1*> \brief \b CTPT05
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 CTPT05( 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*       REAL               BERR( * ), FERR( * ), RESLTS( * )
20*       COMPLEX            AP( * ), B( LDB, * ), X( LDX, * ),
21*      $                   XACT( LDXACT, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> CTPT05 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*> triangular matrix in packed storage format.
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 / ( (n+1)*EPS + (*) ), where
41*>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
42*> \endverbatim
43*
44*  Arguments:
45*  ==========
46*
47*> \param[in] UPLO
48*> \verbatim
49*>          UPLO is CHARACTER*1
50*>          Specifies whether the matrix A is upper or lower triangular.
51*>          = 'U':  Upper triangular
52*>          = 'L':  Lower triangular
53*> \endverbatim
54*>
55*> \param[in] TRANS
56*> \verbatim
57*>          TRANS is CHARACTER*1
58*>          Specifies the form of the system of equations.
59*>          = 'N':  A * X = B  (No transpose)
60*>          = 'T':  A'* X = B  (Transpose)
61*>          = 'C':  A'* X = B  (Conjugate transpose = Transpose)
62*> \endverbatim
63*>
64*> \param[in] DIAG
65*> \verbatim
66*>          DIAG is CHARACTER*1
67*>          Specifies whether or not the matrix A is unit triangular.
68*>          = 'N':  Non-unit triangular
69*>          = 'U':  Unit triangular
70*> \endverbatim
71*>
72*> \param[in] N
73*> \verbatim
74*>          N is INTEGER
75*>          The number of rows of the matrices X, B, and XACT, and the
76*>          order of the matrix A.  N >= 0.
77*> \endverbatim
78*>
79*> \param[in] NRHS
80*> \verbatim
81*>          NRHS is INTEGER
82*>          The number of columns of the matrices X, B, and XACT.
83*>          NRHS >= 0.
84*> \endverbatim
85*>
86*> \param[in] AP
87*> \verbatim
88*>          AP is COMPLEX array, dimension (N*(N+1)/2)
89*>          The upper or lower triangular matrix A, packed columnwise in
90*>          a linear array.  The j-th column of A is stored in the array
91*>          AP as follows:
92*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
93*>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
94*>          If DIAG = 'U', the diagonal elements of A are not referenced
95*>          and are assumed to be 1.
96*> \endverbatim
97*>
98*> \param[in] B
99*> \verbatim
100*>          B is COMPLEX array, dimension (LDB,NRHS)
101*>          The right hand side vectors for the system of linear
102*>          equations.
103*> \endverbatim
104*>
105*> \param[in] LDB
106*> \verbatim
107*>          LDB is INTEGER
108*>          The leading dimension of the array B.  LDB >= max(1,N).
109*> \endverbatim
110*>
111*> \param[in] X
112*> \verbatim
113*>          X is COMPLEX array, dimension (LDX,NRHS)
114*>          The computed solution vectors.  Each vector is stored as a
115*>          column of the matrix X.
116*> \endverbatim
117*>
118*> \param[in] LDX
119*> \verbatim
120*>          LDX is INTEGER
121*>          The leading dimension of the array X.  LDX >= max(1,N).
122*> \endverbatim
123*>
124*> \param[in] XACT
125*> \verbatim
126*>          XACT is COMPLEX array, dimension (LDX,NRHS)
127*>          The exact solution vectors.  Each vector is stored as a
128*>          column of the matrix XACT.
129*> \endverbatim
130*>
131*> \param[in] LDXACT
132*> \verbatim
133*>          LDXACT is INTEGER
134*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
135*> \endverbatim
136*>
137*> \param[in] FERR
138*> \verbatim
139*>          FERR is REAL array, dimension (NRHS)
140*>          The estimated forward error bounds for each solution vector
141*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
142*>          of the largest entry in (X - XTRUE) divided by the magnitude
143*>          of the largest entry in X.
144*> \endverbatim
145*>
146*> \param[in] BERR
147*> \verbatim
148*>          BERR is REAL array, dimension (NRHS)
149*>          The componentwise relative backward error of each solution
150*>          vector (i.e., the smallest relative change in any entry of A
151*>          or B that makes X an exact solution).
152*> \endverbatim
153*>
154*> \param[out] RESLTS
155*> \verbatim
156*>          RESLTS is REAL array, dimension (2)
157*>          The maximum over the NRHS solution vectors of the ratios:
158*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
159*>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
160*> \endverbatim
161*
162*  Authors:
163*  ========
164*
165*> \author Univ. of Tennessee
166*> \author Univ. of California Berkeley
167*> \author Univ. of Colorado Denver
168*> \author NAG Ltd.
169*
170*> \ingroup complex_lin
171*
172*  =====================================================================
173      SUBROUTINE CTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX,
174     $                   XACT, LDXACT, FERR, BERR, RESLTS )
175*
176*  -- LAPACK test routine --
177*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
178*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
179*
180*     .. Scalar Arguments ..
181      CHARACTER          DIAG, TRANS, UPLO
182      INTEGER            LDB, LDX, LDXACT, N, NRHS
183*     ..
184*     .. Array Arguments ..
185      REAL               BERR( * ), FERR( * ), RESLTS( * )
186      COMPLEX            AP( * ), B( LDB, * ), X( LDX, * ),
187     $                   XACT( LDXACT, * )
188*     ..
189*
190*  =====================================================================
191*
192*     .. Parameters ..
193      REAL               ZERO, ONE
194      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
195*     ..
196*     .. Local Scalars ..
197      LOGICAL            NOTRAN, UNIT, UPPER
198      INTEGER            I, IFU, IMAX, J, JC, K
199      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
200      COMPLEX            ZDUM
201*     ..
202*     .. External Functions ..
203      LOGICAL            LSAME
204      INTEGER            ICAMAX
205      REAL               SLAMCH
206      EXTERNAL           LSAME, ICAMAX, SLAMCH
207*     ..
208*     .. Intrinsic Functions ..
209      INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
210*     ..
211*     .. Statement Functions ..
212      REAL               CABS1
213*     ..
214*     .. Statement Function definitions ..
215      CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
216*     ..
217*     .. Executable Statements ..
218*
219*     Quick exit if N = 0 or NRHS = 0.
220*
221      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
222         RESLTS( 1 ) = ZERO
223         RESLTS( 2 ) = ZERO
224         RETURN
225      END IF
226*
227      EPS = SLAMCH( 'Epsilon' )
228      UNFL = SLAMCH( 'Safe minimum' )
229      OVFL = ONE / UNFL
230      UPPER = LSAME( UPLO, 'U' )
231      NOTRAN = LSAME( TRANS, 'N' )
232      UNIT = LSAME( DIAG, 'U' )
233*
234*     Test 1:  Compute the maximum of
235*        norm(X - XACT) / ( norm(X) * FERR )
236*     over all the vectors X and XACT using the infinity-norm.
237*
238      ERRBND = ZERO
239      DO 30 J = 1, NRHS
240         IMAX = ICAMAX( N, X( 1, J ), 1 )
241         XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
242         DIFF = ZERO
243         DO 10 I = 1, N
244            DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
245   10    CONTINUE
246*
247         IF( XNORM.GT.ONE ) THEN
248            GO TO 20
249         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
250            GO TO 20
251         ELSE
252            ERRBND = ONE / EPS
253            GO TO 30
254         END IF
255*
256   20    CONTINUE
257         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
258            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
259         ELSE
260            ERRBND = ONE / EPS
261         END IF
262   30 CONTINUE
263      RESLTS( 1 ) = ERRBND
264*
265*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
266*     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
267*
268      IFU = 0
269      IF( UNIT )
270     $   IFU = 1
271      DO 90 K = 1, NRHS
272         DO 80 I = 1, N
273            TMP = CABS1( B( I, K ) )
274            IF( UPPER ) THEN
275               JC = ( ( I-1 )*I ) / 2
276               IF( .NOT.NOTRAN ) THEN
277                  DO 40 J = 1, I - IFU
278                     TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
279   40             CONTINUE
280                  IF( UNIT )
281     $               TMP = TMP + CABS1( X( I, K ) )
282               ELSE
283                  JC = JC + I
284                  IF( UNIT ) THEN
285                     TMP = TMP + CABS1( X( I, K ) )
286                     JC = JC + I
287                  END IF
288                  DO 50 J = I + IFU, N
289                     TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
290                     JC = JC + J
291   50             CONTINUE
292               END IF
293            ELSE
294               IF( NOTRAN ) THEN
295                  JC = I
296                  DO 60 J = 1, I - IFU
297                     TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
298                     JC = JC + N - J
299   60             CONTINUE
300                  IF( UNIT )
301     $               TMP = TMP + CABS1( X( I, K ) )
302               ELSE
303                  JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2
304                  IF( UNIT )
305     $               TMP = TMP + CABS1( X( I, K ) )
306                  DO 70 J = I + IFU, N
307                     TMP = TMP + CABS1( AP( JC+J-I ) )*
308     $                     CABS1( X( J, K ) )
309   70             CONTINUE
310               END IF
311            END IF
312            IF( I.EQ.1 ) THEN
313               AXBI = TMP
314            ELSE
315               AXBI = MIN( AXBI, TMP )
316            END IF
317   80    CONTINUE
318         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
319     $         MAX( AXBI, ( N+1 )*UNFL ) )
320         IF( K.EQ.1 ) THEN
321            RESLTS( 2 ) = TMP
322         ELSE
323            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
324         END IF
325   90 CONTINUE
326*
327      RETURN
328*
329*     End of CTPT05
330*
331      END
332