1CLAEV2(1)           LAPACK auxiliary routine (version 3.1)           CLAEV2(1)
2
3
4

NAME

6       CLAEV2  -  the eigendecomposition of a 2-by-2 Hermitian matrix  [ A B ]
7       [ CONJG(B) C ]
8

SYNOPSIS

10       SUBROUTINE CLAEV2( A, B, C, RT1, RT2, CS1, SN1 )
11
12           REAL           CS1, RT1, RT2
13
14           COMPLEX        A, B, C, SN1
15

PURPOSE

17       CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
18          [  A         B  ]
19          [  CONJG(B)  C  ].  On return, RT1 is the eigenvalue of larger abso‐
20       lute  value,  RT2  is  the  eigenvalue  of  smaller absolute value, and
21       (CS1,SN1) is the unit right eigenvector for RT1, giving the  decomposi‐
22       tion
23
24       [  CS1   CONJG(SN1)  ] [    A     B ] [ CS1 -CONJG(SN1) ] = [ RT1  0  ]
25       [-SN1     CS1     ] [ CONJG(B) C ] [ SN1     CS1     ]   [  0  RT2 ].
26
27

ARGUMENTS

29       A      (input) COMPLEX
30              The (1,1) element of the 2-by-2 matrix.
31
32       B      (input) COMPLEX
33              The (1,2) element and the conjugate of the (2,1) element of  the
34              2-by-2 matrix.
35
36       C      (input) COMPLEX
37              The (2,2) element of the 2-by-2 matrix.
38
39       RT1    (output) REAL
40              The eigenvalue of larger absolute value.
41
42       RT2    (output) REAL
43              The eigenvalue of smaller absolute value.
44
45       CS1    (output) REAL
46              SN1     (output)  COMPLEX  The vector (CS1, SN1) is a unit right
47              eigenvector for RT1.
48

FURTHER DETAILS

50       RT1 is accurate to a few ulps barring over/underflow.
51
52       RT2 may be inaccurate if there is massive cancellation in the  determi‐
53       nant  A*C-B*B; higher precision or correctly rounded or correctly trun‐
54       cated arithmetic would be needed  to  compute  RT2  accurately  in  all
55       cases.
56
57       CS1 and SN1 are accurate to a few ulps barring over/underflow.
58
59       Overflow  is  possible only if RT1 is within a factor of 5 of overflow.
60       Underflow is harmless if the input data is 0 or exceeds
61          underflow_threshold / macheps.
62
63
64
65
66 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       CLAEV2(1)
Impressum