# 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.
1. Numerical Solutions of Ordinary Differential Equations (ODEs):
• Euler’s method.
• Runge-Kutta methods.
• 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).
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