2nd Year

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Computational Methods +

Syllabus📘
Notes📚 PYQs✅
- Unit 1: Foundations & Root-Finding +
  - Theoretical Foundations:
    - Review of Taylor Series
    - Rolle's Theorem and Mean Value Theorem
  - Numerical Fundamentals:
    - Approximations and Errors
    - Data representation and computer arithmetic
    - Loss of significance in computation
  - Root-Finding Methods:
    - Bisection method (convergence analysis)
    - Newton Method (convergence analysis)
    - Secant Method (convergence analysis)
  - Optimization Techniques:
    - Fibonacci search
    - Golden Section Search
    - Newton's method (univariate)
    - Steepest descent (multivariate)
    - Nelder-Mead Algorithm
- Unit 2: Interpolation & Integration +
  - Interpolation Theory:
    - Assumptions and polynomial interpolation errors
    - Finite differences
  - Interpolation Methods:
    - Gregory-Newton's Forward/Backward
    - Lagrange's Interpolation
    - Newton's divided difference
  - Numerical Integration:
    - Newton-Cote's Quadrature
    - Trapezoidal Rule
    - Simpson's 1/3 and 3/8 rules
    - Error analysis
  - Advanced Techniques:
    - Romberg's Algorithm
    - Gaussian Quadrature
- Unit 3: Linear Systems & Splines +
  - Linear Systems:
    - Existence of solutions
    - Gauss elimination (with pivoting)
    - Gauss-Jordan method
  - Matrix Factorization:
    - Dolittle algorithm
    - Crout's Algorithm
    - Cholesky method
  - Eigenvalue Problems: Power method
  - Spline Approximation:
    - First/Second-Degree Splines
    - Natural Cubic Splines
    - B Splines
- Unit 4: Differential Equations +
  - ODE Solutions:
    - Picard's method
    - Taylor series method
    - Euler's method
    - Runge-Kutta methods
  - Predictor-Corrector Methods:
    - Adams-Bashforth
    - Milne's method
  - PDE Solutions:
    - Parabolic equations
    - Hyperbolic equations
    - Elliptic equations
  - Implementation: C/C++ applications

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