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
|
*> \brief \b DSTECT
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
* Definition:
* ===========
*
* SUBROUTINE DSTECT( N, A, B, SHIFT, NUM )
*
* .. Scalar Arguments ..
* INTEGER N, NUM
* DOUBLE PRECISION SHIFT
* ..
* .. Array Arguments ..
* DOUBLE PRECISION A( * ), B( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DSTECT counts the number NUM of eigenvalues of a tridiagonal
*> matrix T which are less than or equal to SHIFT. T has
*> diagonal entries A(1), ... , A(N), and offdiagonal entries
*> B(1), ..., B(N-1).
*> See W. Kahan "Accurate Eigenvalues of a Symmetric Tridiagonal
*> Matrix", Report CS41, Computer Science Dept., Stanford
*> University, July 21, 1966
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The dimension of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION array, dimension (N)
*> The diagonal entries of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION array, dimension (N-1)
*> The offdiagonal entries of the tridiagonal matrix T.
*> \endverbatim
*>
*> \param[in] SHIFT
*> \verbatim
*> SHIFT is DOUBLE PRECISION
*> The shift, used as described under Purpose.
*> \endverbatim
*>
*> \param[out] NUM
*> \verbatim
*> NUM is INTEGER
*> The number of eigenvalues of T less than or equal
*> to SHIFT.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup double_eig
*
* =====================================================================
SUBROUTINE DSTECT( N, A, B, SHIFT, NUM )
*
* -- 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 ..
INTEGER N, NUM
DOUBLE PRECISION SHIFT
* ..
* .. Array Arguments ..
DOUBLE PRECISION A( * ), B( * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO, ONE, THREE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0, THREE = 3.0D0 )
* ..
* .. Local Scalars ..
INTEGER I
DOUBLE PRECISION M1, M2, MX, OVFL, SOV, SSHIFT, SSUN, SUN, TMP,
$ TOM, U, UNFL
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, SQRT
* ..
* .. Executable Statements ..
*
* Get machine constants
*
UNFL = DLAMCH( 'Safe minimum' )
OVFL = DLAMCH( 'Overflow' )
*
* Find largest entry
*
MX = ABS( A( 1 ) )
DO 10 I = 1, N - 1
MX = MAX( MX, ABS( A( I+1 ) ), ABS( B( I ) ) )
10 CONTINUE
*
* Handle easy cases, including zero matrix
*
IF( SHIFT.GE.THREE*MX ) THEN
NUM = N
RETURN
END IF
IF( SHIFT.LT.-THREE*MX ) THEN
NUM = 0
RETURN
END IF
*
* Compute scale factors as in Kahan's report
* At this point, MX .NE. 0 so we can divide by it
*
SUN = SQRT( UNFL )
SSUN = SQRT( SUN )
SOV = SQRT( OVFL )
TOM = SSUN*SOV
IF( MX.LE.ONE ) THEN
M1 = ONE / MX
M2 = TOM
ELSE
M1 = ONE
M2 = TOM / MX
END IF
*
* Begin counting
*
NUM = 0
SSHIFT = ( SHIFT*M1 )*M2
U = ( A( 1 )*M1 )*M2 - SSHIFT
IF( U.LE.SUN ) THEN
IF( U.LE.ZERO ) THEN
NUM = NUM + 1
IF( U.GT.-SUN )
$ U = -SUN
ELSE
U = SUN
END IF
END IF
DO 20 I = 2, N
TMP = ( B( I-1 )*M1 )*M2
U = ( ( A( I )*M1 )*M2-TMP*( TMP / U ) ) - SSHIFT
IF( U.LE.SUN ) THEN
IF( U.LE.ZERO ) THEN
NUM = NUM + 1
IF( U.GT.-SUN )
$ U = -SUN
ELSE
U = SUN
END IF
END IF
20 CONTINUE
RETURN
*
* End of DSTECT
*
END
|
