LAPACK  3.10.1 LAPACK: Linear Algebra PACKage
sqrt16.f
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1 *> \brief \b SQRT16
2 *
3 * =========== DOCUMENTATION ===========
4 *
5 * Online html documentation available at
6 * http://www.netlib.org/lapack/explore-html/
7 *
8 * Definition:
9 * ===========
10 *
11 * SUBROUTINE SQRT16( TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB,
12 * RWORK, RESID )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER TRANS
16 * INTEGER LDA, LDB, LDX, M, N, NRHS
17 * REAL RESID
18 * ..
19 * .. Array Arguments ..
20 * REAL A( LDA, * ), B( LDB, * ), RWORK( * ),
21 * \$ X( LDX, * )
22 * ..
23 *
24 *
25 *> \par Purpose:
26 * =============
27 *>
28 *> \verbatim
29 *>
30 *> SQRT16 computes the residual for a solution of a system of linear
31 *> equations A*x = b or A'*x = b:
32 *> RESID = norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ),
33 *> where EPS is the machine epsilon.
34 *> \endverbatim
35 *
36 * Arguments:
37 * ==========
38 *
39 *> \param[in] TRANS
40 *> \verbatim
41 *> TRANS is CHARACTER*1
42 *> Specifies the form of the system of equations:
43 *> = 'N': A *x = b
44 *> = 'T': A'*x = b, where A' is the transpose of A
45 *> = 'C': A'*x = b, where A' is the transpose of A
46 *> \endverbatim
47 *>
48 *> \param[in] M
49 *> \verbatim
50 *> M is INTEGER
51 *> The number of rows of the matrix A. M >= 0.
52 *> \endverbatim
53 *>
54 *> \param[in] N
55 *> \verbatim
56 *> N is INTEGER
57 *> The number of columns of the matrix A. N >= 0.
58 *> \endverbatim
59 *>
60 *> \param[in] NRHS
61 *> \verbatim
62 *> NRHS is INTEGER
63 *> The number of columns of B, the matrix of right hand sides.
64 *> NRHS >= 0.
65 *> \endverbatim
66 *>
67 *> \param[in] A
68 *> \verbatim
69 *> A is REAL array, dimension (LDA,N)
70 *> The original M x N matrix A.
71 *> \endverbatim
72 *>
73 *> \param[in] LDA
74 *> \verbatim
75 *> LDA is INTEGER
76 *> The leading dimension of the array A. LDA >= max(1,M).
77 *> \endverbatim
78 *>
79 *> \param[in] X
80 *> \verbatim
81 *> X is REAL array, dimension (LDX,NRHS)
82 *> The computed solution vectors for the system of linear
83 *> equations.
84 *> \endverbatim
85 *>
86 *> \param[in] LDX
87 *> \verbatim
88 *> LDX is INTEGER
89 *> The leading dimension of the array X. If TRANS = 'N',
90 *> LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M).
91 *> \endverbatim
92 *>
93 *> \param[in,out] B
94 *> \verbatim
95 *> B is REAL array, dimension (LDB,NRHS)
96 *> On entry, the right hand side vectors for the system of
97 *> linear equations.
98 *> On exit, B is overwritten with the difference B - A*X.
99 *> \endverbatim
100 *>
101 *> \param[in] LDB
102 *> \verbatim
103 *> LDB is INTEGER
104 *> The leading dimension of the array B. IF TRANS = 'N',
105 *> LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N).
106 *> \endverbatim
107 *>
108 *> \param[out] RWORK
109 *> \verbatim
110 *> RWORK is REAL array, dimension (M)
111 *> \endverbatim
112 *>
113 *> \param[out] RESID
114 *> \verbatim
115 *> RESID is REAL
116 *> The maximum over the number of right hand sides of
117 *> norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ).
118 *> \endverbatim
119 *
120 * Authors:
121 * ========
122 *
123 *> \author Univ. of Tennessee
124 *> \author Univ. of California Berkeley
125 *> \author Univ. of Colorado Denver
126 *> \author NAG Ltd.
127 *
128 *> \ingroup single_lin
129 *
130 * =====================================================================
131  SUBROUTINE sqrt16( TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB,
132  \$ RWORK, RESID )
133 *
134 * -- LAPACK test routine --
135 * -- LAPACK is a software package provided by Univ. of Tennessee, --
136 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
137 *
138 * .. Scalar Arguments ..
139  CHARACTER TRANS
140  INTEGER LDA, LDB, LDX, M, N, NRHS
141  REAL RESID
142 * ..
143 * .. Array Arguments ..
144  REAL A( LDA, * ), B( LDB, * ), RWORK( * ),
145  \$ x( ldx, * )
146 * ..
147 *
148 * =====================================================================
149 *
150 * .. Parameters ..
151  REAL ZERO, ONE
152  parameter( zero = 0.0e+0, one = 1.0e+0 )
153 * ..
154 * .. Local Scalars ..
155  INTEGER J, N1, N2
156  REAL ANORM, BNORM, EPS, XNORM
157 * ..
158 * .. External Functions ..
159  LOGICAL LSAME
160  REAL SASUM, SLAMCH, SLANGE
161  EXTERNAL lsame, sasum, slamch, slange
162 * ..
163 * .. External Subroutines ..
164  EXTERNAL sgemm
165 * ..
166 * .. Intrinsic Functions ..
167  INTRINSIC max
168 * ..
169 * .. Executable Statements ..
170 *
171 * Quick exit if M = 0 or N = 0 or NRHS = 0
172 *
173  IF( m.LE.0 .OR. n.LE.0 .OR. nrhs.EQ.0 ) THEN
174  resid = zero
175  RETURN
176  END IF
177 *
178  IF( lsame( trans, 'T' ) .OR. lsame( trans, 'C' ) ) THEN
179  anorm = slange( 'I', m, n, a, lda, rwork )
180  n1 = n
181  n2 = m
182  ELSE
183  anorm = slange( '1', m, n, a, lda, rwork )
184  n1 = m
185  n2 = n
186  END IF
187 *
188  eps = slamch( 'Epsilon' )
189 *
190 * Compute B - A*X (or B - A'*X ) and store in B.
191 *
192  CALL sgemm( trans, 'No transpose', n1, nrhs, n2, -one, a, lda, x,
193  \$ ldx, one, b, ldb )
194 *
195 * Compute the maximum over the number of right hand sides of
196 * norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ) .
197 *
198  resid = zero
199  DO 10 j = 1, nrhs
200  bnorm = sasum( n1, b( 1, j ), 1 )
201  xnorm = sasum( n2, x( 1, j ), 1 )
202  IF( anorm.EQ.zero .AND. bnorm.EQ.zero ) THEN
203  resid = zero
204  ELSE IF( anorm.LE.zero .OR. xnorm.LE.zero ) THEN
205  resid = one / eps
206  ELSE
207  resid = max( resid, ( ( bnorm / anorm ) / xnorm ) /
208  \$ ( max( m, n )*eps ) )
209  END IF
210  10 CONTINUE
211 *
212  RETURN
213 *
214 * End of SQRT16
215 *
216  END
subroutine sgemm(TRANSA, TRANSB, M, N, K, ALPHA, A, LDA, B, LDB, BETA, C, LDC)
SGEMM
Definition: sgemm.f:187
subroutine sqrt16(TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB, RWORK, RESID)
SQRT16
Definition: sqrt16.f:133