LAPACK  3.8.0
LAPACK: Linear Algebra PACKage
stpt03.f
Go to the documentation of this file.
1 *> \brief \b STPT03
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 STPT03( UPLO, TRANS, DIAG, N, NRHS, AP, SCALE, CNORM,
12 * TSCAL, X, LDX, B, LDB, WORK, RESID )
13 *
14 * .. Scalar Arguments ..
15 * CHARACTER DIAG, TRANS, UPLO
16 * INTEGER LDB, LDX, N, NRHS
17 * REAL RESID, SCALE, TSCAL
18 * ..
19 * .. Array Arguments ..
20 * REAL AP( * ), B( LDB, * ), CNORM( * ), WORK( * ),
21 * $ X( LDX, * )
22 * ..
23 *
24 *
25 *> \par Purpose:
26 * =============
27 *>
28 *> \verbatim
29 *>
30 *> STPT03 computes the residual for the solution to a scaled triangular
31 *> system of equations A*x = s*b or A'*x = s*b when the triangular
32 *> matrix A is stored in packed format. Here A' is the transpose of A,
33 *> s is a scalar, and x and b are N by NRHS matrices. The test ratio is
34 *> the maximum over the number of right hand sides of
35 *> norm(s*b - op(A)*x) / ( norm(op(A)) * norm(x) * EPS ),
36 *> where op(A) denotes A or A' and EPS is the machine epsilon.
37 *> \endverbatim
38 *
39 * Arguments:
40 * ==========
41 *
42 *> \param[in] UPLO
43 *> \verbatim
44 *> UPLO is CHARACTER*1
45 *> Specifies whether the matrix A is upper or lower triangular.
46 *> = 'U': Upper triangular
47 *> = 'L': Lower triangular
48 *> \endverbatim
49 *>
50 *> \param[in] TRANS
51 *> \verbatim
52 *> TRANS is CHARACTER*1
53 *> Specifies the operation applied to A.
54 *> = 'N': A *x = s*b (No transpose)
55 *> = 'T': A'*x = s*b (Transpose)
56 *> = 'C': A'*x = s*b (Conjugate transpose = Transpose)
57 *> \endverbatim
58 *>
59 *> \param[in] DIAG
60 *> \verbatim
61 *> DIAG is CHARACTER*1
62 *> Specifies whether or not the matrix A is unit triangular.
63 *> = 'N': Non-unit triangular
64 *> = 'U': Unit triangular
65 *> \endverbatim
66 *>
67 *> \param[in] N
68 *> \verbatim
69 *> N is INTEGER
70 *> The order of the matrix A. N >= 0.
71 *> \endverbatim
72 *>
73 *> \param[in] NRHS
74 *> \verbatim
75 *> NRHS is INTEGER
76 *> The number of right hand sides, i.e., the number of columns
77 *> of the matrices X and B. NRHS >= 0.
78 *> \endverbatim
79 *>
80 *> \param[in] AP
81 *> \verbatim
82 *> AP is REAL array, dimension (N*(N+1)/2)
83 *> The upper or lower triangular matrix A, packed columnwise in
84 *> a linear array. The j-th column of A is stored in the array
85 *> AP as follows:
86 *> if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
87 *> if UPLO = 'L',
88 *> AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
89 *> \endverbatim
90 *>
91 *> \param[in] SCALE
92 *> \verbatim
93 *> SCALE is REAL
94 *> The scaling factor s used in solving the triangular system.
95 *> \endverbatim
96 *>
97 *> \param[in] CNORM
98 *> \verbatim
99 *> CNORM is REAL array, dimension (N)
100 *> The 1-norms of the columns of A, not counting the diagonal.
101 *> \endverbatim
102 *>
103 *> \param[in] TSCAL
104 *> \verbatim
105 *> TSCAL is REAL
106 *> The scaling factor used in computing the 1-norms in CNORM.
107 *> CNORM actually contains the column norms of TSCAL*A.
108 *> \endverbatim
109 *>
110 *> \param[in] X
111 *> \verbatim
112 *> X is REAL array, dimension (LDX,NRHS)
113 *> The computed solution vectors for the system of linear
114 *> equations.
115 *> \endverbatim
116 *>
117 *> \param[in] LDX
118 *> \verbatim
119 *> LDX is INTEGER
120 *> The leading dimension of the array X. LDX >= max(1,N).
121 *> \endverbatim
122 *>
123 *> \param[in] B
124 *> \verbatim
125 *> B is REAL array, dimension (LDB,NRHS)
126 *> The right hand side vectors for the system of linear
127 *> equations.
128 *> \endverbatim
129 *>
130 *> \param[in] LDB
131 *> \verbatim
132 *> LDB is INTEGER
133 *> The leading dimension of the array B. LDB >= max(1,N).
134 *> \endverbatim
135 *>
136 *> \param[out] WORK
137 *> \verbatim
138 *> WORK is REAL array, dimension (N)
139 *> \endverbatim
140 *>
141 *> \param[out] RESID
142 *> \verbatim
143 *> RESID is REAL
144 *> The maximum over the number of right hand sides of
145 *> norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
146 *> \endverbatim
147 *
148 * Authors:
149 * ========
150 *
151 *> \author Univ. of Tennessee
152 *> \author Univ. of California Berkeley
153 *> \author Univ. of Colorado Denver
154 *> \author NAG Ltd.
155 *
156 *> \date December 2016
157 *
158 *> \ingroup single_lin
159 *
160 * =====================================================================
161  SUBROUTINE stpt03( UPLO, TRANS, DIAG, N, NRHS, AP, SCALE, CNORM,
162  $ TSCAL, X, LDX, B, LDB, WORK, RESID )
163 *
164 * -- LAPACK test routine (version 3.7.0) --
165 * -- LAPACK is a software package provided by Univ. of Tennessee, --
166 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
167 * December 2016
168 *
169 * .. Scalar Arguments ..
170  CHARACTER DIAG, TRANS, UPLO
171  INTEGER LDB, LDX, N, NRHS
172  REAL RESID, SCALE, TSCAL
173 * ..
174 * .. Array Arguments ..
175  REAL AP( * ), B( ldb, * ), CNORM( * ), WORK( * ),
176  $ x( ldx, * )
177 * ..
178 *
179 * =====================================================================
180 *
181 * .. Parameters ..
182  REAL ONE, ZERO
183  parameter( one = 1.0e+0, zero = 0.0e+0 )
184 * ..
185 * .. Local Scalars ..
186  INTEGER IX, J, JJ
187  REAL BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
188 * ..
189 * .. External Functions ..
190  LOGICAL LSAME
191  INTEGER ISAMAX
192  REAL SLAMCH
193  EXTERNAL lsame, isamax, slamch
194 * ..
195 * .. External Subroutines ..
196  EXTERNAL saxpy, scopy, slabad, sscal, stpmv
197 * ..
198 * .. Intrinsic Functions ..
199  INTRINSIC abs, max, real
200 * ..
201 * .. Executable Statements ..
202 *
203 * Quick exit if N = 0.
204 *
205  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
206  resid = zero
207  RETURN
208  END IF
209  eps = slamch( 'Epsilon' )
210  smlnum = slamch( 'Safe minimum' )
211  bignum = one / smlnum
212  CALL slabad( smlnum, bignum )
213 *
214 * Compute the norm of the triangular matrix A using the column
215 * norms already computed by SLATPS.
216 *
217  tnorm = zero
218  IF( lsame( diag, 'N' ) ) THEN
219  IF( lsame( uplo, 'U' ) ) THEN
220  jj = 1
221  DO 10 j = 1, n
222  tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
223  jj = jj + j + 1
224  10 CONTINUE
225  ELSE
226  jj = 1
227  DO 20 j = 1, n
228  tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
229  jj = jj + n - j + 1
230  20 CONTINUE
231  END IF
232  ELSE
233  DO 30 j = 1, n
234  tnorm = max( tnorm, tscal+cnorm( j ) )
235  30 CONTINUE
236  END IF
237 *
238 * Compute the maximum over the number of right hand sides of
239 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
240 *
241  resid = zero
242  DO 40 j = 1, nrhs
243  CALL scopy( n, x( 1, j ), 1, work, 1 )
244  ix = isamax( n, work, 1 )
245  xnorm = max( one, abs( x( ix, j ) ) )
246  xscal = ( one / xnorm ) / REAL( n )
247  CALL sscal( n, xscal, work, 1 )
248  CALL stpmv( uplo, trans, diag, n, ap, work, 1 )
249  CALL saxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
250  ix = isamax( n, work, 1 )
251  err = tscal*abs( work( ix ) )
252  ix = isamax( n, x( 1, j ), 1 )
253  xnorm = abs( x( ix, j ) )
254  IF( err*smlnum.LE.xnorm ) THEN
255  IF( xnorm.GT.zero )
256  $ err = err / xnorm
257  ELSE
258  IF( err.GT.zero )
259  $ err = one / eps
260  END IF
261  IF( err*smlnum.LE.tnorm ) THEN
262  IF( tnorm.GT.zero )
263  $ err = err / tnorm
264  ELSE
265  IF( err.GT.zero )
266  $ err = one / eps
267  END IF
268  resid = max( resid, err )
269  40 CONTINUE
270 *
271  RETURN
272 *
273 * End of STPT03
274 *
275  END
subroutine stpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPMV
Definition: stpmv.f:144
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:91
subroutine slabad(SMALL, LARGE)
SLABAD
Definition: slabad.f:76
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:81
subroutine stpt03(UPLO, TRANS, DIAG, N, NRHS, AP, SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID)
STPT03
Definition: stpt03.f:163
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84