SLAEV2
November 2006
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 ].
[ 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 ].
Arguments
| A |
(input) REAL
The (1,1) element of the 2-by-2 matrix.
|
| B |
(input) REAL
The (1,2) element and the conjugate of the (2,1) element of
the 2-by-2 matrix. |
| C |
(input) REAL
The (2,2) element of the 2-by-2 matrix.
|
| RT1 |
(output) REAL
The eigenvalue of larger absolute value.
|
| RT2 |
(output) REAL
The eigenvalue of smaller absolute value.
|
| CS1 |
(output) REAL
|
| SN1 |
(output) 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.
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.