zlaev2(3P)

NAME

zlaev2 - compute the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]

SYNOPSIS

SUBROUTINE ZLAEV2( A, B, C, RT1, RT2, CS1, SN1 )

DOUBLE PRECISION CS1, RT1, RT2

COMPLEX∗16 A, B, C, SN1

#include <sunperf.h>

void zlaev2(doublecomplex ∗za, doublecomplex ∗zb, doublecomplex ∗zc, double ∗rt1, double ∗rt2, double ∗cs1, doublecomplex ∗sn1) ;

PURPOSE

ZLAEV2 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∗16
The (1,1) element of the 2-by-2 matrix.

B (input) COMPLEX∗16
The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix.

C (input) COMPLEX∗16
The (2,2) element of the 2-by-2 matrix.

RT1 (output) DOUBLE PRECISION
The eigenvalue of larger absolute value.

RT2 (output) DOUBLE PRECISION
The eigenvalue of smaller absolute value.

CS1 (output) DOUBLE PRECISION
SN1 (output) COMPLEX∗16 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.