LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
dptt05.f
Go to the documentation of this file.
1 *> \brief \b DPTT05
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 DPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT,
12 * FERR, BERR, RESLTS )
13 *
14 * .. Scalar Arguments ..
15 * INTEGER LDB, LDX, LDXACT, N, NRHS
16 * ..
17 * .. Array Arguments ..
18 * DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), E( * ),
19 * $ FERR( * ), RESLTS( * ), X( LDX, * ),
20 * $ XACT( LDXACT, * )
21 * ..
22 *
23 *
24 *> \par Purpose:
25 * =============
26 *>
27 *> \verbatim
28 *>
29 *> DPTT05 tests the error bounds from iterative refinement for the
30 *> computed solution to a system of equations A*X = B, where A is a
31 *> symmetric tridiagonal matrix of order n.
32 *>
33 *> RESLTS(1) = test of the error bound
34 *> = norm(X - XACT) / ( norm(X) * FERR )
35 *>
36 *> A large value is returned if this ratio is not less than one.
37 *>
38 *> RESLTS(2) = residual from the iterative refinement routine
39 *> = the maximum of BERR / ( NZ*EPS + (*) ), where
40 *> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
41 *> and NZ = max. number of nonzeros in any row of A, plus 1
42 *> \endverbatim
43 *
44 * Arguments:
45 * ==========
46 *
47 *> \param[in] N
48 *> \verbatim
49 *> N is INTEGER
50 *> The number of rows of the matrices X, B, and XACT, and the
51 *> order of the matrix A. N >= 0.
52 *> \endverbatim
53 *>
54 *> \param[in] NRHS
55 *> \verbatim
56 *> NRHS is INTEGER
57 *> The number of columns of the matrices X, B, and XACT.
58 *> NRHS >= 0.
59 *> \endverbatim
60 *>
61 *> \param[in] D
62 *> \verbatim
63 *> D is DOUBLE PRECISION array, dimension (N)
64 *> The n diagonal elements of the tridiagonal matrix A.
65 *> \endverbatim
66 *>
67 *> \param[in] E
68 *> \verbatim
69 *> E is DOUBLE PRECISION array, dimension (N-1)
70 *> The (n-1) subdiagonal elements of the tridiagonal matrix A.
71 *> \endverbatim
72 *>
73 *> \param[in] B
74 *> \verbatim
75 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
76 *> The right hand side vectors for the system of linear
77 *> equations.
78 *> \endverbatim
79 *>
80 *> \param[in] LDB
81 *> \verbatim
82 *> LDB is INTEGER
83 *> The leading dimension of the array B. LDB >= max(1,N).
84 *> \endverbatim
85 *>
86 *> \param[in] X
87 *> \verbatim
88 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
89 *> The computed solution vectors. Each vector is stored as a
90 *> column of the matrix X.
91 *> \endverbatim
92 *>
93 *> \param[in] LDX
94 *> \verbatim
95 *> LDX is INTEGER
96 *> The leading dimension of the array X. LDX >= max(1,N).
97 *> \endverbatim
98 *>
99 *> \param[in] XACT
100 *> \verbatim
101 *> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
102 *> The exact solution vectors. Each vector is stored as a
103 *> column of the matrix XACT.
104 *> \endverbatim
105 *>
106 *> \param[in] LDXACT
107 *> \verbatim
108 *> LDXACT is INTEGER
109 *> The leading dimension of the array XACT. LDXACT >= max(1,N).
110 *> \endverbatim
111 *>
112 *> \param[in] FERR
113 *> \verbatim
114 *> FERR is DOUBLE PRECISION array, dimension (NRHS)
115 *> The estimated forward error bounds for each solution vector
116 *> X. If XTRUE is the true solution, FERR bounds the magnitude
117 *> of the largest entry in (X - XTRUE) divided by the magnitude
118 *> of the largest entry in X.
119 *> \endverbatim
120 *>
121 *> \param[in] BERR
122 *> \verbatim
123 *> BERR is DOUBLE PRECISION array, dimension (NRHS)
124 *> The componentwise relative backward error of each solution
125 *> vector (i.e., the smallest relative change in any entry of A
126 *> or B that makes X an exact solution).
127 *> \endverbatim
128 *>
129 *> \param[out] RESLTS
130 *> \verbatim
131 *> RESLTS is DOUBLE PRECISION array, dimension (2)
132 *> The maximum over the NRHS solution vectors of the ratios:
133 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
134 *> RESLTS(2) = BERR / ( NZ*EPS + (*) )
135 *> \endverbatim
136 *
137 * Authors:
138 * ========
139 *
140 *> \author Univ. of Tennessee
141 *> \author Univ. of California Berkeley
142 *> \author Univ. of Colorado Denver
143 *> \author NAG Ltd.
144 *
145 *> \date December 2016
146 *
147 *> \ingroup double_lin
148 *
149 * =====================================================================
150  SUBROUTINE dptt05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT,
151  $ FERR, BERR, RESLTS )
152 *
153 * -- LAPACK test routine (version 3.7.0) --
154 * -- LAPACK is a software package provided by Univ. of Tennessee, --
155 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
156 * December 2016
157 *
158 * .. Scalar Arguments ..
159  INTEGER LDB, LDX, LDXACT, N, NRHS
160 * ..
161 * .. Array Arguments ..
162  DOUBLE PRECISION B( ldb, * ), BERR( * ), D( * ), E( * ),
163  $ ferr( * ), reslts( * ), x( ldx, * ),
164  $ xact( ldxact, * )
165 * ..
166 *
167 * =====================================================================
168 *
169 * .. Parameters ..
170  DOUBLE PRECISION ZERO, ONE
171  parameter( zero = 0.0d+0, one = 1.0d+0 )
172 * ..
173 * .. Local Scalars ..
174  INTEGER I, IMAX, J, K, NZ
175  DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
176 * ..
177 * .. External Functions ..
178  INTEGER IDAMAX
179  DOUBLE PRECISION DLAMCH
180  EXTERNAL idamax, dlamch
181 * ..
182 * .. Intrinsic Functions ..
183  INTRINSIC abs, max, min
184 * ..
185 * .. Executable Statements ..
186 *
187 * Quick exit if N = 0 or NRHS = 0.
188 *
189  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
190  reslts( 1 ) = zero
191  reslts( 2 ) = zero
192  RETURN
193  END IF
194 *
195  eps = dlamch( 'Epsilon' )
196  unfl = dlamch( 'Safe minimum' )
197  ovfl = one / unfl
198  nz = 4
199 *
200 * Test 1: Compute the maximum of
201 * norm(X - XACT) / ( norm(X) * FERR )
202 * over all the vectors X and XACT using the infinity-norm.
203 *
204  errbnd = zero
205  DO 30 j = 1, nrhs
206  imax = idamax( n, x( 1, j ), 1 )
207  xnorm = max( abs( x( imax, j ) ), unfl )
208  diff = zero
209  DO 10 i = 1, n
210  diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
211  10 CONTINUE
212 *
213  IF( xnorm.GT.one ) THEN
214  GO TO 20
215  ELSE IF( diff.LE.ovfl*xnorm ) THEN
216  GO TO 20
217  ELSE
218  errbnd = one / eps
219  GO TO 30
220  END IF
221 *
222  20 CONTINUE
223  IF( diff / xnorm.LE.ferr( j ) ) THEN
224  errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
225  ELSE
226  errbnd = one / eps
227  END IF
228  30 CONTINUE
229  reslts( 1 ) = errbnd
230 *
231 * Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
232 * (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
233 *
234  DO 50 k = 1, nrhs
235  IF( n.EQ.1 ) THEN
236  axbi = abs( b( 1, k ) ) + abs( d( 1 )*x( 1, k ) )
237  ELSE
238  axbi = abs( b( 1, k ) ) + abs( d( 1 )*x( 1, k ) ) +
239  $ abs( e( 1 )*x( 2, k ) )
240  DO 40 i = 2, n - 1
241  tmp = abs( b( i, k ) ) + abs( e( i-1 )*x( i-1, k ) ) +
242  $ abs( d( i )*x( i, k ) ) + abs( e( i )*x( i+1, k ) )
243  axbi = min( axbi, tmp )
244  40 CONTINUE
245  tmp = abs( b( n, k ) ) + abs( e( n-1 )*x( n-1, k ) ) +
246  $ abs( d( n )*x( n, k ) )
247  axbi = min( axbi, tmp )
248  END IF
249  tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
250  IF( k.EQ.1 ) THEN
251  reslts( 2 ) = tmp
252  ELSE
253  reslts( 2 ) = max( reslts( 2 ), tmp )
254  END IF
255  50 CONTINUE
256 *
257  RETURN
258 *
259 * End of DPTT05
260 *
261  END
subroutine dptt05(N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS)
DPTT05
Definition: dptt05.f:152