1*> \brief \b ZTRT03
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 ZTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE,
12*                          CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID )
13*
14*       .. Scalar Arguments ..
15*       CHARACTER          DIAG, TRANS, UPLO
16*       INTEGER            LDA, LDB, LDX, N, NRHS
17*       DOUBLE PRECISION   RESID, SCALE, TSCAL
18*       ..
19*       .. Array Arguments ..
20*       DOUBLE PRECISION   CNORM( * )
21*       COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
22*      $                   X( LDX, * )
23*       ..
24*
25*
26*> \par Purpose:
27*  =============
28*>
29*> \verbatim
30*>
31*> ZTRT03 computes the residual for the solution to a scaled triangular
32*> system of equations A*x = s*b,  A**T *x = s*b,  or  A**H *x = s*b.
33*> Here A is a triangular matrix, A**T denotes the transpose of A, A**H
34*> denotes the conjugate transpose of A, s is a scalar, and x and b are
35*> N by NRHS matrices.  The test ratio is the maximum over the number of
36*> right hand sides of
37*>    norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
38*> where op(A) denotes A, A**T, or A**H, and EPS is the machine epsilon.
39*> \endverbatim
40*
41*  Arguments:
42*  ==========
43*
44*> \param[in] UPLO
45*> \verbatim
46*>          UPLO is CHARACTER*1
47*>          Specifies whether the matrix A is upper or lower triangular.
48*>          = 'U':  Upper triangular
49*>          = 'L':  Lower triangular
50*> \endverbatim
51*>
52*> \param[in] TRANS
53*> \verbatim
54*>          TRANS is CHARACTER*1
55*>          Specifies the operation applied to A.
56*>          = 'N':  A *x = s*b     (No transpose)
57*>          = 'T':  A**T *x = s*b  (Transpose)
58*>          = 'C':  A**H *x = s*b  (Conjugate transpose)
59*> \endverbatim
60*>
61*> \param[in] DIAG
62*> \verbatim
63*>          DIAG is CHARACTER*1
64*>          Specifies whether or not the matrix A is unit triangular.
65*>          = 'N':  Non-unit triangular
66*>          = 'U':  Unit triangular
67*> \endverbatim
68*>
69*> \param[in] N
70*> \verbatim
71*>          N is INTEGER
72*>          The order of the matrix A.  N >= 0.
73*> \endverbatim
74*>
75*> \param[in] NRHS
76*> \verbatim
77*>          NRHS is INTEGER
78*>          The number of right hand sides, i.e., the number of columns
79*>          of the matrices X and B.  NRHS >= 0.
80*> \endverbatim
81*>
82*> \param[in] A
83*> \verbatim
84*>          A is COMPLEX*16 array, dimension (LDA,N)
85*>          The triangular matrix A.  If UPLO = 'U', the leading n by n
86*>          upper triangular part of the array A contains the upper
87*>          triangular matrix, and the strictly lower triangular part of
88*>          A is not referenced.  If UPLO = 'L', the leading n by n lower
89*>          triangular part of the array A contains the lower triangular
90*>          matrix, and the strictly upper triangular part of A is not
91*>          referenced.  If DIAG = 'U', the diagonal elements of A are
92*>          also not referenced and are assumed to be 1.
93*> \endverbatim
94*>
95*> \param[in] LDA
96*> \verbatim
97*>          LDA is INTEGER
98*>          The leading dimension of the array A.  LDA >= max(1,N).
99*> \endverbatim
100*>
101*> \param[in] SCALE
102*> \verbatim
103*>          SCALE is DOUBLE PRECISION
104*>          The scaling factor s used in solving the triangular system.
105*> \endverbatim
106*>
107*> \param[in] CNORM
108*> \verbatim
109*>          CNORM is DOUBLE PRECISION array, dimension (N)
110*>          The 1-norms of the columns of A, not counting the diagonal.
111*> \endverbatim
112*>
113*> \param[in] TSCAL
114*> \verbatim
115*>          TSCAL is DOUBLE PRECISION
116*>          The scaling factor used in computing the 1-norms in CNORM.
117*>          CNORM actually contains the column norms of TSCAL*A.
118*> \endverbatim
119*>
120*> \param[in] X
121*> \verbatim
122*>          X is COMPLEX*16 array, dimension (LDX,NRHS)
123*>          The computed solution vectors for the system of linear
124*>          equations.
125*> \endverbatim
126*>
127*> \param[in] LDX
128*> \verbatim
129*>          LDX is INTEGER
130*>          The leading dimension of the array X.  LDX >= max(1,N).
131*> \endverbatim
132*>
133*> \param[in] B
134*> \verbatim
135*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
136*>          The right hand side vectors for the system of linear
137*>          equations.
138*> \endverbatim
139*>
140*> \param[in] LDB
141*> \verbatim
142*>          LDB is INTEGER
143*>          The leading dimension of the array B.  LDB >= max(1,N).
144*> \endverbatim
145*>
146*> \param[out] WORK
147*> \verbatim
148*>          WORK is COMPLEX*16 array, dimension (N)
149*> \endverbatim
150*>
151*> \param[out] RESID
152*> \verbatim
153*>          RESID is DOUBLE PRECISION
154*>          The maximum over the number of right hand sides of
155*>          norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
156*> \endverbatim
157*
158*  Authors:
159*  ========
160*
161*> \author Univ. of Tennessee
162*> \author Univ. of California Berkeley
163*> \author Univ. of Colorado Denver
164*> \author NAG Ltd.
165*
166*> \date November 2011
167*
168*> \ingroup complex16_lin
169*
170*  =====================================================================
171      SUBROUTINE ZTRT03( UPLO, TRANS, DIAG, N, NRHS, A, LDA, SCALE,
172     $                   CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID )
173*
174*  -- LAPACK test routine (version 3.4.0) --
175*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
176*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
177*     November 2011
178*
179*     .. Scalar Arguments ..
180      CHARACTER          DIAG, TRANS, UPLO
181      INTEGER            LDA, LDB, LDX, N, NRHS
182      DOUBLE PRECISION   RESID, SCALE, TSCAL
183*     ..
184*     .. Array Arguments ..
185      DOUBLE PRECISION   CNORM( * )
186      COMPLEX*16         A( LDA, * ), B( LDB, * ), WORK( * ),
187     $                   X( LDX, * )
188*     ..
189*
190*  =====================================================================
191*
192*     .. Parameters ..
193      DOUBLE PRECISION   ONE, ZERO
194      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )
195*     ..
196*     .. Local Scalars ..
197      INTEGER            IX, J
198      DOUBLE PRECISION   EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
199*     ..
200*     .. External Functions ..
201      LOGICAL            LSAME
202      INTEGER            IZAMAX
203      DOUBLE PRECISION   DLAMCH
204      EXTERNAL           LSAME, IZAMAX, DLAMCH
205*     ..
206*     .. External Subroutines ..
207      EXTERNAL           ZAXPY, ZCOPY, ZDSCAL, ZTRMV
208*     ..
209*     .. Intrinsic Functions ..
210      INTRINSIC          ABS, DBLE, DCMPLX, MAX
211*     ..
212*     .. Executable Statements ..
213*
214*     Quick exit if N = 0
215*
216      IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
217         RESID = ZERO
218         RETURN
219      END IF
220      EPS = DLAMCH( 'Epsilon' )
221      SMLNUM = DLAMCH( 'Safe minimum' )
222*
223*     Compute the norm of the triangular matrix A using the column
224*     norms already computed by ZLATRS.
225*
226      TNORM = ZERO
227      IF( LSAME( DIAG, 'N' ) ) THEN
228         DO 10 J = 1, N
229            TNORM = MAX( TNORM, TSCAL*ABS( A( J, J ) )+CNORM( J ) )
230   10    CONTINUE
231      ELSE
232         DO 20 J = 1, N
233            TNORM = MAX( TNORM, TSCAL+CNORM( J ) )
234   20    CONTINUE
235      END IF
236*
237*     Compute the maximum over the number of right hand sides of
238*        norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
239*
240      RESID = ZERO
241      DO 30 J = 1, NRHS
242         CALL ZCOPY( N, X( 1, J ), 1, WORK, 1 )
243         IX = IZAMAX( N, WORK, 1 )
244         XNORM = MAX( ONE, ABS( X( IX, J ) ) )
245         XSCAL = ( ONE / XNORM ) / DBLE( N )
246         CALL ZDSCAL( N, XSCAL, WORK, 1 )
247         CALL ZTRMV( UPLO, TRANS, DIAG, N, A, LDA, WORK, 1 )
248         CALL ZAXPY( N, DCMPLX( -SCALE*XSCAL ), B( 1, J ), 1, WORK, 1 )
249         IX = IZAMAX( N, WORK, 1 )
250         ERR = TSCAL*ABS( WORK( IX ) )
251         IX = IZAMAX( N, X( 1, J ), 1 )
252         XNORM = ABS( X( IX, J ) )
253         IF( ERR*SMLNUM.LE.XNORM ) THEN
254            IF( XNORM.GT.ZERO )
255     $         ERR = ERR / XNORM
256         ELSE
257            IF( ERR.GT.ZERO )
258     $         ERR = ONE / EPS
259         END IF
260         IF( ERR*SMLNUM.LE.TNORM ) THEN
261            IF( TNORM.GT.ZERO )
262     $         ERR = ERR / TNORM
263         ELSE
264            IF( ERR.GT.ZERO )
265     $         ERR = ONE / EPS
266         END IF
267         RESID = MAX( RESID, ERR )
268   30 CONTINUE
269*
270      RETURN
271*
272*     End of ZTRT03
273*
274      END
275