1*> \brief \b DTPT05
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 DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
12*                          XACT, LDXACT, FERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          DIAG, TRANS, UPLO
16*       INTEGER            LDB, LDX, LDXACT, N, NRHS
17*       ..
18*       .. Array Arguments ..
19*       DOUBLE PRECISION   AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
20*      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
21*       ..
22*
23*
24*> \par Purpose:
25*  =============
26*>
27*> \verbatim
28*>
29*> DTPT05 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 matrix in packed storage format.
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] AP
86*> \verbatim
87*>          AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
88*>          The upper or lower triangular matrix A, packed columnwise in
89*>          a linear array.  The j-th column of A is stored in the array
90*>          AP as follows:
91*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
92*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
93*>          If DIAG = 'U', the diagonal elements of A are not referenced
94*>          and are assumed to be 1.
95*> \endverbatim
96*>
97*> \param[in] B
98*> \verbatim
99*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
100*>          The right hand side vectors for the system of linear
101*>          equations.
102*> \endverbatim
103*>
104*> \param[in] LDB
105*> \verbatim
106*>          LDB is INTEGER
107*>          The leading dimension of the array B.  LDB >= max(1,N).
108*> \endverbatim
109*>
110*> \param[in] X
111*> \verbatim
112*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
113*>          The computed solution vectors.  Each vector is stored as a
114*>          column of the matrix X.
115*> \endverbatim
116*>
117*> \param[in] LDX
118*> \verbatim
119*>          LDX is INTEGER
120*>          The leading dimension of the array X.  LDX >= max(1,N).
121*> \endverbatim
122*>
123*> \param[in] XACT
124*> \verbatim
125*>          XACT is DOUBLE PRECISION array, dimension (LDX,NRHS)
126*>          The exact solution vectors.  Each vector is stored as a
127*>          column of the matrix XACT.
128*> \endverbatim
129*>
130*> \param[in] LDXACT
131*> \verbatim
132*>          LDXACT is INTEGER
133*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
134*> \endverbatim
135*>
136*> \param[in] FERR
137*> \verbatim
138*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
139*>          The estimated forward error bounds for each solution vector
140*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
141*>          of the largest entry in (X - XTRUE) divided by the magnitude
142*>          of the largest entry in X.
143*> \endverbatim
144*>
145*> \param[in] BERR
146*> \verbatim
147*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
148*>          The componentwise relative backward error of each solution
149*>          vector (i.e., the smallest relative change in any entry of A
150*>          or B that makes X an exact solution).
151*> \endverbatim
152*>
153*> \param[out] RESLTS
154*> \verbatim
155*>          RESLTS is DOUBLE PRECISION array, dimension (2)
156*>          The maximum over the NRHS solution vectors of the ratios:
157*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
158*>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
159*> \endverbatim
160*
161*  Authors:
162*  ========
163*
164*> \author Univ. of Tennessee
165*> \author Univ. of California Berkeley
166*> \author Univ. of Colorado Denver
167*> \author NAG Ltd.
168*
169*> \ingroup double_lin
170*
171*  =====================================================================
172      SUBROUTINE DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
173     $                   XACT, LDXACT, FERR, BERR, RESLTS )
174*
175*  -- LAPACK test routine --
176*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
177*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
178*
179*     .. Scalar Arguments ..
180      CHARACTER          DIAG, TRANS, UPLO
181      INTEGER            LDB, LDX, LDXACT, N, NRHS
182*     ..
183*     .. Array Arguments ..
184      DOUBLE PRECISION   AP( * ), B( LDB, * ), BERR( * ), FERR( * ),
185     $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
186*     ..
187*
188*  =====================================================================
189*
190*     .. Parameters ..
191      DOUBLE PRECISION   ZERO, ONE
192      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
193*     ..
194*     .. Local Scalars ..
195      LOGICAL            NOTRAN, UNIT, UPPER
196      INTEGER            I, IFU, IMAX, J, JC, K
197      DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
198*     ..
199*     .. External Functions ..
200      LOGICAL            LSAME
201      INTEGER            IDAMAX
202      DOUBLE PRECISION   DLAMCH
203      EXTERNAL           LSAME, IDAMAX, DLAMCH
204*     ..
205*     .. Intrinsic Functions ..
206      INTRINSIC          ABS, MAX, MIN
207*     ..
208*     .. Executable Statements ..
209*
210*     Quick exit if N = 0 or NRHS = 0.
211*
212      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
213         RESLTS( 1 ) = ZERO
214         RESLTS( 2 ) = ZERO
215         RETURN
216      END IF
217*
218      EPS = DLAMCH( 'Epsilon' )
219      UNFL = DLAMCH( 'Safe minimum' )
220      OVFL = ONE / UNFL
221      UPPER = LSAME( UPLO, 'U' )
222      NOTRAN = LSAME( TRANS, 'N' )
223      UNIT = LSAME( DIAG, 'U' )
224*
225*     Test 1:  Compute the maximum of
226*        norm(X - XACT) / ( norm(X) * FERR )
227*     over all the vectors X and XACT using the infinity-norm.
228*
229      ERRBND = ZERO
230      DO 30 J = 1, NRHS
231         IMAX = IDAMAX( N, X( 1, J ), 1 )
232         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
233         DIFF = ZERO
234         DO 10 I = 1, N
235            DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
236   10    CONTINUE
237*
238         IF( XNORM.GT.ONE ) THEN
239            GO TO 20
240         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
241            GO TO 20
242         ELSE
243            ERRBND = ONE / EPS
244            GO TO 30
245         END IF
246*
247   20    CONTINUE
248         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
249            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
250         ELSE
251            ERRBND = ONE / EPS
252         END IF
253   30 CONTINUE
254      RESLTS( 1 ) = ERRBND
255*
256*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
257*     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
258*
259      IFU = 0
260      IF( UNIT )
261     $   IFU = 1
262      DO 90 K = 1, NRHS
263         DO 80 I = 1, N
264            TMP = ABS( B( I, K ) )
265            IF( UPPER ) THEN
266               JC = ( ( I-1 )*I ) / 2
267               IF( .NOT.NOTRAN ) THEN
268                  DO 40 J = 1, I - IFU
269                     TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) )
270   40             CONTINUE
271                  IF( UNIT )
272     $               TMP = TMP + ABS( X( I, K ) )
273               ELSE
274                  JC = JC + I
275                  IF( UNIT ) THEN
276                     TMP = TMP + ABS( X( I, K ) )
277                     JC = JC + I
278                  END IF
279                  DO 50 J = I + IFU, N
280                     TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
281                     JC = JC + J
282   50             CONTINUE
283               END IF
284            ELSE
285               IF( NOTRAN ) THEN
286                  JC = I
287                  DO 60 J = 1, I - IFU
288                     TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) )
289                     JC = JC + N - J
290   60             CONTINUE
291                  IF( UNIT )
292     $               TMP = TMP + ABS( X( I, K ) )
293               ELSE
294                  JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
295                  IF( UNIT )
296     $               TMP = TMP + ABS( X( I, K ) )
297                  DO 70 J = I + IFU, N
298                     TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) )
299   70             CONTINUE
300               END IF
301            END IF
302            IF( I.EQ.1 ) THEN
303               AXBI = TMP
304            ELSE
305               AXBI = MIN( AXBI, TMP )
306            END IF
307   80    CONTINUE
308         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
309     $         MAX( AXBI, ( N+1 )*UNFL ) )
310         IF( K.EQ.1 ) THEN
311            RESLTS( 2 ) = TMP
312         ELSE
313            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
314         END IF
315   90 CONTINUE
316*
317      RETURN
318*
319*     End of DTPT05
320*
321      END
322