK( N+J )
IF( IL2BY2 )
$ TEMP = MAX( TEMP, ACOEFA*WORK( J+1 )+BCOEFA*
$ WORK( N+J+1 ) )
TEMP = MAX( TEMP, ACOEFA, BCOEFA )
IF( TEMP.GT.BIGNUM*XSCALE ) THEN
*
DO 330 JW = 0, NW - 1
DO 320 JR = 1, JE
WORK( ( JW+2 )*N+JR ) = XSCALE*
$ WORK( ( JW+2 )*N+JR )
320 CONTINUE
330 CONTINUE
XMAX = XMAX*XSCALE
END IF
*
* Compute the contributions of the off-diagonals of
* column j (and j+1, if 2-by-2 block) of A and B to the
* sums.
*
*
DO 360 JA = 1, NA
IF( ILCPLX ) THEN
CREALA = ACOEF*WORK( 2*N+J+JA-1 )
CIMAGA = ACOEF*WORK( 3*N+J+JA-1 )
CREALB = BCOEFR*WORK( 2*N+J+JA-1 ) -
$ BCOEFI*WORK( 3*N+J+JA-1 )
CIMAGB = BCOEFI*WORK( 2*N+J+JA-1 ) +
$ BCOEFR*WORK( 3*N+J+JA-1 )
DO 340 JR = 1, J - 1
WORK( 2*N+JR ) = WORK( 2*N+JR ) -
$ CREALA*S( JR, J+JA-1 ) +
$ CREALB*P( JR, J+JA-1 )
WORK( 3*N+JR ) = WORK( 3*N+JR ) -
$ CIMAGA*S( JR, J+JA-1 ) +
$ CIMAGB*P( JR, J+JA-1 )
340 CONTINUE
ELSE
CREALA = ACOEF*WORK( 2*N+J+JA-1 )
CREALB = BCOEFR*WORK( 2*N+J+JA-1 )
DO 350 JR = 1, J - 1
WORK( 2*N+JR ) = WORK( 2*N+JR ) -
$ CREALA*S( JR, J+JA-1 ) +
$ CREALB*P( JR, J+JA-1 )
350 CONTINUE
END IF
360 CONTINUE
END IF
*
IL2BY2 = .FALSE.
370 CONTINUE
*
* Copy eigenvector to VR, back transforming if
* HOWMNY='B'.
*
IEIG = IEIG - NW
IF( ILBACK ) THEN
*
DO 410 JW = 0, NW - 1
DO 380 JR = 1, N
WORK( ( JW+4 )*N+JR ) = WORK( ( JW+2 )*N+1 )*
$ VR( JR, 1 )
380 CONTINUE
*
* A series of compiler directives to defeat
* vectorization for the next loop
*
*
DO 400 JC = 2, JE
DO 390 JR = 1, N
WORK( ( JW+4 )*N+JR ) = WORK( ( JW+4 )*N+JR ) +
$ WORK( ( JW+2 )*N+JC )*VR( JR, JC )
390 CONTINUE
400 CONTINUE
410 CONTINUE
*
DO 430 JW = 0, NW - 1
DO 420 JR = 1, N
VR( JR, IEIG+JW ) = WORK( ( JW+4 )*N+JR )
420 CONTINUE
430 CONTINUE
*
IEND = N
ELSE
DO 450 JW = 0, NW - 1
DO 440 JR = 1, N
VR( JR, IEIG+JW ) = WORK( ( JW+2 )*N+JR )
440 CONTINUE
450 CONTINUE
*
IEND = JE
END IF
*
* Scale eigenvector
*
XMAX = ZERO
IF( ILCPLX ) THEN
DO 460 J = 1, IEND
XMAX = MAX( XMAX, ABS( VR( J, IEIG ) )+
$ ABS( VR( J, IEIG+1 ) ) )
460 CONTINUE
ELSE
DO 470 J = 1, IEND
XMAX = MAX( XMAX, ABS( VR( J, IEIG ) ) )
470 CONTINUE
END IF
*
IF( XMAX.GT.SAFMIN ) THEN
XSCALE = ONE / XMAX
DO 490 JW = 0, NW - 1
DO 480 JR = 1, IEND
VR( JR, IEIG+JW ) = XSCALE*VR( JR, IEIG+JW )
480 CONTINUE
490 CONTINUE
END IF
500 CONTINUE
END IF
*
RETURN
*
* End of STGEVC
*
END