.TH "gges" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
gges \- gges: Schur form, unblocked
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcgges\fP (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)"
.br
.RI "\fB CGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.ti -1c
.RI "subroutine \fBdgges\fP (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info)"
.br
.RI "\fB DGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.ti -1c
.RI "subroutine \fBsgges\fP (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, 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 "
.ti -1c
.RI "subroutine \fBzgges\fP (jobvsl, jobvsr, sort, selctg, n, a, lda, b, ldb, sdim, alpha, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, rwork, bwork, info)"
.br
.RI "\fB ZGGES computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer sdim, complex, dimension( * ) alpha, complex, dimension( * ) beta, complex, dimension( ldvsl, * ) vsl, integer ldvsl, complex, dimension( ldvsr, * ) vsr, integer ldvsr, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)"

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

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

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

 where (VSR)**H is the conjugate-transpose of VSR\&.

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

 (If only the generalized eigenvalues are needed, use the driver
 CGGEV 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, and even for both being zero\&.

 A pair of matrices (S,T) is in generalized complex Schur form if S
 and T are upper triangular and, in addition, the diagonal elements
 of T are non-negative real numbers\&.
.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 two COMPLEX 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 ALPHA(j)/BETA(j) is selected if
          SELCTG(ALPHA(j),BETA(j)) is true\&.

          Note that a selected complex eigenvalue may no longer satisfy
          SELCTG(ALPHA(j),BETA(j)) = \&.TRUE\&. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned), in this
          case INFO is set to N+2 (See INFO below)\&.
.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 COMPLEX 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 COMPLEX 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\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is COMPLEX array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is COMPLEX array, dimension (N)
          On exit,  ALPHA(j)/BETA(j), j=1,\&.\&.\&.,N, will be the
          generalized eigenvalues\&.  ALPHA(j), j=1,\&.\&.\&.,N  and  BETA(j),
          j=1,\&.\&.\&.,N  are the diagonals of the complex Schur form (A,B)
          output by CGGES\&. The  BETA(j) will be non-negative real\&.

          Note: the quotients ALPHA(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, ALPHA 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 COMPLEX 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 COMPLEX 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 COMPLEX 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\&.  LWORK >= max(1,2*N)\&.
          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
\fIRWORK\fP 
.PP
.nf
          RWORK is REAL array, dimension (8*N)
.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 ALPHA(j) and BETA(j) should be correct for
                j=INFO+1,\&.\&.\&.,N\&.
          > N:  =N+1: other than QZ iteration failed in CHGEQZ
                =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 CTGSEN\&.
.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
.PP

.SS "subroutine dgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer sdim, double precision, dimension( * ) alphar, double precision, dimension( * ) alphai, double precision, dimension( * ) beta, double precision, dimension( ldvsl, * ) vsl, integer ldvsl, double precision, dimension( ldvsr, * ) vsr, integer ldvsr, double precision, dimension( * ) work, integer lwork, logical, dimension( * ) bwork, integer info)"

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

.PP
.nf
 DGGES 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
 DGGEV 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIALPHAI\fP 
.PP
.nf
          ALPHAI is DOUBLE PRECISION array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 >= 8*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 DHGEQZ\&.
                =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 DTGSEN\&.
.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
.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
.PP

.SS "subroutine zgges (character jobvsl, character jobvsr, character sort, external selctg, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer sdim, complex*16, dimension( * ) alpha, complex*16, dimension( * ) beta, complex*16, dimension( ldvsl, * ) vsl, integer ldvsl, complex*16, dimension( ldvsr, * ) vsr, integer ldvsr, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, logical, dimension( * ) bwork, integer info)"

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

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

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

 where (VSR)**H is the conjugate-transpose of VSR\&.

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

 (If only the generalized eigenvalues are needed, use the driver
 ZGGEV 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, and even for both being zero\&.

 A pair of matrices (S,T) is in generalized complex Schur form if S
 and T are upper triangular and, in addition, the diagonal elements
 of T are non-negative real numbers\&.
.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 two COMPLEX*16 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 ALPHA(j)/BETA(j) is selected if
          SELCTG(ALPHA(j),BETA(j)) is true\&.

          Note that a selected complex eigenvalue may no longer satisfy
          SELCTG(ALPHA(j),BETA(j)) = \&.TRUE\&. after ordering, since
          ordering may change the value of complex eigenvalues
          (especially if the eigenvalue is ill-conditioned), in this
          case INFO is set to N+2 (See INFO below)\&.
.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 COMPLEX*16 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 COMPLEX*16 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\&.
.fi
.PP
.br
\fIALPHA\fP 
.PP
.nf
          ALPHA is COMPLEX*16 array, dimension (N)
.fi
.PP
.br
\fIBETA\fP 
.PP
.nf
          BETA is COMPLEX*16 array, dimension (N)
          On exit,  ALPHA(j)/BETA(j), j=1,\&.\&.\&.,N, will be the
          generalized eigenvalues\&.  ALPHA(j), j=1,\&.\&.\&.,N  and  BETA(j),
          j=1,\&.\&.\&.,N  are the diagonals of the complex Schur form (A,B)
          output by ZGGES\&. The  BETA(j) will be non-negative real\&.

          Note: the quotients ALPHA(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, ALPHA 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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\&.  LWORK >= max(1,2*N)\&.
          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
\fIRWORK\fP 
.PP
.nf
          RWORK is DOUBLE PRECISION array, dimension (8*N)
.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 ALPHA(j) and BETA(j) should be correct for
                j=INFO+1,\&.\&.\&.,N\&.
          > N:  =N+1: other than QZ iteration failed in ZHGEQZ
                =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 ZTGSEN\&.
.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
.PP

.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.