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SUBROUTINE <a name="DGGHRD.1"></a><a href="dgghrd.f.html#DGGHRD.1">DGGHRD</a>( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
$ LDQ, Z, LDZ, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
$ Z( LDZ, * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="DGGHRD.20"></a><a href="dgghrd.f.html#DGGHRD.1">DGGHRD</a> reduces a pair of real matrices (A,B) to generalized upper
</span><span class="comment">*</span><span class="comment"> Hessenberg form using orthogonal transformations, where A is a
</span><span class="comment">*</span><span class="comment"> general matrix and B is upper triangular. The form of the
</span><span class="comment">*</span><span class="comment"> generalized eigenvalue problem is
</span><span class="comment">*</span><span class="comment"> A*x = lambda*B*x,
</span><span class="comment">*</span><span class="comment"> and B is typically made upper triangular by computing its QR
</span><span class="comment">*</span><span class="comment"> factorization and moving the orthogonal matrix Q to the left side
</span><span class="comment">*</span><span class="comment"> of the equation.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This subroutine simultaneously reduces A to a Hessenberg matrix H:
</span><span class="comment">*</span><span class="comment"> Q**T*A*Z = H
</span><span class="comment">*</span><span class="comment"> and transforms B to another upper triangular matrix T:
</span><span class="comment">*</span><span class="comment"> Q**T*B*Z = T
</span><span class="comment">*</span><span class="comment"> in order to reduce the problem to its standard form
</span><span class="comment">*</span><span class="comment"> H*y = lambda*T*y
</span><span class="comment">*</span><span class="comment"> where y = Z**T*x.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The orthogonal matrices Q and Z are determined as products of Givens
</span><span class="comment">*</span><span class="comment"> rotations. They may either be formed explicitly, or they may be
</span><span class="comment">*</span><span class="comment"> postmultiplied into input matrices Q1 and Z1, so that
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If Q1 is the orthogonal matrix from the QR factorization of B in the
</span><span class="comment">*</span><span class="comment"> original equation A*x = lambda*B*x, then <a name="DGGHRD.46"></a><a href="dgghrd.f.html#DGGHRD.1">DGGHRD</a> reduces the original
</span><span class="comment">*</span><span class="comment"> problem to generalized Hessenberg form.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> COMPQ (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': do not compute Q;
</span><span class="comment">*</span><span class="comment"> = 'I': Q is initialized to the unit matrix, and the
</span><span class="comment">*</span><span class="comment"> orthogonal matrix Q is returned;
</span><span class="comment">*</span><span class="comment"> = 'V': Q must contain an orthogonal matrix Q1 on entry,
</span><span class="comment">*</span><span class="comment"> and the product Q1*Q is returned.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> COMPZ (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment"> = 'N': do not compute Z;
</span><span class="comment">*</span><span class="comment"> = 'I': Z is initialized to the unit matrix, and the
</span><span class="comment">*</span><span class="comment"> orthogonal matrix Z is returned;
</span><span class="comment">*</span><span class="comment"> = 'V': Z must contain an orthogonal matrix Z1 on entry,
</span><span class="comment">*</span><span class="comment"> and the product Z1*Z is returned.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrices A and B. N >= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ILO (input) INTEGER
</span><span class="comment">*</span><span class="comment"> IHI (input) INTEGER
</span><span class="comment">*</span><span class="comment"> ILO and IHI mark the rows and columns of A which are to be
</span><span class="comment">*</span><span class="comment"> reduced. It is assumed that A is already upper triangular
</span><span class="comment">*</span><span class="comment"> in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
</span><span class="comment">*</span><span class="comment"> normally set by a previous call to <a name="SGGBAL.74"></a><a href="sggbal.f.html#SGGBAL.1">SGGBAL</a>; otherwise they
</span><span class="comment">*</span><span class="comment"> should be set to 1 and N respectively.
</span><span class="comment">*</span><span class="comment"> 1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
</span><span class="comment">*</span><span class="comment"> On entry, the N-by-N general matrix to be reduced.
</span><span class="comment">*</span><span class="comment"> On exit, the upper triangle and the first subdiagonal of A
</span><span class="comment">*</span><span class="comment"> are overwritten with the upper Hessenberg matrix H, and the
</span><span class="comment">*</span><span class="comment"> rest is set to zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDA (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array A. LDA >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
</span><span class="comment">*</span><span class="comment"> On entry, the N-by-N upper triangular matrix B.
</span><span class="comment">*</span><span class="comment"> On exit, the upper triangular matrix T = Q**T B Z. The
</span><span class="comment">*</span><span class="comment"> elements below the diagonal are set to zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array B. LDB >= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
</span><span class="comment">*</span><span class="comment"> On entry, if COMPQ = 'V', the orthogonal matrix Q1,
</span><span class="comment">*</span><span class="comment"> typically from the QR factorization of B.
</span><span class="comment">*</span><span class="comment"> On exit, if COMPQ='I', the orthogonal matrix Q, and if
</span><span class="comment">*</span><span class="comment"> COMPQ = 'V', the product Q1*Q.
</span><span class="comment">*</span><span class="comment"> Not referenced if COMPQ='N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDQ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Q.
</span><span class="comment">*</span><span class="comment"> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
</span><span class="comment">*</span><span class="comment"> On entry, if COMPZ = 'V', the orthogonal matrix Z1.
</span><span class="comment">*</span><span class="comment"> On exit, if COMPZ='I', the orthogonal matrix Z, and if
</span><span class="comment">*</span><span class="comment"> COMPZ = 'V', the product Z1*Z.
</span><span class="comment">*</span><span class="comment"> Not referenced if COMPZ='N'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDZ (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array Z.
</span><span class="comment">*</span><span class="comment"> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> INFO (output) INTEGER
</span><span class="comment">*</span><span class="comment"> = 0: successful exit.
</span><span class="comment">*</span><span class="comment"> < 0: if INFO = -i, the i-th argument had an illegal value.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This routine reduces A to Hessenberg and B to triangular form by
</span><span class="comment">*</span><span class="comment"> an unblocked reduction, as described in _Matrix_Computations_,
</span><span class="comment">*</span><span class="comment"> by Golub and Van Loan (Johns Hopkins Press.)
</span><span class="comment">*</span><span class="comment">
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