 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ slaev2()

 subroutine slaev2 ( real A, real B, real C, real RT1, real RT2, real CS1, real SN1 )

SLAEV2 computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.

Purpose:
``` SLAEV2 computes the eigendecomposition of a 2-by-2 symmetric matrix
[  A   B  ]
[  B   C  ].
On return, RT1 is the eigenvalue of larger absolute value, RT2 is the
eigenvalue of smaller absolute value, and (CS1,SN1) is the unit right
eigenvector for RT1, giving the decomposition

[ CS1  SN1 ] [  A   B  ] [ CS1 -SN1 ]  =  [ RT1  0  ]
[-SN1  CS1 ] [  B   C  ] [ SN1  CS1 ]     [  0  RT2 ].```
Parameters
 [in] A ``` A is REAL The (1,1) element of the 2-by-2 matrix.``` [in] B ``` B is REAL The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.``` [in] C ``` C is REAL The (2,2) element of the 2-by-2 matrix.``` [out] RT1 ``` RT1 is REAL The eigenvalue of larger absolute value.``` [out] RT2 ``` RT2 is REAL The eigenvalue of smaller absolute value.``` [out] CS1 ` CS1 is REAL` [out] SN1 ``` SN1 is REAL The vector (CS1, SN1) is a unit right eigenvector for RT1.```
Further Details:
```  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.

CS1 and SN1 are accurate to a few ulps barring over/underflow.

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.```

Definition at line 119 of file slaev2.f.

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 *
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