Engineering Mathematics

Numerical Methods

Root finding, interpolation and numerical integration — approximate techniques for problems with no closed form.

PART 1

Topic Breakdown & Traps

Numerical methods produce approximate answers iteratively. Newton–Raphson refines a root with

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

and converges quadratically near a simple root. The trapezoidal and Simpson's rules approximate

\int f d x

from sampled values; Simpson's is exact for cubics and needs an even number of intervals.

Newton–Raphson:

x_{n + 1} = x_{n} - \frac{f ( x _{n} )}{f ^{'} ( x _{n} )}

Bisection: root bracket halved each step (linear convergence)

Trapezoidal rule:

\int_{a}^{b} f d x \approx \frac{h}{2} [f_{0} + 2 (f_{1} + \dots + f_{n - 1}) + f_{n}]

Simpson's 1/3 rule:

\frac{h}{3} [f_{0} + 4 (f_{1} + f_{3} + \dots) + 2 (f_{2} + f_{4} + \dots) + f_{n}]

⚠
Newton–Raphson can diverge if $f^{'} (x_{n}) \approx 0$ or the start point is poor — it is not unconditionally stable.
⚠
Simpson's 1/3 rule needs an even number of sub-intervals (odd number of ordinates).
⚠
**Use $f^{'}$ , not $f$ , in the denominator** of the Newton update — a frequent sign/term slip.

PART 2

Q1BASIC1 Mark · NAT

Using Newton–Raphson on

f (x) = x^{2} - 2

with

x_{0} = 1.5

, the next estimate

x_{1}

is _____.

Q2MEDIUM2 Marks · NAT

Using the trapezoidal rule with one interval,

\int_{0}^{2} x^{2} d x

is estimated as _____.

Q3HARD2 Marks · MCQ

The order of convergence of the Newton–Raphson method near a simple root is: