NAME¶

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

SYNOPSIS¶

SUBROUTINE CLAEV2(

A, B, C, RT1, RT2, CS1, SN1 )

REAL CS1, RT1, RT2 COMPLEX A, B, C, SN1

PURPOSE¶

CLAEV2 computes the eigendecomposition of a 2-by-2 Hermitian matrix
[ A B ]
[ CONJG(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 CONJG(SN1) ] [ A B ] [ CS1 -CONJG(SN1) ] = [ RT1 0 ] [-SN1 CS1 ] [ CONJG(B) C ] [ SN1 CS1 ] [ 0 RT2 ].

ARGUMENTS¶

A (input) COMPLEX

The (1,1) element of the 2-by-2 matrix.

B (input) COMPLEX

The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.

C (input) COMPLEX

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) COMPLEX 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.