Numerical differentiation
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The document discusses numerical methods for estimating the derivative of a function f(x) at a point x=xi. It introduces three approaches: 1) the forward difference approximation calculates the slope between xi and xi+h, 2) the backward difference approximation calculates the slope between xi-h and xi, and 3) the centered difference approximation calculates the average of the forward and backward slopes. Each method has an error term that approaches zero as h approaches zero.
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Numerical differentiation
- 2. We like to estimate the value of f '(x) for a given function f(x).
- 3. The derivative represents the rate of change of a dependent variable with respect to an independent variable.
- 5. If x is allowed to approach zero, the difference becomes a derivative:f (Xi + ∆X)∆yf(Xi) X∆XiXi + ∆X∆x
- 6. Yf (Xi + ∆X)∆yf(Xi) ∆XiXi + ∆XX∆x
- 7. Yf’(Xi)XiX
- 8. The Taylor series expansion of f(x) about xi is
- 9. From this:
- 10. This formula is called the first forward divided difference formula and the error is of order O(h).Or equivalently, the Taylor series expansion of f(x) about xi can be written as
- 11. From this:
- 12. This formula is called the first backward divided difference formula and the error is of order O(h).A third way to approximate the first derivative is to subtract the backward from the forward Taylor series expansions:
- 13. This yields to
- 14. This formula is called the centered divided difference formula and the error is of order O(h2).FORWARDTrue derivateYApproximationhXXiXi+1
- 17. THANK YOU!SOURCE: Dr Muhammad Al-Salamah, Industrial Engineering KFUPM