1*> \brief \b STRT05
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 STRT05( 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*       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
20*      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> STRT05 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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*> \ingroup single_lin
177*
178*  =====================================================================
179      SUBROUTINE STRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X,
180     $                   LDX, XACT, LDXACT, FERR, BERR, RESLTS )
181*
182*  -- LAPACK test routine --
183*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
184*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
185*
186*     .. Scalar Arguments ..
187      CHARACTER          DIAG, TRANS, UPLO
188      INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
189*     ..
190*     .. Array Arguments ..
191      REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
192     $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
193*     ..
194*
195*  =====================================================================
196*
197*     .. Parameters ..
198      REAL               ZERO, ONE
199      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
200*     ..
201*     .. Local Scalars ..
202      LOGICAL            NOTRAN, UNIT, UPPER
203      INTEGER            I, IFU, IMAX, J, K
204      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
205*     ..
206*     .. External Functions ..
207      LOGICAL            LSAME
208      INTEGER            ISAMAX
209      REAL               SLAMCH
210      EXTERNAL           LSAME, ISAMAX, SLAMCH
211*     ..
212*     .. Intrinsic Functions ..
213      INTRINSIC          ABS, MAX, MIN
214*     ..
215*     .. Executable Statements ..
216*
217*     Quick exit if N = 0 or NRHS = 0.
218*
219      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
220         RESLTS( 1 ) = ZERO
221         RESLTS( 2 ) = ZERO
222         RETURN
223      END IF
224*
225      EPS = SLAMCH( 'Epsilon' )
226      UNFL = SLAMCH( 'Safe minimum' )
227      OVFL = ONE / UNFL
228      UPPER = LSAME( UPLO, 'U' )
229      NOTRAN = LSAME( TRANS, 'N' )
230      UNIT = LSAME( DIAG, 'U' )
231*
232*     Test 1:  Compute the maximum of
233*        norm(X - XACT) / ( norm(X) * FERR )
234*     over all the vectors X and XACT using the infinity-norm.
235*
236      ERRBND = ZERO
237      DO 30 J = 1, NRHS
238         IMAX = ISAMAX( N, X( 1, J ), 1 )
239         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
240         DIFF = ZERO
241         DO 10 I = 1, N
242            DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
243   10    CONTINUE
244*
245         IF( XNORM.GT.ONE ) THEN
246            GO TO 20
247         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
248            GO TO 20
249         ELSE
250            ERRBND = ONE / EPS
251            GO TO 30
252         END IF
253*
254   20    CONTINUE
255         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
256            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
257         ELSE
258            ERRBND = ONE / EPS
259         END IF
260   30 CONTINUE
261      RESLTS( 1 ) = ERRBND
262*
263*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
264*     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
265*
266      IFU = 0
267      IF( UNIT )
268     $   IFU = 1
269      DO 90 K = 1, NRHS
270         DO 80 I = 1, N
271            TMP = ABS( B( I, K ) )
272            IF( UPPER ) THEN
273               IF( .NOT.NOTRAN ) THEN
274                  DO 40 J = 1, I - IFU
275                     TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
276   40             CONTINUE
277                  IF( UNIT )
278     $               TMP = TMP + ABS( X( I, K ) )
279               ELSE
280                  IF( UNIT )
281     $               TMP = TMP + ABS( X( I, K ) )
282                  DO 50 J = I + IFU, N
283                     TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
284   50             CONTINUE
285               END IF
286            ELSE
287               IF( NOTRAN ) THEN
288                  DO 60 J = 1, I - IFU
289                     TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
290   60             CONTINUE
291                  IF( UNIT )
292     $               TMP = TMP + ABS( X( I, K ) )
293               ELSE
294                  IF( UNIT )
295     $               TMP = TMP + ABS( X( I, K ) )
296                  DO 70 J = I + IFU, N
297                     TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
298   70             CONTINUE
299               END IF
300            END IF
301            IF( I.EQ.1 ) THEN
302               AXBI = TMP
303            ELSE
304               AXBI = MIN( AXBI, TMP )
305            END IF
306   80    CONTINUE
307         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
308     $         MAX( AXBI, ( N+1 )*UNFL ) )
309         IF( K.EQ.1 ) THEN
310            RESLTS( 2 ) = TMP
311         ELSE
312            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
313         END IF
314   90 CONTINUE
315*
316      RETURN
317*
318*     End of STRT05
319*
320      END
321