.TH "sgges.f" 3 "Tue Nov 14 2017" "Version 3.8.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
sgges.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBsgges\fP (JOBVSL, JOBVSR, SORT, SELCTG, \fBN\fP, A, \fBLDA\fP, B, \fBLDB\fP, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR, WORK, LWORK, BWORK, INFO)"
.br
.RI "\fB SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine sgges (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, integer SDIM, real, dimension( * ) ALPHAR, real, dimension( * ) ALPHAI, real, dimension( * ) BETA, real, dimension( ldvsl, * ) VSL, integer LDVSL, real, dimension( ldvsr, * ) VSR, integer LDVSR, real, dimension( * ) WORK, integer LWORK, logical, dimension( * ) BWORK, integer INFO)"

.PP
\fB SGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP  
.PP
\fBPurpose: \fP
.RS 4

.PP
.nf
 SGGES computes for a pair of N-by-N real nonsymmetric matrices (A,B),
 the generalized eigenvalues, the generalized real Schur form (S,T),
 optionally, the left and/or right matrices of Schur vectors (VSL and
 VSR). This gives the generalized Schur factorization

          (A,B) = ( (VSL)*S*(VSR)**T, (VSL)*T*(VSR)**T )

 Optionally, it also orders the eigenvalues so that a selected cluster
 of eigenvalues appears in the leading diagonal blocks of the upper
 quasi-triangular matrix S and the upper triangular matrix T.The
 leading columns of VSL and VSR then form an orthonormal basis for the
 corresponding left and right eigenspaces (deflating subspaces).

 (If only the generalized eigenvalues are needed, use the driver
 SGGEV instead, which is faster.)

 A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
 or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
 usually represented as the pair (alpha,beta), as there is a
 reasonable interpretation for beta=0 or both being zero.

 A pair of matrices (S,T) is in generalized real Schur form if T is
 upper triangular with non-negative diagonal and S is block upper
 triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
 to real generalized eigenvalues, while 2-by-2 blocks of S will be
 "standardized" by making the corresponding elements of T have the
 form:
         [  a  0  ]
         [  0  b  ]

 and the pair of corresponding 2-by-2 blocks in S and T will have a
 complex conjugate pair of generalized eigenvalues.
.fi
.PP
 
.RE
.PP
\fBParameters:\fP
.RS 4
\fIJOBVSL\fP 
.PP
.nf
          JOBVSL is CHARACTER*1
          = 'N':  do not compute the left Schur vectors;
          = 'V':  compute the left Schur vectors.
.fi
.PP
.br
\fIJOBVSR\fP 
.PP
.nf
          JOBVSR is CHARACTER*1
          = 'N':  do not compute the right Schur vectors;
          = 'V':  compute the right Schur vectors.
.fi
.PP
.br
\fISORT\fP 
.PP
.nf
          SORT is CHARACTER*1
          Specifies whether or not to order the eigenvalues on the
          diagonal of the generalized Schur form.
          = 'N':  Eigenvalues are not ordered;
          = 'S':  Eigenvalues are ordered (see SELCTG);
.fi
.PP
.br
\fISELCTG\fP 
.PP
.nf
          SELCTG is a LOGICAL FUNCTION of three REAL arguments
          SELCTG must be declared EXTERNAL in the calling subroutine.
          If SORT = 'N', SELCTG is not referenced.
          If SORT = 'S', SELCTG is used to select eigenvalues to sort
          to the top left of the Schur form.
          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
          SELCTG(ALPHAR(j),ALPHAI(j),BETA(j)) is true; i.e. if either
          one of a complex conjugate pair of eigenvalues is selected,
          then both complex eigenvalues are selected.

          Note that in the ill-conditioned case, a selected complex
          eigenvalue may no longer satisfy SELCTG(ALPHAR(j),ALPHAI(j),
          BETA(j)) = .TRUE. after ordering. INFO is to be set to N+2
          in this case.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A, B, VSL, and VSR.  N >= 0.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA, N)
          On entry, the first of the pair of matrices.
          On exit, A has been overwritten by its generalized Schur
          form S.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of A.  LDA >= max(1,N).
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is REAL array, dimension (LDB, N)
          On entry, the second of the pair of matrices.
          On exit, B has been overwritten by its generalized Schur
          form T.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of B.  LDB >= max(1,N).
.fi
.PP
.br
\fISDIM\fP 
.PP
.nf
          SDIM is INTEGER
          If SORT = 'N', SDIM = 0.
          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
          for which SELCTG is true.  (Complex conjugate pairs for which
          SELCTG is true for either eigenvalue count as 2.)
.fi
.PP
.br
\fIALPHAR\fP 
.PP
.nf
          ALPHAR is REAL array, dimension (N)
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          ALPHAI is REAL array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is REAL array, dimension (N)
          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i,
          and  BETA(j),j=1,...,N are the diagonals of the complex Schur
          form (S,T) that would result if the 2-by-2 diagonal blocks of
          the real Schur form of (A,B) were further reduced to
          triangular form using 2-by-2 complex unitary transformations.
          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.
          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).
.fi
.PP
.br
\fIVSL\fP 
.PP
.nf
          VSL is REAL array, dimension (LDVSL,N)
          If JOBVSL = 'V', VSL will contain the left Schur vectors.
          Not referenced if JOBVSL = 'N'.
.fi
.PP
.br
\fILDVSL\fP 
.PP
.nf
          LDVSL is INTEGER
          The leading dimension of the matrix VSL. LDVSL >=1, and
          if JOBVSL = 'V', LDVSL >= N.
.fi
.PP
.br
\fIVSR\fP 
.PP
.nf
          VSR is REAL array, dimension (LDVSR,N)
          If JOBVSR = 'V', VSR will contain the right Schur vectors.
          Not referenced if JOBVSR = 'N'.
.fi
.PP
.br
\fILDVSR\fP 
.PP
.nf
          LDVSR is INTEGER
          The leading dimension of the matrix VSR. LDVSR >= 1, and
          if JOBVSR = 'V', LDVSR >= N.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.fi
.PP
.br
\fILWORK\fP 
.PP
.nf
          LWORK is INTEGER
          The dimension of the array WORK.
          If N = 0, LWORK >= 1, else LWORK >= max(8*N,6*N+16).
          For good performance , LWORK must generally be larger.

          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.
.fi
.PP
.br
\fIBWORK\fP 
.PP
.nf
          BWORK is LOGICAL array, dimension (N)
          Not referenced if SORT = 'N'.
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
          INFO is INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value.
          = 1,...,N:
                The QZ iteration failed.  (A,B) are not in Schur
                form, 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: after reordering, roundoff changed values of
                      some complex eigenvalues so that leading
                      eigenvalues in the Generalized Schur form no
                      longer satisfy SELCTG=.TRUE.  This could also
                      be caused due to scaling.
                =N+3: reordering failed in STGSEN.
.fi
.PP
 
.RE
.PP
\fBAuthor:\fP
.RS 4
Univ\&. of Tennessee 
.PP
Univ\&. of California Berkeley 
.PP
Univ\&. of Colorado Denver 
.PP
NAG Ltd\&. 
.RE
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\fBDate:\fP
.RS 4
December 2016 
.RE
.PP

.PP
Definition at line 286 of file sgges\&.f\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.
