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SUBROUTINE <a name="CGEHRD.1"></a><a href="cgehrd.f.html#CGEHRD.1">CGEHRD</a>( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, 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> INTEGER IHI, ILO, INFO, LDA, LWORK, N
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> COMPLEX A( LDA, * ), TAU( * ), WORK( * )
<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="CGEHRD.17"></a><a href="cgehrd.f.html#CGEHRD.1">CGEHRD</a> reduces a complex general matrix A to upper Hessenberg form H by
</span><span class="comment">*</span><span class="comment"> an unitary similarity transformation: Q' * A * Q = H .
</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"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix A. 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"> It is assumed that A is already upper triangular in rows
</span><span class="comment">*</span><span class="comment"> and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
</span><span class="comment">*</span><span class="comment"> set by a previous call to <a name="CGEBAL.30"></a><a href="cgebal.f.html#CGEBAL.1">CGEBAL</a>; otherwise they should be
</span><span class="comment">*</span><span class="comment"> set to 1 and N respectively. See Further Details.
</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) COMPLEX 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"> elements below the first subdiagonal, with the array TAU,
</span><span class="comment">*</span><span class="comment"> represent the unitary matrix Q as a product of elementary
</span><span class="comment">*</span><span class="comment"> reflectors. See Further Details.
</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"> TAU (output) COMPLEX array, dimension (N-1)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors (see Further
</span><span class="comment">*</span><span class="comment"> Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to
</span><span class="comment">*</span><span class="comment"> zero.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace/output) COMPLEX array, dimension (LWORK)
</span><span class="comment">*</span><span class="comment"> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LWORK (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The length of the array WORK. LWORK >= max(1,N).
</span><span class="comment">*</span><span class="comment"> For optimum performance LWORK >= N*NB, where NB is the
</span><span class="comment">*</span><span class="comment"> optimal blocksize.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment"> only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment"> this value as the first entry of the WORK array, and no error
</span><span class="comment">*</span><span class="comment"> message related to LWORK is issued by <a name="XERBLA.61"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>.
</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"> The matrix Q is represented as a product of (ihi-ilo) elementary
</span><span class="comment">*</span><span class="comment"> reflectors
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Q = H(ilo) H(ilo+1) . . . H(ihi-1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Each H(i) has the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> H(i) = I - tau * v * v'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where tau is a complex scalar, and v is a complex vector with
</span><span class="comment">*</span><span class="comment"> v(1:i) = 0, v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on
</span><span class="comment">*</span><span class="comment"> exit in A(i+2:ihi,i), and tau in TAU(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> The contents of A are illustrated by the following example, with
</span><span class="comment">*</span><span class="comment"> n = 7, ilo = 2 and ihi = 6:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> on entry, on exit,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ( a a a a a a a ) ( a a h h h h a )
</span><span class="comment">*</span><span class="comment"> ( a a a a a a ) ( a h h h h a )
</span><span class="comment">*</span><span class="comment"> ( a a a a a a ) ( h h h h h h )
</span><span class="comment">*</span><span class="comment"> ( a a a a a a ) ( v2 h h h h h )
</span><span class="comment">*</span><span class="comment"> ( a a a a a a ) ( v2 v3 h h h h )
</span><span class="comment">*</span><span class="comment"> ( a a a a a a ) ( v2 v3 v4 h h h )
</span><span class="comment">*</span><span class="comment"> ( a ) ( a )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> where a denotes an element of the original matrix A, h denotes a
</span><span class="comment">*</span><span class="comment"> modified element of the upper Hessenberg matrix H, and vi denotes an
</span><span class="comment">*</span><span class="comment"> element of the vector defining H(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> This file is a slight modification of LAPACK-3.0's <a name="CGEHRD.100"></a><a href="cgehrd.f.html#CGEHRD.1">CGEHRD</a>
</span><span class="comment">*</span><span class="comment"> subroutine incorporating improvements proposed by Quintana-Orti and
</span><span class="comment">*</span><span class="comment"> Van de Geijn (2005).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> INTEGER NBMAX, LDT
PARAMETER ( NBMAX = 64, LDT = NBMAX+1 )
COMPLEX ZERO, ONE
PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ),
$ ONE = ( 1.0E+0, 0.0E+0 ) )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> LOGICAL LQUERY
INTEGER I, IB, IINFO, IWS, J, LDWORK, LWKOPT, NB,
$ NBMIN, NH, NX
COMPLEX EI
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Arrays ..
</span> COMPLEX T( LDT, NBMAX )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL CAXPY, <a name="CGEHD2.123"></a><a href="cgehd2.f.html#CGEHD2.1">CGEHD2</a>, CGEMM, <a name="CLAHR2.123"></a><a href="clahr2.f.html#CLAHR2.1">CLAHR2</a>, <a name="CLARFB.123"></a><a href="clarfb.f.html#CLARFB.1">CLARFB</a>, CTRMM,
$ <a name="XERBLA.124"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC MAX, MIN
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> INTEGER <a name="ILAENV.130"></a><a href="hfy-index.html#ILAENV">ILAENV</a>
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