 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ sgeqpf()

 subroutine sgeqpf ( integer M, integer N, real, dimension( lda, * ) A, integer LDA, integer, dimension( * ) JPVT, real, dimension( * ) TAU, real, dimension( * ) WORK, integer INFO )

SGEQPF

Purpose:
``` This routine is deprecated and has been replaced by routine SGEQP3.

SGEQPF computes a QR factorization with column pivoting of a
real M-by-N matrix A: A*P = Q*R.```
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,out] A ``` A is REAL array, dimension (LDA,N) On entry, the M-by-N matrix A. On exit, the upper triangle of the array contains the min(M,N)-by-N upper triangular matrix R; the elements below the diagonal, together with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors.``` [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.``` [out] WORK ` WORK is REAL array, dimension (3*N)` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value```
Further Details:
```  The matrix Q is represented as a product of elementary reflectors

Q = H(1) H(2) . . . H(n)

Each H(i) has the form

H = I - tau * v * v**T

where tau is a real scalar, and v is a real vector with
v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i).

The matrix P is represented in jpvt as follows: If
jpvt(j) = i
then the jth column of P is the ith canonical unit vector.

Partial column norm updating strategy modified by
Z. Drmac and Z. Bujanovic, Dept. of Mathematics,
University of Zagreb, Croatia.
-- April 2011                                                      --
For more details see LAPACK Working Note 176.```

Definition at line 141 of file sgeqpf.f.

142 *
143 * -- LAPACK computational routine --
144 * -- LAPACK is a software package provided by Univ. of Tennessee, --
145 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
146 *
147 * .. Scalar Arguments ..
148  INTEGER INFO, LDA, M, N
149 * ..
150 * .. Array Arguments ..
151  INTEGER JPVT( * )
152  REAL A( LDA, * ), TAU( * ), WORK( * )
153 * ..
154 *
155 * =====================================================================
156 *
157 * .. Parameters ..
158  REAL ZERO, ONE
159  parameter( zero = 0.0e+0, one = 1.0e+0 )
160 * ..
161 * .. Local Scalars ..
162  INTEGER I, ITEMP, J, MA, MN, PVT
163  REAL AII, TEMP, TEMP2, TOL3Z
164 * ..
165 * .. External Subroutines ..
166  EXTERNAL sgeqr2, slarf, slarfg, sorm2r, sswap, xerbla
167 * ..
168 * .. Intrinsic Functions ..
169  INTRINSIC abs, max, min, sqrt
170 * ..
171 * .. External Functions ..
172  INTEGER ISAMAX
173  REAL SLAMCH, SNRM2
174  EXTERNAL isamax, slamch, snrm2
175 * ..
176 * .. Executable Statements ..
177 *
178 * Test the input arguments
179 *
180  info = 0
181  IF( m.LT.0 ) THEN
182  info = -1
183  ELSE IF( n.LT.0 ) THEN
184  info = -2
185  ELSE IF( lda.LT.max( 1, m ) ) THEN
186  info = -4
187  END IF
188  IF( info.NE.0 ) THEN
189  CALL xerbla( 'SGEQPF', -info )
190  RETURN
191  END IF
192 *
193  mn = min( m, n )
194  tol3z = sqrt(slamch('Epsilon'))
195 *
196 * Move initial columns up front
197 *
198  itemp = 1
199  DO 10 i = 1, n
200  IF( jpvt( i ).NE.0 ) THEN
201  IF( i.NE.itemp ) THEN
202  CALL sswap( m, a( 1, i ), 1, a( 1, itemp ), 1 )
203  jpvt( i ) = jpvt( itemp )
204  jpvt( itemp ) = i
205  ELSE
206  jpvt( i ) = i
207  END IF
208  itemp = itemp + 1
209  ELSE
210  jpvt( i ) = i
211  END IF
212  10 CONTINUE
213  itemp = itemp - 1
214 *
215 * Compute the QR factorization and update remaining columns
216 *
217  IF( itemp.GT.0 ) THEN
218  ma = min( itemp, m )
219  CALL sgeqr2( m, ma, a, lda, tau, work, info )
220  IF( ma.LT.n ) THEN
221  CALL sorm2r( 'Left', 'Transpose', m, n-ma, ma, a, lda, tau,
222  \$ a( 1, ma+1 ), lda, work, info )
223  END IF
224  END IF
225 *
226  IF( itemp.LT.mn ) THEN
227 *
228 * Initialize partial column norms. The first n elements of
229 * work store the exact column norms.
230 *
231  DO 20 i = itemp + 1, n
232  work( i ) = snrm2( m-itemp, a( itemp+1, i ), 1 )
233  work( n+i ) = work( i )
234  20 CONTINUE
235 *
236 * Compute factorization
237 *
238  DO 40 i = itemp + 1, mn
239 *
240 * Determine ith pivot column and swap if necessary
241 *
242  pvt = ( i-1 ) + isamax( n-i+1, work( i ), 1 )
243 *
244  IF( pvt.NE.i ) THEN
245  CALL sswap( m, a( 1, pvt ), 1, a( 1, i ), 1 )
246  itemp = jpvt( pvt )
247  jpvt( pvt ) = jpvt( i )
248  jpvt( i ) = itemp
249  work( pvt ) = work( i )
250  work( n+pvt ) = work( n+i )
251  END IF
252 *
253 * Generate elementary reflector H(i)
254 *
255  IF( i.LT.m ) THEN
256  CALL slarfg( m-i+1, a( i, i ), a( i+1, i ), 1, tau( i ) )
257  ELSE
258  CALL slarfg( 1, a( m, m ), a( m, m ), 1, tau( m ) )
259  END IF
260 *
261  IF( i.LT.n ) THEN
262 *
263 * Apply H(i) to A(i:m,i+1:n) from the left
264 *
265  aii = a( i, i )
266  a( i, i ) = one
267  CALL slarf( 'LEFT', m-i+1, n-i, a( i, i ), 1, tau( i ),
268  \$ a( i, i+1 ), lda, work( 2*n+1 ) )
269  a( i, i ) = aii
270  END IF
271 *
272 * Update partial column norms
273 *
274  DO 30 j = i + 1, n
275  IF( work( j ).NE.zero ) THEN
276 *
277 * NOTE: The following 4 lines follow from the analysis in
278 * Lapack Working Note 176.
279 *
280  temp = abs( a( i, j ) ) / work( j )
281  temp = max( zero, ( one+temp )*( one-temp ) )
282  temp2 = temp*( work( j ) / work( n+j ) )**2
283  IF( temp2 .LE. tol3z ) THEN
284  IF( m-i.GT.0 ) THEN
285  work( j ) = snrm2( m-i, a( i+1, j ), 1 )
286  work( n+j ) = work( j )
287  ELSE
288  work( j ) = zero
289  work( n+j ) = zero
290  END IF
291  ELSE
292  work( j ) = work( j )*sqrt( temp )
293  END IF
294  END IF
295  30 CONTINUE
296 *
297  40 CONTINUE
298  END IF
299  RETURN
300 *
301 * End of SGEQPF
302 *
integer function isamax(N, SX, INCX)
ISAMAX
Definition: isamax.f:71
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgeqr2(M, N, A, LDA, TAU, WORK, INFO)
SGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
Definition: sgeqr2.f:130
subroutine slarfg(N, ALPHA, X, INCX, TAU)
SLARFG generates an elementary reflector (Householder matrix).
Definition: slarfg.f:106
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 sorm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sge...
Definition: sorm2r.f:159
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
real(wp) function snrm2(n, x, incx)
SNRM2
Definition: snrm2.f90:89
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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