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Stability Convergence and Accuracy
as special cases methods based on numerical quadrature as well as methods based
We now consider a class of methods denned
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algorithm assume called characteristic equation coefficients column compute condition number consider consistent with L[u convergence coordinate vector Corollary corresponding defined degree of approximation determine diagonal difference equation differential equation discrete operator eigenvalues eigenvectors elementary divisors elements Euler method exact solution example exercise exists function Gaussian elimination given h sufficiently Hence Hermitian Hermitian matrix Heun method inner product integer interpolation error iterative Jacobi method Lemma Lh[u linear combination linear mapping linear N-step method linearly independent matrix norm method of order midpoint method modulo multiplicative inverse non-singular nonnegative obtain ordered basis partial pivoting polynomial positive definite procedure proof is left Prove Theorem relative degree residue representation result roots satisfies Section sequence similarity transformation SOR method stable starting values subspace sufficiently small symmetric system of equations triangular unique solution vanish vector norm vector space Verify write zero