.TH CLAEV2 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME CLAEV2 - the eigendecomposition of a 2-by-2 Hermitian matrix [ A B ] [ CONJG(B) C ] .SH SYNOPSIS .TP 19 SUBROUTINE CLAEV2( A, B, C, RT1, RT2, CS1, SN1 ) .TP 19 .ti +4 REAL CS1, RT1, RT2 .TP 19 .ti +4 COMPLEX A, B, C, SN1 .SH 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 .br [ CS1 CONJG(SN1) ] [ A B ] [ CS1 -CONJG(SN1) ] = [ RT1 0 ] [-SN1 CS1 ] [ CONJG(B) C ] [ SN1 CS1 ] [ 0 RT2 ]. .SH ARGUMENTS .TP 7 A (input) COMPLEX The (1,1) element of the 2-by-2 matrix. .TP 7 B (input) COMPLEX The (1,2) element and the conjugate of the (2,1) element of the 2-by-2 matrix. .TP 7 C (input) COMPLEX The (2,2) element of the 2-by-2 matrix. .TP 7 RT1 (output) REAL The eigenvalue of larger absolute value. .TP 7 RT2 (output) REAL The eigenvalue of smaller absolute value. .TP 7 CS1 (output) REAL SN1 (output) COMPLEX The vector (CS1, SN1) is a unit right eigenvector for RT1. .SH FURTHER DETAILS RT1 is accurate to a few ulps barring over/underflow. .br 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. .br 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 .br underflow_threshold / macheps. .br