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Taylors Polynomial and Series
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accuracy Adams-Bashforth algorithm assume calculate Chapter Chebyshev polynomials Chebyshev series choose coefficients column compute consider construct contraction mapping converges corrector decimal places deduce defined denote derivatives diagonal difference equation differential equation digits divided differences elements estimate Euler's method evaluations Example exists formula function Gauss-Seidel method given initial value problem integral interchanges interpolating polynomial interval least squares approximations Lemma linear equations Lipschitz condition matrix mean value theorem minimax approximation Newton's method non-singular non-zero norms obtain orthogonal pn(x polynomial of degree proof real numbers recurrence relation replace result right side root rounding errors satisfies second order Section Show Simpson's rule solve straight line sub-intervals Suppose Table Taylor polynomial Taylor series Theorem trapezoidal rule truncation error unique solution vector verify write xn+i xr+i xu x2 y(xn yn+i zero