1*> \brief \b CQRT16
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 CQRT16( 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*       REAL               RESID
18*       ..
19*       .. Array Arguments ..
20*       REAL               RWORK( * )
21*       COMPLEX            A( LDA, * ), B( LDB, * ), X( LDX, * )
22*       ..
23*
24*
25*> \par Purpose:
26*  =============
27*>
28*> \verbatim
29*>
30*> CQRT16 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^T*x = b, where A^T is the transpose of A
45*>          = 'C':  A^H*x = b, where A^H is the conjugate 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 COMPLEX 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 COMPLEX 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 COMPLEX 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 REAL array, dimension (M)
111*> \endverbatim
112*>
113*> \param[out] RESID
114*> \verbatim
115*>          RESID is REAL
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 complex_lin
129*
130*  =====================================================================
131      SUBROUTINE CQRT16( 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      REAL               RESID
142*     ..
143*     .. Array Arguments ..
144      REAL               RWORK( * )
145      COMPLEX            A( LDA, * ), B( LDB, * ), X( LDX, * )
146*     ..
147*
148*  =====================================================================
149*
150*     .. Parameters ..
151      REAL               ZERO, ONE
152      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
153      COMPLEX            CONE
154      PARAMETER          ( CONE = ( 1.0E+0, 0.0E+0 ) )
155*     ..
156*     .. Local Scalars ..
157      INTEGER            J, N1, N2
158      REAL               ANORM, BNORM, EPS, XNORM
159*     ..
160*     .. External Functions ..
161      LOGICAL            LSAME
162      REAL               CLANGE, SCASUM, SLAMCH
163      EXTERNAL           LSAME, CLANGE, SCASUM, SLAMCH
164*     ..
165*     .. External Subroutines ..
166      EXTERNAL           CGEMM
167*     ..
168*     .. Intrinsic Functions ..
169      INTRINSIC          MAX
170*     ..
171*     .. Executable Statements ..
172*
173*     Quick exit if M = 0 or N = 0 or NRHS = 0
174*
175      IF( M.LE.0 .OR. N.LE.0 .OR. NRHS.EQ.0 ) THEN
176         RESID = ZERO
177         RETURN
178      END IF
179*
180      IF( LSAME( TRANS, 'T' ) .OR. LSAME( TRANS, 'C' ) ) THEN
181         ANORM = CLANGE( 'I', M, N, A, LDA, RWORK )
182         N1 = N
183         N2 = M
184      ELSE
185         ANORM = CLANGE( '1', M, N, A, LDA, RWORK )
186         N1 = M
187         N2 = N
188      END IF
189*
190      EPS = SLAMCH( 'Epsilon' )
191*
192*     Compute  B - A*X  (or  B - A'*X ) and store in B.
193*
194      CALL CGEMM( TRANS, 'No transpose', N1, NRHS, N2, -CONE, A, LDA, X,
195     $            LDX, CONE, B, LDB )
196*
197*     Compute the maximum over the number of right hand sides of
198*        norm(B - A*X) / ( max(m,n) * norm(A) * norm(X) * EPS ) .
199*
200      RESID = ZERO
201      DO 10 J = 1, NRHS
202         BNORM = SCASUM( N1, B( 1, J ), 1 )
203         XNORM = SCASUM( N2, X( 1, J ), 1 )
204         IF( ANORM.EQ.ZERO .AND. BNORM.EQ.ZERO ) THEN
205            RESID = ZERO
206         ELSE IF( ANORM.LE.ZERO .OR. XNORM.LE.ZERO ) THEN
207            RESID = ONE / EPS
208         ELSE
209            RESID = MAX( RESID, ( ( BNORM / ANORM ) / XNORM ) /
210     $              ( MAX( M, N )*EPS ) )
211         END IF
212   10 CONTINUE
213*
214      RETURN
215*
216*     End of CQRT16
217*
218      END
219