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Numerical Techniques
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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
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)
3. Numerical Differentiation
Background of Differentiation
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
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
6. Matrices
Diagonal and Identity Matrices
What is a Diagonally Dominant Matrix
Upper and Lower Triangular Matrices
Subtracting Two Matrices
Multiplying Two Matrices
Inverse of a Matrix
Setting up Equations in a Matrix Form
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
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
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
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
11. Polynomial Interpolation
Polynomial Interpolation Method
Uniqueness of Polynomial Interpolant: Part 1 of 2
Uniqueness of Polynomial Interpolant: Part 2 of 2
Linear 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
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
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
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
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
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
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
18. Richardson Extrapolation
Richardsons Extrapolation of Trapezoidal Rule: Theory
Richardsons Extrapolation of Trapezoidal Rule: Example