Numerical Analysis

Premise

Numerical analysis is the study of methods to solve mathematical problems involving real valued quantities. In particular, numerical methods are necessary when an analytical solution to a problem cannot be found or is too cumbersome in practice. Numerical methods are also necessary when analytical solutions exist, but one needs to obtain specific numerical values.

• 1. Python
  • 1.1. Python versions and installation
  • 1.2. Integrated Development Environment (IDE)
  • 1.3. Variables
    • 1.3.1. Numbers
    • 1.3.2. Booleans
    • 1.3.3. Strings
  • 1.4. Data containers
    • 1.4.1. Lists
    • 1.4.2. Dictionaries
    • 1.4.3. Sets
    • 1.4.4. Tuples
  • 1.5. Functions
  • 1.6. Object-Oriented Programming
  • 1.7. Control flow statements
  • 1.8. Recursion
  • 1.9. Reading and writing files
  • 1.10. Numpy
    • 1.10.1. Numpy arrays
    • 1.10.2. Array indexing
    • 1.10.3. Array operations
  • 1.11. Plotting
  • 1.12. Exercises
• 2. Solution of Nonlinear Equations
  • 2.1. Bisection method
    • 2.1.1. Example
  • 2.2. Fixed point method
    • 2.2.1. Example
  • 2.3. Newton’s method
    • 2.3.1. Example
  • 2.4. Horner’s method
    • 2.4.1. Example
  • 2.5. Horner-Newton’s method
    • 2.5.1. Example
• 3. Linear Systems
  • 3.1. Forward substitution
    • 3.1.1. Example 1
  • 3.2. Backward substitution
    • 3.2.1. Example 1
  • 3.3. Swapping rows
    • 3.3.1. Example
  • 3.4. Gaussian elimination
    • 3.4.1. Example 1
    • 3.4.2. Example 2
  • 3.5. LU decomposition
    • 3.5.1. Example 1
  • 3.6. Cholesky decomposition
    • 3.6.1. Example 1
  • 3.7. Jacobi Iteration Method
    • 3.7.1. Example 1
  • 3.8. Exercises
• 4. Finding Eigenvalues and Eigenvectors
  • 4.1. Norms
  • 4.2. Power method
    • 4.2.1. Example 1
  • 4.3. Inverse power method
    • 4.3.1. Adjusted code for linear systems
    • 4.3.2. Inverse power method code
    • 4.3.3. Example
  • 4.4. Exercises
• 5. Polynomial Interpolation
  • 5.1. Polynomials in Newton’s form
    • 5.1.1. Example 1
  • 5.2. Computation of the coefficients by nested multiplication
    • 5.2.1. Example 1
  • 5.3. Computation of the coefficients by the divided differences method
    • 5.3.1. Example 1
  • 5.4. Exercises