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
|
SUBROUTINE DSTT21( N, KBAND, AD, AE, SD, SE, U, LDU, WORK,
$ RESULT )
*
* -- LAPACK test routine (version 3.0) --
* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,
* Courant Institute, Argonne National Lab, and Rice University
* September 30, 1994
*
* .. Scalar Arguments ..
INTEGER KBAND, LDU, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION AD( * ), AE( * ), RESULT( 2 ), SD( * ),
$ SE( * ), U( LDU, * ), WORK( * )
* ..
*
* Purpose
* =======
*
* DSTT21 checks a decomposition of the form
*
* A = U S U'
*
* where ' means transpose, A is symmetric tridiagonal, U is orthogonal,
* and S is diagonal (if KBAND=0) or symmetric tridiagonal (if KBAND=1).
* Two tests are performed:
*
* RESULT(1) = | A - U S U' | / ( |A| n ulp )
*
* RESULT(2) = | I - UU' | / ( n ulp )
*
* Arguments
* =========
*
* N (input) INTEGER
* The size of the matrix. If it is zero, DSTT21 does nothing.
* It must be at least zero.
*
* KBAND (input) INTEGER
* The bandwidth of the matrix S. It may only be zero or one.
* If zero, then S is diagonal, and SE is not referenced. If
* one, then S is symmetric tri-diagonal.
*
* AD (input) DOUBLE PRECISION array, dimension (N)
* The diagonal of the original (unfactored) matrix A. A is
* assumed to be symmetric tridiagonal.
*
* AE (input) DOUBLE PRECISION array, dimension (N-1)
* The off-diagonal of the original (unfactored) matrix A. A
* is assumed to be symmetric tridiagonal. AE(1) is the (1,2)
* and (2,1) element, AE(2) is the (2,3) and (3,2) element, etc.
*
* SD (input) DOUBLE PRECISION array, dimension (N)
* The diagonal of the (symmetric tri-) diagonal matrix S.
*
* SE (input) DOUBLE PRECISION array, dimension (N-1)
* The off-diagonal of the (symmetric tri-) diagonal matrix S.
* Not referenced if KBSND=0. If KBAND=1, then AE(1) is the
* (1,2) and (2,1) element, SE(2) is the (2,3) and (3,2)
* element, etc.
*
* U (input) DOUBLE PRECISION array, dimension (LDU, N)
* The orthogonal matrix in the decomposition.
*
* LDU (input) INTEGER
* The leading dimension of U. LDU must be at least N.
*
* WORK (workspace) DOUBLE PRECISION array, dimension (N*(N+1))
*
* RESULT (output) DOUBLE PRECISION array, dimension (2)
* The values computed by the two tests described above. The
* values are currently limited to 1/ulp, to avoid overflow.
* RESULT(1) is always modified.
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
* ..
* .. Local Scalars ..
INTEGER J
DOUBLE PRECISION ANORM, TEMP1, TEMP2, ULP, UNFL, WNORM
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH, DLANGE, DLANSY
EXTERNAL DLAMCH, DLANGE, DLANSY
* ..
* .. External Subroutines ..
EXTERNAL DGEMM, DLASET, DSYR, DSYR2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, DBLE, MAX, MIN
* ..
* .. Executable Statements ..
*
* 1) Constants
*
RESULT( 1 ) = ZERO
RESULT( 2 ) = ZERO
IF( N.LE.0 )
$ RETURN
*
UNFL = DLAMCH( 'Safe minimum' )
ULP = DLAMCH( 'Precision' )
*
* Do Test 1
*
* Copy A & Compute its 1-Norm:
*
CALL DLASET( 'Full', N, N, ZERO, ZERO, WORK, N )
*
ANORM = ZERO
TEMP1 = ZERO
*
DO 10 J = 1, N - 1
WORK( ( N+1 )*( J-1 )+1 ) = AD( J )
WORK( ( N+1 )*( J-1 )+2 ) = AE( J )
TEMP2 = ABS( AE( J ) )
ANORM = MAX( ANORM, ABS( AD( J ) )+TEMP1+TEMP2 )
TEMP1 = TEMP2
10 CONTINUE
*
WORK( N**2 ) = AD( N )
ANORM = MAX( ANORM, ABS( AD( N ) )+TEMP1, UNFL )
*
* Norm of A - USU'
*
DO 20 J = 1, N
CALL DSYR( 'L', N, -SD( J ), U( 1, J ), 1, WORK, N )
20 CONTINUE
*
IF( N.GT.1 .AND. KBAND.EQ.1 ) THEN
DO 30 J = 1, N - 1
CALL DSYR2( 'L', N, -SE( J ), U( 1, J ), 1, U( 1, J+1 ), 1,
$ WORK, N )
30 CONTINUE
END IF
*
WNORM = DLANSY( '1', 'L', N, WORK, N, WORK( N**2+1 ) )
*
IF( ANORM.GT.WNORM ) THEN
RESULT( 1 ) = ( WNORM / ANORM ) / ( N*ULP )
ELSE
IF( ANORM.LT.ONE ) THEN
RESULT( 1 ) = ( MIN( WNORM, N*ANORM ) / ANORM ) / ( N*ULP )
ELSE
RESULT( 1 ) = MIN( WNORM / ANORM, DBLE( N ) ) / ( N*ULP )
END IF
END IF
*
* Do Test 2
*
* Compute UU' - I
*
CALL DGEMM( 'N', 'C', N, N, N, ONE, U, LDU, U, LDU, ZERO, WORK,
$ N )
*
DO 40 J = 1, N
WORK( ( N+1 )*( J-1 )+1 ) = WORK( ( N+1 )*( J-1 )+1 ) - ONE
40 CONTINUE
*
RESULT( 2 ) = MIN( DBLE( N ), DLANGE( '1', N, N, WORK, N,
$ WORK( N**2+1 ) ) ) / ( N*ULP )
*
RETURN
*
* End of DSTT21
*
END
|
