Interdisciplinary Applied Mathematics

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Krylov Subspace Technique Based on the Arnoldi Method

C onsider a nonlinear system of the form

x = f (x) + bu(t),    y = lT x,    (17.13)

where x is a vector of length n, f is a nonlinear vector function, u(t) is the input of the system, and y(t) is the output. Taylor series expansion of the function f about the origin (the equilibrium point) to second order yields a quadratic approximation of the form

x = Jf x + xT Wx + bu(t),    y = lT x,    (17.14)

where Jf is the Jacobian of f evaluated at the origin and W is an N xN xN Hessian tensor. The matrices Jf and W are given by

We assume that Jf is nonsingular. Let A = Jf 1 be the inverse of the Jacobian. Multiplying equation (17.14) by A yields

The orthogonal basis for the Krylov subspace span[Ab, A2b,…, Aqb], where q ^    N,    is    the    size    of the    reduced    system    that    will    be generated

by using the Arnoldi method (Chen and White, 2000) for numerical stability. The Arnoldi process generates V, an n x q orthonormal matrix whose columns span the Krylov subspace. Using the change of variables x = Vz in equation (17.15), we have

AVZ = Vz + AzT VT WVz + Abu(t),    y = lT Vz.

Left-multiplying by VT, and defining H = VT AV, we have

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