Polynomial Interpolation & Spline Interpolation With Matlab

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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 аbout: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
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