1*> \brief \b CGET07
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 CGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
12*                          LDXACT, FERR, CHKFERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          TRANS
16*       LOGICAL            CHKFERR
17*       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
18*       ..
19*       .. Array Arguments ..
20*       REAL               BERR( * ), FERR( * ), RESLTS( * )
21*       COMPLEX            A( LDA, * ), B( LDB, * ), X( LDX, * ),
22*      $                   XACT( LDXACT, * )
23*       ..
24*
25*
26*> \par Purpose:
27*  =============
28*>
29*> \verbatim
30*>
31*> CGET07 tests the error bounds from iterative refinement for the
32*> computed solution to a system of equations op(A)*X = B, where A is a
33*> general n by n matrix and op(A) = A or A**T, 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 / ( (n+1)*EPS + (*) ), where
42*>             (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
43*> \endverbatim
44*
45*  Arguments:
46*  ==========
47*
48*> \param[in] TRANS
49*> \verbatim
50*>          TRANS is CHARACTER*1
51*>          Specifies the form of the system of equations.
52*>          = 'N':  A * X = B     (No transpose)
53*>          = 'T':  A**T * X = B  (Transpose)
54*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
55*> \endverbatim
56*>
57*> \param[in] N
58*> \verbatim
59*>          N is INTEGER
60*>          The number of rows of the matrices X and XACT.  N >= 0.
61*> \endverbatim
62*>
63*> \param[in] NRHS
64*> \verbatim
65*>          NRHS is INTEGER
66*>          The number of columns of the matrices X and XACT.  NRHS >= 0.
67*> \endverbatim
68*>
69*> \param[in] A
70*> \verbatim
71*>          A is COMPLEX array, dimension (LDA,N)
72*>          The original n by n matrix A.
73*> \endverbatim
74*>
75*> \param[in] LDA
76*> \verbatim
77*>          LDA is INTEGER
78*>          The leading dimension of the array A.  LDA >= max(1,N).
79*> \endverbatim
80*>
81*> \param[in] B
82*> \verbatim
83*>          B is COMPLEX array, dimension (LDB,NRHS)
84*>          The right hand side vectors for the system of linear
85*>          equations.
86*> \endverbatim
87*>
88*> \param[in] LDB
89*> \verbatim
90*>          LDB is INTEGER
91*>          The leading dimension of the array B.  LDB >= max(1,N).
92*> \endverbatim
93*>
94*> \param[in] X
95*> \verbatim
96*>          X is COMPLEX array, dimension (LDX,NRHS)
97*>          The computed solution vectors.  Each vector is stored as a
98*>          column of the matrix X.
99*> \endverbatim
100*>
101*> \param[in] LDX
102*> \verbatim
103*>          LDX is INTEGER
104*>          The leading dimension of the array X.  LDX >= max(1,N).
105*> \endverbatim
106*>
107*> \param[in] XACT
108*> \verbatim
109*>          XACT is COMPLEX array, dimension (LDX,NRHS)
110*>          The exact solution vectors.  Each vector is stored as a
111*>          column of the matrix XACT.
112*> \endverbatim
113*>
114*> \param[in] LDXACT
115*> \verbatim
116*>          LDXACT is INTEGER
117*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
118*> \endverbatim
119*>
120*> \param[in] FERR
121*> \verbatim
122*>          FERR is REAL array, dimension (NRHS)
123*>          The estimated forward error bounds for each solution vector
124*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
125*>          of the largest entry in (X - XTRUE) divided by the magnitude
126*>          of the largest entry in X.
127*> \endverbatim
128*>
129*> \param[in] CHKFERR
130*> \verbatim
131*>          CHKFERR is LOGICAL
132*>          Set to .TRUE. to check FERR, .FALSE. not to check FERR.
133*>          When the test system is ill-conditioned, the "true"
134*>          solution in XACT may be incorrect.
135*> \endverbatim
136*>
137*> \param[in] BERR
138*> \verbatim
139*>          BERR is REAL array, dimension (NRHS)
140*>          The componentwise relative backward error of each solution
141*>          vector (i.e., the smallest relative change in any entry of A
142*>          or B that makes X an exact solution).
143*> \endverbatim
144*>
145*> \param[out] RESLTS
146*> \verbatim
147*>          RESLTS is REAL array, dimension (2)
148*>          The maximum over the NRHS solution vectors of the ratios:
149*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
150*>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
151*> \endverbatim
152*
153*  Authors:
154*  ========
155*
156*> \author Univ. of Tennessee
157*> \author Univ. of California Berkeley
158*> \author Univ. of Colorado Denver
159*> \author NAG Ltd.
160*
161*> \ingroup complex_lin
162*
163*  =====================================================================
164      SUBROUTINE CGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
165     $                   LDXACT, FERR, CHKFERR, BERR, RESLTS )
166*
167*  -- LAPACK test routine --
168*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
169*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
170*
171*     .. Scalar Arguments ..
172      CHARACTER          TRANS
173      LOGICAL            CHKFERR
174      INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
175*     ..
176*     .. Array Arguments ..
177      REAL               BERR( * ), FERR( * ), RESLTS( * )
178      COMPLEX            A( LDA, * ), B( LDB, * ), X( LDX, * ),
179     $                   XACT( LDXACT, * )
180*     ..
181*
182*  =====================================================================
183*
184*     .. Parameters ..
185      REAL               ZERO, ONE
186      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
187*     ..
188*     .. Local Scalars ..
189      LOGICAL            NOTRAN
190      INTEGER            I, IMAX, J, K
191      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
192      COMPLEX            ZDUM
193*     ..
194*     .. External Functions ..
195      LOGICAL            LSAME
196      INTEGER            ICAMAX
197      REAL               SLAMCH
198      EXTERNAL           LSAME, ICAMAX, SLAMCH
199*     ..
200*     .. Intrinsic Functions ..
201      INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
202*     ..
203*     .. Statement Functions ..
204      REAL               CABS1
205*     ..
206*     .. Statement Function definitions ..
207      CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
208*     ..
209*     .. Executable Statements ..
210*
211*     Quick exit if N = 0 or NRHS = 0.
212*
213      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
214         RESLTS( 1 ) = ZERO
215         RESLTS( 2 ) = ZERO
216         RETURN
217      END IF
218*
219      EPS = SLAMCH( 'Epsilon' )
220      UNFL = SLAMCH( 'Safe minimum' )
221      OVFL = ONE / UNFL
222      NOTRAN = LSAME( TRANS, 'N' )
223*
224*     Test 1:  Compute the maximum of
225*        norm(X - XACT) / ( norm(X) * FERR )
226*     over all the vectors X and XACT using the infinity-norm.
227*
228      ERRBND = ZERO
229      IF( CHKFERR ) THEN
230         DO 30 J = 1, NRHS
231            IMAX = ICAMAX( N, X( 1, J ), 1 )
232            XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
233            DIFF = ZERO
234            DO 10 I = 1, N
235               DIFF = MAX( DIFF, CABS1( 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      END IF
255      RESLTS( 1 ) = ERRBND
256*
257*     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
258*     (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
259*
260      DO 70 K = 1, NRHS
261         DO 60 I = 1, N
262            TMP = CABS1( B( I, K ) )
263            IF( NOTRAN ) THEN
264               DO 40 J = 1, N
265                  TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) )
266   40          CONTINUE
267            ELSE
268               DO 50 J = 1, N
269                  TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) )
270   50          CONTINUE
271            END IF
272            IF( I.EQ.1 ) THEN
273               AXBI = TMP
274            ELSE
275               AXBI = MIN( AXBI, TMP )
276            END IF
277   60    CONTINUE
278         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
279     $         MAX( AXBI, ( N+1 )*UNFL ) )
280         IF( K.EQ.1 ) THEN
281            RESLTS( 2 ) = TMP
282         ELSE
283            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
284         END IF
285   70 CONTINUE
286*
287      RETURN
288*
289*     End of CGET07
290*
291      END
292