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Truncation error results from using an approximation in place of an exact mathematical procedure.

Non-elementary functions such as trigonometric, exponential, and others are expressed in an approximate fashion using Taylor series when their values, derivatives, and integrals are computed.

Any smooth function can be approximated as a polynomial. Taylor series provides a means to predict the value of a function at one point in terms of the function value and its derivatives at another point.

***Taylor Theorem**

If the function f and its first n+1 derivatives are continuous on an interval containing a & x, then the value of x is given by

•Any smooth function can be approximated as a polynomial.

f(xi+1) ≈ f(xi) zero order approximation, only true if xi+1 and xi are very close to each other.

f(xi+1) ≈ f(xi) + f′(xi) (xi+1-xi) first order approximation, in form of a straight line

nth order approximation:

(xi+1-xi)= h step size (define first)

Reminder term, Rn, accounts for all terms from (n+1) to infinity. - x is not known exactly, lies somewhere between xi+1>x >xi .
Need to determine f ^(n+1)(x), to do this you need f'(x).

If we knew f(x), there wouldn’t be any need to perform the Taylor series expansion.

However, R=O(h^(n+1)), (n+1)th order, the order of truncation error is hn+1.

O(h), halving the step size will halve the error.

O(h2), halving the step size will quarter the error.

Truncation error is decreased by addition of terms to the Taylor series.

If h is sufficiently small, only a few terms may be required to obtain an approximation close enough to the actual value for practical purposes.

•Numerical Differentiation: Finite Divided Differences

•Forward Difference

•Backward Difference

•Centered Difference