1*> \brief \b SGBT05
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 SGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
12*                          LDX, XACT, LDXACT, FERR, BERR, RESLTS )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          TRANS
16*       INTEGER            KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
17*       ..
18*       .. Array Arguments ..
19*       REAL               AB( LDAB, * ), B( LDB, * ), BERR( * ),
20*      $                   FERR( * ), RESLTS( * ), X( LDX, * ),
21*      $                   XACT( LDXACT, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> SGBT05 tests the error bounds from iterative refinement for the
31*> computed solution to a system of equations op(A)*X = B, where A is a
32*> general band matrix of order n with kl subdiagonals and ku
33*> superdiagonals 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 / ( 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, B, and XACT, and the
62*>          order of the matrix A.  N >= 0.
63*> \endverbatim
64*>
65*> \param[in] KL
66*> \verbatim
67*>          KL is INTEGER
68*>          The number of subdiagonals within the band of A.  KL >= 0.
69*> \endverbatim
70*>
71*> \param[in] KU
72*> \verbatim
73*>          KU is INTEGER
74*>          The number of superdiagonals within the band of A.  KU >= 0.
75*> \endverbatim
76*>
77*> \param[in] NRHS
78*> \verbatim
79*>          NRHS is INTEGER
80*>          The number of columns of the matrices X, B, and XACT.
81*>          NRHS >= 0.
82*> \endverbatim
83*>
84*> \param[in] AB
85*> \verbatim
86*>          AB is REAL array, dimension (LDAB,N)
87*>          The original band matrix A, stored in rows 1 to KL+KU+1.
88*>          The j-th column of A is stored in the j-th column of the
89*>          array AB as follows:
90*>          AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl).
91*> \endverbatim
92*>
93*> \param[in] LDAB
94*> \verbatim
95*>          LDAB is INTEGER
96*>          The leading dimension of the array AB.  LDAB >= KL+KU+1.
97*> \endverbatim
98*>
99*> \param[in] B
100*> \verbatim
101*>          B is REAL array, dimension (LDB,NRHS)
102*>          The right hand side vectors for the system of linear
103*>          equations.
104*> \endverbatim
105*>
106*> \param[in] LDB
107*> \verbatim
108*>          LDB is INTEGER
109*>          The leading dimension of the array B.  LDB >= max(1,N).
110*> \endverbatim
111*>
112*> \param[in] X
113*> \verbatim
114*>          X is REAL array, dimension (LDX,NRHS)
115*>          The computed solution vectors.  Each vector is stored as a
116*>          column of the matrix X.
117*> \endverbatim
118*>
119*> \param[in] LDX
120*> \verbatim
121*>          LDX is INTEGER
122*>          The leading dimension of the array X.  LDX >= max(1,N).
123*> \endverbatim
124*>
125*> \param[in] XACT
126*> \verbatim
127*>          XACT is REAL array, dimension (LDX,NRHS)
128*>          The exact solution vectors.  Each vector is stored as a
129*>          column of the matrix XACT.
130*> \endverbatim
131*>
132*> \param[in] LDXACT
133*> \verbatim
134*>          LDXACT is INTEGER
135*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
136*> \endverbatim
137*>
138*> \param[in] FERR
139*> \verbatim
140*>          FERR is REAL array, dimension (NRHS)
141*>          The estimated forward error bounds for each solution vector
142*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
143*>          of the largest entry in (X - XTRUE) divided by the magnitude
144*>          of the largest entry in X.
145*> \endverbatim
146*>
147*> \param[in] BERR
148*> \verbatim
149*>          BERR is REAL array, dimension (NRHS)
150*>          The componentwise relative backward error of each solution
151*>          vector (i.e., the smallest relative change in any entry of A
152*>          or B that makes X an exact solution).
153*> \endverbatim
154*>
155*> \param[out] RESLTS
156*> \verbatim
157*>          RESLTS is REAL array, dimension (2)
158*>          The maximum over the NRHS solution vectors of the ratios:
159*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
160*>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
161*> \endverbatim
162*
163*  Authors:
164*  ========
165*
166*> \author Univ. of Tennessee
167*> \author Univ. of California Berkeley
168*> \author Univ. of Colorado Denver
169*> \author NAG Ltd.
170*
171*> \ingroup single_lin
172*
173*  =====================================================================
174      SUBROUTINE SGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X,
175     $                   LDX, XACT, LDXACT, FERR, BERR, RESLTS )
176*
177*  -- LAPACK test routine --
178*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
179*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
180*
181*     .. Scalar Arguments ..
182      CHARACTER          TRANS
183      INTEGER            KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS
184*     ..
185*     .. Array Arguments ..
186      REAL               AB( LDAB, * ), B( LDB, * ), BERR( * ),
187     $                   FERR( * ), RESLTS( * ), X( LDX, * ),
188     $                   XACT( LDXACT, * )
189*     ..
190*
191*  =====================================================================
192*
193*     .. Parameters ..
194      REAL               ZERO, ONE
195      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
196*     ..
197*     .. Local Scalars ..
198      LOGICAL            NOTRAN
199      INTEGER            I, IMAX, J, K, NZ
200      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
201*     ..
202*     .. External Functions ..
203      LOGICAL            LSAME
204      INTEGER            ISAMAX
205      REAL               SLAMCH
206      EXTERNAL           LSAME, ISAMAX, SLAMCH
207*     ..
208*     .. Intrinsic Functions ..
209      INTRINSIC          ABS, MAX, MIN
210*     ..
211*     .. Executable Statements ..
212*
213*     Quick exit if N = 0 or NRHS = 0.
214*
215      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
216         RESLTS( 1 ) = ZERO
217         RESLTS( 2 ) = ZERO
218         RETURN
219      END IF
220*
221      EPS = SLAMCH( 'Epsilon' )
222      UNFL = SLAMCH( 'Safe minimum' )
223      OVFL = ONE / UNFL
224      NOTRAN = LSAME( TRANS, 'N' )
225      NZ = MIN( KL+KU+2, N+1 )
226*
227*     Test 1:  Compute the maximum of
228*        norm(X - XACT) / ( norm(X) * FERR )
229*     over all the vectors X and XACT using the infinity-norm.
230*
231      ERRBND = ZERO
232      DO 30 J = 1, NRHS
233         IMAX = ISAMAX( N, X( 1, J ), 1 )
234         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
235         DIFF = ZERO
236         DO 10 I = 1, N
237            DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
238   10    CONTINUE
239*
240         IF( XNORM.GT.ONE ) THEN
241            GO TO 20
242         ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
243            GO TO 20
244         ELSE
245            ERRBND = ONE / EPS
246            GO TO 30
247         END IF
248*
249   20    CONTINUE
250         IF( DIFF / XNORM.LE.FERR( J ) ) THEN
251            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
252         ELSE
253            ERRBND = ONE / EPS
254         END IF
255   30 CONTINUE
256      RESLTS( 1 ) = ERRBND
257*
258*     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
259*     (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
260*
261      DO 70 K = 1, NRHS
262         DO 60 I = 1, N
263            TMP = ABS( B( I, K ) )
264            IF( NOTRAN ) THEN
265               DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N )
266                  TMP = TMP + ABS( AB( KU+1+I-J, J ) )*ABS( X( J, K ) )
267   40          CONTINUE
268            ELSE
269               DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N )
270                  TMP = TMP + ABS( AB( KU+1+J-I, I ) )*ABS( X( J, K ) )
271   50          CONTINUE
272            END IF
273            IF( I.EQ.1 ) THEN
274               AXBI = TMP
275            ELSE
276               AXBI = MIN( AXBI, TMP )
277            END IF
278   60    CONTINUE
279         TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*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 SGBT05
290*
291      END
292