LAPACK  3.10.0 LAPACK: Linear Algebra PACKage
sqrt11.f
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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 *> \ingroup single_lin
95 *
96 * =====================================================================
97  REAL function sqrt11( m, k, a, lda, tau, work, lwork )
98 *
99 * -- LAPACK test routine --
100 * -- LAPACK is a software package provided by Univ. of Tennessee, --
101 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
102 *
103 * .. Scalar Arguments ..
104  INTEGER k, lda, lwork, m
105 * ..
106 * .. Array Arguments ..
107  REAL a( lda, * ), tau( * ), work( lwork )
108 * ..
109 *
110 * =====================================================================
111 *
112 * .. Parameters ..
113  REAL zero, one
114  parameter( zero = 0.0e0, one = 1.0e0 )
115 * ..
116 * .. Local Scalars ..
117  INTEGER info, j
118 * ..
119 * .. External Functions ..
120  REAL slamch, slange
121  EXTERNAL slamch, slange
122 * ..
123 * .. External Subroutines ..
124  EXTERNAL slaset, sorm2r, xerbla
125 * ..
126 * .. Intrinsic Functions ..
127  INTRINSIC real
128 * ..
129 * .. Local Arrays ..
130  REAL rdummy( 1 )
131 * ..
132 * .. Executable Statements ..
133 *
134  sqrt11 = zero
135 *
136 * Test for sufficient workspace
137 *
138  IF( lwork.LT.m*m+m ) THEN
139  CALL xerbla( 'SQRT11', 7 )
140  RETURN
141  END IF
142 *
143 * Quick return if possible
144 *
145  IF( m.LE.0 )
146  \$ RETURN
147 *
148  CALL slaset( 'Full', m, m, zero, one, work, m )
149 *
150 * Form Q
151 *
152  CALL sorm2r( 'Left', 'No transpose', m, m, k, a, lda, tau, work,
153  \$ m, work( m*m+1 ), info )
154 *
155 * Form Q'*Q
156 *
157  CALL sorm2r( 'Left', 'Transpose', m, m, k, a, lda, tau, work, m,
158  \$ work( m*m+1 ), info )
159 *
160  DO 10 j = 1, m
161  work( ( j-1 )*m+j ) = work( ( j-1 )*m+j ) - one
162  10 CONTINUE
163 *
164  sqrt11 = slange( 'One-norm', m, m, work, m, rdummy ) /
165  \$ ( real( m )*slamch( 'Epsilon' ) )
166 *
167  RETURN
168 *
169 * End of SQRT11
170 *
171  END
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:110
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 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:159
real function sqrt11(M, K, A, LDA, TAU, WORK, LWORK)
SQRT11
Definition: sqrt11.f:98
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68