Numerical Analysis
Numerical Analysis in GATE MA tests interpolation, numerical integration, root finding, linear-system solvers, and numerical ODE methods, with emphasis on error orders and convergence.
Core rules
- Lagrange interpolation through n+1 points has error f^{(n+1)}(ξ)/(n+1)! ∏(x−x_i).
- Trapezoidal rule error O(h²); Simpson's rule is exact for cubics and has error O(h⁴).
- Newton's method x_{n+1}=x_n−f(x_n)/f′(x_n) converges quadratically near a simple root.
- Jacobi and Gauss–Seidel iterations converge for strictly diagonally dominant matrices; Gauss–Seidel is usually faster.
- Euler's method is order 1; classical Runge–Kutta (RK4) is order 4.
- Gaussian elimination with partial pivoting controls round-off.
Worked example
Approximate √2 by Newton's method with f(x)=x²−2. Then x_{n+1}=x_n−(x_n²−2)/(2x_n)= (x_n+2/x_n)/2. Starting at x_0=1: x_1=1.5, x_2≈1.4167, x_3≈1.41422. The number of correct digits roughly doubles per step, illustrating quadratic convergence.
Traps
- High-degree polynomial interpolation can oscillate (Runge phenomenon); equidistant nodes are not always best.
- Simpson's rule requires an even number of subintervals; using an odd count silently changes the formula.
- Newton's method can diverge or cycle if f′(x_n) is near zero or the initial guess is poor.
- A small residual does not guarantee a small error when the condition number is large.