1*> \brief \b DQRT16
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 DQRT16( TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB,
12*                          RWORK, RESID )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          TRANS
16*       INTEGER            LDA, LDB, LDX, M, N, NRHS
17*       DOUBLE PRECISION   RESID
18*       ..
19*       .. Array Arguments ..
20*       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), RWORK( * ),
21*      $                   X( LDX, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> DQRT16 computes the residual for a solution of a system of linear
31*> equations  A*x = b  or  A'*x = b:
32*>    RESID = norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ),
33*> where EPS is the machine epsilon.
34*> \endverbatim
35*
36*  Arguments:
37*  ==========
38*
39*> \param[in] TRANS
40*> \verbatim
41*>          TRANS is CHARACTER*1
42*>          Specifies the form of the system of equations:
43*>          = 'N':  A *x = b
44*>          = 'T':  A'*x = b, where A' is the transpose of A
45*>          = 'C':  A'*x = b, where A' is the transpose of A
46*> \endverbatim
47*>
48*> \param[in] M
49*> \verbatim
50*>          M is INTEGER
51*>          The number of rows of the matrix A.  M >= 0.
52*> \endverbatim
53*>
54*> \param[in] N
55*> \verbatim
56*>          N is INTEGER
57*>          The number of columns of the matrix A.  N >= 0.
58*> \endverbatim
59*>
60*> \param[in] NRHS
61*> \verbatim
62*>          NRHS is INTEGER
63*>          The number of columns of B, the matrix of right hand sides.
64*>          NRHS >= 0.
65*> \endverbatim
66*>
67*> \param[in] A
68*> \verbatim
69*>          A is DOUBLE PRECISION array, dimension (LDA,N)
70*>          The original M x N matrix A.
71*> \endverbatim
72*>
73*> \param[in] LDA
74*> \verbatim
75*>          LDA is INTEGER
76*>          The leading dimension of the array A.  LDA >= max(1,M).
77*> \endverbatim
78*>
79*> \param[in] X
80*> \verbatim
81*>          X is DOUBLE PRECISION array, dimension (LDX,NRHS)
82*>          The computed solution vectors for the system of linear
83*>          equations.
84*> \endverbatim
85*>
86*> \param[in] LDX
87*> \verbatim
88*>          LDX is INTEGER
89*>          The leading dimension of the array X.  If TRANS = 'N',
90*>          LDX >= max(1,N); if TRANS = 'T' or 'C', LDX >= max(1,M).
91*> \endverbatim
92*>
93*> \param[in,out] B
94*> \verbatim
95*>          B is DOUBLE PRECISION array, dimension (LDB,NRHS)
96*>          On entry, the right hand side vectors for the system of
97*>          linear equations.
98*>          On exit, B is overwritten with the difference B - A*X.
99*> \endverbatim
100*>
101*> \param[in] LDB
102*> \verbatim
103*>          LDB is INTEGER
104*>          The leading dimension of the array B.  IF TRANS = 'N',
105*>          LDB >= max(1,M); if TRANS = 'T' or 'C', LDB >= max(1,N).
106*> \endverbatim
107*>
108*> \param[out] RWORK
109*> \verbatim
110*>          RWORK is DOUBLE PRECISION array, dimension (M)
111*> \endverbatim
112*>
113*> \param[out] RESID
114*> \verbatim
115*>          RESID is DOUBLE PRECISION
116*>          The maximum over the number of right hand sides of
117*>          norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ).
118*> \endverbatim
119*
120*  Authors:
121*  ========
122*
123*> \author Univ. of Tennessee
124*> \author Univ. of California Berkeley
125*> \author Univ. of Colorado Denver
126*> \author NAG Ltd.
127*
128*> \ingroup double_lin
129*
130*  =====================================================================
131      SUBROUTINE DQRT16( TRANS, M, N, NRHS, A, LDA, X, LDX, B, LDB,
132     $                   RWORK, RESID )
133*
134*  -- LAPACK test routine --
135*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
136*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
137*
138*     .. Scalar Arguments ..
139      CHARACTER          TRANS
140      INTEGER            LDA, LDB, LDX, M, N, NRHS
141      DOUBLE PRECISION   RESID
142*     ..
143*     .. Array Arguments ..
144      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), RWORK( * ),
145     $                   X( LDX, * )
146*     ..
147*
148*  =====================================================================
149*
150*     .. Parameters ..
151      DOUBLE PRECISION   ZERO, ONE
152      PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
153*     ..
154*     .. Local Scalars ..
155      INTEGER            J, N1, N2
156      DOUBLE PRECISION   ANORM, BNORM, EPS, XNORM
157*     ..
158*     .. External Functions ..
159      LOGICAL            LSAME
160      DOUBLE PRECISION   DASUM, DLAMCH, DLANGE
161      EXTERNAL           LSAME, DASUM, DLAMCH, DLANGE
162*     ..
163*     .. External Subroutines ..
164      EXTERNAL           DGEMM
165*     ..
166*     .. Intrinsic Functions ..
167      INTRINSIC          MAX
168*     ..
169*     .. Executable Statements ..
170*
171*     Quick exit if M = 0 or N = 0 or NRHS = 0
172*
173      IF( M.LE.0 .OR. N.LE.0 .OR. NRHS.EQ.0 ) THEN
174         RESID = ZERO
175         RETURN
176      END IF
177*
178      IF( LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' ) ) THEN
179         ANORM = DLANGE( 'I', M, N, A, LDA, RWORK )
180         N1 = N
181         N2 = M
182      ELSE
183         ANORM = DLANGE( '1', M, N, A, LDA, RWORK )
184         N1 = M
185         N2 = N
186      END IF
187*
188      EPS = DLAMCH( 'Epsilon' )
189*
190*     Compute  B - A*X  (or  B - A'*X ) and store in B.
191*
192      CALL DGEMM( TRANS, 'No transpose', N1, NRHS, N2, -ONE, A, LDA, X,
193     $            LDX, ONE, B, LDB )
194*
195*     Compute the maximum over the number of right hand sides of
196*        norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ) .
197*
198      RESID = ZERO
199      DO 10 J = 1, NRHS
200         BNORM = DASUM( N1, B( 1, J ), 1 )
201         XNORM = DASUM( N2, X( 1, J ), 1 )
202         IF( ANORM.EQ.ZERO .AND. BNORM.EQ.ZERO ) THEN
203            RESID = ZERO
204         ELSE IF( ANORM.LE.ZERO .OR. XNORM.LE.ZERO ) THEN
205            RESID = ONE / EPS
206         ELSE
207            RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) /
208     $              ( MAX( M, N )*EPS ) )
209         END IF
210   10 CONTINUE
211*
212      RETURN
213*
214*     End of DQRT16
215*
216      END
217