The Islamic University of Gaza Faculty of Engineering Civil Engineering Department

1. The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter4 Truncation Errors and Taylor Series

2. Introduction Truncation errors • Result when approximations are used to represent exact mathematical procedure • For example:

3. Taylor Series - Definition Mathematical Formulation used widely in numerical methods to express functions in an approximate fashion……. Taylor Series. It is of great value in the study of numerical methods. It provides means to predict a functional value at one point in terms of: - the function value - its derivatives at another point

4. Taylor’s Theorem General Expression Where: Rnis the remainder term to account for all terms from n+1 to infinity. And is a value of x that lies somewhere between xi and xi+1

5. Taylor’s Theorem Any smooth function can be approximated as a polynomial • Zero- order approximation: only true if xi+1 and xi are very close to each other. • First- order approximation: in form of a straight line • Second- order approximation:

6. Taylor’s Theorem - Remainder Term Remainder Term: What is ξ? If Zero- order approximation:

7. Taylor Series - Example Use zero-order to fourth-order Taylor series expansions to approximate the function. f(x)= -0.1x4 – 0.15x3 – 0.5x2 – 0.25x +1.2 Fromxi = 0 with h =1. Predict the function’s value at xi+1 =1. Solution • f(xi)= f(0)= 1.2 , f(xi+1)= f(1) =0.2 ………exact solution • Zero- order approx. (n=0)f(xi+1)=1.2 Et = 0.2 – 1.2 = -1.0 • First- order approx. (n=1)f(xi+1)=0.95 f(x)= -0.4x3 – 0.45x2 – x – 0.25, f’(0)= -0.25 f( xi+1)= 1.2- 0.25h = 0.95 Et = 0.2 - 0.95 = -0.75

8. Taylor Series - Example • Second- order approximation (n=2) f(xi+1)= 0.45 f’’(x) = -1.2 x2 – 0.9x -1 , f’’(0)= -1 f( xi+1)= 1.2 - 0.25h - 0.5 h2 = 0.45 Et = 0.2 – 0.45 = -0.25 • Third-order approximation (n=3) f(xi+1)= 0.3 f( xi+1)= 1.2 - 0.25h - 0.5 h2 – 0.15h3 = 0.3 Et = 0.2 – 0.3 = -0.1

9. Taylor Series - Example • Fourth-order approximation(n = 4) f(xi+1)= 0.2 f( xi+1)= 1.2 - 0.25h - 0.5 h2 – 0.15h3 – 0.1h 4= 0.2 Et = 0.2 – 0.2 = 0 The remainder term (R4) = 0 because the fifth derivative of the fourth-order polynomial is zero.

10. Approximation using Taylor Series Expansion The nth-order Approximation

11. Taylor Series • In General, the n-th order Taylor Series will be exact for n-th order polynomial. • For other differentiable and continuous functions, such as exponentials and sinusoids, a finite number of terms will not yield an exact estimate. Each additional term will contribute some improvement. (see example 4.2)

12. Example 4.2

14. Effect of non-linearity

17. Taylor Series • 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.

18. Effect of step size