Numerical Techniques

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Numerical Techniques

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About course

This course covers numerical analysis course for Computational and Applied Mathematics mainly for the University of the Witwatersrand

Syllabus

Section 1. Errors

True Error: Definition & Example

Relative True Error: Definition & Example

Approximate Error Definition & Example

Relative Approximate Error: Definition & Example

Round-off Error: Definition and Examples

Effect of Carrying Significant Digits

Truncation Error: Definition

Truncation Error : Example : Series

Truncation Error : Example : Integration

Truncation Error : Example : Differentiation

Floating Point Representation Background: Part 1 of 3

Floating Point Representation Background: Part 2 of 3

Floating Point Representation Background: Part 3 of 3

Floating Point Representation: Example: Part 1 of 1

Floating Point Representation: Example: Part 2 of 2

Section 2. Taylor Series

Introduction to Taylor Series

Taylor Series: Example

Taylor Series: Deriving the Series for exp(x)

Taylor Series: Example: Finding sin(2.0)

Section 3. Numerical Differentiation

Background of Differentiation

Section 4. Interpolation

Forward Divided Difference: Part 1 of 2

Forward Divided Difference: Part 2 of 2

Backward Divided Difference: Part 1 of 2

Backward Divided Difference: Part 2 of 2

Central Divided Difference

Higher Order Derivative Divided Difference: Theory

Higher Order Derivative Divided Difference: Example

Accuracy of Divided Difference Formulas: Part 1 of 2

Accuracy of Divided Difference Formulas Part 2 of 2

Divided Difference Approach

Newton's Divided Difference Polynomial Method: Theory

Newton's Divided Difference Polynomial Method: Example

Section 5. Approximations

Background of Bisection Method

Algorithm of Bisection Method

Example of Bisection Method

Advantages & Drawbacks of Bisection Method

Derivation of Newton-Raphson Method

Example for Newton-Raphson Method

Advantages & Drawbacks for Newton-Raphson Method: Part 1 of 2

Advantages & Drawbacks for Newton-Raphson Method: Part 2 of 2

Derivation from Taylor Series of Newton-Raphson Method

Supercomputers have No Divide Unit - A Newton-Raphson Method Approach

Supercomputers have No Divide Unit - Example

Finding Square Root of a Number - A Newton-Raphson Method Approach

Finding Square Root of a Number - Example

Derivation of Secant Method: Approach 1 of 2

Derivation of Secant Method: Approach 2 of 2

Algorithm of Secant Method

Example of Secant Method

False-Position Method: Part 1 of 3

False-Position Method: Part 2 of 3

False-Position Method: Part 3 of 3

Section 6. Matrices

Diagonal and Identity Matrices

What is a Diagonally Dominant Matrix

Upper and Lower Triangular Matrices

Adding Two Matrices

Subtracting Two Matrices

Multiplying Two Matrices

Inverse of a Matrix

Setting up Equations in a Matrix Form

Section 7. Naive-Gauss Elimination

Naive Gaussian elimination: Theory: Part 1 of 2

Naive Gaussian elimination: Theory: Part 2 of 2

Naive Gauss Elimination Method: Example: Part 1 of 2 (Forward Elimination)

Naive Gauss Elimination Method: Example: Part 2 of 2 (Back Substitution)

Pitfalls of Naive Gauss Elimination Method

Naive Gauss Elimination: Round-off Error Issues: Example: Part 1 of 3

Naive Gauss Elimination: Round-off Error Issues: Example: Part 2 of 3

Naive Gauss Elimination: Round-off Error Issues: Example: Part 3 of 3

Section 8. Gaussian Elimination with Partial Pivoting

Gaussian Elimination With Partial Pivoting: Theory

Gaussian Elimination With Partial Pivoting: Example: Part 1 of 3 (Forward Elimination)

Gaussian Elimination With Partial Pivoting: Example: Part 2 of 3 (Forward Elimination)

Gaussian Elimination With Partial Pivoting: Example: Part 3 of 3 (Back Substitution)

Gaussian Elimination With Partial Pivoting: Round-off Error Issues: Example: Part 1 of 3

Gaussian Elimination With Partial Pivoting: Round-off Error Issues: Example: Part 2 of 3

Gaussian Elimination With Partial Pivoting: Round-off Error Issues: Example: Part 3 of 3

Section 9. Gauss-Seidel Method

Gauss-Seidel Method of Solving Simul Linear Eqns: Theory: Part 1 of 2

Gauss-Seidel Method of Solving Simul Linear Eqns: Theory: Part 2 of 2

Gauss-Seidel Method of Solving Simul Linear Eqns: Example: Part 1 of 2

Gauss-Seidel Method of Solving Simul Linear Eqns: Example: Part 2 of 2

Gauss-Seidel Method of Solving Simul Linear Eqns: Pitfalls and Advantages: Part 1 of 2

Gauss-Seidel Method of Solving Simul Linear Eqns: Pitfalls and Advantages: Part 2 of 2

Section 10. LU Decomposition

LU Decomposition: Basis

LU Decomposition Method: Example

Why LU Decomposition: Part 1

Why LU Decomposition: Part 2

Decomposing a Square Matrix: Part 1 of 2

Decomposing a Square Matrix: Part 2 of 2

Finding Inverse of a Matrix Using LU Decomposition: Background

Finding Inverse of a Matrix Using LU Decomposition: Example

Section 11. Polynomial Interpolation

Polynomial Interpolation Method

Uniqueness of Polynomial Interpolant: Part 1 of 2

Uniqueness of Polynomial Interpolant: Part 2 of 2

Linear Interpolation

Quadratic Interpolation

Cubic Interpolation - Part 1 of 2

Cubic Interpolation - Part 2 of 2

Newton's Divided Difference Polynomial: Linear Interpolation: Theory

Newton's Divided Difference Polynomial: Linear Interpolation: Example

Newton's Divided Difference Polynomial: Quadratic Interpolation: Theory

Newtons Divided Difference Polynomial Interpolation: Quadratic Interpolation: Example Part 1 of 2

Newtons Divided Difference Polynomial Interpolation: Quadratic Interpolation: Example Part 2 of 2

General Order: Newton's Divided Difference Polynomial: Theory: Part 1 of 2

General Order: Newton's Divided Difference Polynomial: Theory: Part 2 of 2

General Order: Newton's Divided Difference Polynomial: Example: Part 1 of 2

General Order: Newton's Divided Difference Polynomial: Example: Part 2 of 2

Section 12. Lagrangian Interpolation

Lagrangian Interpolation: Theory

Lagrangian Interpolation: Linear Interpolation: Example

Lagrangian Interpolation: Quadratic Interpolation: Example: Part 1 of 2

Lagrangian Interpolation: Quadratic Interpolation: Example: Part 2 of 2

Lagrangian Interpolation: Cubic Interpolation: Example: Part 1 of 2

Lagrangian Interpolation: Cubic Interpolation: Example: Part 2 of 2

Section 13. Trapezoidal Rule

Trapezoidal Rule Derivation

Trapezoidal Rule: Example

Trapezoidal Rule Multiple Segment Rule: Motivation

Trapezoidal Rule Multiple Segment Rule: Derivation

Trapezoidal Rule Multiple Segment Rule: Example: Part 1 of 2

Trapezoidal Rule Multiple Segment Rule: Example: Part 2 of 2

Trapezoidal Rule Multiple Segment Rule: Error Derivation

Trapezoidal Rule Multiple Segment Rule: Error Example

Method of Undetermined Coefficients: Trapezoidal Rule Derivation

Section 14. Simpson's 1/3 Rule

Simpson's One Third Rule Derivation

Simpsons 1/3 Rule of Integration: Example

Multiple Segment Simpson 1/3 Rule Derivation Part 1 of 2

Multiple Segment Simpson 1/3 Rule Derivation Part 2 of 2

Multiple Segment Simpson 1/3 Rule Example: Part 1 of 2

Multiple Segment Simpson 1/3 Rule Example Part 2 of 2

Section 15. Euler's Methods

Euler's Method of Solving ODEs: Derivation

Euler's Method of Solving ODEs: Example

Euler's Method of Estimating Integrals: Theory

Euler's Method of Estimating Integrals: Example

Section 16. Runge-Kutta 2nd Order Method

Runge-Kutta 2nd Order Method: Background

Runge-Kutta 2nd Order Method: Formulas

Runge-Kutta 2nd Order Method: Heun's Method

Runge-Kutta 2nd Order Method: Midpoint Method

Runge-Kutta 2nd Order Method: Ralston's Method Part 1 of 2

Runge-Kutta 2nd Order Method: Ralston's Method Part 2 of 2

Runge-Kutta 2nd Order Method: Derivation Part 1 of 2

Runge-Kutta 2nd Order Method: Derivation Part 2 of 2

Section 17. Runge-Kutta 4th Order Method

Runge Kutta 4th Order Method: Formulas

Runge Kutta 4th Order Method: Example: Part 1 of 2

Runge Kutta 4th Order Method: Example: Part 2 of 2

Section 18. Richardson Extrapolation

Richardsons Extrapolation of Trapezoidal Rule: Theory

Richardsons Extrapolation of Trapezoidal Rule: Example

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