Polynomial Interpolation & Spline Interpolation With Matlab

Understand the math and the code behind the most common interpolation methods

What you’ll learn
Fast evaluation of polynomial functions, including Horner’s method and Horner’s method for derivatives
Polynomial interpolation, including interpolation via monomial basis, Lagrange interpolation and Newton interpolation
Spline interpolation, including linear spline interpolation and cubic spline interpolation
How to implement all discussed algorithms in MATLAB
The math behind all algorithms
How to use MATLAB to present your results graphically
Requirements
Calculus and linear algebra knowledge at a beginner level (polynomials, continuous functions, linearity of functions, derivatives of functions, systems of linear equations, polynomial long division, …)
You should know the low level basics of MATLAB (Syntax, functions, scripts, etc.)
MATLAB (already installed on your PC)
Description
Through the study of mathematics I have acquired the method to questionize any algorithm down to the smallest detail before I start with the implementation. On the one hand it is a great feeling to fully understand an algorithm (The Aha! moment). On the other hand one can avoid many errors whose cause lie in the misunderstanding of the algorithm in this way.Thus it is a big part of this course to enlight the math behind the treated algorithms. Nevertheless programming techniques behind algorithms are also very important and will be treated as accurate as the the math on which the algorithms are based on. Here we will consider the code for any algorithm for an implementation in MATLAB.MATLAB is a fundamental and enormously powerful programming language. This language is nearly unavoidable if one consider a career in engineering, science or related fields. That is why this course is based on MATLAB.This course is for everyone who wants to learn about interpolation. In this course you will learn about:Horner’s method Horner’s method for derivativesPolynomial interpolation, including interpolation via monomial basis, Lagrange interpolation and Newton interpolationThe interpolation error (How can one minimize or estimate this error? How do we proof the formula for the error bound?)Spline interpolation, including linear spline interpolation and natural cubic spline interpolation Derivation of the formula for the linear and the natural cubic spline interpolationThe code behind all methods The math on which these algorithms are based onHow to use MATLAB to visualize your own examplesWith this course you will obtain also two Matlab Live Scripts with all implementations and all discussed examples.

Overview

Section 1: Introduction

Lecture 1 Preview

Lecture 2 Overview

Lecture 3 Definition of a polynomial function

Lecture 4 The monomial basis

Lecture 5 Basics about polynomial functions

Lecture 6 Quiz

Section 2: Evaluation of a polynomial function

Lecture 7 Usual evaluation & Horner’s method

Lecture 8 Horner’s method – Examples

Lecture 9 Horner’s method – Code

Lecture 10 Polynomial decomposition/ factorization

Lecture 11 Decomposition with Horner’ method – Proof

Lecture 12 Repeated application of Horner’s method

Lecture 13 Horner’s method for derivatives

Lecture 14 Horner’s method for derivatives – Example

Lecture 15 Horner’s method for derivatives – Code

Section 3: Polynomial interpolation

Lecture 16 Introduction to interpolation

Lecture 17 Polynomial interpolation

Lecture 18 Reasons for the usage of polynomials for interpolation

Lecture 19 Interpolation via monomial basis

Lecture 20 Interpolation via monomial basis – Example 1

Lecture 21 Interpolation via monomial basis – Example 2

Lecture 22 Interpolation via monomial basis – Code

Lecture 23 Interpolation via monomial basis – Advantages/disadvantages

Lecture 24 Uniqueness of the interpolating polynomial function

Lecture 25 Lagrange interpolation

Lecture 26 Lagrange interpolation – Example 1

Lecture 27 Lagrange interpolation – Example 2

Lecture 28 Lagrange interpolation – Code

Lecture 29 Lagrange interpolation – Advantages/disadvantages

Lecture 30 Newton interpolation

Lecture 31 Newton interpolation – Example 1

Lecture 32 Newton interpolation – Example 2

Lecture 33 Newton interpolation – Code

Lecture 34 Horner’s method for a Newton polynomial

Lecture 35 Newton interpolation – Advantages/disadvantages

Section 4: Interpolation error for an interpolating polynomial

Lecture 36 Formula & estimate for the interpolation error

Lecture 37 Chebyshev points & Runge’s phenomenon

Lecture 38 Proof of the theorem for the interpolation error

Lecture 39 Exercise 1: Polynomial interpolation & interpolation error

Lecture 40 Exercise 2: Piecewise linear interpolation

Section 5: Spline interpolation

Lecture 41 Definition & Motivation

Lecture 42 Linear spline interpolation

Lecture 43 Exercise 3: Linear spline interpolation

Lecture 44 Linear spline interpolation – Code

Lecture 45 Cubic spline interpolation

Lecture 46 Derivation of the formula for a natural cubic interpolating spline

Lecture 47 Exercise 4: Natural cubic spline interpolation

Lecture 48 Natural cubic spline – Code

Lecture 49 Remarks

Section 6: Matlab

Lecture 50 Matlab Live Script – Horner’s method & polynomial interpolation

Lecture 51 Matlab Live Script – Spline interpolation

Section 7: Conclusion

Lecture 52 Conclusion

Engineering students,Science students,Anyone who has interest in interpolation