 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ sgeqrt()

 subroutine sgeqrt ( integer M, integer N, integer NB, real, dimension( lda, * ) A, integer LDA, real, dimension( ldt, * ) T, integer LDT, real, dimension( * ) WORK, integer INFO )

SGEQRT

Purpose:
``` SGEQRT computes a blocked QR factorization of a real M-by-N matrix A
using the compact WY representation of Q.```
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] NB ``` NB is INTEGER The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.``` [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 are the columns of V.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).``` [out] T ``` T is REAL array, dimension (LDT,MIN(M,N)) The upper triangular block reflectors stored in compact form as a sequence of upper triangular blocks. See below for further details.``` [in] LDT ``` LDT is INTEGER The leading dimension of the array T. LDT >= NB.``` [out] WORK ` WORK is REAL array, dimension (NB*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 V stores the elementary reflectors H(i) in the i-th column
below the diagonal. For example, if M=5 and N=3, the matrix V is

V = (  1       )
( v1  1    )
( v1 v2  1 )
( v1 v2 v3 )
( v1 v2 v3 )

where the vi's represent the vectors which define H(i), which are returned
in the matrix A.  The 1's along the diagonal of V are not stored in A.

Let K=MIN(M,N).  The number of blocks is B = ceiling(K/NB), where each
block is of order NB except for the last block, which is of order
IB = K - (B-1)*NB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The NB-by-NB (and IB-by-IB
for the last block) T's are stored in the NB-by-K matrix T as

T = (T1 T2 ... TB).```

Definition at line 140 of file sgeqrt.f.

141 *
142 * -- LAPACK computational routine --
143 * -- LAPACK is a software package provided by Univ. of Tennessee, --
144 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
145 *
146 * .. Scalar Arguments ..
147  INTEGER INFO, LDA, LDT, M, N, NB
148 * ..
149 * .. Array Arguments ..
150  REAL A( LDA, * ), T( LDT, * ), WORK( * )
151 * ..
152 *
153 * =====================================================================
154 *
155 * ..
156 * .. Local Scalars ..
157  INTEGER I, IB, IINFO, K
158  LOGICAL USE_RECURSIVE_QR
159  parameter( use_recursive_qr=.true. )
160 * ..
161 * .. External Subroutines ..
162  EXTERNAL sgeqrt2, sgeqrt3, slarfb, xerbla
163 * ..
164 * .. Executable Statements ..
165 *
166 * Test the input arguments
167 *
168  info = 0
169  IF( m.LT.0 ) THEN
170  info = -1
171  ELSE IF( n.LT.0 ) THEN
172  info = -2
173  ELSE IF( nb.LT.1 .OR. ( nb.GT.min(m,n) .AND. min(m,n).GT.0 ) )THEN
174  info = -3
175  ELSE IF( lda.LT.max( 1, m ) ) THEN
176  info = -5
177  ELSE IF( ldt.LT.nb ) THEN
178  info = -7
179  END IF
180  IF( info.NE.0 ) THEN
181  CALL xerbla( 'SGEQRT', -info )
182  RETURN
183  END IF
184 *
185 * Quick return if possible
186 *
187  k = min( m, n )
188  IF( k.EQ.0 ) RETURN
189 *
190 * Blocked loop of length K
191 *
192  DO i = 1, k, nb
193  ib = min( k-i+1, nb )
194 *
195 * Compute the QR factorization of the current block A(I:M,I:I+IB-1)
196 *
197  IF( use_recursive_qr ) THEN
198  CALL sgeqrt3( m-i+1, ib, a(i,i), lda, t(1,i), ldt, iinfo )
199  ELSE
200  CALL sgeqrt2( m-i+1, ib, a(i,i), lda, t(1,i), ldt, iinfo )
201  END IF
202  IF( i+ib.LE.n ) THEN
203 *
204 * Update by applying H**T to A(I:M,I+IB:N) from the left
205 *
206  CALL slarfb( 'L', 'T', 'F', 'C', m-i+1, n-i-ib+1, ib,
207  \$ a( i, i ), lda, t( 1, i ), ldt,
208  \$ a( i, i+ib ), lda, work , n-i-ib+1 )
209  END IF
210  END DO
211  RETURN
212 *
213 * End of SGEQRT
214 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
recursive subroutine sgeqrt3(M, N, A, LDA, T, LDT, INFO)
SGEQRT3 recursively computes a QR factorization of a general real or complex matrix using the compact...
Definition: sgeqrt3.f:132
subroutine sgeqrt2(M, N, A, LDA, T, LDT, INFO)
SGEQRT2 computes a QR factorization of a general real or complex matrix using the compact WY represen...
Definition: sgeqrt2.f:127
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
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