LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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dgbt05.f
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1*> \brief \b DGBT05
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 DGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
12* LDX, XACT, LDXACT, FERR, BERR, RESLTS )
13*
14* .. Scalar Arguments ..
15* CHARACTER TRANS
16* INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
17* ..
18* .. Array Arguments ..
19* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
20* $ FERR( * ), RESLTS( * ), X( LDX, * ),
21* $ XACT( LDXACT, * )
22* ..
23*
24*
25*> \par Purpose:
26* =============
27*>
28*> \verbatim
29*>
30*> DGBT05 tests the error bounds from iterative refinement for the
31*> computed solution to a system of equations op(A)*X = B, where A is a
32*> general band matrix of order n with kl subdiagonals and ku
33*> superdiagonals and op(A) = A or A**T, 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, B, and XACT, and the
62*> order of the matrix A. N >= 0.
63*> \endverbatim
64*>
65*> \param[in] KL
66*> \verbatim
67*> KL is INTEGER
68*> The number of subdiagonals within the band of A. KL >= 0.
69*> \endverbatim
70*>
71*> \param[in] KU
72*> \verbatim
73*> KU is INTEGER
74*> The number of superdiagonals within the band of A. KU >= 0.
75*> \endverbatim
76*>
77*> \param[in] NRHS
78*> \verbatim
79*> NRHS is INTEGER
80*> The number of columns of the matrices X, B, and XACT.
81*> NRHS >= 0.
82*> \endverbatim
83*>
84*> \param[in] AB
85*> \verbatim
86*> AB is DOUBLE PRECISION array, dimension (LDAB,N)
87*> The original band matrix A, stored in rows 1 to KL+KU+1.
88*> The j-th column of A is stored in the j-th column of the
89*> array AB as follows:
90*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
91*> \endverbatim
92*>
93*> \param[in] LDAB
94*> \verbatim
95*> LDAB is INTEGER
96*> The leading dimension of the array AB. LDAB >= KL+KU+1.
97*> \endverbatim
98*>
99*> \param[in] B
100*> \verbatim
101*> B is DOUBLE PRECISION array, dimension (LDB,NRHS)
102*> The right hand side vectors for the system of linear
103*> equations.
104*> \endverbatim
105*>
106*> \param[in] LDB
107*> \verbatim
108*> LDB is INTEGER
109*> The leading dimension of the array B. LDB >= max(1,N).
110*> \endverbatim
111*>
112*> \param[in] X
113*> \verbatim
114*> X is DOUBLE PRECISION array, dimension (LDX,NRHS)
115*> The computed solution vectors. Each vector is stored as a
116*> column of the matrix X.
117*> \endverbatim
118*>
119*> \param[in] LDX
120*> \verbatim
121*> LDX is INTEGER
122*> The leading dimension of the array X. LDX >= max(1,N).
123*> \endverbatim
124*>
125*> \param[in] XACT
126*> \verbatim
127*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
128*> The exact solution vectors. Each vector is stored as a
129*> column of the matrix XACT.
130*> \endverbatim
131*>
132*> \param[in] LDXACT
133*> \verbatim
134*> LDXACT is INTEGER
135*> The leading dimension of the array XACT. LDXACT >= max(1,N).
136*> \endverbatim
137*>
138*> \param[in] FERR
139*> \verbatim
140*> FERR is DOUBLE PRECISION array, dimension (NRHS)
141*> The estimated forward error bounds for each solution vector
142*> X. If XTRUE is the true solution, FERR bounds the magnitude
143*> of the largest entry in (X - XTRUE) divided by the magnitude
144*> of the largest entry in X.
145*> \endverbatim
146*>
147*> \param[in] BERR
148*> \verbatim
149*> BERR is DOUBLE PRECISION array, dimension (NRHS)
150*> The componentwise relative backward error of each solution
151*> vector (i.e., the smallest relative change in any entry of A
152*> or B that makes X an exact solution).
153*> \endverbatim
154*>
155*> \param[out] RESLTS
156*> \verbatim
157*> RESLTS is DOUBLE PRECISION array, dimension (2)
158*> The maximum over the NRHS solution vectors of the ratios:
159*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
160*> RESLTS(2) = BERR / ( NZ*EPS + (*) )
161*> \endverbatim
162*
163* Authors:
164* ========
165*
166*> \author Univ. of Tennessee
167*> \author Univ. of California Berkeley
168*> \author Univ. of Colorado Denver
169*> \author NAG Ltd.
170*
171*> \ingroup double_lin
172*
173* =====================================================================
174 SUBROUTINE dgbt05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
175 $ LDX, XACT, LDXACT, FERR, BERR, RESLTS )
176*
177* -- LAPACK test routine --
178* -- LAPACK is a software package provided by Univ. of Tennessee, --
179* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180*
181* .. Scalar Arguments ..
182 CHARACTER TRANS
183 INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
184* ..
185* .. Array Arguments ..
186 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ),
187 $ ferr( * ), reslts( * ), x( ldx, * ),
188 $ xact( ldxact, * )
189* ..
190*
191* =====================================================================
192*
193* .. Parameters ..
194 DOUBLE PRECISION ZERO, ONE
195 parameter( zero = 0.0d+0, one = 1.0d+0 )
196* ..
197* .. Local Scalars ..
198 LOGICAL NOTRAN
199 INTEGER I, IMAX, J, K, NZ
200 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
201* ..
202* .. External Functions ..
203 LOGICAL LSAME
204 INTEGER IDAMAX
205 DOUBLE PRECISION DLAMCH
206 EXTERNAL lsame, idamax, dlamch
207* ..
208* .. Intrinsic Functions ..
209 INTRINSIC abs, max, min
210* ..
211* .. Executable Statements ..
212*
213* Quick exit if N = 0 or NRHS = 0.
214*
215 IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
216 reslts( 1 ) = zero
217 reslts( 2 ) = zero
218 RETURN
219 END IF
220*
221 eps = dlamch( 'Epsilon' )
222 unfl = dlamch( 'Safe minimum' )
223 ovfl = one / unfl
224 notran = lsame( trans, 'N' )
225 nz = min( kl+ku+2, n+1 )
226*
227* Test 1: Compute the maximum of
228* norm(X - XACT) / ( norm(X) * FERR )
229* over all the vectors X and XACT using the infinity-norm.
230*
231 errbnd = zero
232 DO 30 j = 1, nrhs
233 imax = idamax( n, x( 1, j ), 1 )
234 xnorm = max( abs( x( imax, j ) ), unfl )
235 diff = zero
236 DO 10 i = 1, n
237 diff = max( diff, abs( x( i, j )-xact( i, j ) ) )
238 10 CONTINUE
239*
240 IF( xnorm.GT.one ) THEN
241 GO TO 20
242 ELSE IF( diff.LE.ovfl*xnorm ) THEN
243 GO TO 20
244 ELSE
245 errbnd = one / eps
246 GO TO 30
247 END IF
248*
249 20 CONTINUE
250 IF( diff / xnorm.LE.ferr( j ) ) THEN
251 errbnd = max( errbnd, ( diff / xnorm ) / ferr( j ) )
252 ELSE
253 errbnd = one / eps
254 END IF
255 30 CONTINUE
256 reslts( 1 ) = errbnd
257*
258* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where
259* (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
260*
261 DO 70 k = 1, nrhs
262 DO 60 i = 1, n
263 tmp = abs( b( i, k ) )
264 IF( notran ) THEN
265 DO 40 j = max( i-kl, 1 ), min( i+ku, n )
266 tmp = tmp + abs( ab( ku+1+i-j, j ) )*abs( x( j, k ) )
267 40 CONTINUE
268 ELSE
269 DO 50 j = max( i-ku, 1 ), min( i+kl, n )
270 tmp = tmp + abs( ab( ku+1+j-i, i ) )*abs( x( j, k ) )
271 50 CONTINUE
272 END IF
273 IF( i.EQ.1 ) THEN
274 axbi = tmp
275 ELSE
276 axbi = min( axbi, tmp )
277 END IF
278 60 CONTINUE
279 tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
280 IF( k.EQ.1 ) THEN
281 reslts( 2 ) = tmp
282 ELSE
283 reslts( 2 ) = max( reslts( 2 ), tmp )
284 END IF
285 70 CONTINUE
286*
287 RETURN
288*
289* End of DGBT05
290*
291 END
subroutine dgbt05(TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS)
DGBT05
Definition: dgbt05.f:176