1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
|
*> \brief \b DTPT01
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DTPT01( UPLO, DIAG, N, AP, AINVP, RCOND, WORK, RESID )
*
* .. Scalar Arguments ..
* CHARACTER DIAG, UPLO
* INTEGER N
* DOUBLE PRECISION RCOND, RESID
* ..
* .. Array Arguments ..
* DOUBLE PRECISION AINVP( * ), AP( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DTPT01 computes the residual for a triangular matrix A times its
*> inverse when A is stored in packed format:
*> RESID = norm(A*AINV - I) / ( N * norm(A) * norm(AINV) * EPS ),
*> where EPS is the machine epsilon.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> Specifies whether the matrix A is upper or lower triangular.
*> = 'U': Upper triangular
*> = 'L': Lower triangular
*> \endverbatim
*>
*> \param[in] DIAG
*> \verbatim
*> DIAG is CHARACTER*1
*> Specifies whether or not the matrix A is unit triangular.
*> = 'N': Non-unit triangular
*> = 'U': Unit triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in] AP
*> \verbatim
*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> The original upper or lower triangular matrix A, packed
*> columnwise in a linear array. The j-th column of A is stored
*> in the array AP as follows:
*> if UPLO = 'U', AP((j-1)*j/2 + i) = A(i,j) for 1<=i<=j;
*> if UPLO = 'L',
*> AP((j-1)*(n-j) + j*(j+1)/2 + i-j) = A(i,j) for j<=i<=n.
*> \endverbatim
*>
*> \param[in,out] AINVP
*> \verbatim
*> AINVP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
*> On entry, the (triangular) inverse of the matrix A, packed
*> columnwise in a linear array as in AP.
*> On exit, the contents of AINVP are destroyed.
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The reciprocal condition number of A, computed as
*> 1/(norm(A) * norm(AINV)).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] RESID
*> \verbatim
*> RESID is DOUBLE PRECISION
*> norm(A*AINV - I) / ( N * norm(A) * norm(AINV) * EPS )
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup double_lin
*
* =====================================================================
SUBROUTINE DTPT01( UPLO, DIAG, N, AP, AINVP, RCOND, WORK, RESID )
*
* -- LAPACK test routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIAG, UPLO
INTEGER N
DOUBLE PRECISION RCOND, RESID
* ..
* .. Array Arguments ..
DOUBLE PRECISION AINVP( * ), AP( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL UNITD
INTEGER J, JC
DOUBLE PRECISION AINVNM, ANORM, EPS
* ..
* .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH, DLANTP
EXTERNAL LSAME, DLAMCH, DLANTP
* ..
* .. External Subroutines ..
EXTERNAL DTPMV
* ..
* .. Intrinsic Functions ..
INTRINSIC DBLE
* ..
* .. Executable Statements ..
*
* Quick exit if N = 0.
*
IF( N.LE.0 ) THEN
RCOND = ONE
RESID = ZERO
RETURN
END IF
*
* Exit with RESID = 1/EPS if ANORM = 0 or AINVNM = 0.
*
EPS = DLAMCH( 'Epsilon' )
ANORM = DLANTP( '1', UPLO, DIAG, N, AP, WORK )
AINVNM = DLANTP( '1', UPLO, DIAG, N, AINVP, WORK )
IF( ANORM.LE.ZERO .OR. AINVNM.LE.ZERO ) THEN
RCOND = ZERO
RESID = ONE / EPS
RETURN
END IF
RCOND = ( ONE / ANORM ) / AINVNM
*
* Compute A * AINV, overwriting AINV.
*
UNITD = LSAME( DIAG, 'U' )
IF( LSAME( UPLO, 'U' ) ) THEN
JC = 1
DO 10 J = 1, N
IF( UNITD )
$ AINVP( JC+J-1 ) = ONE
*
* Form the j-th column of A*AINV
*
CALL DTPMV( 'Upper', 'No transpose', DIAG, J, AP,
$ AINVP( JC ), 1 )
*
* Subtract 1 from the diagonal
*
AINVP( JC+J-1 ) = AINVP( JC+J-1 ) - ONE
JC = JC + J
10 CONTINUE
ELSE
JC = 1
DO 20 J = 1, N
IF( UNITD )
$ AINVP( JC ) = ONE
*
* Form the j-th column of A*AINV
*
CALL DTPMV( 'Lower', 'No transpose', DIAG, N-J+1, AP( JC ),
$ AINVP( JC ), 1 )
*
* Subtract 1 from the diagonal
*
AINVP( JC ) = AINVP( JC ) - ONE
JC = JC + N - J + 1
20 CONTINUE
END IF
*
* Compute norm(A*AINV - I) / (N * norm(A) * norm(AINV) * EPS)
*
RESID = DLANTP( '1', UPLO, 'Non-unit', N, AINVP, WORK )
*
RESID = ( ( RESID*RCOND ) / DBLE( N ) ) / EPS
*
RETURN
*
* End of DTPT01
*
END
|
