On Efficient Numerical Approximation of the Bilinear Form $c^*A^{-1}b$
SIAM Journal on Scientific Computing
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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.