Applied Numerical Methods

1. Algebraic and Transcendental Equations

• Overview: This section introduces methods for solving algebraic and transcendental equations.
• Key Topics:
  • Iteration, Secant, Newton-Raphson, and Regula-Falsi Methods: Methods for iterative solutions.
  • Error Analysis: Discusses errors in numerical calculations.
  • Bisection Method: A root-finding method for continuous functions.

• Overview: Focuses on solving systems of linear equations and eigenvalue problems.
• Key Topics:
  • Solving Linear Equations: Gauss-Seidel iteration and LU-Decomposition.
  • Special Matrices: Tridiagonal systems and the Thomas algorithm.
  • Eigenvalue/Eigenvector Computation: Jacobi and Power methods for eigenvalues.

• Overview: Explains interpolation techniques for estimating unknown values.
• Key Topics:
  • Newton's Interpolation: Forward and backward interpolation formulas.
  • Other Formulas: Central difference, Lagrange, and divided difference formulas.
  • Spline Interpolation: Linear and cubic spline methods.

• Overview: Covers techniques for differentiation and integration of tabulated functions.
• Key Topics:
  • Numerical Differentiation: Derivatives from discrete data.
  • Numerical Integration: Newton-Cotes, Romberg's method, and Gaussian integer methods.

• Overview: Methods for solving ODEs numerically.
• Key Topics:
  • Runge-Kutta Methods: For initial value problems.
  • Predictor-Corrector Methods: Including Adams-Bashforth-Moulton.
  • Gaussian Quadrature: For integral approximation within ODE solutions.