LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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slaev2.f
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1*> \brief \b SLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
2*
3* =========== DOCUMENTATION ===========
4*
5* Online html documentation available at
6* http://www.netlib.org/lapack/explore-html/
7*
8*> \htmlonly
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13*> [ZIP]</a>
14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slaev2.f">
15*> [TXT]</a>
16*> \endhtmlonly
17*
18* Definition:
19* ===========
20*
21* SUBROUTINE SLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
22*
23* .. Scalar Arguments ..
24* REAL A, B, C, CS1, RT1, RT2, SN1
25* ..
26*
27*
28*> \par Purpose:
29* =============
30*>
31*> \verbatim
32*>
33*> SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
34*> [ A B ]
35*> [ B C ].
36*> On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
37*> eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
38*> eigenvector for RT1, giving the decomposition
39*>
40*> [ CS1 SN1 ] [ A B ] [ CS1 -SN1 ] = [ RT1 0 ]
41*> [-SN1 CS1 ] [ B C ] [ SN1 CS1 ] [ 0 RT2 ].
42*> \endverbatim
43*
44* Arguments:
45* ==========
46*
47*> \param[in] A
48*> \verbatim
49*> A is REAL
50*> The (1,1) element of the 2-by-2 matrix.
51*> \endverbatim
52*>
53*> \param[in] B
54*> \verbatim
55*> B is REAL
56*> The (1,2) element and the conjugate of the (2,1) element of
57*> the 2-by-2 matrix.
58*> \endverbatim
59*>
60*> \param[in] C
61*> \verbatim
62*> C is REAL
63*> The (2,2) element of the 2-by-2 matrix.
64*> \endverbatim
65*>
66*> \param[out] RT1
67*> \verbatim
68*> RT1 is REAL
69*> The eigenvalue of larger absolute value.
70*> \endverbatim
71*>
72*> \param[out] RT2
73*> \verbatim
74*> RT2 is REAL
75*> The eigenvalue of smaller absolute value.
76*> \endverbatim
77*>
78*> \param[out] CS1
79*> \verbatim
80*> CS1 is REAL
81*> \endverbatim
82*>
83*> \param[out] SN1
84*> \verbatim
85*> SN1 is REAL
86*> The vector (CS1, SN1) is a unit right eigenvector for RT1.
87*> \endverbatim
88*
89* Authors:
90* ========
91*
92*> \author Univ. of Tennessee
93*> \author Univ. of California Berkeley
94*> \author Univ. of Colorado Denver
95*> \author NAG Ltd.
96*
97*> \ingroup laev2
98*
99*> \par Further Details:
100* =====================
101*>
102*> \verbatim
103*>
104*> RT1 is accurate to a few ulps barring over/underflow.
105*>
106*> RT2 may be inaccurate if there is massive cancellation in the
107*> determinant A*C-B*B; higher precision or correctly rounded or
108*> correctly truncated arithmetic would be needed to compute RT2
109*> accurately in all cases.
110*>
111*> CS1 and SN1 are accurate to a few ulps barring over/underflow.
112*>
113*> Overflow is possible only if RT1 is within a factor of 5 of overflow.
114*> Underflow is harmless if the input data is 0 or exceeds
115*> underflow_threshold / macheps.
116*> \endverbatim
117*>
118* =====================================================================
119 SUBROUTINE slaev2( A, B, C, RT1, RT2, CS1, SN1 )
120*
121* -- LAPACK auxiliary routine --
122* -- LAPACK is a software package provided by Univ. of Tennessee, --
123* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
124*
125* .. Scalar Arguments ..
126 REAL A, B, C, CS1, RT1, RT2, SN1
127* ..
128*
129* =====================================================================
130*
131* .. Parameters ..
132 REAL ONE
133 parameter( one = 1.0e0 )
134 REAL TWO
135 parameter( two = 2.0e0 )
136 REAL ZERO
137 parameter( zero = 0.0e0 )
138 REAL HALF
139 parameter( half = 0.5e0 )
140* ..
141* .. Local Scalars ..
142 INTEGER SGN1, SGN2
143 REAL AB, ACMN, ACMX, ACS, ADF, CS, CT, DF, RT, SM,
144 \$ TB, TN
145* ..
146* .. Intrinsic Functions ..
147 INTRINSIC abs, sqrt
148* ..
149* .. Executable Statements ..
150*
151* Compute the eigenvalues
152*
153 sm = a + c
154 df = a - c
155 adf = abs( df )
156 tb = b + b
157 ab = abs( tb )
158 IF( abs( a ).GT.abs( c ) ) THEN
159 acmx = a
160 acmn = c
161 ELSE
162 acmx = c
163 acmn = a
164 END IF
167 ELSE IF( adf.LT.ab ) THEN
168 rt = ab*sqrt( one+( adf / ab )**2 )
169 ELSE
170*
172*
173 rt = ab*sqrt( two )
174 END IF
175 IF( sm.LT.zero ) THEN
176 rt1 = half*( sm-rt )
177 sgn1 = -1
178*
179* Order of execution important.
180* To get fully accurate smaller eigenvalue,
181* next line needs to be executed in higher precision.
182*
183 rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
184 ELSE IF( sm.GT.zero ) THEN
185 rt1 = half*( sm+rt )
186 sgn1 = 1
187*
188* Order of execution important.
189* To get fully accurate smaller eigenvalue,
190* next line needs to be executed in higher precision.
191*
192 rt2 = ( acmx / rt1 )*acmn - ( b / rt1 )*b
193 ELSE
194*
195* Includes case RT1 = RT2 = 0
196*
197 rt1 = half*rt
198 rt2 = -half*rt
199 sgn1 = 1
200 END IF
201*
202* Compute the eigenvector
203*
204 IF( df.GE.zero ) THEN
205 cs = df + rt
206 sgn2 = 1
207 ELSE
208 cs = df - rt
209 sgn2 = -1
210 END IF
211 acs = abs( cs )
212 IF( acs.GT.ab ) THEN
213 ct = -tb / cs
214 sn1 = one / sqrt( one+ct*ct )
215 cs1 = ct*sn1
216 ELSE
217 IF( ab.EQ.zero ) THEN
218 cs1 = one
219 sn1 = zero
220 ELSE
221 tn = -cs / tb
222 cs1 = one / sqrt( one+tn*tn )
223 sn1 = tn*cs1
224 END IF
225 END IF
226 IF( sgn1.EQ.sgn2 ) THEN
227 tn = cs1
228 cs1 = -sn1
229 sn1 = tn
230 END IF
231 RETURN
232*
233* End of SLAEV2
234*
235 END
subroutine slaev2(a, b, c, rt1, rt2, cs1, sn1)
SLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
Definition slaev2.f:120