Computes the eigenvalues and eigenvectors of a 2-by-2 symmetric/Hermitian matrix.
Syntax
call slaev2( a, b, c, rt1, rt2, cs1, sn1 )
call dlaev2( a, b, c, rt1, rt2, cs1, sn1 )
call claev2( a, b, c, rt1, rt2, cs1, sn1 )
call zlaev2( a, b, c, rt1, rt2, cs1, sn1 )
Include Files
The FORTRAN 77 interfaces are specified in the mkl_lapack.fi include file (to be used in Fortran programs) and in the mkl_lapack.h include file (to be used in C programs).
Description
The routine performs the eigendecomposition of a 2-by-2 symmetric matrix
(for claev2/zlaev2).
On return, rt1 is the eigenvalue of larger absolute value, rt2 of smaller absolute value, and (cs1, sn1) is the unit right eigenvector for rt1, giving the decomposition
(for slaev2/dlaev2),
or
(for claev2/zlaev2).
Input Parameters
- a, b, c
REAL for slaev2
DOUBLE PRECISION for dlaev2
COMPLEX for claev2
COMPLEX*16 for zlaev2.
Elements of the input matrix.
Output Parameters
- rt1, rt2
REAL for slaev2/claev2
DOUBLE PRECISION for dlaev2/zlaev2.
Eigenvalues of larger and smaller absolute value, respectively.
- cs1
REAL for slaev2/claev2
DOUBLE PRECISION for dlaev2/zlaev2.
- sn1
REAL for slaev2
DOUBLE PRECISION for dlaev2
COMPLEX for claev2
COMPLEX*16 for zlaev2.
The vector (cs1, sn1) is the unit right eigenvector for rt1.
Application Notes
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.