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Differentiation
Difference Formulas
Finite Difference Formulas for Numerical Differentiation
Finite difference formulas are commonly used to approximate derivatives. Here, we discuss the 2-point centered difference formula , 3-point centered difference formula , and a non-symmetric 3-point formula.
Taylor series:
1. 2-Point Centered Difference Formula
The 2-point centered difference formula approximates the first derivative at a point xx using the values of the function on either side of xx. It is given by:
where is the step size.
Advantages :
f ′( x ) ≈ f^ ( x + h )− 2 hf^ ( x − h )
h
Simple and easy to implement. Second-order accurate (error term proportional to ) Disadvantages : Requires two function evaluations. Accuracy is limited by the size of
2. 3-Point Centered Difference Formula
The 3-point centered difference formula provides a more accurate approximation of the first derivative by including a higher-order term. The formula is:
Advantages : Higher accuracy: Fourth-order accurate (error term proportional to ) Useful for applications requiring greater precision. Disadvantages : Requires four function evaluations, increasing computational cost. May introduce larger rounding errors for very small
3. Non-Symmetric 3-Point Difference Formula
The non-symmetric 3-point formula is used when data points are not symmetrically distributed. For example, to approximate using , the formula is:
Advantages :
h^2
h
h^4
f ′( x ) f ( x ), f ( x +
h ), andf ( x + 2 h )
f ′( x ) ≈ −3 f^ ( x )+4 f^ ( x 2 + hh )− f^ ( x +2 h )
