1*> \brief \b DPBT05
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 DPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
12*                          XACT, LDXACT, FERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          UPLO
16*       INTEGER            KD, 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*> DPBT05 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*> symmetric band 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 / ( NZ*EPS + (*) ), where
41*>             (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
42*>             and NZ = max. number of nonzeros in any row of A, plus 1
43*> \endverbatim
44*
45*  Arguments:
46*  ==========
47*
48*> \param[in] UPLO
49*> \verbatim
50*>          UPLO is CHARACTER*1
51*>          Specifies whether the upper or lower triangular part of the
52*>          symmetric matrix A is stored.
53*>          = 'U':  Upper triangular
54*>          = 'L':  Lower triangular
55*> \endverbatim
56*>
57*> \param[in] N
58*> \verbatim
59*>          N is INTEGER
60*>          The number of rows of the matrices X, B, and XACT, and the
61*>          order of the matrix A.  N >= 0.
62*> \endverbatim
63*>
64*> \param[in] KD
65*> \verbatim
66*>          KD is INTEGER
67*>          The number of super-diagonals of the matrix A if UPLO = 'U',
68*>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
69*> \endverbatim
70*>
71*> \param[in] NRHS
72*> \verbatim
73*>          NRHS is INTEGER
74*>          The number of columns of the matrices X, B, and XACT.
75*>          NRHS >= 0.
76*> \endverbatim
77*>
78*> \param[in] AB
79*> \verbatim
80*>          AB is DOUBLE PRECISION array, dimension (LDAB,N)
81*>          The upper or lower triangle of the symmetric band matrix A,
82*>          stored in the first KD+1 rows of the array.  The j-th column
83*>          of A is stored in the j-th column of the array AB as follows:
84*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
85*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
86*> \endverbatim
87*>
88*> \param[in] LDAB
89*> \verbatim
90*>          LDAB is INTEGER
91*>          The leading dimension of the array AB.  LDAB >= KD+1.
92*> \endverbatim
93*>
94*> \param[in] B
95*> \verbatim
96*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
97*>          The right hand side vectors for the system of linear
98*>          equations.
99*> \endverbatim
100*>
101*> \param[in] LDB
102*> \verbatim
103*>          LDB is INTEGER
104*>          The leading dimension of the array B.  LDB >= max(1,N).
105*> \endverbatim
106*>
107*> \param[in] X
108*> \verbatim
109*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
110*>          The computed solution vectors.  Each vector is stored as a
111*>          column of the matrix X.
112*> \endverbatim
113*>
114*> \param[in] LDX
115*> \verbatim
116*>          LDX is INTEGER
117*>          The leading dimension of the array X.  LDX >= max(1,N).
118*> \endverbatim
119*>
120*> \param[in] XACT
121*> \verbatim
122*>          XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
123*>          The exact solution vectors.  Each vector is stored as a
124*>          column of the matrix XACT.
125*> \endverbatim
126*>
127*> \param[in] LDXACT
128*> \verbatim
129*>          LDXACT is INTEGER
130*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
131*> \endverbatim
132*>
133*> \param[in] FERR
134*> \verbatim
135*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
136*>          The estimated forward error bounds for each solution vector
137*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
138*>          of the largest entry in (X - XTRUE) divided by the magnitude
139*>          of the largest entry in X.
140*> \endverbatim
141*>
142*> \param[in] BERR
143*> \verbatim
144*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
145*>          The componentwise relative backward error of each solution
146*>          vector (i.e., the smallest relative change in any entry of A
147*>          or B that makes X an exact solution).
148*> \endverbatim
149*>
150*> \param[out] RESLTS
151*> \verbatim
152*>          RESLTS is DOUBLE PRECISION array, dimension (2)
153*>          The maximum over the NRHS solution vectors of the ratios:
154*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
155*>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
156*> \endverbatim
157*
158*  Authors:
159*  ========
160*
161*> \author Univ. of Tennessee
162*> \author Univ. of California Berkeley
163*> \author Univ. of Colorado Denver
164*> \author NAG Ltd.
165*
166*> \ingroup double_lin
167*
168*  =====================================================================
169      SUBROUTINE DPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
170     $                   XACT, LDXACT, FERR, BERR, RESLTS )
171*
172*  -- LAPACK test routine --
173*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
174*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
175*
176*     .. Scalar Arguments ..
177      CHARACTER          UPLO
178      INTEGER            KD, LDAB, LDB, LDX, LDXACT, N, NRHS
179*     ..
180*     .. Array Arguments ..
181      DOUBLE PRECISION   AB( LDAB, * ), B( LDB, * ), BERR( * ),
182     $                   FERR( * ), RESLTS( * ), X( LDX, * ),
183     $                   XACT( LDXACT, * )
184*     ..
185*
186*  =====================================================================
187*
188*     .. Parameters ..
189      DOUBLE PRECISION   ZERO, ONE
190      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
191*     ..
192*     .. Local Scalars ..
193      LOGICAL            UPPER
194      INTEGER            I, IMAX, J, K, NZ
195      DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
196*     ..
197*     .. External Functions ..
198      LOGICAL            LSAME
199      INTEGER            IDAMAX
200      DOUBLE PRECISION   DLAMCH
201      EXTERNAL           LSAME, IDAMAX, DLAMCH
202*     ..
203*     .. Intrinsic Functions ..
204      INTRINSIC          ABS, MAX, MIN
205*     ..
206*     .. Executable Statements ..
207*
208*     Quick exit if N = 0 or NRHS = 0.
209*
210      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
211         RESLTS( 1 ) = ZERO
212         RESLTS( 2 ) = ZERO
213         RETURN
214      END IF
215*
216      EPS = DLAMCH( 'Epsilon' )
217      UNFL = DLAMCH( 'Safe minimum' )
218      OVFL = ONE / UNFL
219      UPPER = LSAME( UPLO, 'U' )
220      NZ = 2*MAX( KD, N-1 ) + 1
221*
222*     Test 1:  Compute the maximum of
223*        norm(X - XACT) / ( norm(X) * FERR )
224*     over all the vectors X and XACT using the infinity-norm.
225*
226      ERRBND = ZERO
227      DO 30 J = 1, NRHS
228         IMAX = IDAMAX( N, X( 1, J ), 1 )
229         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
230         DIFF = ZERO
231         DO 10 I = 1, N
232            DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
233   10    CONTINUE
234*
235         IF( XNORM.GT.ONE ) THEN
236            GO TO 20
237         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
238            GO TO 20
239         ELSE
240            ERRBND = ONE / EPS
241            GO TO 30
242         END IF
243*
244   20    CONTINUE
245         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
246            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
247         ELSE
248            ERRBND = ONE / EPS
249         END IF
250   30 CONTINUE
251      RESLTS( 1 ) = ERRBND
252*
253*     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
254*     (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
255*
256      DO 90 K = 1, NRHS
257         DO 80 I = 1, N
258            TMP = ABS( B( I, K ) )
259            IF( UPPER ) THEN
260               DO 40 J = MAX( I-KD, 1 ), I
261                  TMP = TMP + ABS( AB( KD+1-I+J, I ) )*ABS( X( J, K ) )
262   40          CONTINUE
263               DO 50 J = I + 1, MIN( I+KD, N )
264                  TMP = TMP + ABS( AB( KD+1+I-J, J ) )*ABS( X( J, K ) )
265   50          CONTINUE
266            ELSE
267               DO 60 J = MAX( I-KD, 1 ), I - 1
268                  TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) )
269   60          CONTINUE
270               DO 70 J = I, MIN( I+KD, N )
271                  TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) )
272   70          CONTINUE
273            END IF
274            IF( I.EQ.1 ) THEN
275               AXBI = TMP
276            ELSE
277               AXBI = MIN( AXBI, TMP )
278            END IF
279   80    CONTINUE
280         TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
281         IF( K.EQ.1 ) THEN
282            RESLTS( 2 ) = TMP
283         ELSE
284            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
285         END IF
286   90 CONTINUE
287*
288      RETURN
289*
290*     End of DPBT05
291*
292      END
293