Write down algorithm and program for the differentiating continuous function using three point formula. [5]
2.
How Simpson's 1/3 rule differs from Trapezoidal rule? Drive the formula for Simpson's 1/3 rule. [5]
Solution of Nonlinear Equations
1.
Derive the formula for Newton Raphson Method. Solve the equation using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.
$x^2 + 4x - 9 = 0$
[10]
2.
Define the terms true error and relative error? Use Horner' method to evaluate polynomial at x = 3 and write down its algorithm.
$2x^3 - 3x^2 + 5x - 2$
[5]
Solution of Ordinary Differential Equations
1.
Appropriate the solution of $y' = 2x + y$, $y(0) = 1$ using Euler’s method with step size 0.1. Approximate the value of $y(0.4)$. [5]
2.
A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at $100^\circ C$, right side at $200^\circ C$. Upper part at $50^\circ C$, and lower at $150^\circ C$. If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes? [5]
3.
How boundary value problems differs from initial value problems? Discuss shooting method for solving boundary value problem. [5]
Solving System of Linear Equations
1.
Why partial pivoting is used with Naive Gauss Elimination method? Solve the following system of equations using Gauss Elimination method with partial pivoting? How Gauss Jordan method differs from Gauss elimination method?
$2x + 2y - z = 6$
$4x + 2y + 3z = 4$
$x + y + z = 0$
[10]
2.
Discuss the Doolittle LU decomposition method for matrix factorization. [5]