LAPACK  3.7.0 LAPACK: Linear Algebra PACKage
dgeqr2.f
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1 *> \brief \b DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DGEQR2( M, N, A, LDA, TAU, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, M, N
25 * ..
26 * .. Array Arguments ..
27 * DOUBLE PRECISION A( LDA, * ), TAU( * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> DGEQR2 computes a QR factorization of a real m by n matrix A:
37 *> A = Q * R.
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,out] A
56 *> \verbatim
57 *> A is DOUBLE PRECISION array, dimension (LDA,N)
58 *> On entry, the m by n matrix A.
59 *> On exit, the elements on and above the diagonal of the array
60 *> contain the min(m,n) by n upper trapezoidal matrix R (R is
61 *> upper triangular if m >= n); the elements below the diagonal,
62 *> with the array TAU, represent the orthogonal matrix Q as a
63 *> product of elementary reflectors (see Further Details).
64 *> \endverbatim
65 *>
66 *> \param[in] LDA
67 *> \verbatim
68 *> LDA is INTEGER
69 *> The leading dimension of the array A. LDA >= max(1,M).
70 *> \endverbatim
71 *>
72 *> \param[out] TAU
73 *> \verbatim
74 *> TAU is DOUBLE PRECISION array, dimension (min(M,N))
75 *> The scalar factors of the elementary reflectors (see Further
76 *> Details).
77 *> \endverbatim
78 *>
79 *> \param[out] WORK
80 *> \verbatim
81 *> WORK is DOUBLE PRECISION array, dimension (N)
82 *> \endverbatim
83 *>
84 *> \param[out] INFO
85 *> \verbatim
86 *> INFO is INTEGER
87 *> = 0: successful exit
88 *> < 0: if INFO = -i, the i-th argument had an illegal value
89 *> \endverbatim
90 *
91 * Authors:
92 * ========
93 *
94 *> \author Univ. of Tennessee
95 *> \author Univ. of California Berkeley
96 *> \author Univ. of Colorado Denver
97 *> \author NAG Ltd.
98 *
99 *> \date December 2016
100 *
101 *> \ingroup doubleGEcomputational
102 *
103 *> \par Further Details:
104 * =====================
105 *>
106 *> \verbatim
107 *>
108 *> The matrix Q is represented as a product of elementary reflectors
109 *>
110 *> Q = H(1) H(2) . . . H(k), where k = min(m,n).
111 *>
112 *> Each H(i) has the form
113 *>
114 *> H(i) = I - tau * v * v**T
115 *>
116 *> where tau is a real scalar, and v is a real vector with
117 *> v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is stored on exit in A(i+1:m,i),
118 *> and tau in TAU(i).
119 *> \endverbatim
120 *>
121 * =====================================================================
122  SUBROUTINE dgeqr2( M, N, A, LDA, TAU, WORK, INFO )
123 *
124 * -- LAPACK computational routine (version 3.7.0) --
125 * -- LAPACK is a software package provided by Univ. of Tennessee, --
126 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127 * December 2016
128 *
129 * .. Scalar Arguments ..
130  INTEGER INFO, LDA, M, N
131 * ..
132 * .. Array Arguments ..
133  DOUBLE PRECISION A( lda, * ), TAU( * ), WORK( * )
134 * ..
135 *
136 * =====================================================================
137 *
138 * .. Parameters ..
139  DOUBLE PRECISION ONE
140  parameter ( one = 1.0d+0 )
141 * ..
142 * .. Local Scalars ..
143  INTEGER I, K
144  DOUBLE PRECISION AII
145 * ..
146 * .. External Subroutines ..
147  EXTERNAL dlarf, dlarfg, xerbla
148 * ..
149 * .. Intrinsic Functions ..
150  INTRINSIC max, min
151 * ..
152 * .. Executable Statements ..
153 *
154 * Test the input arguments
155 *
156  info = 0
157  IF( m.LT.0 ) THEN
158  info = -1
159  ELSE IF( n.LT.0 ) THEN
160  info = -2
161  ELSE IF( lda.LT.max( 1, m ) ) THEN
162  info = -4
163  END IF
164  IF( info.NE.0 ) THEN
165  CALL xerbla( 'DGEQR2', -info )
166  RETURN
167  END IF
168 *
169  k = min( m, n )
170 *
171  DO 10 i = 1, k
172 *
173 * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
174 *
175  CALL dlarfg( m-i+1, a( i, i ), a( min( i+1, m ), i ), 1,
176  \$ tau( i ) )
177  IF( i.LT.n ) THEN
178 *
179 * Apply H(i) to A(i:m,i+1:n) from the left
180 *
181  aii = a( i, i )
182  a( i, i ) = one
183  CALL dlarf( 'Left', m-i+1, n-i, a( i, i ), 1, tau( i ),
184  \$ a( i, i+1 ), lda, work )
185  a( i, i ) = aii
186  END IF
187  10 CONTINUE
188  RETURN
189 *
190 * End of DGEQR2
191 *
192  END
subroutine dlarf(SIDE, M, N, V, INCV, TAU, C, LDC, WORK)
DLARF applies an elementary reflector to a general rectangular matrix.
Definition: dlarf.f:126
subroutine dgeqr2(M, N, A, LDA, TAU, WORK, INFO)
DGEQR2 computes the QR factorization of a general rectangular matrix using an unblocked algorithm...
Definition: dgeqr2.f:123
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine dlarfg(N, ALPHA, X, INCX, TAU)
DLARFG generates an elementary reflector (Householder matrix).
Definition: dlarfg.f:108