LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
dlae2.f
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1 *> \brief \b DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix.
2 *
3 * =========== DOCUMENTATION ===========
4 *
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17 *
18 * Definition:
19 * ===========
20 *
21 * SUBROUTINE DLAE2( A, B, C, RT1, RT2 )
22 *
23 * .. Scalar Arguments ..
24 * DOUBLE PRECISION A, B, C, RT1, RT2
25 * ..
26 *
27 *
28 *> \par Purpose:
29 * =============
30 *>
31 *> \verbatim
32 *>
33 *> DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix
34 *> [ A B ]
35 *> [ B C ].
36 *> On return, RT1 is the eigenvalue of larger absolute value, and RT2
37 *> is the eigenvalue of smaller absolute value.
38 *> \endverbatim
39 *
40 * Arguments:
41 * ==========
42 *
43 *> \param[in] A
44 *> \verbatim
45 *> A is DOUBLE PRECISION
46 *> The (1,1) element of the 2-by-2 matrix.
47 *> \endverbatim
48 *>
49 *> \param[in] B
50 *> \verbatim
51 *> B is DOUBLE PRECISION
52 *> The (1,2) and (2,1) elements of the 2-by-2 matrix.
53 *> \endverbatim
54 *>
55 *> \param[in] C
56 *> \verbatim
57 *> C is DOUBLE PRECISION
58 *> The (2,2) element of the 2-by-2 matrix.
59 *> \endverbatim
60 *>
61 *> \param[out] RT1
62 *> \verbatim
63 *> RT1 is DOUBLE PRECISION
64 *> The eigenvalue of larger absolute value.
65 *> \endverbatim
66 *>
67 *> \param[out] RT2
68 *> \verbatim
69 *> RT2 is DOUBLE PRECISION
70 *> The eigenvalue of smaller absolute value.
71 *> \endverbatim
72 *
73 * Authors:
74 * ========
75 *
76 *> \author Univ. of Tennessee
77 *> \author Univ. of California Berkeley
78 *> \author Univ. of Colorado Denver
79 *> \author NAG Ltd.
80 *
81 *> \date September 2012
82 *
83 *> \ingroup auxOTHERauxiliary
84 *
85 *> \par Further Details:
86 * =====================
87 *>
88 *> \verbatim
89 *>
90 *> RT1 is accurate to a few ulps barring over/underflow.
91 *>
92 *> RT2 may be inaccurate if there is massive cancellation in the
93 *> determinant A*C-B*B; higher precision or correctly rounded or
94 *> correctly truncated arithmetic would be needed to compute RT2
95 *> accurately in all cases.
96 *>
97 *> Overflow is possible only if RT1 is within a factor of 5 of overflow.
98 *> Underflow is harmless if the input data is 0 or exceeds
99 *> underflow_threshold / macheps.
100 *> \endverbatim
101 *>
102 * =====================================================================
103  SUBROUTINE dlae2( A, B, C, RT1, RT2 )
104 *
105 * -- LAPACK auxiliary routine (version 3.4.2) --
106 * -- LAPACK is a software package provided by Univ. of Tennessee, --
107 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
108 * September 2012
109 *
110 * .. Scalar Arguments ..
111  DOUBLE PRECISION A, B, C, RT1, RT2
112 * ..
113 *
114 * =====================================================================
115 *
116 * .. Parameters ..
117  DOUBLE PRECISION ONE
118  parameter ( one = 1.0d0 )
119  DOUBLE PRECISION TWO
120  parameter ( two = 2.0d0 )
121  DOUBLE PRECISION ZERO
122  parameter ( zero = 0.0d0 )
123  DOUBLE PRECISION HALF
124  parameter ( half = 0.5d0 )
125 * ..
126 * .. Local Scalars ..
127  DOUBLE PRECISION AB, ACMN, ACMX, ADF, DF, RT, SM, TB
128 * ..
129 * .. Intrinsic Functions ..
130  INTRINSIC abs, sqrt
131 * ..
132 * .. Executable Statements ..
133 *
134 * Compute the eigenvalues
135 *
136  sm = a + c
137  df = a - c
138  adf = abs( df )
139  tb = b + b
140  ab = abs( tb )
141  IF( abs( a ).GT.abs( c ) ) THEN
142  acmx = a
143  acmn = c
144  ELSE
145  acmx = c
146  acmn = a
147  END IF
150  ELSE IF( adf.LT.ab ) THEN
151  rt = ab*sqrt( one+( adf / ab )**2 )
152  ELSE
153 *
155 *
156  rt = ab*sqrt( two )
157  END IF
158  IF( sm.LT.zero ) THEN
159  rt1 = half*( sm-rt )
160 *
161 * Order of execution important.
162 * To get fully accurate smaller eigenvalue,
163 * next line needs to be executed in higher precision.
164 *
165  rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
166  ELSE IF( sm.GT.zero ) THEN
167  rt1 = half*( sm+rt )
168 *
169 * Order of execution important.
170 * To get fully accurate smaller eigenvalue,
171 * next line needs to be executed in higher precision.
172 *
173  rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
174  ELSE
175 *
176 * Includes case RT1 = RT2 = 0
177 *
178  rt1 = half*rt
179  rt2 = -half*rt
180  END IF
181  RETURN
182 *
183 * End of DLAE2
184 *
185  END
subroutine dlae2(A, B, C, RT1, RT2)
DLAE2 computes the eigenvalues of a 2-by-2 symmetric matrix.
Definition: dlae2.f:104