MCA-247
Numerical and Scientific Computing
Getting usable answers out of intractable mathematics — root finding, linear systems and interpolation, numerical integration and differential equations, then curve fitting and inference.
- Semester
- Semester 3
- Credits
- 3
- Scheme
- 3 lectures a week
- Contact hours
- 40 across four units
- Category
- Core Elective - III
What the paper is
Numerical and Scientific Computing is about approximation done responsibly. Every method in it produces an answer that is wrong, and the paper is really about two questions: how wrong, and does the error shrink as you iterate?
That is why Unit I starts with floating point representation and the taxonomy of error — absolute, relative, truncation, rounding — before any method appears. Rate of convergence then recurs for every technique that follows, and it is the criterion by which methods are compared.
Units II and III are the working core: solving linear systems directly and iteratively, interpolating between known points, integrating numerically, and solving ordinary differential equations with Euler and Runge-Kutta. Unit IV turns statistical with curve fitting, sampling and hypothesis testing.
What to concentrate on
- Newton-Raphson is the most examined method in the paper: apply it, derive its quadratic rate of convergence, and say when it fails. Bisection and false position are asked alongside it as the comparison.
- Gauss elimination with partial pivoting and Gauss-Seidel iteration are the Unit II certainties. Do both on the same 3×3 system once and the difference between direct and iterative becomes obvious.
- Simpson's rule and the fourth-order Runge-Kutta method are the standard computations in Unit III, and both are formula-driven. Know the formulae exactly, including the error terms for the integration rules.
Syllabus, unit by unit
Summarised from the approved scheme. Hours are the contact hours the scheme allots, which is a fair guide to where the marks sit.
Unit I · Floating point arithmetic and root finding
10 hours- Representation of floating point numbers, operations, normalisation, pitfalls of the representation
- Errors in numerical computation: absolute, relative, truncation and rounding error
- Zeros of transcendental equations and polynomials: bisection, secant, false position, Newton-Raphson and Ramanujan's method
- Rate of convergence of bisection, Newton-Raphson and iterative methods; methods for finding complex roots
Unit II · Linear systems and interpolation
10 hours- Gauss elimination: direct method and pivoting; LU decomposition
- Gauss-Seidel and Gauss-Jacobi iterative methods and their rate of convergence
- Finding a matrix inverse by Gauss-Jordan; ill-conditioned systems and refinement of solutions
- Lagrange interpolation and Newton divided differences; finite differences; Newton's forward and backward formulae
- Central difference formulae: Gauss forward and backward, Bessel, Stirling and Everett; Hermite interpolation
- Approximation by Taylor series and Chebyshev polynomials
Unit III · Integration and differential equations
10 hours- Numerical differentiation
- Numerical integration: trapezoidal rule, Simpson's rules, Boole's rule, Weddle's rule, Gauss quadrature, Romberg integration, and the error in each
- Ordinary differential equations: Picard's method, Taylor series method, Euler's and modified Euler's methods
- Runge-Kutta methods of second and fourth order, automatic error monitoring, stability
- Boundary value problems by the finite difference method
Unit IV · Curve fitting, sampling and inference
10 hours- Method of least squares; fitting straight lines, polynomials and exponential curves
- Linear and non-linear regression, multiple regression, moving averages, smoothing of curves
- Random sampling and sampling distributions: mean, variance, difference of means and proportions
- Parameter estimation: method of moments, maximum likelihood estimation, interval estimation
- Hypothesis testing: level of significance, Type I and Type II errors; chi-square, t and F tests
How it is examined
- Question 1 is compulsory, covers the whole syllabus, and is made of objective or short-answer questions. It carries 12 marks.
- The rest of the paper is four units matching the four above. Each unit sets two questions and you attempt one, so there are four more answers at 12 marks each.
- Examiners are asked to map every question to a course outcome using Bloom's taxonomy, which is why questions are phrased as “apply”, “analyse” or “evaluate” rather than only “explain”.
The pass mark is 40 out of 100 in every paper.
Learning objectives
- An overview of the problems arising in scientific computation — non-linear systems, integration, differential equations and simulation
- Effective numerical methods for obtaining approximate results
- Problems in applied mathematics, physics and engineering requiring numerical results from raw data
- Solving complex mathematical problems using only simple arithmetic operations
- Root finding, linear systems, interpolation and regression, integration and differentiation, differential and boundary value problems, and matrix problems
Assumed beforehand
- Basics of mathematics
Paired in the scheme with MCA-267 Lab based on Core Elective III.
Books it is taught from
- TB1Curtis F. Gerald and Patrick O. Wheatley, “Applied Numerical Analysis”, Pearson, 7th edition, 2003
- TB2David Kincaid and Ward Cheney, “Numerical Analysis: Mathematics of Scientific Computing”, Brooks/Cole, 6th edition, 2002
- TB3C. R. Kothari and Gaurav Garg, “Research Methodology”, New Age International, 4th edition, 2019
Notes for MCA-247
Lecture notes, question papers and assignments filed against MCA-247 are in the library. Everything there has been checked by a moderator before it appears, and an account is free — new ones are approved by hand, which is what keeps the shelf clean.