 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sgeqrf()

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

SGEQRF

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Purpose:
``` SGEQRF computes a QR factorization of a real M-by-N matrix A:

A = Q * ( R ),
( 0 )

where:

Q is a M-by-M orthogonal matrix;
R is an upper-triangular N-by-N matrix;
0 is a (M-N)-by-N zero matrix, if M > N.```
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 elements on and above the diagonal of the array contain the min(M,N)-by-N upper trapezoidal matrix R (R is upper triangular if m >= n); the elements below the diagonal, with the array TAU, represent the orthogonal matrix Q as a product of min(m,n) elementary reflectors (see Further Details).``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] TAU ``` TAU is REAL array, dimension (min(M,N)) The scalar factors of the elementary reflectors (see Further Details).``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= 1, if MIN(M,N) = 0, and LWORK >= N, otherwise. For optimum performance LWORK >= N*NB, where NB is the optimal blocksize. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [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(k), where k = min(m,n).

Each H(i) has the form

H(i) = 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),
and tau in TAU(i).```

Definition at line 145 of file sgeqrf.f.

146 *
147 * -- LAPACK computational routine --
148 * -- LAPACK is a software package provided by Univ. of Tennessee, --
149 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
150 *
151 * .. Scalar Arguments ..
152  INTEGER INFO, LDA, LWORK, M, N
153 * ..
154 * .. Array Arguments ..
155  REAL A( LDA, * ), TAU( * ), WORK( * )
156 * ..
157 *
158 * =====================================================================
159 *
160 * .. Local Scalars ..
161  LOGICAL LQUERY
162  INTEGER I, IB, IINFO, IWS, K, LDWORK, LWKOPT, NB,
163  \$ NBMIN, NX
164 * ..
165 * .. External Subroutines ..
166  EXTERNAL sgeqr2, slarfb, slarft, xerbla
167 * ..
168 * .. Intrinsic Functions ..
169  INTRINSIC max, min
170 * ..
171 * .. External Functions ..
172  INTEGER ILAENV
173  EXTERNAL ilaenv
174 * ..
175 * .. Executable Statements ..
176 *
177 * Test the input arguments
178 *
179  k = min( m, n )
180  info = 0
181  nb = ilaenv( 1, 'SGEQRF', ' ', m, n, -1, -1 )
182  lquery = ( lwork.EQ.-1 )
183  IF( m.LT.0 ) THEN
184  info = -1
185  ELSE IF( n.LT.0 ) THEN
186  info = -2
187  ELSE IF( lda.LT.max( 1, m ) ) THEN
188  info = -4
189  ELSE IF( .NOT.lquery ) THEN
190  IF( lwork.LE.0 .OR. ( m.GT.0 .AND. lwork.LT.max( 1, n ) ) )
191  \$ info = -7
192  END IF
193  IF( info.NE.0 ) THEN
194  CALL xerbla( 'SGEQRF', -info )
195  RETURN
196  ELSE IF( lquery ) THEN
197  IF( k.EQ.0 ) THEN
198  lwkopt = 1
199  ELSE
200  lwkopt = n*nb
201  END IF
202  work( 1 ) = lwkopt
203  RETURN
204  END IF
205 *
206 * Quick return if possible
207 *
208  IF( k.EQ.0 ) THEN
209  work( 1 ) = 1
210  RETURN
211  END IF
212 *
213  nbmin = 2
214  nx = 0
215  iws = n
216  IF( nb.GT.1 .AND. nb.LT.k ) THEN
217 *
218 * Determine when to cross over from blocked to unblocked code.
219 *
220  nx = max( 0, ilaenv( 3, 'SGEQRF', ' ', m, n, -1, -1 ) )
221  IF( nx.LT.k ) THEN
222 *
223 * Determine if workspace is large enough for blocked code.
224 *
225  ldwork = n
226  iws = ldwork*nb
227  IF( lwork.LT.iws ) THEN
228 *
229 * Not enough workspace to use optimal NB: reduce NB and
230 * determine the minimum value of NB.
231 *
232  nb = lwork / ldwork
233  nbmin = max( 2, ilaenv( 2, 'SGEQRF', ' ', m, n, -1,
234  \$ -1 ) )
235  END IF
236  END IF
237  END IF
238 *
239  IF( nb.GE.nbmin .AND. nb.LT.k .AND. nx.LT.k ) THEN
240 *
241 * Use blocked code initially
242 *
243  DO 10 i = 1, k - nx, nb
244  ib = min( k-i+1, nb )
245 *
246 * Compute the QR factorization of the current block
247 * A(i:m,i:i+ib-1)
248 *
249  CALL sgeqr2( m-i+1, ib, a( i, i ), lda, tau( i ), work,
250  \$ iinfo )
251  IF( i+ib.LE.n ) THEN
252 *
253 * Form the triangular factor of the block reflector
254 * H = H(i) H(i+1) . . . H(i+ib-1)
255 *
256  CALL slarft( 'Forward', 'Columnwise', m-i+1, ib,
257  \$ a( i, i ), lda, tau( i ), work, ldwork )
258 *
259 * Apply H**T to A(i:m,i+ib:n) from the left
260 *
261  CALL slarfb( 'Left', 'Transpose', 'Forward',
262  \$ 'Columnwise', m-i+1, n-i-ib+1, ib,
263  \$ a( i, i ), lda, work, ldwork, a( i, i+ib ),
264  \$ lda, work( ib+1 ), ldwork )
265  END IF
266  10 CONTINUE
267  ELSE
268  i = 1
269  END IF
270 *
271 * Use unblocked code to factor the last or only block.
272 *
273  IF( i.LE.k )
274  \$ CALL sgeqr2( m-i+1, n-i+1, a( i, i ), lda, tau( i ), work,
275  \$ iinfo )
276 *
277  work( 1 ) = iws
278  RETURN
279 *
280 * End of SGEQRF
281 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
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 slarfb(SIDE, TRANS, DIRECT, STOREV, M, N, K, V, LDV, T, LDT, C, LDC, WORK, LDWORK)
SLARFB applies a block reflector or its transpose to a general rectangular matrix.
Definition: slarfb.f:197
subroutine slarft(DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT)
SLARFT forms the triangular factor T of a block reflector H = I - vtvH
Definition: slarft.f:163
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