LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ slaqp2()

 subroutine slaqp2 ( integer m, integer n, integer offset, real, dimension( lda, * ) a, integer lda, integer, dimension( * ) jpvt, real, dimension( * ) tau, real, dimension( * ) vn1, real, dimension( * ) vn2, real, dimension( * ) work )

SLAQP2 computes a QR factorization with column pivoting of the matrix block.

Purpose:
``` SLAQP2 computes a QR factorization with column pivoting of
the block A(OFFSET+1:M,1:N).
The block A(1:OFFSET,1:N) is accordingly pivoted, but not factorized.```
Parameters
 [in] M ``` M is INTEGER The number of rows of the matrix A. M >= 0.``` [in] N ``` N is INTEGER The number of columns of the matrix A. N >= 0.``` [in] OFFSET ``` OFFSET is INTEGER The number of rows of the matrix A that must be pivoted but no factorized. OFFSET >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the upper triangle of block A(OFFSET+1:M,1:N) is the triangular factor obtained; the elements in block A(OFFSET+1:M,1:N) below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of elementary reflectors. Block A(1:OFFSET,1:N) has been accordingly pivoted, but no factorized.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [in,out] JPVT ``` JPVT is INTEGER array, dimension (N) On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted to the front of A*P (a leading column); if JPVT(i) = 0, the i-th column of A is a free column. On exit, if JPVT(i) = k, then the i-th column of A*P was the k-th column of A.``` [out] TAU ``` TAU is REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors.``` [in,out] VN1 ``` VN1 is REAL array, dimension (N) The vector with the partial column norms.``` [in,out] VN2 ``` VN2 is REAL array, dimension (N) The vector with the exact column norms.``` [out] WORK ` WORK is REAL array, dimension (N)`
Contributors:
G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain X. Sun, Computer Science Dept., Duke University, USA
Partial column norm updating strategy modified on April 2011 Z. Drmac and Z. Bujanovic, Dept. of Mathematics, University of Zagreb, Croatia.
References:
LAPACK Working Note 176 [PDF]

Definition at line 147 of file slaqp2.f.

149*
150* -- LAPACK auxiliary routine --
151* -- LAPACK is a software package provided by Univ. of Tennessee, --
152* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
153*
154* .. Scalar Arguments ..
155 INTEGER LDA, M, N, OFFSET
156* ..
157* .. Array Arguments ..
158 INTEGER JPVT( * )
159 REAL A( LDA, * ), TAU( * ), VN1( * ), VN2( * ),
160 \$ WORK( * )
161* ..
162*
163* =====================================================================
164*
165* .. Parameters ..
166 REAL ZERO, ONE
167 parameter( zero = 0.0e+0, one = 1.0e+0 )
168* ..
169* .. Local Scalars ..
170 INTEGER I, ITEMP, J, MN, OFFPI, PVT
171 REAL AII, TEMP, TEMP2, TOL3Z
172* ..
173* .. External Subroutines ..
174 EXTERNAL slarf, slarfg, sswap
175* ..
176* .. Intrinsic Functions ..
177 INTRINSIC abs, max, min, sqrt
178* ..
179* .. External Functions ..
180 INTEGER ISAMAX
181 REAL SLAMCH, SNRM2
182 EXTERNAL isamax, slamch, snrm2
183* ..
184* .. Executable Statements ..
185*
186 mn = min( m-offset, n )
187 tol3z = sqrt(slamch('Epsilon'))
188*
189* Compute factorization.
190*
191 DO 20 i = 1, mn
192*
193 offpi = offset + i
194*
195* Determine ith pivot column and swap if necessary.
196*
197 pvt = ( i-1 ) + isamax( n-i+1, vn1( i ), 1 )
198*
199 IF( pvt.NE.i ) THEN
200 CALL sswap( m, a( 1, pvt ), 1, a( 1, i ), 1 )
201 itemp = jpvt( pvt )
202 jpvt( pvt ) = jpvt( i )
203 jpvt( i ) = itemp
204 vn1( pvt ) = vn1( i )
205 vn2( pvt ) = vn2( i )
206 END IF
207*
208* Generate elementary reflector H(i).
209*
210 IF( offpi.LT.m ) THEN
211 CALL slarfg( m-offpi+1, a( offpi, i ), a( offpi+1, i ), 1,
212 \$ tau( i ) )
213 ELSE
214 CALL slarfg( 1, a( m, i ), a( m, i ), 1, tau( i ) )
215 END IF
216*
217 IF( i.LT.n ) THEN
218*
219* Apply H(i)**T to A(offset+i:m,i+1:n) from the left.
220*
221 aii = a( offpi, i )
222 a( offpi, i ) = one
223 CALL slarf( 'Left', m-offpi+1, n-i, a( offpi, i ), 1,
224 \$ tau( i ), a( offpi, i+1 ), lda, work( 1 ) )
225 a( offpi, i ) = aii
226 END IF
227*
228* Update partial column norms.
229*
230 DO 10 j = i + 1, n
231 IF( vn1( j ).NE.zero ) THEN
232*
233* NOTE: The following 4 lines follow from the analysis in
234* Lapack Working Note 176.
235*
236 temp = one - ( abs( a( offpi, j ) ) / vn1( j ) )**2
237 temp = max( temp, zero )
238 temp2 = temp*( vn1( j ) / vn2( j ) )**2
239 IF( temp2 .LE. tol3z ) THEN
240 IF( offpi.LT.m ) THEN
241 vn1( j ) = snrm2( m-offpi, a( offpi+1, j ), 1 )
242 vn2( j ) = vn1( j )
243 ELSE
244 vn1( j ) = zero
245 vn2( j ) = zero
246 END IF
247 ELSE
248 vn1( j ) = vn1( j )*sqrt( temp )
249 END IF
250 END IF
251 10 CONTINUE
252*
253 20 CONTINUE
254*
255 RETURN
256*
257* End of SLAQP2
258*
integer function isamax(n, sx, incx)
ISAMAX
Definition isamax.f:71
real function slamch(cmach)
SLAMCH
Definition slamch.f:68
subroutine slarf(side, m, n, v, incv, tau, c, ldc, work)
SLARF applies an elementary reflector to a general rectangular matrix.
Definition slarf.f:124
subroutine slarfg(n, alpha, x, incx, tau)
SLARFG generates an elementary reflector (Householder matrix).
Definition slarfg.f:106
real(wp) function snrm2(n, x, incx)
SNRM2
Definition snrm2.f90:89
subroutine sswap(n, sx, incx, sy, incy)
SSWAP
Definition sswap.f:82
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