.TH "sgghrd.f" 3 "Wed Oct 15 2014" "Version 3.4.2" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
sgghrd.f \- 
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBsgghrd\fP (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)"
.br
.RI "\fI\fBSGGHRD\fP \fP"
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine sgghrd (characterCOMPQ, characterCOMPZ, integerN, integerILO, integerIHI, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, real, dimension( ldq, * )Q, integerLDQ, real, dimension( ldz, * )Z, integerLDZ, integerINFO)"

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

.PP
.nf
 SGGHRD reduces a pair of real matrices (A,B) to generalized upper
 Hessenberg form using orthogonal transformations, where A is a
 general matrix and B is upper triangular.  The form of the
 generalized eigenvalue problem is
    A*x = lambda*B*x,
 and B is typically made upper triangular by computing its QR
 factorization and moving the orthogonal matrix Q to the left side
 of the equation.

 This subroutine simultaneously reduces A to a Hessenberg matrix H:
    Q**T*A*Z = H
 and transforms B to another upper triangular matrix T:
    Q**T*B*Z = T
 in order to reduce the problem to its standard form
    H*y = lambda*T*y
 where y = Z**T*x.

 The orthogonal matrices Q and Z are determined as products of Givens
 rotations.  They may either be formed explicitly, or they may be
 postmultiplied into input matrices Q1 and Z1, so that

      Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T

      Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T

 If Q1 is the orthogonal matrix from the QR factorization of B in the
 original equation A*x = lambda*B*x, then SGGHRD reduces the original
 problem to generalized Hessenberg form.
.fi
.PP
 
.RE
.PP
\fBParameters:\fP
.RS 4
\fICOMPQ\fP 
.PP
.nf
          COMPQ is CHARACTER*1
          = 'N': do not compute Q;
          = 'I': Q is initialized to the unit matrix, and the
                 orthogonal matrix Q is returned;
          = 'V': Q must contain an orthogonal matrix Q1 on entry,
                 and the product Q1*Q is returned.
.fi
.PP
.br
\fICOMPZ\fP 
.PP
.nf
          COMPZ is CHARACTER*1
          = 'N': do not compute Z;
          = 'I': Z is initialized to the unit matrix, and the
                 orthogonal matrix Z is returned;
          = 'V': Z must contain an orthogonal matrix Z1 on entry,
                 and the product Z1*Z is returned.
.fi
.PP
.br
\fIN\fP 
.PP
.nf
          N is INTEGER
          The order of the matrices A and B.  N >= 0.
.fi
.PP
.br
\fIILO\fP 
.PP
.nf
          ILO is INTEGER
.fi
.PP
.br
\fIIHI\fP 
.PP
.nf
          IHI is INTEGER

          ILO and IHI mark the rows and columns of A which are to be
          reduced.  It is assumed that A is already upper triangular
          in rows and columns 1:ILO-1 and IHI+1:N.  ILO and IHI are
          normally set by a previous call to SGGBAL; otherwise they
          should be set to 1 and N respectively.
          1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
.fi
.PP
.br
\fIA\fP 
.PP
.nf
          A is REAL array, dimension (LDA, N)
          On entry, the N-by-N general matrix to be reduced.
          On exit, the upper triangle and the first subdiagonal of A
          are overwritten with the upper Hessenberg matrix H, and the
          rest is set to zero.
.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 N-by-N upper triangular matrix B.
          On exit, the upper triangular matrix T = Q**T B Z.  The
          elements below the diagonal are set to zero.
.fi
.PP
.br
\fILDB\fP 
.PP
.nf
          LDB is INTEGER
          The leading dimension of the array B.  LDB >= max(1,N).
.fi
.PP
.br
\fIQ\fP 
.PP
.nf
          Q is REAL array, dimension (LDQ, N)
          On entry, if COMPQ = 'V', the orthogonal matrix Q1,
          typically from the QR factorization of B.
          On exit, if COMPQ='I', the orthogonal matrix Q, and if
          COMPQ = 'V', the product Q1*Q.
          Not referenced if COMPQ='N'.
.fi
.PP
.br
\fILDQ\fP 
.PP
.nf
          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
.fi
.PP
.br
\fIZ\fP 
.PP
.nf
          Z is REAL array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the orthogonal matrix Z1.
          On exit, if COMPZ='I', the orthogonal matrix Z, and if
          COMPZ = 'V', the product Z1*Z.
          Not referenced if COMPZ='N'.
.fi
.PP
.br
\fILDZ\fP 
.PP
.nf
          LDZ is INTEGER
          The leading dimension of the array Z.
          LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
.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
\fBDate:\fP
.RS 4
November 2011 
.RE
.PP
\fBFurther Details: \fP
.RS 4

.PP
.nf
  This routine reduces A to Hessenberg and B to triangular form by
  an unblocked reduction, as described in _Matrix_Computations_,
  by Golub and Van Loan (Johns Hopkins Press.)
.fi
.PP
 
.RE
.PP

.PP
Definition at line 207 of file sgghrd\&.f\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.