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LAPACK-3

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LAPACK-3: computes the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ]

NAME
LAPACK-3 - computes 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

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. LAPACK auxiliary routine (versionMarch 2011 LAPACK-3(3)
 
 
 

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