 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ claesy()

 subroutine claesy ( complex A, complex B, complex C, complex RT1, complex RT2, complex EVSCAL, complex CS1, complex SN1 )

CLAESY computes the eigenvalues and eigenvectors of a 2-by-2 complex symmetric matrix.

Purpose:
``` CLAESY computes the eigendecomposition of a 2-by-2 symmetric matrix
( ( A, B );( B, C ) )
provided the norm of the matrix of eigenvectors is larger than
some threshold value.

RT1 is the eigenvalue of larger absolute value, and RT2 of
smaller absolute value.  If the eigenvectors are computed, then
on return ( CS1, SN1 ) is the unit eigenvector for RT1, hence

[  CS1     SN1   ] . [ A  B ] . [ CS1    -SN1   ] = [ RT1  0  ]
[ -SN1     CS1   ]   [ B  C ]   [ SN1     CS1   ]   [  0  RT2 ]```
Parameters
 [in] A ``` A is COMPLEX The ( 1, 1 ) element of input matrix.``` [in] B ``` B is COMPLEX The ( 1, 2 ) element of input matrix. The ( 2, 1 ) element is also given by B, since the 2-by-2 matrix is symmetric.``` [in] C ``` C is COMPLEX The ( 2, 2 ) element of input matrix.``` [out] RT1 ``` RT1 is COMPLEX The eigenvalue of larger modulus.``` [out] RT2 ``` RT2 is COMPLEX The eigenvalue of smaller modulus.``` [out] EVSCAL ``` EVSCAL is COMPLEX The complex value by which the eigenvector matrix was scaled to make it orthonormal. If EVSCAL is zero, the eigenvectors were not computed. This means one of two things: the 2-by-2 matrix could not be diagonalized, or the norm of the matrix of eigenvectors before scaling was larger than the threshold value THRESH (set below).``` [out] CS1 ` CS1 is COMPLEX` [out] SN1 ``` SN1 is COMPLEX If EVSCAL .NE. 0, ( CS1, SN1 ) is the unit right eigenvector for RT1.```

Definition at line 114 of file claesy.f.

115*
116* -- LAPACK auxiliary routine --
117* -- LAPACK is a software package provided by Univ. of Tennessee, --
118* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
119*
120* .. Scalar Arguments ..
121 COMPLEX A, B, C, CS1, EVSCAL, RT1, RT2, SN1
122* ..
123*
124* =====================================================================
125*
126* .. Parameters ..
127 REAL ZERO
128 parameter( zero = 0.0e0 )
129 REAL ONE
130 parameter( one = 1.0e0 )
131 COMPLEX CONE
132 parameter( cone = ( 1.0e0, 0.0e0 ) )
133 REAL HALF
134 parameter( half = 0.5e0 )
135 REAL THRESH
136 parameter( thresh = 0.1e0 )
137* ..
138* .. Local Scalars ..
139 REAL BABS, EVNORM, TABS, Z
140 COMPLEX S, T, TMP
141* ..
142* .. Intrinsic Functions ..
143 INTRINSIC abs, max, sqrt
144* ..
145* .. Executable Statements ..
146*
147*
148* Special case: The matrix is actually diagonal.
149* To avoid divide by zero later, we treat this case separately.
150*
151 IF( abs( b ).EQ.zero ) THEN
152 rt1 = a
153 rt2 = c
154 IF( abs( rt1 ).LT.abs( rt2 ) ) THEN
155 tmp = rt1
156 rt1 = rt2
157 rt2 = tmp
158 cs1 = zero
159 sn1 = one
160 ELSE
161 cs1 = one
162 sn1 = zero
163 END IF
164 ELSE
165*
166* Compute the eigenvalues and eigenvectors.
167* The characteristic equation is
168* lambda **2 - (A+C) lambda + (A*C - B*B)
169* and we solve it using the quadratic formula.
170*
171 s = ( a+c )*half
172 t = ( a-c )*half
173*
174* Take the square root carefully to avoid over/under flow.
175*
176 babs = abs( b )
177 tabs = abs( t )
178 z = max( babs, tabs )
179 IF( z.GT.zero )
180 \$ t = z*sqrt( ( t / z )**2+( b / z )**2 )
181*
182* Compute the two eigenvalues. RT1 and RT2 are exchanged
183* if necessary so that RT1 will have the greater magnitude.
184*
185 rt1 = s + t
186 rt2 = s - t
187 IF( abs( rt1 ).LT.abs( rt2 ) ) THEN
188 tmp = rt1
189 rt1 = rt2
190 rt2 = tmp
191 END IF
192*
193* Choose CS1 = 1 and SN1 to satisfy the first equation, then
194* scale the components of this eigenvector so that the matrix
195* of eigenvectors X satisfies X * X**T = I . (No scaling is
196* done if the norm of the eigenvalue matrix is less than THRESH.)
197*
198 sn1 = ( rt1-a ) / b
199 tabs = abs( sn1 )
200 IF( tabs.GT.one ) THEN
201 t = tabs*sqrt( ( one / tabs )**2+( sn1 / tabs )**2 )
202 ELSE
203 t = sqrt( cone+sn1*sn1 )
204 END IF
205 evnorm = abs( t )
206 IF( evnorm.GE.thresh ) THEN
207 evscal = cone / t
208 cs1 = evscal
209 sn1 = sn1*evscal
210 ELSE
211 evscal = zero
212 END IF
213 END IF
214 RETURN
215*
216* End of CLAESY
217*