LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
sqpt01.f
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1 *> \brief \b SQPT01
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * REAL FUNCTION SQPT01( M, N, K, A, AF, LDA, TAU, JPVT,
12 * WORK, LWORK )
13 *
14 * .. Scalar Arguments ..
15 * INTEGER K, LDA, LWORK, M, N
16 * ..
17 * .. Array Arguments ..
18 * INTEGER JPVT( * )
19 * REAL A( LDA, * ), AF( LDA, * ), TAU( * ),
20 * $ WORK( LWORK )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> SQPT01 tests the QR-factorization with pivoting of a matrix A. The
30 *> array AF contains the (possibly partial) QR-factorization of A, where
31 *> the upper triangle of AF(1:k,1:k) is a partial triangular factor,
32 *> the entries below the diagonal in the first k columns are the
33 *> Householder vectors, and the rest of AF contains a partially updated
34 *> matrix.
35 *>
36 *> This function returns ||A*P - Q*R||/(||norm(A)||*eps*M)
37 *> \endverbatim
38 *
39 * Arguments:
40 * ==========
41 *
42 *> \param[in] M
43 *> \verbatim
44 *> M is INTEGER
45 *> The number of rows of the matrices A and AF.
46 *> \endverbatim
47 *>
48 *> \param[in] N
49 *> \verbatim
50 *> N is INTEGER
51 *> The number of columns of the matrices A and AF.
52 *> \endverbatim
53 *>
54 *> \param[in] K
55 *> \verbatim
56 *> K is INTEGER
57 *> The number of columns of AF that have been reduced
58 *> to upper triangular form.
59 *> \endverbatim
60 *>
61 *> \param[in] A
62 *> \verbatim
63 *> A is REAL array, dimension (LDA, N)
64 *> The original matrix A.
65 *> \endverbatim
66 *>
67 *> \param[in] AF
68 *> \verbatim
69 *> AF is REAL array, dimension (LDA,N)
70 *> The (possibly partial) output of SGEQPF. The upper triangle
71 *> of AF(1:k,1:k) is a partial triangular factor, the entries
72 *> below the diagonal in the first k columns are the Householder
73 *> vectors, and the rest of AF contains a partially updated
74 *> matrix.
75 *> \endverbatim
76 *>
77 *> \param[in] LDA
78 *> \verbatim
79 *> LDA is INTEGER
80 *> The leading dimension of the arrays A and AF.
81 *> \endverbatim
82 *>
83 *> \param[in] TAU
84 *> \verbatim
85 *> TAU is REAL array, dimension (K)
86 *> Details of the Householder transformations as returned by
87 *> SGEQPF.
88 *> \endverbatim
89 *>
90 *> \param[in] JPVT
91 *> \verbatim
92 *> JPVT is INTEGER array, dimension (N)
93 *> Pivot information as returned by SGEQPF.
94 *> \endverbatim
95 *>
96 *> \param[out] WORK
97 *> \verbatim
98 *> WORK is REAL array, dimension (LWORK)
99 *> \endverbatim
100 *>
101 *> \param[in] LWORK
102 *> \verbatim
103 *> LWORK is INTEGER
104 *> The length of the array WORK. LWORK >= M*N+N.
105 *> \endverbatim
106 *
107 * Authors:
108 * ========
109 *
110 *> \author Univ. of Tennessee
111 *> \author Univ. of California Berkeley
112 *> \author Univ. of Colorado Denver
113 *> \author NAG Ltd.
114 *
115 *> \ingroup single_lin
116 *
117 * =====================================================================
118  REAL function sqpt01( m, n, k, a, af, lda, tau, jpvt,
119  $ work, lwork )
120 *
121 * -- LAPACK test routine --
122 * -- LAPACK is a software package provided by Univ. of Tennessee, --
123 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
124 *
125 * .. Scalar Arguments ..
126  INTEGER k, lda, lwork, m, n
127 * ..
128 * .. Array Arguments ..
129  INTEGER jpvt( * )
130  REAL a( lda, * ), af( lda, * ), tau( * ),
131  $ work( lwork )
132 * ..
133 *
134 * =====================================================================
135 *
136 * .. Parameters ..
137  REAL zero, one
138  parameter( zero = 0.0e0, one = 1.0e0 )
139 * ..
140 * .. Local Scalars ..
141  INTEGER i, info, j
142  REAL norma
143 * ..
144 * .. Local Arrays ..
145  REAL rwork( 1 )
146 * ..
147 * .. External Functions ..
148  REAL slamch, slange
149  EXTERNAL slamch, slange
150 * ..
151 * .. External Subroutines ..
152  EXTERNAL saxpy, scopy, sormqr, xerbla
153 * ..
154 * .. Intrinsic Functions ..
155  INTRINSIC max, min, real
156 * ..
157 * .. Executable Statements ..
158 *
159  sqpt01 = zero
160 *
161 * Test if there is enough workspace
162 *
163  IF( lwork.LT.m*n+n ) THEN
164  CALL xerbla( 'SQPT01', 10 )
165  RETURN
166  END IF
167 *
168 * Quick return if possible
169 *
170  IF( m.LE.0 .OR. n.LE.0 )
171  $ RETURN
172 *
173  norma = slange( 'One-norm', m, n, a, lda, rwork )
174 *
175  DO 30 j = 1, k
176  DO 10 i = 1, min( j, m )
177  work( ( j-1 )*m+i ) = af( i, j )
178  10 CONTINUE
179  DO 20 i = j + 1, m
180  work( ( j-1 )*m+i ) = zero
181  20 CONTINUE
182  30 CONTINUE
183  DO 40 j = k + 1, n
184  CALL scopy( m, af( 1, j ), 1, work( ( j-1 )*m+1 ), 1 )
185  40 CONTINUE
186 *
187  CALL sormqr( 'Left', 'No transpose', m, n, k, af, lda, tau, work,
188  $ m, work( m*n+1 ), lwork-m*n, info )
189 *
190  DO 50 j = 1, n
191 *
192 * Compare i-th column of QR and jpvt(i)-th column of A
193 *
194  CALL saxpy( m, -one, a( 1, jpvt( j ) ), 1, work( ( j-1 )*m+1 ),
195  $ 1 )
196  50 CONTINUE
197 *
198  sqpt01 = slange( 'One-norm', m, n, work, m, rwork ) /
199  $ ( real( max( m, n ) )*slamch( 'Epsilon' ) )
200  IF( norma.NE.zero )
201  $ sqpt01 = sqpt01 / norma
202 *
203  RETURN
204 *
205 * End of SQPT01
206 *
207  END
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
subroutine sormqr(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMQR
Definition: sormqr.f:168
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
real function sqpt01(M, N, K, A, AF, LDA, TAU, JPVT, WORK, LWORK)
SQPT01
Definition: sqpt01.f:120
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68