LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
ctrt05.f
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1 *> \brief \b CTRT05
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 CTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
12 * LDX, XACT, LDXACT, FERR, BERR, RESLTS )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER DIAG, TRANS, UPLO
16 * INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
17 * ..
18 * .. Array Arguments ..
19 * REAL BERR( * ), FERR( * ), RESLTS( * )
20 * COMPLEX A( LDA, * ), B( LDB, * ), X( LDX, * ),
21 * $ XACT( LDXACT, * )
22 * ..
23 *
24 *
25 *> \par Purpose:
26 * =============
27 *>
28 *> \verbatim
29 *>
30 *> CTRT05 tests the error bounds from iterative refinement for the
31 *> computed solution to a system of equations A*X = B, where A is a
32 *> triangular n by n matrix.
33 *>
34 *> RESLTS(1) = test of the error bound
35 *> = norm(X - XACT) / ( norm(X) * FERR )
36 *>
37 *> A large value is returned if this ratio is not less than one.
38 *>
39 *> RESLTS(2) = residual from the iterative refinement routine
40 *> = the maximum of BERR / ( (n+1)*EPS + (*) ), where
41 *> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
42 *> \endverbatim
43 *
44 * Arguments:
45 * ==========
46 *
47 *> \param[in] UPLO
48 *> \verbatim
49 *> UPLO is CHARACTER*1
50 *> Specifies whether the matrix A is upper or lower triangular.
51 *> = 'U': Upper triangular
52 *> = 'L': Lower triangular
53 *> \endverbatim
54 *>
55 *> \param[in] TRANS
56 *> \verbatim
57 *> TRANS is CHARACTER*1
58 *> Specifies the form of the system of equations.
59 *> = 'N': A * X = B (No transpose)
60 *> = 'T': A'* X = B (Transpose)
61 *> = 'C': A'* X = B (Conjugate transpose = Transpose)
62 *> \endverbatim
63 *>
64 *> \param[in] DIAG
65 *> \verbatim
66 *> DIAG is CHARACTER*1
67 *> Specifies whether or not the matrix A is unit triangular.
68 *> = 'N': Non-unit triangular
69 *> = 'U': Unit triangular
70 *> \endverbatim
71 *>
72 *> \param[in] N
73 *> \verbatim
74 *> N is INTEGER
75 *> The number of rows of the matrices X, B, and XACT, and the
76 *> order of the matrix A. N >= 0.
77 *> \endverbatim
78 *>
79 *> \param[in] NRHS
80 *> \verbatim
81 *> NRHS is INTEGER
82 *> The number of columns of the matrices X, B, and XACT.
83 *> NRHS >= 0.
84 *> \endverbatim
85 *>
86 *> \param[in] A
87 *> \verbatim
88 *> A is COMPLEX array, dimension (LDA,N)
89 *> The triangular matrix A. If UPLO = 'U', the leading n by n
90 *> upper triangular part of the array A contains the upper
91 *> triangular matrix, and the strictly lower triangular part of
92 *> A is not referenced. If UPLO = 'L', the leading n by n lower
93 *> triangular part of the array A contains the lower triangular
94 *> matrix, and the strictly upper triangular part of A is not
95 *> referenced. If DIAG = 'U', the diagonal elements of A are
96 *> also not referenced and are assumed to be 1.
97 *> \endverbatim
98 *>
99 *> \param[in] LDA
100 *> \verbatim
101 *> LDA is INTEGER
102 *> The leading dimension of the array A. LDA >= max(1,N).
103 *> \endverbatim
104 *>
105 *> \param[in] B
106 *> \verbatim
107 *> B is COMPLEX array, dimension (LDB,NRHS)
108 *> The right hand side vectors for the system of linear
109 *> equations.
110 *> \endverbatim
111 *>
112 *> \param[in] LDB
113 *> \verbatim
114 *> LDB is INTEGER
115 *> The leading dimension of the array B. LDB >= max(1,N).
116 *> \endverbatim
117 *>
118 *> \param[in] X
119 *> \verbatim
120 *> X is COMPLEX array, dimension (LDX,NRHS)
121 *> The computed solution vectors. Each vector is stored as a
122 *> column of the matrix X.
123 *> \endverbatim
124 *>
125 *> \param[in] LDX
126 *> \verbatim
127 *> LDX is INTEGER
128 *> The leading dimension of the array X. LDX >= max(1,N).
129 *> \endverbatim
130 *>
131 *> \param[in] XACT
132 *> \verbatim
133 *> XACT is COMPLEX array, dimension (LDX,NRHS)
134 *> The exact solution vectors. Each vector is stored as a
135 *> column of the matrix XACT.
136 *> \endverbatim
137 *>
138 *> \param[in] LDXACT
139 *> \verbatim
140 *> LDXACT is INTEGER
141 *> The leading dimension of the array XACT. LDXACT >= max(1,N).
142 *> \endverbatim
143 *>
144 *> \param[in] FERR
145 *> \verbatim
146 *> FERR is REAL array, dimension (NRHS)
147 *> The estimated forward error bounds for each solution vector
148 *> X. If XTRUE is the true solution, FERR bounds the magnitude
149 *> of the largest entry in (X - XTRUE) divided by the magnitude
150 *> of the largest entry in X.
151 *> \endverbatim
152 *>
153 *> \param[in] BERR
154 *> \verbatim
155 *> BERR is REAL array, dimension (NRHS)
156 *> The componentwise relative backward error of each solution
157 *> vector (i.e., the smallest relative change in any entry of A
158 *> or B that makes X an exact solution).
159 *> \endverbatim
160 *>
161 *> \param[out] RESLTS
162 *> \verbatim
163 *> RESLTS is REAL array, dimension (2)
164 *> The maximum over the NRHS solution vectors of the ratios:
165 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
166 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
167 *> \endverbatim
168 *
169 * Authors:
170 * ========
171 *
172 *> \author Univ. of Tennessee
173 *> \author Univ. of California Berkeley
174 *> \author Univ. of Colorado Denver
175 *> \author NAG Ltd.
176 *
177 *> \ingroup complex_lin
178 *
179 * =====================================================================
180  SUBROUTINE ctrt05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
181  $ LDX, XACT, LDXACT, FERR, BERR, RESLTS )
182 *
183 * -- LAPACK test routine --
184 * -- LAPACK is a software package provided by Univ. of Tennessee, --
185 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
186 *
187 * .. Scalar Arguments ..
188  CHARACTER DIAG, TRANS, UPLO
189  INTEGER LDA, LDB, LDX, LDXACT, N, NRHS
190 * ..
191 * .. Array Arguments ..
192  REAL BERR( * ), FERR( * ), RESLTS( * )
193  COMPLEX A( LDA, * ), B( LDB, * ), X( LDX, * ),
194  $ xact( ldxact, * )
195 * ..
196 *
197 * =====================================================================
198 *
199 * .. Parameters ..
200  REAL ZERO, ONE
201  parameter( zero = 0.0e+0, one = 1.0e+0 )
202 * ..
203 * .. Local Scalars ..
204  LOGICAL NOTRAN, UNIT, UPPER
205  INTEGER I, IFU, IMAX, J, K
206  REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
207  COMPLEX ZDUM
208 * ..
209 * .. External Functions ..
210  LOGICAL LSAME
211  INTEGER ICAMAX
212  REAL SLAMCH
213  EXTERNAL lsame, icamax, slamch
214 * ..
215 * .. Intrinsic Functions ..
216  INTRINSIC abs, aimag, max, min, real
217 * ..
218 * .. Statement Functions ..
219  REAL CABS1
220 * ..
221 * .. Statement Function definitions ..
222  cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
223 * ..
224 * .. Executable Statements ..
225 *
226 * Quick exit if N = 0 or NRHS = 0.
227 *
228  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
229  reslts( 1 ) = zero
230  reslts( 2 ) = zero
231  RETURN
232  END IF
233 *
234  eps = slamch( 'Epsilon' )
235  unfl = slamch( 'Safe minimum' )
236  ovfl = one / unfl
237  upper = lsame( uplo, 'U' )
238  notran = lsame( trans, 'N' )
239  unit = lsame( diag, 'U' )
240 *
241 * Test 1: Compute the maximum of
242 * norm(X - XACT) / ( norm(X) * FERR )
243 * over all the vectors X and XACT using the infinity-norm.
244 *
245  errbnd = zero
246  DO 30 j = 1, nrhs
247  imax = icamax( n, x( 1, j ), 1 )
248  xnorm = max( cabs1( x( imax, j ) ), unfl )
249  diff = zero
250  DO 10 i = 1, n
251  diff = max( diff, cabs1( x( i, j )-xact( i, j ) ) )
252  10 CONTINUE
253 *
254  IF( xnorm.GT.one ) THEN
255  GO TO 20
256  ELSE IF( diff.LE.ovfl*xnorm ) THEN
257  GO TO 20
258  ELSE
259  errbnd = one / eps
260  GO TO 30
261  END IF
262 *
263  20 CONTINUE
264  IF( diff / xnorm.LE.ferr( j ) ) THEN
265  errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
266  ELSE
267  errbnd = one / eps
268  END IF
269  30 CONTINUE
270  reslts( 1 ) = errbnd
271 *
272 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
273 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
274 *
275  ifu = 0
276  IF( unit )
277  $ ifu = 1
278  DO 90 k = 1, nrhs
279  DO 80 i = 1, n
280  tmp = cabs1( b( i, k ) )
281  IF( upper ) THEN
282  IF( .NOT.notran ) THEN
283  DO 40 j = 1, i - ifu
284  tmp = tmp + cabs1( a( j, i ) )*cabs1( x( j, k ) )
285  40 CONTINUE
286  IF( unit )
287  $ tmp = tmp + cabs1( x( i, k ) )
288  ELSE
289  IF( unit )
290  $ tmp = tmp + cabs1( x( i, k ) )
291  DO 50 j = i + ifu, n
292  tmp = tmp + cabs1( a( i, j ) )*cabs1( x( j, k ) )
293  50 CONTINUE
294  END IF
295  ELSE
296  IF( notran ) THEN
297  DO 60 j = 1, i - ifu
298  tmp = tmp + cabs1( a( i, j ) )*cabs1( x( j, k ) )
299  60 CONTINUE
300  IF( unit )
301  $ tmp = tmp + cabs1( x( i, k ) )
302  ELSE
303  IF( unit )
304  $ tmp = tmp + cabs1( x( i, k ) )
305  DO 70 j = i + ifu, n
306  tmp = tmp + cabs1( a( j, i ) )*cabs1( x( j, k ) )
307  70 CONTINUE
308  END IF
309  END IF
310  IF( i.EQ.1 ) THEN
311  axbi = tmp
312  ELSE
313  axbi = min( axbi, tmp )
314  END IF
315  80 CONTINUE
316  tmp = berr( k ) / ( ( n+1 )*eps+( n+1 )*unfl /
317  $ max( axbi, ( n+1 )*unfl ) )
318  IF( k.EQ.1 ) THEN
319  reslts( 2 ) = tmp
320  ELSE
321  reslts( 2 ) = max( reslts( 2 ), tmp )
322  END IF
323  90 CONTINUE
324 *
325  RETURN
326 *
327 * End of CTRT05
328 *
329  END
subroutine ctrt05(UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS)
CTRT05
Definition: ctrt05.f:182