LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
cgtt05.f
Go to the documentation of this file.
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 *> \date December 2016
161 *
162 *> \ingroup complex_lin
163 *
164 * =====================================================================
165  SUBROUTINE cgtt05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
166  $ XACT, LDXACT, FERR, BERR, RESLTS )
167 *
168 * -- LAPACK test routine (version 3.7.0) --
169 * -- LAPACK is a software package provided by Univ. of Tennessee, --
170 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
171 * December 2016
172 *
173 * .. Scalar Arguments ..
174  CHARACTER TRANS
175  INTEGER LDB, LDX, LDXACT, N, NRHS
176 * ..
177 * .. Array Arguments ..
178  REAL BERR( * ), FERR( * ), RESLTS( * )
179  COMPLEX B( ldb, * ), D( * ), DL( * ), DU( * ),
180  $ x( ldx, * ), xact( ldxact, * )
181 * ..
182 *
183 * =====================================================================
184 *
185 * .. Parameters ..
186  REAL ZERO, ONE
187  parameter( zero = 0.0e+0, one = 1.0e+0 )
188 * ..
189 * .. Local Scalars ..
190  LOGICAL NOTRAN
191  INTEGER I, IMAX, J, K, NZ
192  REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
193  COMPLEX ZDUM
194 * ..
195 * .. External Functions ..
196  LOGICAL LSAME
197  INTEGER ICAMAX
198  REAL SLAMCH
199  EXTERNAL lsame, icamax, slamch
200 * ..
201 * .. Intrinsic Functions ..
202  INTRINSIC abs, aimag, max, min, real
203 * ..
204 * .. Statement Functions ..
205  REAL CABS1
206 * ..
207 * .. Statement Function definitions ..
208  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( AIMAG( zdum ) )
209 * ..
210 * .. Executable Statements ..
211 *
212 * Quick exit if N = 0 or NRHS = 0.
213 *
214  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
215  reslts( 1 ) = zero
216  reslts( 2 ) = zero
217  RETURN
218  END IF
219 *
220  eps = slamch( 'Epsilon' )
221  unfl = slamch( 'Safe minimum' )
222  ovfl = one / unfl
223  notran = lsame( trans, 'N' )
224  nz = 4
225 *
226 * Test 1: Compute the maximum of
227 * norm(X - XACT) / ( norm(X) * FERR )
228 * over all the vectors X and XACT using the infinity-norm.
229 *
230  errbnd = zero
231  DO 30 j = 1, nrhs
232  imax = icamax( n, x( 1, j ), 1 )
233  xnorm = max( cabs1( x( imax, j ) ), unfl )
234  diff = zero
235  DO 10 i = 1, n
236  diff = max( diff, cabs1( x( i, j )-xact( i, j ) ) )
237  10 CONTINUE
238 *
239  IF( xnorm.GT.one ) THEN
240  GO TO 20
241  ELSE IF( diff.LE.ovfl*xnorm ) THEN
242  GO TO 20
243  ELSE
244  errbnd = one / eps
245  GO TO 30
246  END IF
247 *
248  20 CONTINUE
249  IF( diff / xnorm.LE.ferr( j ) ) THEN
250  errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
251  ELSE
252  errbnd = one / eps
253  END IF
254  30 CONTINUE
255  reslts( 1 ) = errbnd
256 *
257 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
258 * (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
259 *
260  DO 60 k = 1, nrhs
261  IF( notran ) THEN
262  IF( n.EQ.1 ) THEN
263  axbi = cabs1( b( 1, k ) ) +
264  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) )
265  ELSE
266  axbi = cabs1( b( 1, k ) ) +
267  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) ) +
268  $ cabs1( du( 1 ) )*cabs1( x( 2, k ) )
269  DO 40 i = 2, n - 1
270  tmp = cabs1( b( i, k ) ) +
271  $ cabs1( dl( i-1 ) )*cabs1( x( i-1, k ) ) +
272  $ cabs1( d( i ) )*cabs1( x( i, k ) ) +
273  $ cabs1( du( i ) )*cabs1( x( i+1, k ) )
274  axbi = min( axbi, tmp )
275  40 CONTINUE
276  tmp = cabs1( b( n, k ) ) + cabs1( dl( n-1 ) )*
277  $ cabs1( x( n-1, k ) ) + cabs1( d( n ) )*
278  $ cabs1( x( n, k ) )
279  axbi = min( axbi, tmp )
280  END IF
281  ELSE
282  IF( n.EQ.1 ) THEN
283  axbi = cabs1( b( 1, k ) ) +
284  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) )
285  ELSE
286  axbi = cabs1( b( 1, k ) ) +
287  $ cabs1( d( 1 ) )*cabs1( x( 1, k ) ) +
288  $ cabs1( dl( 1 ) )*cabs1( x( 2, k ) )
289  DO 50 i = 2, n - 1
290  tmp = cabs1( b( i, k ) ) +
291  $ cabs1( du( i-1 ) )*cabs1( x( i-1, k ) ) +
292  $ cabs1( d( i ) )*cabs1( x( i, k ) ) +
293  $ cabs1( dl( i ) )*cabs1( x( i+1, k ) )
294  axbi = min( axbi, tmp )
295  50 CONTINUE
296  tmp = cabs1( b( n, k ) ) + cabs1( du( n-1 ) )*
297  $ cabs1( x( n-1, k ) ) + cabs1( d( n ) )*
298  $ cabs1( x( n, k ) )
299  axbi = min( axbi, tmp )
300  END IF
301  END IF
302  tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
303  IF( k.EQ.1 ) THEN
304  reslts( 2 ) = tmp
305  ELSE
306  reslts( 2 ) = max( reslts( 2 ), tmp )
307  END IF
308  60 CONTINUE
309 *
310  RETURN
311 *
312 * End of CGTT05
313 *
314  END
subroutine cgtt05(TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS)
CGTT05
Definition: cgtt05.f:167