LAPACK  3.10.1
LAPACK: Linear Algebra PACKage
sgeqrt.f
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1 *> \brief \b SGEQRT
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 *> \htmlonly
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15 *> [TXT]</a>
16 *> \endhtmlonly
17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE SGEQRT( M, N, NB, A, LDA, T, LDT, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, LDT, M, N, NB
25 * ..
26 * .. Array Arguments ..
27 * REAL A( LDA, * ), T( LDT, * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> SGEQRT computes a blocked QR factorization of a real M-by-N matrix A
37 *> using the compact WY representation of Q.
38 *> \endverbatim
39 *
40 * Arguments:
41 * ==========
42 *
43 *> \param[in] M
44 *> \verbatim
45 *> M is INTEGER
46 *> The number of rows of the matrix A. M >= 0.
47 *> \endverbatim
48 *>
49 *> \param[in] N
50 *> \verbatim
51 *> N is INTEGER
52 *> The number of columns of the matrix A. N >= 0.
53 *> \endverbatim
54 *>
55 *> \param[in] NB
56 *> \verbatim
57 *> NB is INTEGER
58 *> The block size to be used in the blocked QR. MIN(M,N) >= NB >= 1.
59 *> \endverbatim
60 *>
61 *> \param[in,out] A
62 *> \verbatim
63 *> A is REAL array, dimension (LDA,N)
64 *> On entry, the M-by-N matrix A.
65 *> On exit, the elements on and above the diagonal of the array
66 *> contain the min(M,N)-by-N upper trapezoidal matrix R (R is
67 *> upper triangular if M >= N); the elements below the diagonal
68 *> are the columns of V.
69 *> \endverbatim
70 *>
71 *> \param[in] LDA
72 *> \verbatim
73 *> LDA is INTEGER
74 *> The leading dimension of the array A. LDA >= max(1,M).
75 *> \endverbatim
76 *>
77 *> \param[out] T
78 *> \verbatim
79 *> T is REAL array, dimension (LDT,MIN(M,N))
80 *> The upper triangular block reflectors stored in compact form
81 *> as a sequence of upper triangular blocks. See below
82 *> for further details.
83 *> \endverbatim
84 *>
85 *> \param[in] LDT
86 *> \verbatim
87 *> LDT is INTEGER
88 *> The leading dimension of the array T. LDT >= NB.
89 *> \endverbatim
90 *>
91 *> \param[out] WORK
92 *> \verbatim
93 *> WORK is REAL array, dimension (NB*N)
94 *> \endverbatim
95 *>
96 *> \param[out] INFO
97 *> \verbatim
98 *> INFO is INTEGER
99 *> = 0: successful exit
100 *> < 0: if INFO = -i, the i-th argument had an illegal value
101 *> \endverbatim
102 *
103 * Authors:
104 * ========
105 *
106 *> \author Univ. of Tennessee
107 *> \author Univ. of California Berkeley
108 *> \author Univ. of Colorado Denver
109 *> \author NAG Ltd.
110 *
111 *> \ingroup realGEcomputational
112 *
113 *> \par Further Details:
114 * =====================
115 *>
116 *> \verbatim
117 *>
118 *> The matrix V stores the elementary reflectors H(i) in the i-th column
119 *> below the diagonal. For example, if M=5 and N=3, the matrix V is
120 *>
121 *> V = ( 1 )
122 *> ( v1 1 )
123 *> ( v1 v2 1 )
124 *> ( v1 v2 v3 )
125 *> ( v1 v2 v3 )
126 *>
127 *> where the vi's represent the vectors which define H(i), which are returned
128 *> in the matrix A. The 1's along the diagonal of V are not stored in A.
129 *>
130 *> Let K=MIN(M,N). The number of blocks is B = ceiling(K/NB), where each
131 *> block is of order NB except for the last block, which is of order
132 *> IB = K - (B-1)*NB. For each of the B blocks, a upper triangular block
133 *> reflector factor is computed: T1, T2, ..., TB. The NB-by-NB (and IB-by-IB
134 *> for the last block) T's are stored in the NB-by-K matrix T as
135 *>
136 *> T = (T1 T2 ... TB).
137 *> \endverbatim
138 *>
139 * =====================================================================
140  SUBROUTINE sgeqrt( M, N, NB, A, LDA, T, LDT, WORK, INFO )
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 *
215  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgeqrt(M, N, NB, A, LDA, T, LDT, WORK, INFO)
SGEQRT
Definition: sgeqrt.f:141
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