Application of the singular value decomposition to the numerical computation of the coefficients of amplitude equations and normal forms

The Fredholm alternative is a standard procedure by which one generates the coefficients of amplitude equations and normal forms. The alternative requires that the inhomogeneous terms in the underlying system of differential equations, which contain the unknown coefficients, be orthogonal to the independent eigenvectors spanning the null space of the adjoint system of differential equations. The numerical computation of the adjoint eigenvectors and their application to solvability is frequently difficult and inefficient. Typically the underlying system of the inhomogeneous differential equation is discretized and solved as an inhomogeneous matrix-valued eigenvalue problem. We find that the solvability conditions which lead to values of the unknown coefficients are conviniently and economically computed by application of the singular value decomposition directly to the matrix formulation, avoiding completely the computation of an adjoint system of differential equations.