Question

Search 5 pages about ( truncation errors and Taylor series ) contains Abstract, Introduction, Discussion, Conclusion, and References

Answer 1

Truncation Errors and the Taylor Series

Truncation errors are those that result from using an approximation in place of an exact mathematical procedure. For example, in Chap. 1 we approximated the derivative of velocity of a falling parachutist by a finite-divided-difference equation of the form

(4.1)

A truncation error was introduced into the numerical solution because the difference equation only approximates the true value of the derivative (recall Fig. 1.4). In order to gain insight into the properties of such errors, we now turn to a mathematical formulation that is used widely in numerical methods to express functions in an approximate fashion—the Taylor series.

THE TAYLOR SERIES

Taylor’s theorem (Box 4.1) and its associated formula, the Taylor series, is of great value in the study of numerical methods. In essence, the Taylor series provides a means to predict a function value at one point in terms of the function value and its derivatives at another point. In particular, the theorem states that any smooth function can be approximated as a polynomial.

A useful way to gain insight into the Taylor series is to build it term by term. For example, the first term in the series is

f(xi+1) ∼= f(xi) (4.2)

This relationship, called the zero-order approximation, indicates that the value of f at the new point is the same as its value at the old point. This result makes intuitive sense because if xi and xi+1 are close to each other, it is likely that the new value is probably similar to the old value. Equation (4.2) provides a perfect estimate if the function being approximated is, in fact, a constant. However, if the function changes at all over the interval, additional terms of the Taylor series are required to provide a better estimate. For example, the first-order approximation is developed by adding another term to yield