Determine the spacing h in a table of equally spaced values for the function

 so that the quadratic interpolation in this table satisfies

The iteration method

where N is positive constant, converges to some quantity. Determine this quantity. Also find the rate of convergence of this method.

Find the solution of the difference equation   the particular solution when  

Use Runge-Kutta method of order four to solve  carry to  x =1.5 with h = 0.1.

Suppose n f denotes the value of  then find the value of  

For the table of values of x f(x) = xe ^xgiven by

  Find f"(2.0) using the central difference formula of 0(h²) for h = 0.1 and h = 0.2 . Calculate T.E. and actual error

Calculate T.E. and actual error

Take 10 figure logarithm to base 10 from by unit increment. Calculate the first derivative of log

A solid of revolution is formed by rotating about the x-axis the area bounded between   the curve given by the table   

Find the volume of the solid so formed using

i) Trapezodial rule    ii) Simpson’s rule

The position f(x) of a particle moving in a line at various times xk is given in the following

table. Estimate the velocity and acceleration of the particle at  x =1.2

Solve the  using R-K method of 0(h⁴ )  with

h = 0.1 and obtain the value of y(0.2) . Also find the error at t = 0.2 , if the exact solution  is   

Consider the following data

Use Stirling’s formula to approximate  

Using sin( 0.1) = 0.09983 and sin( 0.2) = 0.19867 , find an approximate value of sin( 0.15) by using Lagrange interpolation. Obtain a bound on the truncation error.

Use the polynomial in part (a) to approximate 

Use the polynomial in part (a) to approximate and find a bound for the error involved. 

Calculate the third-degree Taylor polynomial about 

From the following table, find the number of students who obtained less than 45 marks.

Find the minimum number of intervals required to evaluate   with an accuracy of

by using the Trapezoidal rule.

Obtain an approximate value of  using composite Simpson’s rule with h = 0.25 and

h = 0.125  Find also the improved value using Romberg integration.

The solution of the system of equations is attempted by the Gauss

Jacobi and Gauss Seidel iteration schemes. Set up the two schemes in matrix form. Will the iteration schemes converge? Justify your answer.

Starting with  find the dominant eigenvalue and corresponding eigenvector for the matrix using the power method.