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cgeevx (3)
  • >> cgeevx (3) ( Solaris man: Библиотечные вызовы )
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    NAME
         cgeevx - compute for an N-by-N complex  nonsymmetric  matrix
         A,  the  eigenvalues  and, optionally, the left and/or right
         eigenvectors
    
    SYNOPSIS
         SUBROUTINE CGEEVX( BALANC, JOBVL, JOBVR, SENSE, N,  A,  LDA,
                   W,  VL,  LDVL,  VR,  LDVR, ILO, IHI, SCALE, ABNRM,
                   RCONDE, RCONDV, WORK, LWORK, RWORK, INFO )
    
         CHARACTER BALANC, JOBVL, JOBVR, SENSE
    
         INTEGER IHI, ILO, INFO, LDA, LDVL, LDVR, LWORK, N
    
         REAL ABNRM
    
         REAL RCONDE( * ), RCONDV( * ), RWORK( * ), SCALE( * )
    
         COMPLEX A( LDA, * ), VL( LDVL, * ), VR( LDVR, * ), W(  *  ),
                   WORK( * )
    
    
    
         #include <sunperf.h>
    
         void cgeevx(char balanc, char jobvl, char jobvr, char sense,
                   int  n,  complex *ca, int lda, complex *w, complex
                   *vl, int ldvl, complex *vr, int  ldvr,  int  *ilo,
                   int  *ihi,  float  *sscale,  float  *abnrm,  float
                   *rconde, float *rcondv, int *info);
    
    PURPOSE
         CGEEVX computes for an N-by-N complex nonsymmetric matrix A,
         the  eigenvalues  and,  optionally,  the  left  and/or right
         eigenvectors.
    
         Optionally also, it computes a balancing  transformation  to
         improve the conditioning of the eigenvalues and eigenvectors
         (ILO, IHI, SCALE, and ABNRM), reciprocal  condition  numbers
         for  the  eigenvalues  (RCONDE),  and  reciprocal  condition
         numbers for the right
         eigenvectors (RCONDV).
    
         The right eigenvector v(j) of A satisfies
                          A * v(j) = lambda(j) * v(j)
         where lambda(j) is its eigenvalue.
         The left eigenvector u(j) of A satisfies
                       u(j)**H * A = lambda(j) * u(j)**H
         where u(j)**H denotes the conjugate transpose of u(j).
    
         The computed eigenvectors are normalized to  have  Euclidean
         norm equal to 1 and largest component real.
         Balancing a matrix means permuting the rows and  columns  to
         make  it more nearly upper triangular, and applying a diago-
         nal similarity transformation D * A * D**(-1), where D is  a
         diagonal matrix, to make its rows and columns closer in norm
         and the condition numbers of its eigenvalues  and  eigenvec-
         tors  smaller.   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 explana-
         tion  of  balancing, see section 4.10.2 of the LAPACK Users'
         Guide.
    
    
    ARGUMENTS
         BALANC    (input) CHARACTER*1
                   Indicates how the input matrix  should  be  diago-
                   nally scaled and/or permuted to improve the condi-
                   tioning of its eigenvalues.  = 'N': Do not  diago-
                   nally scale or permute;
                   = 'P': Perform permutations  to  make  the  matrix
                   more  nearly  upper  triangular. Do not diagonally
                   scale; = 'S': Diagonally  scale  the  matrix,  ie.
                   replace  A  by  D*A*D**(-1), where D is a diagonal
                   matrix chosen to make the rows and  columns  of  A
                   more  equal  in  norm. Do not permute; = 'B': Both
                   diagonally scale and permute A.
    
                   Computed reciprocal condition numbers will be  for
                   the  matrix after balancing and/or permuting. Per-
                   muting does not change condition numbers (in exact
                   arithmetic), but balancing does.
    
         JOBVL     (input) CHARACTER*1
                   = 'N': left eigenvectors of A are not computed;
                   = 'V': left eigenvectors of A  are  computed.   If
                   SENSE = 'E' or 'B', JOBVL must = 'V'.
    
         JOBVR     (input) CHARACTER*1
                   = 'N': right eigenvectors of A are not computed;
                   = 'V': right eigenvectors of A are  computed.   If
                   SENSE = 'E' or 'B', JOBVR must = 'V'.
    
         SENSE     (input) CHARACTER*1
                   Determines which reciprocal condition numbers  are
                   computed.  = 'N': None are computed;
                   = 'E': Computed for eigenvalues only;
                   = 'V': Computed for right eigenvectors only;
                   = 'B': Computed for eigenvalues and  right  eigen-
                   vectors.
    
                   If SENSE = 'E' or 'B', both left and right  eigen-
                   vectors  must  also  be  computed (JOBVL = 'V' and
                   JOBVR = 'V').
    
         N         (input) INTEGER
                   The order of the matrix A. N >= 0.
    
         A         (input/output) COMPLEX array, dimension (LDA,N)
                   On entry, the N-by-N matrix A.   On  exit,  A  has
                   been  overwritten.  If JOBVL = 'V' or JOBVR = 'V',
                   A contains the Schur form of the balanced  version
                   of the matrix A.
    
         LDA       (input) INTEGER
                   The leading dimension of  the  array  A.   LDA  >=
                   max(1,N).
    
         W         (output) COMPLEX array, dimension (N)
                   W contains the computed eigenvalues.
    
         VL        (output) COMPLEX 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  JOBVL  =
                   'N',  VL  is  not referenced.  u(j) = VL(:,j), the
                   j-th column of VL.
    
         LDVL      (input) INTEGER
                   The leading dimension of the array VL.  LDVL >= 1;
                   if JOBVL = 'V', LDVL >= N.
    
         VR        (output) COMPLEX 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  JOBVR  =
                   'N',  VR  is  not referenced.  v(j) = VR(:,j), the
                   j-th column of VR.
    
         LDVR      (input) INTEGER
                   The leading dimension of the array VR.  LDVR >= 1;
                   if JOBVR = 'V', LDVR >= N.
    
                   ILO,IHI (output) INTEGER ILO and IHI  are  integer
                   values  determined  when A was balanced.  The bal-
                   anced A(i,j) = 0 if I > J and J = 1,...,ILO-1 or I
                   = IHI+1,...,N.
    
         SCALE     (output) REAL array, dimension (N)
                   Details of the permutations  and  scaling  factors
                   applied when balancing A.  If P(j) is the index of
                   the row  and  column  interchanged  with  row  and
                   column  j,  and D(j) is the scaling factor applied
                   to row and column j, then SCALE(J) = P(J),     for
                   J  =  1,...,ILO-1 = D(J),    for J = ILO,...,IHI =
                   P(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  (the  maximum
                   of  the  sum of absolute values of elements of any
                   column).
    
         RCONDE    (output) REAL array, dimension (N)
                   RCONDE(j) is the reciprocal  condition  number  of
                   the j-th eigenvalue.
    
         RCONDV    (output) REAL array, dimension (N)
                   RCONDV(j) is the reciprocal  condition  number  of
                   the j-th right eigenvector.
    
         WORK      (workspace/output)   COMPLEX   array,    dimension
                   (LWORK)
                   On exit, if INFO = 0, WORK(1) returns the  optimal
                   LWORK.
    
         LWORK     (input) INTEGER
                   The dimension of the array WORK.  If SENSE  =  'N'
                   or 'E', LWORK >= max(1,2*N), and if SENSE = 'V' or
                   'B', LWORK  >=  N*N+2*N.   For  good  performance,
                   LWORK must generally be larger.
    
         RWORK     (workspace) REAL array, dimension (2*N)
    
         INFO      (output) INTEGER
                   = 0:  successful exit
                   < 0:  if INFO = -i, the i-th argument had an ille-
                   gal value.
                   > 0:  if INFO = i, the QR algorithm failed to com-
                   pute  all  the eigenvalues, and no eigenvectors or
                   condition numbers  have  been  computed;  elements
                   1:ILO-1  and  i+1:N of W contain eigenvalues which
                   have converged.
    
    
    
    


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