LAPACK  3.8.0 LAPACK: Linear Algebra PACKage
sqrt11.f
Go to the documentation of this file.
1 *> \brief \b SQRT11
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 SQRT11( M, K, A, LDA, TAU, WORK, LWORK )
12 *
13 * .. Scalar Arguments ..
14 * INTEGER K, LDA, LWORK, M
15 * ..
16 * .. Array Arguments ..
17 * REAL A( LDA, * ), TAU( * ), WORK( LWORK )
18 * ..
19 *
20 *
21 *> \par Purpose:
22 * =============
23 *>
24 *> \verbatim
25 *>
26 *> SQRT11 computes the test ratio
27 *>
28 *> || Q'*Q - I || / (eps * m)
29 *>
30 *> where the orthogonal matrix Q is represented as a product of
31 *> elementary transformations. Each transformation has the form
32 *>
33 *> H(k) = I - tau(k) v(k) v(k)'
34 *>
35 *> where tau(k) is stored in TAU(k) and v(k) is an m-vector of the form
36 *> [ 0 ... 0 1 x(k) ]', where x(k) is a vector of length m-k stored
37 *> in A(k+1:m,k).
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.
47 *> \endverbatim
48 *>
49 *> \param[in] K
50 *> \verbatim
51 *> K is INTEGER
52 *> The number of columns of A whose subdiagonal entries
53 *> contain information about orthogonal transformations.
54 *> \endverbatim
55 *>
56 *> \param[in] A
57 *> \verbatim
58 *> A is REAL array, dimension (LDA,K)
59 *> The (possibly partial) output of a QR reduction routine.
60 *> \endverbatim
61 *>
62 *> \param[in] LDA
63 *> \verbatim
64 *> LDA is INTEGER
65 *> The leading dimension of the array A.
66 *> \endverbatim
67 *>
68 *> \param[in] TAU
69 *> \verbatim
70 *> TAU is REAL array, dimension (K)
71 *> The scaling factors tau for the elementary transformations as
72 *> computed by the QR factorization routine.
73 *> \endverbatim
74 *>
75 *> \param[out] WORK
76 *> \verbatim
77 *> WORK is REAL array, dimension (LWORK)
78 *> \endverbatim
79 *>
80 *> \param[in] LWORK
81 *> \verbatim
82 *> LWORK is INTEGER
83 *> The length of the array WORK. LWORK >= M*M + M.
84 *> \endverbatim
85 *
86 * Authors:
87 * ========
88 *
89 *> \author Univ. of Tennessee
90 *> \author Univ. of California Berkeley
91 *> \author Univ. of Colorado Denver
92 *> \author NAG Ltd.
93 *
94 *> \date December 2016
95 *
96 *> \ingroup single_lin
97 *
98 * =====================================================================
99  REAL FUNCTION sqrt11( M, K, A, LDA, TAU, WORK, LWORK )
100 *
101 * -- LAPACK test routine (version 3.7.0) --
102 * -- LAPACK is a software package provided by Univ. of Tennessee, --
103 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
104 * December 2016
105 *
106 * .. Scalar Arguments ..
107  INTEGER K, LDA, LWORK, M
108 * ..
109 * .. Array Arguments ..
110  REAL A( lda, * ), TAU( * ), WORK( lwork )
111 * ..
112 *
113 * =====================================================================
114 *
115 * .. Parameters ..
116  REAL ZERO, ONE
117  parameter( zero = 0.0e0, one = 1.0e0 )
118 * ..
119 * .. Local Scalars ..
120  INTEGER INFO, J
121 * ..
122 * .. External Functions ..
123  REAL SLAMCH, SLANGE
124  EXTERNAL slamch, slange
125 * ..
126 * .. External Subroutines ..
127  EXTERNAL slaset, sorm2r, xerbla
128 * ..
129 * .. Intrinsic Functions ..
130  INTRINSIC real
131 * ..
132 * .. Local Arrays ..
133  REAL RDUMMY( 1 )
134 * ..
135 * .. Executable Statements ..
136 *
137  sqrt11 = zero
138 *
139 * Test for sufficient workspace
140 *
141  IF( lwork.LT.m*m+m ) THEN
142  CALL xerbla( 'SQRT11', 7 )
143  RETURN
144  END IF
145 *
146 * Quick return if possible
147 *
148  IF( m.LE.0 )
149  \$ RETURN
150 *
151  CALL slaset( 'Full', m, m, zero, one, work, m )
152 *
153 * Form Q
154 *
155  CALL sorm2r( 'Left', 'No transpose', m, m, k, a, lda, tau, work,
156  \$ m, work( m*m+1 ), info )
157 *
158 * Form Q'*Q
159 *
160  CALL sorm2r( 'Left', 'Transpose', m, m, k, a, lda, tau, work, m,
161  \$ work( m*m+1 ), info )
162 *
163  DO 10 j = 1, m
164  work( ( j-1 )*m+j ) = work( ( j-1 )*m+j ) - one
165  10 CONTINUE
166 *
167  sqrt11 = slange( 'One-norm', m, m, work, m, rdummy ) /
168  \$ ( REAL( m )*SLAMCH( 'Epsilon' ) )
169 *
170  RETURN
171 *
172 * End of SQRT11
173 *
174  END
real function sqrt11(M, K, A, LDA, TAU, WORK, LWORK)
SQRT11
Definition: sqrt11.f:100
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
subroutine sorm2r(SIDE, TRANS, M, N, K, A, LDA, TAU, C, LDC, WORK, INFO)
SORM2R multiplies a general matrix by the orthogonal matrix from a QR factorization determined by sge...
Definition: sorm2r.f:161