LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
cgtt05.f
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1 *> \brief \b CGTT05
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 CGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
12 * XACT, LDXACT, FERR, BERR, RESLTS )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER TRANS
16 * INTEGER LDB, LDX, LDXACT, N, NRHS
17 * ..
18 * .. Array Arguments ..
19 * REAL BERR( * ), FERR( * ), RESLTS( * )
20 * COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ),
21 * $ X( LDX, * ), XACT( LDXACT, * )
22 * ..
23 *
24 *
25 *> \par Purpose:
26 * =============
27 *>
28 *> \verbatim
29 *>
30 *> CGTT05 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 *> general tridiagonal matrix of order n and op(A) = A or A**T,
33 *> depending on TRANS.
34 *>
35 *> RESLTS(1) = test of the error bound
36 *> = norm(X - XACT) / ( norm(X) * FERR )
37 *>
38 *> A large value is returned if this ratio is not less than one.
39 *>
40 *> RESLTS(2) = residual from the iterative refinement routine
41 *> = the maximum of BERR / ( NZ*EPS + (*) ), where
42 *> (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
43 *> and NZ = max. number of nonzeros in any row of A, plus 1
44 *> \endverbatim
45 *
46 * Arguments:
47 * ==========
48 *
49 *> \param[in] TRANS
50 *> \verbatim
51 *> TRANS is CHARACTER*1
52 *> Specifies the form of the system of equations.
53 *> = 'N': A * X = B (No transpose)
54 *> = 'T': A**T * X = B (Transpose)
55 *> = 'C': A**H * X = B (Conjugate transpose = Transpose)
56 *> \endverbatim
57 *>
58 *> \param[in] N
59 *> \verbatim
60 *> N is INTEGER
61 *> The number of rows of the matrices X and XACT. N >= 0.
62 *> \endverbatim
63 *>
64 *> \param[in] NRHS
65 *> \verbatim
66 *> NRHS is INTEGER
67 *> The number of columns of the matrices X and XACT. NRHS >= 0.
68 *> \endverbatim
69 *>
70 *> \param[in] DL
71 *> \verbatim
72 *> DL is COMPLEX array, dimension (N-1)
73 *> The (n-1) sub-diagonal elements of A.
74 *> \endverbatim
75 *>
76 *> \param[in] D
77 *> \verbatim
78 *> D is COMPLEX array, dimension (N)
79 *> The diagonal elements of A.
80 *> \endverbatim
81 *>
82 *> \param[in] DU
83 *> \verbatim
84 *> DU is COMPLEX array, dimension (N-1)
85 *> The (n-1) super-diagonal elements of A.
86 *> \endverbatim
87 *>
88 *> \param[in] B
89 *> \verbatim
90 *> B is COMPLEX array, dimension (LDB,NRHS)
91 *> The right hand side vectors for the system of linear
92 *> equations.
93 *> \endverbatim
94 *>
95 *> \param[in] LDB
96 *> \verbatim
97 *> LDB is INTEGER
98 *> The leading dimension of the array B. LDB >= max(1,N).
99 *> \endverbatim
100 *>
101 *> \param[in] X
102 *> \verbatim
103 *> X is COMPLEX array, dimension (LDX,NRHS)
104 *> The computed solution vectors. Each vector is stored as a
105 *> column of the matrix X.
106 *> \endverbatim
107 *>
108 *> \param[in] LDX
109 *> \verbatim
110 *> LDX is INTEGER
111 *> The leading dimension of the array X. LDX >= max(1,N).
112 *> \endverbatim
113 *>
114 *> \param[in] XACT
115 *> \verbatim
116 *> XACT is COMPLEX array, dimension (LDX,NRHS)
117 *> The exact solution vectors. Each vector is stored as a
118 *> column of the matrix XACT.
119 *> \endverbatim
120 *>
121 *> \param[in] LDXACT
122 *> \verbatim
123 *> LDXACT is INTEGER
124 *> The leading dimension of the array XACT. LDXACT >= max(1,N).
125 *> \endverbatim
126 *>
127 *> \param[in] FERR
128 *> \verbatim
129 *> FERR is REAL array, dimension (NRHS)
130 *> The estimated forward error bounds for each solution vector
131 *> X. If XTRUE is the true solution, FERR bounds the magnitude
132 *> of the largest entry in (X - XTRUE) divided by the magnitude
133 *> of the largest entry in X.
134 *> \endverbatim
135 *>
136 *> \param[in] BERR
137 *> \verbatim
138 *> BERR is REAL array, dimension (NRHS)
139 *> The componentwise relative backward error of each solution
140 *> vector (i.e., the smallest relative change in any entry of A
141 *> or B that makes X an exact solution).
142 *> \endverbatim
143 *>
144 *> \param[out] RESLTS
145 *> \verbatim
146 *> RESLTS is REAL array, dimension (2)
147 *> The maximum over the NRHS solution vectors of the ratios:
148 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
149 *> RESLTS(2) = BERR / ( NZ*EPS + (*) )
150 *> \endverbatim
151 *
152 * Authors:
153 * ========
154 *
155 *> \author Univ. of Tennessee
156 *> \author Univ. of California Berkeley
157 *> \author Univ. of Colorado Denver
158 *> \author NAG Ltd.
159 *
160 *> \ingroup complex_lin
161 *
162 * =====================================================================
163  SUBROUTINE cgtt05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
164  $ XACT, LDXACT, FERR, BERR, RESLTS )
165 *
166 * -- LAPACK test routine --
167 * -- LAPACK is a software package provided by Univ. of Tennessee, --
168 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169 *
170 * .. Scalar Arguments ..
171  CHARACTER TRANS
172  INTEGER LDB, LDX, LDXACT, N, NRHS
173 * ..
174 * .. Array Arguments ..
175  REAL BERR( * ), FERR( * ), RESLTS( * )
176  COMPLEX B( LDB, * ), D( * ), DL( * ), DU( * ),
177  $ x( ldx, * ), xact( ldxact, * )
178 * ..
179 *
180 * =====================================================================
181 *
182 * .. Parameters ..
183  REAL ZERO, ONE
184  parameter( zero = 0.0e+0, one = 1.0e+0 )
185 * ..
186 * .. Local Scalars ..
187  LOGICAL NOTRAN
188  INTEGER I, IMAX, J, K, NZ
189  REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
190  COMPLEX ZDUM
191 * ..
192 * .. External Functions ..
193  LOGICAL LSAME
194  INTEGER ICAMAX
195  REAL SLAMCH
196  EXTERNAL lsame, icamax, slamch
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC abs, aimag, max, min, real
200 * ..
201 * .. Statement Functions ..
202  REAL CABS1
203 * ..
204 * .. Statement Function definitions ..
205  cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
206 * ..
207 * .. Executable Statements ..
208 *
209 * Quick exit if N = 0 or NRHS = 0.
210 *
211  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
212  reslts( 1 ) = zero
213  reslts( 2 ) = zero
214  RETURN
215  END IF
216 *
217  eps = slamch( 'Epsilon' )
218  unfl = slamch( 'Safe minimum' )
219  ovfl = one / unfl
220  notran = lsame( trans, 'N' )
221  nz = 4
222 *
223 * Test 1: Compute the maximum of
224 * norm(X - XACT) / ( norm(X) * FERR )
225 * over all the vectors X and XACT using the infinity-norm.
226 *
227  errbnd = zero
228  DO 30 j = 1, nrhs
229  imax = icamax( n, x( 1, j ), 1 )
230  xnorm = max( cabs1( x( imax, j ) ), unfl )
231  diff = zero
232  DO 10 i = 1, n
233  diff = max( diff, cabs1( x( i, j )-xact( i, j ) ) )
234  10 CONTINUE
235 *
236  IF( xnorm.GT.one ) THEN
237  GO TO 20
238  ELSE IF( diff.LE.ovfl*xnorm ) THEN
239  GO TO 20
240  ELSE
241  errbnd = one / eps
242  GO TO 30
243  END IF
244 *
245  20 CONTINUE
246  IF( diff / xnorm.LE.ferr( j ) ) THEN
247  errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
248  ELSE
249  errbnd = one / eps
250  END IF
251  30 CONTINUE
252  reslts( 1 ) = errbnd
253 *
254 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
255 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
256 *
257  DO 60 k = 1, nrhs
258  IF( notran ) THEN
259  IF( n.EQ.1 ) THEN
260  axbi = cabs1( b( 1, k ) ) +
261  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) )
262  ELSE
263  axbi = cabs1( b( 1, k ) ) +
264  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) ) +
265  $ cabs1( du( 1 ) )*cabs1( x( 2, k ) )
266  DO 40 i = 2, n - 1
267  tmp = cabs1( b( i, k ) ) +
268  $ cabs1( dl( i-1 ) )*cabs1( x( i-1, k ) ) +
269  $ cabs1( d( i ) )*cabs1( x( i, k ) ) +
270  $ cabs1( du( i ) )*cabs1( x( i+1, k ) )
271  axbi = min( axbi, tmp )
272  40 CONTINUE
273  tmp = cabs1( b( n, k ) ) + cabs1( dl( n-1 ) )*
274  $ cabs1( x( n-1, k ) ) + cabs1( d( n ) )*
275  $ cabs1( x( n, k ) )
276  axbi = min( axbi, tmp )
277  END IF
278  ELSE
279  IF( n.EQ.1 ) THEN
280  axbi = cabs1( b( 1, k ) ) +
281  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) )
282  ELSE
283  axbi = cabs1( b( 1, k ) ) +
284  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) ) +
285  $ cabs1( dl( 1 ) )*cabs1( x( 2, k ) )
286  DO 50 i = 2, n - 1
287  tmp = cabs1( b( i, k ) ) +
288  $ cabs1( du( i-1 ) )*cabs1( x( i-1, k ) ) +
289  $ cabs1( d( i ) )*cabs1( x( i, k ) ) +
290  $ cabs1( dl( i ) )*cabs1( x( i+1, k ) )
291  axbi = min( axbi, tmp )
292  50 CONTINUE
293  tmp = cabs1( b( n, k ) ) + cabs1( du( n-1 ) )*
294  $ cabs1( x( n-1, k ) ) + cabs1( d( n ) )*
295  $ cabs1( x( n, k ) )
296  axbi = min( axbi, tmp )
297  END IF
298  END IF
299  tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
300  IF( k.EQ.1 ) THEN
301  reslts( 2 ) = tmp
302  ELSE
303  reslts( 2 ) = max( reslts( 2 ), tmp )
304  END IF
305  60 CONTINUE
306 *
307  RETURN
308 *
309 * End of CGTT05
310 *
311  END
subroutine cgtt05(TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS)
CGTT05
Definition: cgtt05.f:165