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shgeqz (3)
  • >> shgeqz (3) ( Solaris man: Библиотечные вызовы )
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    NAME
         shgeqz - implement a  single-shift/double-shift  version  of
         the  QZ  method  for  finding  the  generalized  eigenvalues
         w(j)=(ALPHAR(j)  +  i*ALPHAI(j))/BETAR(j)  of  the  equation
         det(  A-w(i)  B  )  =  0   In  addition, the pair A,B may be
         reduced to generalized Schur form
    
    SYNOPSIS
         SUBROUTINE SHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI,  A,  LDA,
                   B,  LDB,  ALPHAR,  ALPHAI,  BETA,  Q, LDQ, Z, LDZ,
                   WORK, LWORK, INFO )
    
         CHARACTER COMPQ, COMPZ, JOB
    
         INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, LWORK, N
    
         REAL A( LDA, * ), ALPHAI( * ), ALPHAR( * ),  B(  LDB,  *  ),
                   BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
    
    
    
         #include <sunperf.h>
    
         void shgeqz(char job, char compq, char  compz,  int  n,  int
                   ilo,  int  ihi, float *sa, int lda, float *sb, int
                   ldb, float *alphar, float *alphai,  float  *sbeta,
                   float *q, int ldq, float *sz, int ldz, int *info);
    
    PURPOSE
         SHGEQZ implements a single-/double-shift version of  the  QZ
         method  for  finding  the generalized eigenvalues B is upper
         triangular, and A is block upper triangular, where the diag-
         onal  blocks  are either 1-by-1 or 2-by-2, the 2-by-2 blocks
         having complex generalized eigenvalues (see the  description
         of the argument JOB.)
    
         If JOB='S', then the pair (A,B) is simultaneously reduced to
         Schur form by applying one orthogonal tranformation (usually
         called Q) on the left and another (usually called Z) on  the
         right.   The  2-by-2  upper-triangular  diagonal blocks of B
         corresponding to 2-by-2 blocks of A will be reduced to posi-
         tive  diagonal  matrices.   (I.e.,  if A(j+1,j) is non-zero,
         then B(j+1,j)=B(j,j+1)=0 and B(j,j) and B(j+1,j+1)  will  be
         positive.)
    
         If JOB='E', then at each iteration, the same transformations
         are  computed, but they are only applied to those parts of A
         and B which are needed to compute ALPHAR, ALPHAI, and BETAR.
    
         If JOB='S' and COMPQ and COMPZ are  'V'  or  'I',  then  the
         orthogonal  transformations used to reduce (A,B) are accumu-
         lated into the arrays Q and Z s.t.:
              Q(in) A(in) Z(in)* = Q(out) A(out) Z(out)*
              Q(in) B(in) Z(in)* = Q(out) B(out) Z(out)*
    
         Ref: C.B. Moler & G.W. Stewart, "An Algorithm  for  General-
         ized Matrix
              Eigenvalue Problems", SIAM J. Numer. Anal., 10(1973),
              pp. 241--256.
    
    
    ARGUMENTS
         JOB       (input) CHARACTER*1
                   = 'E': compute only ALPHAR, ALPHAI, and  BETA.   A
                   and B will not necessarily be put into generalized
                   Schur form.  = 'S': put A and B  into  generalized
                   Schur  form,  as well as computing ALPHAR, ALPHAI,
                   and BETA.
    
         COMPQ     (input) CHARACTER*1
                   = 'N': do not modify Q.
                   = 'V': multiply the array Q on the  right  by  the
                   transpose  of the orthogonal tranformation that is
                   applied to the left side of A and B to reduce them
                   to Schur form.  = 'I': like COMPQ='V', except that
                   Q will be initialized to the identity first.
    
         COMPZ     (input) CHARACTER*1
                   = 'N': do not modify Z.
                   = 'V': multiply the array Z on the  right  by  the
                   orthogonal  tranformation  that  is applied to the
                   right side of A and B  to  reduce  them  to  Schur
                   form.   =  'I': like COMPZ='V', except that Z will
                   be initialized to the identity first.
    
         N         (input) INTEGER
                   The order of the matrices A, B, Q, and Z.  N >= 0.
    
         ILO       (input) INTEGER
                   IHI     (input) INTEGER It is assumed  that  A  is
                   already  upper  triangular  in  rows  and  columns
                   1:ILO-1 and IHI+1:N.  1 <= ILO <= IHI <= N, if N >
                   0; ILO=1 and IHI=0, if N=0.
    
         A         (input/output) REAL array, dimension (LDA, N)
                   On entry, the N-by-N upper  Hessenberg  matrix  A.
                   Elements  below  the subdiagonal must be zero.  If
                   JOB='S', then on exit  A  and  B  will  have  been
                   simultaneously  reduced to generalized Schur form.
                   If JOB='E', then on exit A  will  have  been  des-
                   troyed.   The diagonal blocks will be correct, but
                   the off-diagonal portion will be meaningless.
    
         LDA       (input) INTEGER
                   The leading dimension of the array A.  LDA >= max(
                   1, N ).
    
         B         (input/output) REAL array, dimension (LDB, N)
                   On entry, the N-by-N upper  triangular  matrix  B.
                   Elements  below the diagonal must be zero.  2-by-2
                   blocks in B corresponding to 2-by-2  blocks  in  A
                   will be reduced to positive diagonal form.  (I.e.,
                   if A(j+1,j) is non-zero, then  B(j+1,j)=B(j,j+1)=0
                   and  B(j,j)  and B(j+1,j+1) will be positive.)  If
                   JOB='S', then on exit  A  and  B  will  have  been
                   simultaneously reduced to Schur form.  If JOB='E',
                   then on exit B will have been destroyed.  Elements
                   corresponding  to  diagonal  blocks  of  A will be
                   correct, but  the  off-diagonal  portion  will  be
                   meaningless.
    
         LDB       (input) INTEGER
                   The leading dimension of the array B.  LDB >= max(
                   1, N ).
    
         ALPHAR    (output) REAL array, dimension (N)
                   ALPHAR(1:N) will be set to real parts of the diag-
                   onal elements of A that would result from reducing
                   A and B to Schur form and  then  further  reducing
                   them   both   to  triangular  form  using  unitary
                   transformations s.t. the diagonal of  B  was  non-
                   negative  real.   Thus,  if  A(j,j) is in a 1-by-1
                   block    (i.e.,     A(j+1,j)=A(j,j+1)=0),     then
                   ALPHAR(j)=A(j,j).  Note that the (real or complex)
                   values    (ALPHAR(j)    +    i*ALPHAI(j))/BETA(j),
                   j=1,...,N,  are the generalized eigenvalues of the
                   matrix pencil A - wB.
    
         ALPHAI    (output) REAL array, dimension (N)
                   ALPHAI(1:N) will be set to imaginary parts of  the
                   diagonal  elements  of  A  that  would result from
                   reducing A and B to Schur form  and  then  further
                   reducing  them  both to triangular form using uni-
                   tary transformations s.t. the diagonal  of  B  was
                   non-negative real.  Thus, if A(j,j) is in a 1-by-1
                   block    (i.e.,     A(j+1,j)=A(j,j+1)=0),     then
                   ALPHAR(j)=0.   Note  that  the  (real  or complex)
                   values    (ALPHAR(j)    +    i*ALPHAI(j))/BETA(j),
                   j=1,...,N,  are the generalized eigenvalues of the
                   matrix pencil A - wB.
    
         BETA      (output) REAL array, dimension (N)
                   BETA(1:N) will be set to the (real) diagonal  ele-
                   ments of B that would result from reducing A and B
                   to Schur form and then further reducing them  both
                   to  triangular  form using unitary transformations
                   s.t. the diagonal  of  B  was  non-negative  real.
                   Thus,  if  A(j,j)  is  in  a  1-by-1  block (i.e.,
                   A(j+1,j)=A(j,j+1)=0), then  BETA(j)=B(j,j).   Note
                   that  the  (real  or  complex) values (ALPHAR(j) +
                   i*ALPHAI(j))/BETA(j), j=1,...,N, are the  general-
                   ized  eigenvalues  of  the  matrix  pencil A - wB.
                   (Note that BETA(1:N) will always be  non-negative,
                   and no BETAI is necessary.)
    
         Q         (input/output) REAL array, dimension (LDQ, N)
                   If COMPQ='N', then Q will not be  referenced.   If
                   COMPQ='V'  or  'I',  then  the  transpose  of  the
                   orthogonal transformations which are applied to  A
                   and  B  on the left will be applied to the array Q
                   on the right.
    
         LDQ       (input) INTEGER
                   The leading dimension of the array Q.  LDQ  >=  1.
                   If COMPQ='V' or 'I', then LDQ >= N.
    
         Z         (input/output) REAL array, dimension (LDZ, N)
                   If COMPZ='N', then Z will not be  referenced.   If
                   COMPZ='V'  or 'I', then the orthogonal transforma-
                   tions which are applied to A and B  on  the  right
                   will be applied to the array Z on the right.
    
         LDZ       (input) INTEGER
                   The leading dimension of the array Z.  LDZ  >=  1.
                   If COMPZ='V' or 'I', then LDZ >= N.
    
         WORK      (workspace/output) REAL array, dimension (LWORK)
                   On exit, if INFO >= 0, WORK(1) returns the optimal
                   LWORK.
    
         LWORK     (input) INTEGER
                   The  dimension  of  the  array  WORK.   LWORK   >=
                   max(1,N).
    
         INFO      (output) INTEGER
                   = 0: successful exit
                   < 0: if INFO = -i, the i-th argument had an  ille-
                   gal value
                   = 1,...,N: the  QZ  iteration  did  not  converge.
                   (A,B)   is  not  in  Schur  form,  but  ALPHAR(i),
                   ALPHAI(i), and BETA(i), i=INFO+1,...,N  should  be
                   correct.   =  N+1,...,2*N:  the  shift calculation
                   failed.   (A,B)  is  not  in   Schur   form,   but
                   ALPHAR(i),   ALPHAI(i),   and   BETA(i),   i=INFO-
                   N+1,...,N should be correct.  >  2*N:      various
                   "impossible" errors.
    
    
    FURTHER DETAILS
         Iteration counters:
    
         JITER  -- counts iterations.
         IITER  -- counts iterations run since ILAST was last
                   changed.  This is therefore reset only when  a  1-
         by-1 or
                   2-by-2 block deflates off the bottom.
    
    
    
    


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