1*> \brief \b CGTT05
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 CGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
12*                          XACT, LDXACT, FERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          TRANS
16*       INTEGER            LDB, LDX, LDXACT, N, NRHS
17*       ..
18*       .. Array Arguments ..
19*       REAL               BERR( * ), FERR( * ), RESLTS( * )
20*       COMPLEX            B( LDB, * ), D( * ), DL( * ), DU( * ),
21*      $                   X( LDX, * ), XACT( LDXACT, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> CGTT05 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*> general tridiagonal matrix of order n and op(A) = A or A**T,
33*> 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 / ( NZ*EPS + (*) ), where
42*>             (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
43*>             and NZ = max. number of nonzeros in any row of A, plus 1
44*> \endverbatim
45*
46*  Arguments:
47*  ==========
48*
49*> \param[in] TRANS
50*> \verbatim
51*>          TRANS is CHARACTER*1
52*>          Specifies the form of the system of equations.
53*>          = 'N':  A * X = B     (No transpose)
54*>          = 'T':  A**T * X = B  (Transpose)
55*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
56*> \endverbatim
57*>
58*> \param[in] N
59*> \verbatim
60*>          N is INTEGER
61*>          The number of rows of the matrices X and XACT.  N >= 0.
62*> \endverbatim
63*>
64*> \param[in] NRHS
65*> \verbatim
66*>          NRHS is INTEGER
67*>          The number of columns of the matrices X and XACT.  NRHS >= 0.
68*> \endverbatim
69*>
70*> \param[in] DL
71*> \verbatim
72*>          DL is COMPLEX array, dimension (N-1)
73*>          The (n-1) sub-diagonal elements of A.
74*> \endverbatim
75*>
76*> \param[in] D
77*> \verbatim
78*>          D is COMPLEX array, dimension (N)
79*>          The diagonal elements of A.
80*> \endverbatim
81*>
82*> \param[in] DU
83*> \verbatim
84*>          DU is COMPLEX array, dimension (N-1)
85*>          The (n-1) super-diagonal elements of A.
86*> \endverbatim
87*>
88*> \param[in] B
89*> \verbatim
90*>          B is COMPLEX array, dimension (LDB,NRHS)
91*>          The right hand side vectors for the system of linear
92*>          equations.
93*> \endverbatim
94*>
95*> \param[in] LDB
96*> \verbatim
97*>          LDB is INTEGER
98*>          The leading dimension of the array B.  LDB >= max(1,N).
99*> \endverbatim
100*>
101*> \param[in] X
102*> \verbatim
103*>          X is COMPLEX array, dimension (LDX,NRHS)
104*>          The computed solution vectors.  Each vector is stored as a
105*>          column of the matrix X.
106*> \endverbatim
107*>
108*> \param[in] LDX
109*> \verbatim
110*>          LDX is INTEGER
111*>          The leading dimension of the array X.  LDX >= max(1,N).
112*> \endverbatim
113*>
114*> \param[in] XACT
115*> \verbatim
116*>          XACT is COMPLEX array, dimension (LDX,NRHS)
117*>          The exact solution vectors.  Each vector is stored as a
118*>          column of the matrix XACT.
119*> \endverbatim
120*>
121*> \param[in] LDXACT
122*> \verbatim
123*>          LDXACT is INTEGER
124*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
125*> \endverbatim
126*>
127*> \param[in] FERR
128*> \verbatim
129*>          FERR is REAL array, dimension (NRHS)
130*>          The estimated forward error bounds for each solution vector
131*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
132*>          of the largest entry in (X - XTRUE) divided by the magnitude
133*>          of the largest entry in X.
134*> \endverbatim
135*>
136*> \param[in] BERR
137*> \verbatim
138*>          BERR is REAL array, dimension (NRHS)
139*>          The componentwise relative backward error of each solution
140*>          vector (i.e., the smallest relative change in any entry of A
141*>          or B that makes X an exact solution).
142*> \endverbatim
143*>
144*> \param[out] RESLTS
145*> \verbatim
146*>          RESLTS is REAL array, dimension (2)
147*>          The maximum over the NRHS solution vectors of the ratios:
148*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
149*>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
150*> \endverbatim
151*
152*  Authors:
153*  ========
154*
155*> \author Univ. of Tennessee
156*> \author Univ. of California Berkeley
157*> \author Univ. of Colorado Denver
158*> \author NAG Ltd.
159*
160*> \ingroup complex_lin
161*
162*  =====================================================================
163      SUBROUTINE CGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
164     $                   XACT, LDXACT, FERR, BERR, RESLTS )
165*
166*  -- LAPACK test routine --
167*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
168*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
169*
170*     .. Scalar Arguments ..
171      CHARACTER          TRANS
172      INTEGER            LDB, LDX, LDXACT, N, NRHS
173*     ..
174*     .. Array Arguments ..
175      REAL               BERR( * ), FERR( * ), RESLTS( * )
176      COMPLEX            B( LDB, * ), D( * ), DL( * ), DU( * ),
177     $                   X( LDX, * ), XACT( LDXACT, * )
178*     ..
179*
180*  =====================================================================
181*
182*     .. Parameters ..
183      REAL               ZERO, ONE
184      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
185*     ..
186*     .. Local Scalars ..
187      LOGICAL            NOTRAN
188      INTEGER            I, IMAX, J, K, NZ
189      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
190      COMPLEX            ZDUM
191*     ..
192*     .. External Functions ..
193      LOGICAL            LSAME
194      INTEGER            ICAMAX
195      REAL               SLAMCH
196      EXTERNAL           LSAME, ICAMAX, SLAMCH
197*     ..
198*     .. Intrinsic Functions ..
199      INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
200*     ..
201*     .. Statement Functions ..
202      REAL               CABS1
203*     ..
204*     .. Statement Function definitions ..
205      CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
206*     ..
207*     .. Executable Statements ..
208*
209*     Quick exit if N = 0 or NRHS = 0.
210*
211      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
212         RESLTS( 1 ) = ZERO
213         RESLTS( 2 ) = ZERO
214         RETURN
215      END IF
216*
217      EPS = SLAMCH( 'Epsilon' )
218      UNFL = SLAMCH( 'Safe minimum' )
219      OVFL = ONE / UNFL
220      NOTRAN = LSAME( TRANS, 'N' )
221      NZ = 4
222*
223*     Test 1:  Compute the maximum of
224*        norm(X - XACT) / ( norm(X) * FERR )
225*     over all the vectors X and XACT using the infinity-norm.
226*
227      ERRBND = ZERO
228      DO 30 J = 1, NRHS
229         IMAX = ICAMAX( N, X( 1, J ), 1 )
230         XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
231         DIFF = ZERO
232         DO 10 I = 1, N
233            DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
234   10    CONTINUE
235*
236         IF( XNORM.GT.ONE ) THEN
237            GO TO 20
238         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
239            GO TO 20
240         ELSE
241            ERRBND = ONE / EPS
242            GO TO 30
243         END IF
244*
245   20    CONTINUE
246         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
247            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
248         ELSE
249            ERRBND = ONE / EPS
250         END IF
251   30 CONTINUE
252      RESLTS( 1 ) = ERRBND
253*
254*     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
255*     (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
256*
257      DO 60 K = 1, NRHS
258         IF( NOTRAN ) THEN
259            IF( N.EQ.1 ) THEN
260               AXBI = CABS1( B( 1, K ) ) +
261     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) )
262            ELSE
263               AXBI = CABS1( B( 1, K ) ) +
264     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
265     $                CABS1( DU( 1 ) )*CABS1( X( 2, K ) )
266               DO 40 I = 2, N - 1
267                  TMP = CABS1( B( I, K ) ) +
268     $                  CABS1( DL( I-1 ) )*CABS1( X( I-1, K ) ) +
269     $                  CABS1( D( I ) )*CABS1( X( I, K ) ) +
270     $                  CABS1( DU( I ) )*CABS1( X( I+1, K ) )
271                  AXBI = MIN( AXBI, TMP )
272   40          CONTINUE
273               TMP = CABS1( B( N, K ) ) + CABS1( DL( N-1 ) )*
274     $               CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
275     $               CABS1( X( N, K ) )
276               AXBI = MIN( AXBI, TMP )
277            END IF
278         ELSE
279            IF( N.EQ.1 ) THEN
280               AXBI = CABS1( B( 1, K ) ) +
281     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) )
282            ELSE
283               AXBI = CABS1( B( 1, K ) ) +
284     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
285     $                CABS1( DL( 1 ) )*CABS1( X( 2, K ) )
286               DO 50 I = 2, N - 1
287                  TMP = CABS1( B( I, K ) ) +
288     $                  CABS1( DU( I-1 ) )*CABS1( X( I-1, K ) ) +
289     $                  CABS1( D( I ) )*CABS1( X( I, K ) ) +
290     $                  CABS1( DL( I ) )*CABS1( X( I+1, K ) )
291                  AXBI = MIN( AXBI, TMP )
292   50          CONTINUE
293               TMP = CABS1( B( N, K ) ) + CABS1( DU( N-1 ) )*
294     $               CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
295     $               CABS1( X( N, K ) )
296               AXBI = MIN( AXBI, TMP )
297            END IF
298         END IF
299         TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
300         IF( K.EQ.1 ) THEN
301            RESLTS( 2 ) = TMP
302         ELSE
303            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
304         END IF
305   60 CONTINUE
306*
307      RETURN
308*
309*     End of CGTT05
310*
311      END
312