LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
sgerq2.f
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1 *> \brief \b SGERQ2 computes the RQ 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 SGERQ2( M, N, A, LDA, TAU, WORK, INFO )
22 *
23 * .. Scalar Arguments ..
24 * INTEGER INFO, LDA, M, N
25 * ..
26 * .. Array Arguments ..
27 * REAL A( LDA, * ), TAU( * ), WORK( * )
28 * ..
29 *
30 *
31 *> \par Purpose:
32 * =============
33 *>
34 *> \verbatim
35 *>
36 *> SGERQ2 computes an RQ factorization of a real m by n matrix A:
37 *> A = R * 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,out] A
56 *> \verbatim
57 *> A is REAL array, dimension (LDA,N)
58 *> On entry, the m by n matrix A.
59 *> On exit, if m <= n, the upper triangle of the subarray
60 *> A(1:m,n-m+1:n) contains the m by m upper triangular matrix R;
61 *> if m >= n, the elements on and above the (m-n)-th subdiagonal
62 *> contain the m by n upper trapezoidal matrix R; the remaining
63 *> elements, with the array TAU, represent the orthogonal matrix
64 *> Q as a product of elementary reflectors (see Further
65 *> Details).
66 *> \endverbatim
67 *>
68 *> \param[in] LDA
69 *> \verbatim
70 *> LDA is INTEGER
71 *> The leading dimension of the array A. LDA >= max(1,M).
72 *> \endverbatim
73 *>
74 *> \param[out] TAU
75 *> \verbatim
76 *> TAU is REAL array, dimension (min(M,N))
77 *> The scalar factors of the elementary reflectors (see Further
78 *> Details).
79 *> \endverbatim
80 *>
81 *> \param[out] WORK
82 *> \verbatim
83 *> WORK is REAL array, dimension (M)
84 *> \endverbatim
85 *>
86 *> \param[out] INFO
87 *> \verbatim
88 *> INFO is INTEGER
89 *> = 0: successful exit
90 *> < 0: if INFO = -i, the i-th argument had an illegal value
91 *> \endverbatim
92 *
93 * Authors:
94 * ========
95 *
96 *> \author Univ. of Tennessee
97 *> \author Univ. of California Berkeley
98 *> \author Univ. of Colorado Denver
99 *> \author NAG Ltd.
100 *
101 *> \ingroup realGEcomputational
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(n-k+i+1:n) = 0 and v(n-k+i) = 1; v(1:n-k+i-1) is stored on exit in
118 *> A(m-k+i,1:n-k+i-1), and tau in TAU(i).
119 *> \endverbatim
120 *>
121 * =====================================================================
122  SUBROUTINE sgerq2( M, N, A, LDA, TAU, WORK, INFO )
123 *
124 * -- LAPACK computational routine --
125 * -- LAPACK is a software package provided by Univ. of Tennessee, --
126 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
127 *
128 * .. Scalar Arguments ..
129  INTEGER INFO, LDA, M, N
130 * ..
131 * .. Array Arguments ..
132  REAL A( LDA, * ), TAU( * ), WORK( * )
133 * ..
134 *
135 * =====================================================================
136 *
137 * .. Parameters ..
138  REAL ONE
139  parameter( one = 1.0e+0 )
140 * ..
141 * .. Local Scalars ..
142  INTEGER I, K
143  REAL AII
144 * ..
145 * .. External Subroutines ..
146  EXTERNAL slarf, slarfg, xerbla
147 * ..
148 * .. Intrinsic Functions ..
149  INTRINSIC max, min
150 * ..
151 * .. Executable Statements ..
152 *
153 * Test the input arguments
154 *
155  info = 0
156  IF( m.LT.0 ) THEN
157  info = -1
158  ELSE IF( n.LT.0 ) THEN
159  info = -2
160  ELSE IF( lda.LT.max( 1, m ) ) THEN
161  info = -4
162  END IF
163  IF( info.NE.0 ) THEN
164  CALL xerbla( 'SGERQ2', -info )
165  RETURN
166  END IF
167 *
168  k = min( m, n )
169 *
170  DO 10 i = k, 1, -1
171 *
172 * Generate elementary reflector H(i) to annihilate
173 * A(m-k+i,1:n-k+i-1)
174 *
175  CALL slarfg( n-k+i, a( m-k+i, n-k+i ), a( m-k+i, 1 ), lda,
176  \$ tau( i ) )
177 *
178 * Apply H(i) to A(1:m-k+i-1,1:n-k+i) from the right
179 *
180  aii = a( m-k+i, n-k+i )
181  a( m-k+i, n-k+i ) = one
182  CALL slarf( 'Right', m-k+i-1, n-k+i, a( m-k+i, 1 ), lda,
183  \$ tau( i ), a, lda, work )
184  a( m-k+i, n-k+i ) = aii
185  10 CONTINUE
186  RETURN
187 *
188 * End of SGERQ2
189 *
190  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
subroutine sgerq2(M, N, A, LDA, TAU, WORK, INFO)
SGERQ2 computes the RQ factorization of a general rectangular matrix using an unblocked algorithm.
Definition: sgerq2.f:123
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