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MACHINE ARITHMETIC AND RELATED MATTERS
APPROXIMATION AND INTERPOLATION
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A-stable algebraic algorithm analysis applied arithmetic assume boundary value problem Chebyshev Chebyshev polynomials coefficients compute condition number Consider convergence defined degree of exactness denote derivative Determine differential equations divided differences eigenvalues endpoints error function estimate Euler Euler's method evaluation exact solution example floating-point follows Gauss Gautschi given grid function hence Hint implicit initial value problem integral interpolation polynomial interval Lagrange least squares approximation linearly Lipschitz mathematical matrix method of order multistep method Newton's method nodes nonlinear equations norm obtain one-step method orthogonal polynomials polynomial of degree positive principal error function proof of Theorem quadratic quadrature formula quadrature rule result root condition Runge-Kutta method satisfying secant method Show solution y(x solving spline stability starting value step theory trapezoidal rule truncation error un+k unique solution variables vector weight function xn-i xn+i y(xn zero