01 Mar 2026

Numerical Methods Syllabus Map

A compact SEM-I map for numerical methods from roots of equations to finite difference methods.

msc semester-i numerical-methods computational-techniques

Numerical methods are used when an exact analytical solution is difficult, unavailable, or inefficient. The aim is to convert a mathematical problem into a sequence of arithmetic steps that can be carried out by hand for small cases and by computer for larger cases.

Flow of ideas

1. Roots of functions locate where $f(x)=0$.
2. Iteration methods improve an approximate answer step by step.
3. Gauss elimination solves simultaneous linear equations.
4. Eigenvalues and eigenvectors describe matrix modes and characteristic values.
5. Interpolation and extrapolation estimate values inside or outside tabulated data.
6. Curve fitting and least squares find the best approximate relation from data.
7. Numerical integration estimates areas and integrals by trapezoid and Simpson rules.
8. Runge-Kutta method solves first-order differential equations numerically.
9. Finite difference method replaces derivatives by differences on a grid.

Basic numerical philosophy

A numerical answer is not only a number. It must be checked for convergence, stability, step-size dependence, and physical reasonableness.

Common sources of error

• Round-off error comes from finite precision arithmetic.
• Truncation error comes from cutting off an infinite or exact process.
• Iteration error comes from stopping an iterative method before the exact limit.
• Modelling error comes from the assumptions used before computation starts.