sggevx.f

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      SUBROUTINE SGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
     $                   ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
     $                   IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
     $                   RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
*  -- LAPACK driver routine (version 3.1) --
*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
*     November 2006
*
*     .. Scalar Arguments ..
      CHARACTER          BALANC, JOBVL, JOBVR, SENSE
      INTEGER            IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
      REAL               ABNRM, BBNRM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      REAL               A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), LSCALE( * ),
     $                   RCONDE( * ), RCONDV( * ), RSCALE( * ),
     $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*     ..
*
*  Purpose
*  =======
*
*  SGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*  the generalized eigenvalues, and optionally, the left and/or right
*  generalized eigenvectors.
*
*  Optionally also, it computes a balancing transformation to improve
*  the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*  LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
*  the eigenvalues (RCONDE), and reciprocal condition numbers for the
*  right eigenvectors (RCONDV).
*
*  A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*  lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*  singular. It is usually represented as the pair (alpha,beta), as
*  there is a reasonable interpretation for beta=0, and even for both
*  being zero.
*
*  The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
*  of (A,B) satisfies
*
*                   A * v(j) = lambda(j) * B * v(j) .
*
*  The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
*  of (A,B) satisfies
*
*                   u(j)**H * A  = lambda(j) * u(j)**H * B.
*
*  where u(j)**H is the conjugate-transpose of u(j).
*
*
*  Arguments
*  =========
*
*  BALANC  (input) CHARACTER*1
*          Specifies the balance option to be performed.
*          = 'N':  do not diagonally scale or permute;
*          = 'P':  permute only;
*          = 'S':  scale only;
*          = 'B':  both permute and scale.
*          Computed reciprocal condition numbers will be for the
*          matrices after permuting and/or balancing. Permuting does
*          not change condition numbers (in exact arithmetic), but
*          balancing does.
*
*  JOBVL   (input) CHARACTER*1
*          = 'N':  do not compute the left generalized eigenvectors;
*          = 'V':  compute the left generalized eigenvectors.
*
*  JOBVR   (input) CHARACTER*1
*          = 'N':  do not compute the right generalized eigenvectors;
*          = 'V':  compute the right generalized eigenvectors.
*
*  SENSE   (input) CHARACTER*1
*          Determines which reciprocal condition numbers are computed.
*          = 'N': none are computed;
*          = 'E': computed for eigenvalues only;
*          = 'V': computed for eigenvectors only;
*          = 'B': computed for eigenvalues and eigenvectors.
*
*  N       (input) INTEGER
*          The order of the matrices A, B, VL, and VR.  N >= 0.
*
*  A       (input/output) REAL array, dimension (LDA, N)
*          On entry, the matrix A in the pair (A,B).
*          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
*          or both, then A contains the first part of the real Schur
*          form of the "balanced" versions of the input A and B.
*
*  LDA     (input) INTEGER
*          The leading dimension of A.  LDA >= max(1,N).
*
*  B       (input/output) REAL array, dimension (LDB, N)
*          On entry, the matrix B in the pair (A,B).
*          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
*          or both, then B contains the second part of the real Schur
*          form of the "balanced" versions of the input A and B.
*
*  LDB     (input) INTEGER
*          The leading dimension of B.  LDB >= max(1,N).
*
*  ALPHAR  (output) REAL array, dimension (N)
*  ALPHAI  (output) REAL array, dimension (N)
*  BETA    (output) REAL array, dimension (N)
*          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*          be the generalized eigenvalues.  If ALPHAI(j) is zero, then
*          the j-th eigenvalue is real; if positive, then the j-th and
*          (j+1)-st eigenvalues are a complex conjugate pair, with
*          ALPHAI(j+1) negative.
*
*          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*          may easily over- or underflow, and BETA(j) may even be zero.
*          Thus, the user should avoid naively computing the ratio
*          ALPHA/BETA. However, ALPHAR and ALPHAI will be always less
*          than and usually comparable with norm(A) in magnitude, and
*          BETA always less than and usually comparable with norm(B).
*
*  VL      (output) REAL array, dimension (LDVL,N)
*          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*          after another in the columns of VL, in the same order as
*          their eigenvalues. If the j-th eigenvalue is real, then
*          u(j) = VL(:,j), the j-th column of VL. If the j-th and
*          (j+1)-th eigenvalues form a complex conjugate pair, then
*          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
*          Each eigenvector will be scaled so the largest component have
*          abs(real part) + abs(imag. part) = 1.
*          Not referenced if JOBVL = 'N'.
*
*  LDVL    (input) INTEGER
*          The leading dimension of the matrix VL. LDVL >= 1, and
*          if JOBVL = 'V', LDVL >= N.
*
*  VR      (output) REAL array, dimension (LDVR,N)
*          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*          after another in the columns of VR, in the same order as
*          their eigenvalues. If the j-th eigenvalue is real, then
*          v(j) = VR(:,j), the j-th column of VR. If the j-th and
*          (j+1)-th eigenvalues form a complex conjugate pair, then
*          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*          Each eigenvector will be scaled so the largest component have
*          abs(real part) + abs(imag. part) = 1.
*          Not referenced if JOBVR = 'N'.
*
*  LDVR    (input) INTEGER
*          The leading dimension of the matrix VR. LDVR >= 1, and
*          if JOBVR = 'V', LDVR >= N.
*
*  ILO     (output) INTEGER
*  IHI     (output) INTEGER
*          ILO and IHI are integer values such that on exit
*          A(i,j) = 0 and B(i,j) = 0 if i > j and
*          j = 1,...,ILO-1 or i = IHI+1,...,N.
*          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
*
*  LSCALE  (output) REAL array, dimension (N)
*          Details of the permutations and scaling factors applied
*          to the left side of A and B.  If PL(j) is the index of the
*          row interchanged with row j, and DL(j) is the scaling
*          factor applied to row j, then
*            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
*                      = DL(j)  for j = ILO,...,IHI
*                      = PL(j)  for j = IHI+1,...,N.
*          The order in which the interchanges are made is N to IHI+1,
*          then 1 to ILO-1.
*
*  RSCALE  (output) REAL array, dimension (N)
*          Details of the permutations and scaling factors applied
*          to the right side of A and B.  If PR(j) is the index of the
*          column interchanged with column j, and DR(j) is the scaling
*          factor applied to column j, then
*            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
*                      = DR(j)  for j = ILO,...,IHI
*                      = PR(j)  for j = IHI+1,...,N
*          The order in which the interchanges are made is N to IHI+1,
*          then 1 to ILO-1.
*
*  ABNRM   (output) REAL
*          The one-norm of the balanced matrix A.
*
*  BBNRM   (output) REAL
*          The one-norm of the balanced matrix B.
*
*  RCONDE  (output) REAL array, dimension (N)
*          If SENSE = 'E' or 'B', the reciprocal condition numbers of
*          the eigenvalues, stored in consecutive elements of the array.
*          For a complex conjugate pair of eigenvalues two consecutive
*          elements of RCONDE are set to the same value. Thus RCONDE(j),
*          RCONDV(j), and the j-th columns of VL and VR all correspond
*          to the j-th eigenpair.
*          If SENSE = 'N' or 'V', RCONDE is not referenced.
*
*  RCONDV  (output) REAL array, dimension (N)
*          If SENSE = 'V' or 'B', the estimated reciprocal condition
*          numbers of the eigenvectors, stored in consecutive elements
*          of the array. For a complex eigenvector two consecutive
*          elements of RCONDV are set to the same value. If the
*          eigenvalues cannot be reordered to compute RCONDV(j),
*          RCONDV(j) is set to 0; this can only occur when the true
*          value would be very small anyway.
*          If SENSE = 'N' or 'E', RCONDV is not referenced.
*
*  WORK    (workspace/output) REAL array, dimension (MAX(1,LWORK))
*          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*
*  LWORK   (input) INTEGER
*          The dimension of the array WORK. LWORK >= max(1,2*N).
*          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
*          LWORK >= max(1,6*N).
*          If SENSE = 'E', LWORK >= max(1,10*N).
*          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
*
*          If LWORK = -1, then a workspace query is assumed; the routine
*          only calculates the optimal size of the WORK array, returns
*          this value as the first entry of the WORK array, and no error
*          message related to LWORK is issued by XERBLA.
*
*  IWORK   (workspace) INTEGER array, dimension (N+6)
*          If SENSE = 'E', IWORK is not referenced.
*
*  BWORK   (workspace) LOGICAL array, dimension (N)
*          If SENSE = 'N', BWORK is not referenced.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value.
*          = 1,...,N:
*                The QZ iteration failed.  No eigenvectors have been
*                calculated, but ALPHAR(j), ALPHAI(j), and BETA(j)
*                should be correct for j=INFO+1,...,N.
*          > N:  =N+1: other than QZ iteration failed in SHGEQZ.
*                =N+2: error return from STGEVC.
*
*  Further Details
*  ===============
*
*  Balancing a matrix pair (A,B) includes, first, permuting rows and
*  columns to isolate eigenvalues, second, applying diagonal similarity
*  transformation to the rows and columns to make the rows and columns
*  as close in norm as possible. The computed reciprocal condition
*  numbers correspond to the balanced matrix. Permuting rows and columns
*  will not change the condition numbers (in exact arithmetic) but
*  diagonal scaling will.  For further explanation of balancing, see
*  section 4.11.1.2 of LAPACK Users' Guide.
*
*  An approximate error bound on the chordal distance between the i-th
*  computed generalized eigenvalue w and the corresponding exact
*  eigenvalue lambda is
*
*       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
*
*  An approximate error bound for the angle between the i-th computed
*  eigenvector VL(i) or VR(i) is given by
*
*       EPS * norm(ABNRM, BBNRM) / DIF(i).
*
*  For further explanation of the reciprocal condition numbers RCONDE
*  and RCONDV, see section 4.11 of LAPACK User's Guide.
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
     $                   PAIR, WANTSB, WANTSE, WANTSN, WANTSV
      CHARACTER          CHTEMP
      INTEGER            I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
     $                   ITAU, IWRK, IWRK1, J, JC, JR, M, MAXWRK,
     $                   MINWRK, MM
      REAL               ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
     $                   SMLNUM, TEMP
*     ..
*     .. Local Arrays ..
      LOGICAL            LDUMMA( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           SGEQRF, SGGBAK, SGGBAL, SGGHRD, SHGEQZ, SLABAD,
     $                   SLACPY, SLASCL, SLASET, SORGQR, SORMQR, STGEVC,
     $                   STGSNA, XERBLA
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      REAL               SLAMCH, SLANGE
      EXTERNAL           LSAME, ILAENV, SLAMCH, SLANGE
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          ABS, MAX, SQRT
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( LSAME( JOBVL, 'N' ) ) THEN
         IJOBVL = 1
         ILVL = .FALSE.
      ELSE IF( LSAME( JOBVL, 'V' ) ) THEN
         IJOBVL = 2
         ILVL = .TRUE.
      ELSE
         IJOBVL = -1
         ILVL = .FALSE.
      END IF
*
      IF( LSAME( JOBVR, 'N' ) ) THEN
         IJOBVR = 1
         ILVR = .FALSE.
      ELSE IF( LSAME( JOBVR, 'V' ) ) THEN
         IJOBVR = 2
         ILVR = .TRUE.
      ELSE
         IJOBVR = -1
         ILVR = .FALSE.
      END IF
      ILV = ILVL .OR. ILVR
*
      NOSCL  = LSAME( BALANC, 'N' ) .OR. LSAME( BALANC, 'P' )
      WANTSN = LSAME( SENSE, 'N' )
      WANTSE = LSAME( SENSE, 'E' )
      WANTSV = LSAME( SENSE, 'V' )
      WANTSB = LSAME( SENSE, 'B' )
*
*     Test the input arguments
*
      INFO = 0
      LQUERY = ( LWORK.EQ.-1 )
      IF( .NOT.( NOSCL .OR. LSAME( BALANC, 'S' ) .OR.
     $    LSAME( BALANC, 'B' ) ) ) THEN
         INFO = -1
      ELSE IF( IJOBVL.LE.0 ) THEN
         INFO = -2
      ELSE IF( IJOBVR.LE.0 ) THEN
         INFO = -3
      ELSE IF( .NOT.( WANTSN .OR. WANTSE .OR. WANTSB .OR. WANTSV ) )
     $          THEN
         INFO = -4
      ELSE IF( N.LT.0 ) THEN
         INFO = -5
      ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
         INFO = -7
      ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
         INFO = -9
      ELSE IF( LDVL.LT.1 .OR. ( ILVL .AND. LDVL.LT.N ) ) THEN
         INFO = -14
      ELSE IF( LDVR.LT.1 .OR. ( ILVR .AND. LDVR.LT.N ) ) THEN
         INFO = -16
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately