A block-algebra approach to the solution of ∑iaiλ−ci equations

A bordered matrix is proposed, that with block-algebra manipulations rebuilds the form of the equation given in the title. It is shown that the determinant of the bordered matrix results in a Standard Eigenvalue Problem, where the roots are exactly the eigenvalues of the resulting problem. Even more, the block-algebra operations involved are extremely simple. Likewise, a specialized iterative procedure to find one eigenvalue for a Standard Eigenvalue Problem, based on multivariable Newton, is presented. Numerical experiments are presented to show the reliability of the proposed approach taken to find one or all the roots of the equation given in the title.