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SUBROUTINE <a name="DLAQPS.1"></a><a href="dlaqps.f.html#DLAQPS.1">DLAQPS</a>( M, N, OFFSET, NB, KB, A, LDA, JPVT, TAU, VN1,
$ VN2, AUXV, F, LDF )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary 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 KB, LDA, LDF, M, N, NB, OFFSET
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER JPVT( * )
DOUBLE PRECISION A( LDA, * ), AUXV( * ), F( LDF, * ), TAU( * ),
$ VN1( * ), VN2( * )
<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="DLAQPS.20"></a><a href="dlaqps.f.html#DLAQPS.1">DLAQPS</a> computes a step of QR factorization with column pivoting
</span><span class="comment">*</span><span class="comment"> of a real M-by-N matrix A by using Blas-3. It tries to factorize
</span><span class="comment">*</span><span class="comment"> NB columns from A starting from the row OFFSET+1, and updates all
</span><span class="comment">*</span><span class="comment"> of the matrix with Blas-3 xGEMM.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> In some cases, due to catastrophic cancellations, it cannot
</span><span class="comment">*</span><span class="comment"> factorize NB columns. Hence, the actual number of factorized
</span><span class="comment">*</span><span class="comment"> columns is returned in KB.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.
</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"> M (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of the matrix A. M >= 0.
</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 number of columns of the matrix A. N >= 0
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> OFFSET (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of rows of A that have been factorized in
</span><span class="comment">*</span><span class="comment"> previous steps.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NB (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of columns to factorize.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> KB (output) INTEGER
</span><span class="comment">*</span><span class="comment"> The number of columns actually factorized.
</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 M-by-N matrix A.
</span><span class="comment">*</span><span class="comment"> On exit, block A(OFFSET+1:M,1:KB) is the triangular
</span><span class="comment">*</span><span class="comment"> factor obtained and block A(1:OFFSET,1:N) has been
</span><span class="comment">*</span><span class="comment"> accordingly pivoted, but no factorized.
</span><span class="comment">*</span><span class="comment"> The rest of the matrix, block A(OFFSET+1:M,KB+1:N) has
</span><span class="comment">*</span><span class="comment"> been updated.
</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,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> JPVT (input/output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment"> JPVT(I) = K <==> Column K of the full matrix A has been
</span><span class="comment">*</span><span class="comment"> permuted into position I in AP.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> TAU (output) DOUBLE PRECISION array, dimension (KB)
</span><span class="comment">*</span><span class="comment"> The scalar factors of the elementary reflectors.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VN1 (input/output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The vector with the partial column norms.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> VN2 (input/output) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The vector with the exact column norms.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> AUXV (input/output) DOUBLE PRECISION array, dimension (NB)
</span><span class="comment">*</span><span class="comment"> Auxiliar vector.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> F (input/output) DOUBLE PRECISION array, dimension (LDF,NB)
</span><span class="comment">*</span><span class="comment"> Matrix F' = L*Y'*A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LDF (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The leading dimension of the array F. LDF >= max(1,N).
</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"> Based on contributions by
</span><span class="comment">*</span><span class="comment"> G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
</span><span class="comment">*</span><span class="comment"> X. Sun, Computer Science Dept., Duke University, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Partial column norm updating strategy modified by
</span><span class="comment">*</span><span class="comment"> Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
</span><span class="comment">*</span><span class="comment"> University of Zagreb, Croatia.
</span><span class="comment">*</span><span class="comment"> June 2006.
</span><span class="comment">*</span><span class="comment"> For more details see LAPACK Working Note 176.
</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> DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> INTEGER ITEMP, J, K, LASTRK, LSTICC, PVT, RK
DOUBLE PRECISION AKK, TEMP, TEMP2, TOL3Z
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Subroutines ..
</span> EXTERNAL DGEMM, DGEMV, <a name="DLARFG.106"></a><a href="dlarfg.f.html#DLARFG.1">DLARFG</a>, DSWAP
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, DBLE, MAX, MIN, NINT, SQRT
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> INTEGER IDAMAX
DOUBLE PRECISION <a name="DLAMCH.113"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, DNRM2
EXTERNAL IDAMAX, <a name="DLAMCH.114"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>, DNRM2
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span> LASTRK = MIN( M, N+OFFSET )
LSTICC = 0
K = 0
TOL3Z = SQRT(<a name="DLAMCH.121"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>(<span class="string">'Epsilon'</span>))
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Beginning of while loop.
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