1*> \brief \b DGTT05
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 DGTT05( 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*       DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DL( * ),
20*      $                   DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ),
21*      $                   XACT( LDXACT, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> DGTT05 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
79*>          The diagonal elements of A.
80*> \endverbatim
81*>
82*> \param[in] DU
83*> \verbatim
84*>          DU is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 double_lin
161*
162*  =====================================================================
163      SUBROUTINE DGTT05( 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      DOUBLE PRECISION   B( LDB, * ), BERR( * ), D( * ), DL( * ),
176     $                   DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ),
177     $                   XACT( LDXACT, * )
178*     ..
179*
180*  =====================================================================
181*
182*     .. Parameters ..
183      DOUBLE PRECISION   ZERO, ONE
184      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
185*     ..
186*     .. Local Scalars ..
187      LOGICAL            NOTRAN
188      INTEGER            I, IMAX, J, K, NZ
189      DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
190*     ..
191*     .. External Functions ..
192      LOGICAL            LSAME
193      INTEGER            IDAMAX
194      DOUBLE PRECISION   DLAMCH
195      EXTERNAL           LSAME, IDAMAX, DLAMCH
196*     ..
197*     .. Intrinsic Functions ..
198      INTRINSIC          ABS, MAX, MIN
199*     ..
200*     .. Executable Statements ..
201*
202*     Quick exit if N = 0 or NRHS = 0.
203*
204      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
205         RESLTS( 1 ) = ZERO
206         RESLTS( 2 ) = ZERO
207         RETURN
208      END IF
209*
210      EPS = DLAMCH( 'Epsilon' )
211      UNFL = DLAMCH( 'Safe minimum' )
212      OVFL = ONE / UNFL
213      NOTRAN = LSAME( TRANS, 'N' )
214      NZ = 4
215*
216*     Test 1:  Compute the maximum of
217*        norm(X - XACT) / ( norm(X) * FERR )
218*     over all the vectors X and XACT using the infinity-norm.
219*
220      ERRBND = ZERO
221      DO 30 J = 1, NRHS
222         IMAX = IDAMAX( N, X( 1, J ), 1 )
223         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
224         DIFF = ZERO
225         DO 10 I = 1, N
226            DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
227   10    CONTINUE
228*
229         IF( XNORM.GT.ONE ) THEN
230            GO TO 20
231         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
232            GO TO 20
233         ELSE
234            ERRBND = ONE / EPS
235            GO TO 30
236         END IF
237*
238   20    CONTINUE
239         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
240            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
241         ELSE
242            ERRBND = ONE / EPS
243         END IF
244   30 CONTINUE
245      RESLTS( 1 ) = ERRBND
246*
247*     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
248*     (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
249*
250      DO 60 K = 1, NRHS
251         IF( NOTRAN ) THEN
252            IF( N.EQ.1 ) THEN
253               AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
254            ELSE
255               AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
256     $                ABS( DU( 1 )*X( 2, K ) )
257               DO 40 I = 2, N - 1
258                  TMP = ABS( B( I, K ) ) + ABS( DL( I-1 )*X( I-1, K ) )
259     $                   + ABS( D( I )*X( I, K ) ) +
260     $                  ABS( DU( I )*X( I+1, K ) )
261                  AXBI = MIN( AXBI, TMP )
262   40          CONTINUE
263               TMP = ABS( B( N, K ) ) + ABS( DL( N-1 )*X( N-1, K ) ) +
264     $               ABS( D( N )*X( N, K ) )
265               AXBI = MIN( AXBI, TMP )
266            END IF
267         ELSE
268            IF( N.EQ.1 ) THEN
269               AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
270            ELSE
271               AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
272     $                ABS( DL( 1 )*X( 2, K ) )
273               DO 50 I = 2, N - 1
274                  TMP = ABS( B( I, K ) ) + ABS( DU( I-1 )*X( I-1, K ) )
275     $                   + ABS( D( I )*X( I, K ) ) +
276     $                  ABS( DL( I )*X( I+1, K ) )
277                  AXBI = MIN( AXBI, TMP )
278   50          CONTINUE
279               TMP = ABS( B( N, K ) ) + ABS( DU( N-1 )*X( N-1, K ) ) +
280     $               ABS( D( N )*X( N, K ) )
281               AXBI = MIN( AXBI, TMP )
282            END IF
283         END IF
284         TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
285         IF( K.EQ.1 ) THEN
286            RESLTS( 2 ) = TMP
287         ELSE
288            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
289         END IF
290   60 CONTINUE
291*
292      RETURN
293*
294*     End of DGTT05
295*
296      END
297