Questions
1–2 questions per PSU paper
Difficulty
Medium
Importance
Medium yield for ONGC/IOCL
Overview
Numerical Methods provide algorithmic techniques to approximate solutions for mathematical problems that are difficult or impossible to solve analytically. In PSU exams, this topic is critical as it focuses on iterative convergence and error estimation, which are frequently tested via direct application problems.
Root Finding: Newton–Raphson Method
The Newton–Raphson method is an iterative technique used to find the roots of a differentiable function. It is favored in exams due to its quadratic convergence rate, meaning the number of correct digits roughly doubles with each iteration.
- Formula: x(n+1) = x(n) - [f(x(n)) / f'(x(n))]
- Requires: f'(x) ≠ 0 in the neighborhood of the root
- Quadratic convergence: error E(n+1) is proportional to E(n)^2
- Best suited for finding real roots of non-linear equations
- Geometrically represents tangent lines intersecting the x-axis
Numerical Integration
Numerical integration, or quadrature, is used to estimate the definite integral of a function given discrete data points or complex functions. The Trapezoidal and Simpson's rules are the primary methods tested in competitive engineering exams.
- Trapezoidal Rule: I = (h/2) * [y(0) + y(n) + 2*(y(1) + y(2) + ... + y(n-1))]
- Simpson's 1/3 Rule: I = (h/3) * [y(0) + y(n) + 4*sum(odd terms) + 2*sum(even terms)]
- Simpson's 1/3 Rule requires an even number of sub-intervals (odd number of points)
- Simpson's 3/8 Rule requires the number of intervals to be a multiple of 3
- Error in Trapezoidal Rule is proportional to h^2
Solution of ODEs
Solving ordinary differential equations numerically is essential for simulating physical systems. The Euler method is the simplest approach, while the Runge–Kutta (RK) methods provide much higher accuracy for the same step size.
- Euler's method: y(n+1) = y(n) + h * f(x(n), y(n))
- RK2 (Modified Euler): y(n+1) = y(n) + (h/2) * [k1 + k2]
- RK4 is the most widely used industrial and exam-standard method
- RK4 error is of the order O(h^4)
- Always use small step sizes 'h' to reduce truncation error
Formula Sheet
x(n+1) = x(n) - f(x(n))/f'(x(n))
Trapezoidal: (h/2) * [y0 + yn + 2*sum(yi)]
Simpson 1/3: (h/3) * [y0 + yn + 4*sum(y_odd) + 2*sum(y_even)]
Simpson 3/8: (3h/8) * [y0 + yn + 3*sum(y_not_multiples_of_3) + 2*sum(y_multiples_of_3)]
Euler: y(n+1) = y(n) + h*f(xn, yn)
RK4: y(n+1) = y(n) + (1/6)*(k1 + 2k2 + 2k3 + k4)
Exam Tip
Always prioritize calculating the derivative correctly in Newton-Raphson, as a single sign error in f'(x) will lead to divergence or an incorrect final value.
Common Mistakes
- Mixing up the weights in Simpson's rules (4 for odd vs 2 for even indices).
- Failing to check if the interval count is valid for Simpson's 1/3 or 3/8 rule.
- Forgetting to switch the calculator to Radian mode when trigonometric functions are involved.
More Revision Notes
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