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.

**Interpolation and Approximation:**

- Lagrange interpolation.
- Newton interpolation.
- Polynomial and spline interpolation.
- Least squares approximation.

**Numerical Differentiation and Integration:**

- Finite difference approximations.
- Trapezoidal rule.
- Simpson’s rule.
- Gaussian quadrature.

**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.

**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.

**Linear Algebraic Equations:**

- Direct methods (e.g., Gaussian elimination, LU decomposition).
- Iterative methods (e.g., Jacobi, Gauss-Seidel).
- Conjugate gradient method.

**Eigenvalue Problems:**

- Power iteration method.
- QR algorithm.
- Singular value decomposition.

**Numerical Linear Algebra:**

- Matrix factorizations (e.g., Cholesky, QR).
- Condition number of a matrix.
- Stability and accuracy in solving linear systems.

**Optimization:**- Unconstrained optimization methods (e.g., gradient descent, Newton’s method).
- Constrained optimization and Lagrange multipliers.
- Linear programming.

**Error Analysis:**- Absolute and relative errors.
- Floating-point arithmetic.
- Conditioning and stability of algorithms.

**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