Week 09-10: Numerical Differentiation.

 

Learning Outcomes:

We have already introduced the notion of numerical differentiation in previous sections. Recall that we employed Taylor series expansions to derive fi nite-divided-difference approximations of derivatives. We developed forward, backward, and centered difference approximations of first and higher derivatives. Recall that, at best, these estimates had errors that were O(h^2)—that is, their errors were proportional to the square of the step size. This level of accuracy is due to the number of terms of the Taylor series that were retained during the derivation of these formulas. We will now illustrate how to develop more accurate formulas by retaining more terms.

 

Lesson Plan:

Lecture 01: Discussion on Derivatives of Equally Spaced Data.

Lecture 02: Studying (a) First Order Derivative by a Two-Point Formula, (b) Derivation of the same.

Lecture 03: Discussion on First Order Derivative by a Three-Point Formula for Derivative at:

  • First Point

  • Third Point, and

  • Central Point

Lecture 04: Learning the Error Estimation Techniques.

Lecture 05: Discussion on Second and Higher level Differentiation.

Lecture 06: Solving the Exercises Related to the Numerical Differentiation and their Writing C++ Programs.

 

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