1*> \brief \b DTRT05
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 DTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
12*                          LDX, XACT, LDXACT, FERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          DIAG, TRANS, UPLO
16*       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
17*       ..
18*       .. Array Arguments ..
19*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
20*      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> DTRT05 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*> triangular n by n matrix.
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 / ( (n+1)*EPS + (*) ), where
40*>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
41*> \endverbatim
42*
43*  Arguments:
44*  ==========
45*
46*> \param[in] UPLO
47*> \verbatim
48*>          UPLO is CHARACTER*1
49*>          Specifies whether the matrix A is upper or lower triangular.
50*>          = 'U':  Upper triangular
51*>          = 'L':  Lower triangular
52*> \endverbatim
53*>
54*> \param[in] TRANS
55*> \verbatim
56*>          TRANS is CHARACTER*1
57*>          Specifies the form of the system of equations.
58*>          = 'N':  A * X = B  (No transpose)
59*>          = 'T':  A'* X = B  (Transpose)
60*>          = 'C':  A'* X = B  (Conjugate transpose = Transpose)
61*> \endverbatim
62*>
63*> \param[in] DIAG
64*> \verbatim
65*>          DIAG is CHARACTER*1
66*>          Specifies whether or not the matrix A is unit triangular.
67*>          = 'N':  Non-unit triangular
68*>          = 'U':  Unit triangular
69*> \endverbatim
70*>
71*> \param[in] N
72*> \verbatim
73*>          N is INTEGER
74*>          The number of rows of the matrices X, B, and XACT, and the
75*>          order of the matrix A.  N >= 0.
76*> \endverbatim
77*>
78*> \param[in] NRHS
79*> \verbatim
80*>          NRHS is INTEGER
81*>          The number of columns of the matrices X, B, and XACT.
82*>          NRHS >= 0.
83*> \endverbatim
84*>
85*> \param[in] A
86*> \verbatim
87*>          A is DOUBLE PRECISION array, dimension (LDA,N)
88*>          The triangular matrix A.  If UPLO = 'U', the leading n by n
89*>          upper triangular part of the array A contains the upper
90*>          triangular matrix, and the strictly lower triangular part of
91*>          A is not referenced.  If UPLO = 'L', the leading n by n lower
92*>          triangular part of the array A contains the lower triangular
93*>          matrix, and the strictly upper triangular part of A is not
94*>          referenced.  If DIAG = 'U', the diagonal elements of A are
95*>          also not referenced and are assumed to be 1.
96*> \endverbatim
97*>
98*> \param[in] LDA
99*> \verbatim
100*>          LDA is INTEGER
101*>          The leading dimension of the array A.  LDA >= max(1,N).
102*> \endverbatim
103*>
104*> \param[in] B
105*> \verbatim
106*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
107*>          The right hand side vectors for the system of linear
108*>          equations.
109*> \endverbatim
110*>
111*> \param[in] LDB
112*> \verbatim
113*>          LDB is INTEGER
114*>          The leading dimension of the array B.  LDB >= max(1,N).
115*> \endverbatim
116*>
117*> \param[in] X
118*> \verbatim
119*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
120*>          The computed solution vectors.  Each vector is stored as a
121*>          column of the matrix X.
122*> \endverbatim
123*>
124*> \param[in] LDX
125*> \verbatim
126*>          LDX is INTEGER
127*>          The leading dimension of the array X.  LDX >= max(1,N).
128*> \endverbatim
129*>
130*> \param[in] XACT
131*> \verbatim
132*>          XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
133*>          The exact solution vectors.  Each vector is stored as a
134*>          column of the matrix XACT.
135*> \endverbatim
136*>
137*> \param[in] LDXACT
138*> \verbatim
139*>          LDXACT is INTEGER
140*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
141*> \endverbatim
142*>
143*> \param[in] FERR
144*> \verbatim
145*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
146*>          The estimated forward error bounds for each solution vector
147*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
148*>          of the largest entry in (X - XTRUE) divided by the magnitude
149*>          of the largest entry in X.
150*> \endverbatim
151*>
152*> \param[in] BERR
153*> \verbatim
154*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
155*>          The componentwise relative backward error of each solution
156*>          vector (i.e., the smallest relative change in any entry of A
157*>          or B that makes X an exact solution).
158*> \endverbatim
159*>
160*> \param[out] RESLTS
161*> \verbatim
162*>          RESLTS is DOUBLE PRECISION array, dimension (2)
163*>          The maximum over the NRHS solution vectors of the ratios:
164*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
165*>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
166*> \endverbatim
167*
168*  Authors:
169*  ========
170*
171*> \author Univ. of Tennessee
172*> \author Univ. of California Berkeley
173*> \author Univ. of Colorado Denver
174*> \author NAG Ltd.
175*
176*> \date November 2011
177*
178*> \ingroup double_lin
179*
180*  =====================================================================
181      SUBROUTINE DTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
182     $                   LDX, XACT, LDXACT, FERR, BERR, RESLTS )
183*
184*  -- LAPACK test routine (version 3.4.0) --
185*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
186*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
187*     November 2011
188*
189*     .. Scalar Arguments ..
190      CHARACTER          DIAG, TRANS, UPLO
191      INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
192*     ..
193*     .. Array Arguments ..
194      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
195     $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
196*     ..
197*
198*  =====================================================================
199*
200*     .. Parameters ..
201      DOUBLE PRECISION   ZERO, ONE
202      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
203*     ..
204*     .. Local Scalars ..
205      LOGICAL            NOTRAN, UNIT, UPPER
206      INTEGER            I, IFU, IMAX, J, K
207      DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
208*     ..
209*     .. External Functions ..
210      LOGICAL            LSAME
211      INTEGER            IDAMAX
212      DOUBLE PRECISION   DLAMCH
213      EXTERNAL           LSAME, IDAMAX, DLAMCH
214*     ..
215*     .. Intrinsic Functions ..
216      INTRINSIC          ABS, MAX, MIN
217*     ..
218*     .. Executable Statements ..
219*
220*     Quick exit if N = 0 or NRHS = 0.
221*
222      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
223         RESLTS( 1 ) = ZERO
224         RESLTS( 2 ) = ZERO
225         RETURN
226      END IF
227*
228      EPS = DLAMCH( 'Epsilon' )
229      UNFL = DLAMCH( 'Safe minimum' )
230      OVFL = ONE / UNFL
231      UPPER = LSAME( UPLO, 'U' )
232      NOTRAN = LSAME( TRANS, 'N' )
233      UNIT = LSAME( DIAG, 'U' )
234*
235*     Test 1:  Compute the maximum of
236*        norm(X - XACT) / ( norm(X) * FERR )
237*     over all the vectors X and XACT using the infinity-norm.
238*
239      ERRBND = ZERO
240      DO 30 J = 1, NRHS
241         IMAX = IDAMAX( N, X( 1, J ), 1 )
242         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
243         DIFF = ZERO
244         DO 10 I = 1, N
245            DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
246   10    CONTINUE
247*
248         IF( XNORM.GT.ONE ) THEN
249            GO TO 20
250         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
251            GO TO 20
252         ELSE
253            ERRBND = ONE / EPS
254            GO TO 30
255         END IF
256*
257   20    CONTINUE
258         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
259            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
260         ELSE
261            ERRBND = ONE / EPS
262         END IF
263   30 CONTINUE
264      RESLTS( 1 ) = ERRBND
265*
266*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
267*     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
268*
269      IFU = 0
270      IF( UNIT )
271     $   IFU = 1
272      DO 90 K = 1, NRHS
273         DO 80 I = 1, N
274            TMP = ABS( B( I, K ) )
275            IF( UPPER ) THEN
276               IF( .NOT.NOTRAN ) THEN
277                  DO 40 J = 1, I - IFU
278                     TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
279   40             CONTINUE
280                  IF( UNIT )
281     $               TMP = TMP + ABS( X( I, K ) )
282               ELSE
283                  IF( UNIT )
284     $               TMP = TMP + ABS( X( I, K ) )
285                  DO 50 J = I + IFU, N
286                     TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
287   50             CONTINUE
288               END IF
289            ELSE
290               IF( NOTRAN ) THEN
291                  DO 60 J = 1, I - IFU
292                     TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
293   60             CONTINUE
294                  IF( UNIT )
295     $               TMP = TMP + ABS( X( I, K ) )
296               ELSE
297                  IF( UNIT )
298     $               TMP = TMP + ABS( X( I, K ) )
299                  DO 70 J = I + IFU, N
300                     TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
301   70             CONTINUE
302               END IF
303            END IF
304            IF( I.EQ.1 ) THEN
305               AXBI = TMP
306            ELSE
307               AXBI = MIN( AXBI, TMP )
308            END IF
309   80    CONTINUE
310         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
311     $         MAX( AXBI, ( N+1 )*UNFL ) )
312         IF( K.EQ.1 ) THEN
313            RESLTS( 2 ) = TMP
314         ELSE
315            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
316         END IF
317   90 CONTINUE
318*
319      RETURN
320*
321*     End of DTRT05
322*
323      END
324