Numerical Solution of a Secular Equation

Introduction The following equation arises when modifying symmetric eigenvalue problems (Golub (1973)) : 1 + oe n X j=1 b 2 j d j \Gamma = 0 : (1) Assuming that the b j 's are all nonzero and that the d j 's are distinct, this function has n roots, separated by the n values d j . These roots are the eigenvalues of the real symmetric matrix D+oebb T , where oe 2 R, b = [b 1 ; b 2 ; :::; b n ] T 2 R n and D = diagfd 1 ; d 2 ; :::; dn g. This equation is therefore a "secular equation&quot