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algorithm analyze applied approximating function Assume bisection method coefficients column computational error constant data points defined derivative det(A diagonal differential equations digits discrete eigenvalues eigenvectors elements equal error bound error propagation estimate Euler's method evaluate exact-matching example exercise false position formula Fourier transform Gauss-Seidel method Gaussian elimination given by equation hyperplane inner product input integration interpolation interval iterative method iterative step Jacobi method least-squares approximation linear combination linear equations linearly independent magnitude matrix maximum method of false multiplications n x n Newton's method nonlinear equations nonzero norm normal numerical methods obtain operation order of convergence orthogonal Picard iteration pivoting polynomial probability distribution problem produce propagated error quadratic random variables relative error result root secant method sequence Show solution solve subtraction symmetric symmetric matrix tabular arguments theorem triangularization truncation error vector space xi+1 zero