The taylor series and Numerical Differentiation

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Presentation transcript:

The taylor series and Numerical Differentiation Fisika Komputasi The taylor series and Numerical Differentiation Computational Physics

The Taylor Series (1) f (x) f (x0) x x0

The Taylor Series Nilai fungsi pada titik x0+Dx (2) (3)

The Taylor Series

The Taylor Series

EXAMPLE 1

EXAMPLE 1

Numerical Differentiation Numerical differentiation is the process of finding the numerical value of a derivative of a given function at a given point. In numerical analysis, numerical differen-tiation describes algorithms for estimating the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function.

Numerical Differentiation Forward Difference Approximation of the First Derivative h is called the step size

Numerical Differentiation Backward Difference Approximation of the First Derivative

Numerical Differentiation Centered Difference Approximation of the First Derivative (-)

Numerical Differentiation Finite Difference Approximations of Higher Derivatives +

Numerical Differentiation Finite Difference Approximations of Higher Derivatives (1) (2) Equation (2) can be multiplied by 2 and subtracted from Eq. (1) to give The second forward finite divided difference.

Numerical Differentiation The second backward finite divided difference. The second centered finite divided difference.

Numerical Differentiation

Numerical Differentiation

Numerical Differentiation

Numerical Differentiation

Derivative from difference tables

Derivative from difference tables

Derivative from difference tables

NUMERICAL DIFFERENTIATION PROBLEMS

NUMERICAL DIFFERENTIATION PROBLEMS

NUMERICAL DIFFERENTIATION PROBLEMS

NUMERICAL DIFFERENTIATION PROBLEMS

Numerical Differentiation

Numerical Differentiation

Numerical Differentiation