LAPACK  3.10.0
LAPACK: Linear Algebra PACKage
dtpt03.f
Go to the documentation of this file.
1 *> \brief \b DTPT03
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 DTPT03( 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 * DOUBLE PRECISION RESID, SCALE, TSCAL
18 * ..
19 * .. Array Arguments ..
20 * DOUBLE PRECISION AP( * ), B( LDB, * ), CNORM( * ), WORK( * ),
21 * $ X( LDX, * )
22 * ..
23 *
24 *
25 *> \par Purpose:
26 * =============
27 *>
28 *> \verbatim
29 *>
30 *> DTPT03 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 DOUBLE PRECISION 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 DOUBLE PRECISION
94 *> The scaling factor s used in solving the triangular system.
95 *> \endverbatim
96 *>
97 *> \param[in] CNORM
98 *> \verbatim
99 *> CNORM is DOUBLE PRECISION 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 DOUBLE PRECISION
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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)
139 *> \endverbatim
140 *>
141 *> \param[out] RESID
142 *> \verbatim
143 *> RESID is DOUBLE PRECISION
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 *> \ingroup double_lin
157 *
158 * =====================================================================
159  SUBROUTINE dtpt03( UPLO, TRANS, DIAG, N, NRHS, AP, SCALE, CNORM,
160  $ TSCAL, X, LDX, B, LDB, WORK, RESID )
161 *
162 * -- LAPACK test routine --
163 * -- LAPACK is a software package provided by Univ. of Tennessee, --
164 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
165 *
166 * .. Scalar Arguments ..
167  CHARACTER DIAG, TRANS, UPLO
168  INTEGER LDB, LDX, N, NRHS
169  DOUBLE PRECISION RESID, SCALE, TSCAL
170 * ..
171 * .. Array Arguments ..
172  DOUBLE PRECISION AP( * ), B( LDB, * ), CNORM( * ), WORK( * ),
173  $ x( ldx, * )
174 * ..
175 *
176 * =====================================================================
177 *
178 * .. Parameters ..
179  DOUBLE PRECISION ONE, ZERO
180  parameter( one = 1.0d+0, zero = 0.0d+0 )
181 * ..
182 * .. Local Scalars ..
183  INTEGER IX, J, JJ
184  DOUBLE PRECISION BIGNUM, EPS, ERR, SMLNUM, TNORM, XNORM, XSCAL
185 * ..
186 * .. External Functions ..
187  LOGICAL LSAME
188  INTEGER IDAMAX
189  DOUBLE PRECISION DLAMCH
190  EXTERNAL lsame, idamax, dlamch
191 * ..
192 * .. External Subroutines ..
193  EXTERNAL daxpy, dcopy, dlabad, dscal, dtpmv
194 * ..
195 * .. Intrinsic Functions ..
196  INTRINSIC abs, dble, max
197 * ..
198 * .. Executable Statements ..
199 *
200 * Quick exit if N = 0.
201 *
202  IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
203  resid = zero
204  RETURN
205  END IF
206  eps = dlamch( 'Epsilon' )
207  smlnum = dlamch( 'Safe minimum' )
208  bignum = one / smlnum
209  CALL dlabad( smlnum, bignum )
210 *
211 * Compute the norm of the triangular matrix A using the column
212 * norms already computed by DLATPS.
213 *
214  tnorm = zero
215  IF( lsame( diag, 'N' ) ) THEN
216  IF( lsame( uplo, 'U' ) ) THEN
217  jj = 1
218  DO 10 j = 1, n
219  tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
220  jj = jj + j + 1
221  10 CONTINUE
222  ELSE
223  jj = 1
224  DO 20 j = 1, n
225  tnorm = max( tnorm, tscal*abs( ap( jj ) )+cnorm( j ) )
226  jj = jj + n - j + 1
227  20 CONTINUE
228  END IF
229  ELSE
230  DO 30 j = 1, n
231  tnorm = max( tnorm, tscal+cnorm( j ) )
232  30 CONTINUE
233  END IF
234 *
235 * Compute the maximum over the number of right hand sides of
236 * norm(op(A)*x - s*b) / ( norm(op(A)) * norm(x) * EPS ).
237 *
238  resid = zero
239  DO 40 j = 1, nrhs
240  CALL dcopy( n, x( 1, j ), 1, work, 1 )
241  ix = idamax( n, work, 1 )
242  xnorm = max( one, abs( x( ix, j ) ) )
243  xscal = ( one / xnorm ) / dble( n )
244  CALL dscal( n, xscal, work, 1 )
245  CALL dtpmv( uplo, trans, diag, n, ap, work, 1 )
246  CALL daxpy( n, -scale*xscal, b( 1, j ), 1, work, 1 )
247  ix = idamax( n, work, 1 )
248  err = tscal*abs( work( ix ) )
249  ix = idamax( n, x( 1, j ), 1 )
250  xnorm = abs( x( ix, j ) )
251  IF( err*smlnum.LE.xnorm ) THEN
252  IF( xnorm.GT.zero )
253  $ err = err / xnorm
254  ELSE
255  IF( err.GT.zero )
256  $ err = one / eps
257  END IF
258  IF( err*smlnum.LE.tnorm ) THEN
259  IF( tnorm.GT.zero )
260  $ err = err / tnorm
261  ELSE
262  IF( err.GT.zero )
263  $ err = one / eps
264  END IF
265  resid = max( resid, err )
266  40 CONTINUE
267 *
268  RETURN
269 *
270 * End of DTPT03
271 *
272  END
subroutine dlabad(SMALL, LARGE)
DLABAD
Definition: dlabad.f:74
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine dscal(N, DA, DX, INCX)
DSCAL
Definition: dscal.f:79
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPMV
Definition: dtpmv.f:142
subroutine dtpt03(UPLO, TRANS, DIAG, N, NRHS, AP, SCALE, CNORM, TSCAL, X, LDX, B, LDB, WORK, RESID)
DTPT03
Definition: dtpt03.f:161