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
|
*> \brief \b DLAE2
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> Download DLAE2 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlae2.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlae2.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlae2.f">
*> [TXT]</a>
*
* Definition
* ==========
*
* SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
* .. Scalar Arguments ..
* DOUBLE PRECISION A, B, C, RT1, RT2
* ..
*
* Purpose
* =======
*
*>\details \b Purpose:
*>\verbatim
*>
*> DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix
*> [ A B ]
*> [ B C ].
*> On return, RT1 is the eigenvalue of larger absolute value, and RT2
*> is the eigenvalue of smaller absolute value.
*>
*>\endverbatim
*
* Arguments
* =========
*
*> \param[in] A
*> \verbatim
*> A is DOUBLE PRECISION
*> The (1,1) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is DOUBLE PRECISION
*> The (1,2) and (2,1) elements of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[in] C
*> \verbatim
*> C is DOUBLE PRECISION
*> The (2,2) element of the 2-by-2 matrix.
*> \endverbatim
*>
*> \param[out] RT1
*> \verbatim
*> RT1 is DOUBLE PRECISION
*> The eigenvalue of larger absolute value.
*> \endverbatim
*>
*> \param[out] RT2
*> \verbatim
*> RT2 is DOUBLE PRECISION
*> The eigenvalue of smaller absolute value.
*> \endverbatim
*>
*
* Authors
* =======
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup auxOTHERauxiliary
*
*
* Further Details
* ===============
*>\details \b Further \b Details
*> \verbatim
*>
*> RT1 is accurate to a few ulps barring over/underflow.
*>
*> RT2 may be inaccurate if there is massive cancellation in the
*> determinant A*C-B*B; higher precision or correctly rounded or
*> correctly truncated arithmetic would be needed to compute RT2
*> accurately in all cases.
*>
*> Overflow is possible only if RT1 is within a factor of 5 of overflow.
*> Underflow is harmless if the input data is 0 or exceeds
*> underflow_threshold / macheps.
*>
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
*
* -- LAPACK auxiliary routine (version 3.2) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
DOUBLE PRECISION A, B, C, RT1, RT2
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE
PARAMETER ( ONE = 1.0D0 )
DOUBLE PRECISION TWO
PARAMETER ( TWO = 2.0D0 )
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
DOUBLE PRECISION HALF
PARAMETER ( HALF = 0.5D0 )
* ..
* .. Local Scalars ..
DOUBLE PRECISION AB, ACMN, ACMX, ADF, DF, RT, SM, TB
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, SQRT
* ..
* .. Executable Statements ..
*
* Compute the eigenvalues
*
SM = A + C
DF = A - C
ADF = ABS( DF )
TB = B + B
AB = ABS( TB )
IF( ABS( A ).GT.ABS( C ) ) THEN
ACMX = A
ACMN = C
ELSE
ACMX = C
ACMN = A
END IF
IF( ADF.GT.AB ) THEN
RT = ADF*SQRT( ONE+( AB / ADF )**2 )
ELSE IF( ADF.LT.AB ) THEN
RT = AB*SQRT( ONE+( ADF / AB )**2 )
ELSE
*
* Includes case AB=ADF=0
*
RT = AB*SQRT( TWO )
END IF
IF( SM.LT.ZERO ) THEN
RT1 = HALF*( SM-RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE IF( SM.GT.ZERO ) THEN
RT1 = HALF*( SM+RT )
*
* Order of execution important.
* To get fully accurate smaller eigenvalue,
* next line needs to be executed in higher precision.
*
RT2 = ( ACMX / RT1 )*ACMN - ( B / RT1 )*B
ELSE
*
* Includes case RT1 = RT2 = 0
*
RT1 = HALF*RT
RT2 = -HALF*RT
END IF
RETURN
*
* End of DLAE2
*
END
|
