GATE CS - ENGINEERING MATHEMATICS:Numerical Methods
Mastering numerical methods concepts and implementation.
Numerical Methods for GATE CS
Numerical Methods is a thin slice — often 1–2 marks. Know one iteration of bisection or Newton-Raphson, and the trapezoidal / Simpson formulas. GATE wants the next iterate or the integral estimate, not a programming library.
What is tested
- Root finding: Bisection, Newton-Raphson
- Interpolation: linear / Lagrange when asked
- Numerical integration: Trapezoidal, Simpson’s 1/3
- Error: qualitative (truncation vs round-off) if options require it
Root Finding
Bisection Method
Algorithm:
- Start with interval [a, b] where f(a) and f(b) have opposite signs
- Calculate midpoint c = (a + b) / 2
- If f(c) = 0, c is the root
- If f(a) and f(c) have opposite signs, root is in [a, c]
- Otherwise, root is in [c, b]
- Repeat until convergence
Properties:
- Guaranteed to converge
- Slow but reliable
- Requires continuous function
- Requires opposite signs at endpoints
Convergence:
Error reduces by half each iteration
Newton-Raphson Method
Formula:
x_{n+1} = x_n - f(x_n) / f'(x_n)
Algorithm:
- Start with initial guess x_0
- Calculate x_1 = x_0 - f(x_0) / f'(x_0)
- Repeat until convergence
Properties:
- Faster convergence than bisection
- Requires derivative
- May not converge if initial guess is poor
- Can diverge if f'(x) ≈ 0
Convergence:
Quadratic convergence (very fast)
Interpolation
Linear Interpolation
Formula:
y = y₁ + (y₂ - y₁) × (x - x₁) / (x₂ - x₁)
Use Case:
Finding value between two known points
Polynomial Interpolation
Lagrange Interpolation:
P(x) = Σ y_i × L_i(x)
Where L_i(x) = Π (x - x_j) / (x_i - x_j) for j ≠ i
Use Case:
Finding value using multiple known points
Numerical Integration
Trapezoidal Rule
Formula:
∫[a to b] f(x) dx ≈ (h/2) × [f(a) + 2Σf(x_i) + f(b)]
Where h = (b - a) / n
Error:
O(h²)
Simpson's Rule
Formula (n even):
∫[a to b] f(x) dx ≈ (h/3) × [f(a) + 4Σf(x_odd) + 2Σf(x_even) + f(b)]
Error:
O(h⁴) - more accurate than trapezoidal
Error Analysis
Types of Errors
Truncation Error:
Error due to approximation method
Round-off Error:
Error due to finite precision arithmetic
Total Error:
Sum of truncation and round-off errors
GATE CS Weightage
Numerical Methods typically accounts for:
- 1-2 marks out of 100 in GATE CS
- Questions often involve root finding methods
- Less frequently tested than other chapters