Quadrature Rules Based on the Arnoldi Process

Applying a few steps of the Arnoldi process to a large nonsymmetric matrix A with initial vector v is shown to induce several quadrature rules. Properties of these rules are discussed, and their application to the computation of inexpensive estimates of the quadratic form 〈f, g〉 := v∗(f(A))∗g(A)v and related quadratic and bilinear forms is considered. Under suitable conditions on the functions f and g, the matrix A, and the vector v, the computed estimates provide upper and lower bounds of the quadratic and bilinear forms.