Numerical Analysis Notes for BS/MSc

Numerical Analysis Notes for BS/MSc written by Muhammad Usman Hamid. He is a brilliant teacher from Sargodha.

Numerical analysis is a branch of mathematics that focuses on developing algorithms and methods for solving mathematical problems numerically, typically on a computer. This field is crucial for various applications, including scientific computing, engineering, finance, and many others.

Introduction to Numerical Analysis:

  • Overview of numerical methods and their importance in solving mathematical problems.
  • Sources of error in numerical computation and methods to control and analyze errors.

Root Finding:

  • Bisection method.
  • Newton-Raphson method.
  • Secant method.
  • Convergence and convergence rates.
  1. Interpolation and Approximation:
  • Lagrange interpolation.
  • Newton interpolation.
  • Polynomial and spline interpolation.
  • Least squares approximation.
  1. Numerical Differentiation and Integration:
  • Finite difference approximations.
  • Trapezoidal rule.
  • Simpson’s rule.
  • Gaussian quadrature.
  1. Numerical Solutions of Ordinary Differential Equations (ODEs):
  • Euler’s method.
  • Runge-Kutta methods.
  • Multistep methods (e.g., Adams-Bashforth, Adams-Moulton).
  • Stability and stiffness in ODEs.
  1. Numerical Solutions of Partial Differential Equations (PDEs):
  • Finite difference methods.
  • Finite element methods.
  • Boundary value problems and initial value problems.
  • Applications to heat equation, wave equation, and Laplace equation.
  1. Linear Algebraic Equations:
  • Direct methods (e.g., Gaussian elimination, LU decomposition).
  • Iterative methods (e.g., Jacobi, Gauss-Seidel).
  • Conjugate gradient method.
  1. Eigenvalue Problems:
  • Power iteration method.
  • QR algorithm.
  • Singular value decomposition.
  1. Numerical Linear Algebra:
  • Matrix factorizations (e.g., Cholesky, QR).
  • Condition number of a matrix.
  • Stability and accuracy in solving linear systems.
  1. Optimization:
    • Unconstrained optimization methods (e.g., gradient descent, Newton’s method).
    • Constrained optimization and Lagrange multipliers.
    • Linear programming.
  2. Error Analysis:
    • Absolute and relative errors.
    • Floating-point arithmetic.
    • Conditioning and stability of algorithms.
  3. Introduction to MATLAB, Python, or other Numerical Computing Software:
    • Practical implementation and application of numerical methods using a programming language.

These topics provide a comprehensive foundation in numerical analysis, and students typically gain both theoretical understanding and practical programming skills throughout their coursework. The specific topics covered may vary between institutions and courses.
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Numerical Analysis Notes for BS/MSc

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