.TH "ggbal" 3 "Wed Feb 7 2024 11:30:40" "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
ggbal \- ggbal: balance matrix
.SH SYNOPSIS
.br
.PP
.SS "Functions"

.in +1c
.ti -1c
.RI "subroutine \fBcggbal\fP (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)"
.br
.RI "\fBCGGBAL\fP "
.ti -1c
.RI "subroutine \fBdggbal\fP (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)"
.br
.RI "\fBDGGBAL\fP "
.ti -1c
.RI "subroutine \fBsggbal\fP (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)"
.br
.RI "\fBSGGBAL\fP "
.ti -1c
.RI "subroutine \fBzggbal\fP (job, n, a, lda, b, ldb, ilo, ihi, lscale, rscale, work, info)"
.br
.RI "\fBZGGBAL\fP "
.in -1c
.SH "Detailed Description"
.PP 

.SH "Function Documentation"
.PP 
.SS "subroutine cggbal (character job, integer n, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real, dimension( * ) work, integer info)"

.PP
\fBCGGBAL\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 CGGBAL balances a pair of general complex matrices (A,B)\&.  This
 involves, first, permuting A and B by similarity transformations to
 isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
 elements on the diagonal; and second, applying a diagonal similarity
 transformation to rows and columns ILO to IHI to make the rows
 and columns as close in norm as possible\&. Both steps are optional\&.

 Balancing may reduce the 1-norm of the matrices, and improve the
 accuracy of the computed eigenvalues and/or eigenvectors in the
 generalized eigenvalue problem A*x = lambda*B*x\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1\&.0
                  and RSCALE(I) = 1\&.0 for i=1,\&.\&.\&.,N;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX array, dimension (LDA,N)
          On entry, the input matrix A\&.
          On exit, A is overwritten by the balanced matrix\&.
          If JOB = 'N', A is not referenced\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX array, dimension (LDB,N)
          On entry, the input matrix B\&.
          On exit, B is overwritten by the balanced matrix\&.
          If JOB = 'N', B is not referenced\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI are set to integers 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 JOB = 'N' or 'S', ILO = 1 and IHI = N\&.
.fi
.PP
.br
\fILSCALE\fP 
.PP
.nf
          LSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B\&.  If P(j) is the index of the
          row interchanged with row j, and D(j) is the scaling factor
          applied to row j, then
            LSCALE(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\&.
.fi
.PP
.br
\fIRSCALE\fP 
.PP
.nf
          RSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B\&.  If P(j) is the index of the
          column interchanged with column j, and D(j) is the scaling
          factor applied to column j, then
            RSCALE(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\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
          at least 1 when JOB = 'N' or 'P'\&.
.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\&.
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  See R\&.C\&. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J\&. Sci\&. Stat\&. Comp\&. 2 (1981), 141-152\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine dggbal (character job, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision, dimension( * ) work, integer info)"

.PP
\fBDGGBAL\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 DGGBAL balances a pair of general real matrices (A,B)\&.  This
 involves, first, permuting A and B by similarity transformations to
 isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
 elements on the diagonal; and second, applying a diagonal similarity
 transformation to rows and columns ILO to IHI to make the rows
 and columns as close in norm as possible\&. Both steps are optional\&.

 Balancing may reduce the 1-norm of the matrices, and improve the
 accuracy of the computed eigenvalues and/or eigenvectors in the
 generalized eigenvalue problem A*x = lambda*B*x\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1\&.0
                  and RSCALE(I) = 1\&.0 for i = 1,\&.\&.\&.,N\&.
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the input matrix A\&.
          On exit,  A is overwritten by the balanced matrix\&.
          If JOB = 'N', A is not referenced\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is DOUBLE PRECISION array, dimension (LDB,N)
          On entry, the input matrix B\&.
          On exit,  B is overwritten by the balanced matrix\&.
          If JOB = 'N', B is not referenced\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI are set to integers 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 JOB = 'N' or 'S', ILO = 1 and IHI = N\&.
.fi
.PP
.br
\fILSCALE\fP 
.PP
.nf
          LSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B\&.  If P(j) is the index of the
          row interchanged with row j, and D(j)
          is the scaling factor applied to row j, then
            LSCALE(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\&.
.fi
.PP
.br
\fIRSCALE\fP 
.PP
.nf
          RSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B\&.  If P(j) is the index of the
          column interchanged with column j, and D(j)
          is the scaling factor applied to column j, then
            LSCALE(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\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
          at least 1 when JOB = 'N' or 'P'\&.
.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\&.
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  See R\&.C\&. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J\&. Sci\&. Stat\&. Comp\&. 2 (1981), 141-152\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine sggbal (character job, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension( * ) rscale, real, dimension( * ) work, integer info)"

.PP
\fBSGGBAL\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 SGGBAL balances a pair of general real matrices (A,B)\&.  This
 involves, first, permuting A and B by similarity transformations to
 isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
 elements on the diagonal; and second, applying a diagonal similarity
 transformation to rows and columns ILO to IHI to make the rows
 and columns as close in norm as possible\&. Both steps are optional\&.

 Balancing may reduce the 1-norm of the matrices, and improve the
 accuracy of the computed eigenvalues and/or eigenvectors in the
 generalized eigenvalue problem A*x = lambda*B*x\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1\&.0
                  and RSCALE(I) = 1\&.0 for i = 1,\&.\&.\&.,N\&.
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA,N)
          On entry, the input matrix A\&.
          On exit,  A is overwritten by the balanced matrix\&.
          If JOB = 'N', A is not referenced\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is REAL array, dimension (LDB,N)
          On entry, the input matrix B\&.
          On exit,  B is overwritten by the balanced matrix\&.
          If JOB = 'N', B is not referenced\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI are set to integers 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 JOB = 'N' or 'S', ILO = 1 and IHI = N\&.
.fi
.PP
.br
\fILSCALE\fP 
.PP
.nf
          LSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B\&.  If P(j) is the index of the
          row interchanged with row j, and D(j)
          is the scaling factor applied to row j, then
            LSCALE(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\&.
.fi
.PP
.br
\fIRSCALE\fP 
.PP
.nf
          RSCALE is REAL array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B\&.  If P(j) is the index of the
          column interchanged with column j, and D(j)
          is the scaling factor applied to column j, then
            LSCALE(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\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is REAL array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
          at least 1 when JOB = 'N' or 'P'\&.
.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\&.
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  See R\&.C\&. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J\&. Sci\&. Stat\&. Comp\&. 2 (1981), 141-152\&.
.fi
.PP
 
.RE
.PP

.SS "subroutine zggbal (character job, integer n, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, integer ilo, integer ihi, double precision, dimension( * ) lscale, double precision, dimension( * ) rscale, double precision, dimension( * ) work, integer info)"

.PP
\fBZGGBAL\fP  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
 ZGGBAL balances a pair of general complex matrices (A,B)\&.  This
 involves, first, permuting A and B by similarity transformations to
 isolate eigenvalues in the first 1 to ILO$-$1 and last IHI+1 to N
 elements on the diagonal; and second, applying a diagonal similarity
 transformation to rows and columns ILO to IHI to make the rows
 and columns as close in norm as possible\&. Both steps are optional\&.

 Balancing may reduce the 1-norm of the matrices, and improve the
 accuracy of the computed eigenvalues and/or eigenvectors in the
 generalized eigenvalue problem A*x = lambda*B*x\&.
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIJOB\fP 
.PP
.nf
          JOB is CHARACTER*1
          Specifies the operations to be performed on A and B:
          = 'N':  none:  simply set ILO = 1, IHI = N, LSCALE(I) = 1\&.0
                  and RSCALE(I) = 1\&.0 for i=1,\&.\&.\&.,N;
          = 'P':  permute only;
          = 'S':  scale only;
          = 'B':  both permute and scale\&.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B\&.  N >= 0\&.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is COMPLEX*16 array, dimension (LDA,N)
          On entry, the input matrix A\&.
          On exit, A is overwritten by the balanced matrix\&.
          If JOB = 'N', A is not referenced\&.
.fi
.PP
.br
\fILDA\fP 
.PP
.nf
          LDA is INTEGER
          The leading dimension of the array A\&. LDA >= max(1,N)\&.
.fi
.PP
.br
\fIB\fP 
.PP
.nf
          B is COMPLEX*16 array, dimension (LDB,N)
          On entry, the input matrix B\&.
          On exit, B is overwritten by the balanced matrix\&.
          If JOB = 'N', B is not referenced\&.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B\&. LDB >= max(1,N)\&.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER
          ILO and IHI are set to integers 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 JOB = 'N' or 'S', ILO = 1 and IHI = N\&.
.fi
.PP
.br
\fILSCALE\fP 
.PP
.nf
          LSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the left side of A and B\&.  If P(j) is the index of the
          row interchanged with row j, and D(j) is the scaling factor
          applied to row j, then
            LSCALE(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\&.
.fi
.PP
.br
\fIRSCALE\fP 
.PP
.nf
          RSCALE is DOUBLE PRECISION array, dimension (N)
          Details of the permutations and scaling factors applied
          to the right side of A and B\&.  If P(j) is the index of the
          column interchanged with column j, and D(j) is the scaling
          factor applied to column j, then
            RSCALE(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\&.
.fi
.PP
.br
\fIWORK\fP 
.PP
.nf
          WORK is DOUBLE PRECISION array, dimension (lwork)
          lwork must be at least max(1,6*N) when JOB = 'S' or 'B', and
          at least 1 when JOB = 'N' or 'P'\&.
.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\&.
.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
\fBFurther Details:\fP
.RS 4

.PP
.nf
  See R\&.C\&. WARD, Balancing the generalized eigenvalue problem,
                 SIAM J\&. Sci\&. Stat\&. Comp\&. 2 (1981), 141-152\&.
.fi
.PP
 
.RE
.PP

.SH "Author"
.PP 
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