strevc.f

来自「famous linear algebra library (LAPACK) p」· F 代码 · 共 982 行 · 第 1/3 页

      SUBROUTINE STREVC( 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( * )
      REAL               T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ),
     $                   WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  STREVC 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 SHSEQR.
*  
*  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) REAL 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) 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 Schur vectors returned by SHSEQR).
*          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) REAL 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 SHSEQR).
*          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) REAL 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 ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLV, BOTHV, LEFTV, OVER, PAIR, RIGHTV, SOMEV
      INTEGER            I, IERR, II, IP, IS, J, J1, J2, JNXT, K, KI, N2
      REAL               BETA, BIGNUM, EMAX, OVFL, REC, REMAX, SCALE,
     $                   SMIN, SMLNUM, ULP, UNFL, VCRIT, VMAX, WI, WR,
     $                   XNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SDOT, SLAMCH
      EXTERNAL           LSAME, ISAMAX, SDOT, SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           SAXPY, SCOPY, SGEMV, SLABAD, SLALN2, SSCAL,
     $                   XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Local Arrays ..
      REAL               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( 'STREVC', -INFO )
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( N.EQ.0 )
     $   RETURN
*
*     Set the constants to control overflow.
*
      UNFL = SLAMCH( 'Safe minimum' )
      OVFL = ONE / UNFL
      CALL SLABAD( UNFL, OVFL )
      ULP = SLAMCH( '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