Numerical Analysis
Topics covered so far:
Root findings
1. Derive Bisection method/error bound; Describe bisection method for finding root; error bound;
advantage and disadvantages
Let f ( x)  x3  x 2  1 has exactly one zero in [1,2]. Determine the number of iteration required to
find zeros of f(x), with an absolute error of no more than 0.000001.
2. Derive Newton (Raphson) method/Error analysis/error estimation: Describe Newton’s method
for finding the root of f(a)=0; error bound ; Prove that the Newton’s method converges
quadratically if f’(a) not equal to 0/ compute few iterations of Newton’s method for a given
problem using pencil and paper.
3. Derive Secant method/error analysis; Describe secant method for finding the root 0f f(a)=0;
error bound ; convergence(order 1.62)
4. Derive Fixed point iteration method/ compute few iterations of Fixed point method for a given
problem using pencil and paper.
5. Comparison between different methods
Interpolation/approximation:
Suppose we are given table of (n+1) point’s where x’s are distinct and satisfy x_0<x_1…<x_n. Our
objective is to find a polynomial curve that passes through the given points (x_i, y_i), i=1, 2,…n. Hence
we need to find a polynomial p(x) such that p(x_i)=y_i, i=0, 1, …, n.
1. Linear interpolation
2. Quadratic interpolation/Lagrange interpolation formula/derive Lagrange polynomial that
interpolates a given set of data.
3. Divided differences
4. Properties of divided differences
5. Derive Newton’s divided differences interpolating polynomial formula/ Determine a polynomial
using Newton’s divided difference formula
6. Error in polynomial interpolation/Error formula
Numerical integration:
1. Derive formula for the Trapezoidal rule/error formula
2. Derive formula for the Simpsons rule/error formulas
Least square methods:
1. Derive linear least square method/ use the method of least square to find the linear function
that best fit some data.
2. Derive least square polynomial method/ Find least squares of polynomials that fit some data.
Ordinary differential equation:
1. Derive Forward Euler method/ Give Geometrical interpretation of Euler method/truncation
error/global error/convergence/ use this method to obtain few steps of the approximation of
IVP.
2. Derive Implicit Euler method (Backward Euler method)/
3. Derive third order Taylor method/error
4. Derive Runge -Kutta method of order 2/use this method to obtain few steps of the
approximation of IVP.