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