slaev2(3P)

NAME

slaev2 - compute the eigendecomposition of a 2-by-2 symmetric matrix [ A B ] [ B C ]

SYNOPSIS

SUBROUTINE SLAEV2(
A, B, C, RT1, RT2, CS1, SN1 )

void slaev2(float a, float b, float c, float ∗rt1, float ∗rt2,
float ∗cs1, float ∗sn1)

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

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 ].

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.