Unit-1
2marks:
1) If g(x) is continous in[a,b], then under what condition the iterative method x=g(x) has a unique solution in [a,b] ?
2) What do you mean by error by error analysis ?
3) Explain the term Round off error ; Truncation error .
4) Find an iterative formula to find , where N is positive number.
5) State the order of convergence and convergence condition for Newton-Raphson method.
6) Derive Newton’s algorithm for finding the Pth root of a number N.
7) Derive Newton-Raphson formula to find the cube root of a positive number K.
8) Locate the negative root of , approximately.
9) Show that Newton-Raphson formula to find (or) can be expressed in the form , n=0,1,2………..
10) Establish an iteration formula to find the reciprocal of a positive number N by Newton-Raphson method ?
11) State two difference between direct and iterative methods for solving system of equations
12) State sufficient condition for Gauss-Jacobi method to converge.
13) What are the elementary transforms ?
14) Explain Gauss-Elimination method to solve AX=B.
15) Find the dominant eigenvalue of A= by power method.
8 marks:
1) Newton’s method and Newton-Raphson method.
2) Gauss Elimination method and Gauss Jordan method.
3) Iterative method and Gauss-Seidel Iterative Method.
4) Inverse of matrix using Gauss Jordan Method
5) Eigen values and eigen vectors
6) Jacobi method for symmetric matrices.
Unit-2
2marks:
1) What is the lagrange’s formula to find ‘y’ if three sets of values , are given ?
2) Find the second degree polynomial fitting the following data
| X | 1 | 2 | 4 |
| Y | 4 | 5 | 13 |
3) Give inverse Langrange’s interpolation formula.
4) State Langrange’s interpolation formula.
5) Give Newton’s divded difference interpolation formula.
6) Talking h to be the interval of differencing, find
7) Write the divided difference table for
| X | 30 | 35 | 45 | 55 |
| Y | 148 | 96 | 68 | 34 |
8) Find the divided difference table for the following:
| X | 1 | 1 | 4 | 5 |
| f(x) | 8 | 11 | 78 | 123 |
9) Given =2,=4, =4, =32, find = ?
10) Obtain the interpolation quadratic polynomial for the given data by using Newton’s forward difference formula.
| X | 0 | 2 | 4 | 6 |
| Y | -3 | 5 | 21 | 45 |
11) Find ∆f(x) if +2x+2=f(x) and the interval of differencing as unity.
12) State Newton’s formula on interpolation.
13) Write a polynomial to calculate the value of x when
| X | 3 | 5 | 7 | 9 |
| Y | 6 | 24 | 58 | 108 |
14) A third degree polynomial passes through (0,1),(1,-1),(2,-1) and (3,2).Find its value at x=4.
15) Form the difference table for the following:
| X | 5 | 6 | 9 | 11 |
| f(x) | 12 | 13 | 15 | 18 |
8Marks:
1) Lagrange’s Interpolation and inverse interpolation problem for unequal intervals.
2) Newton’s Divided Difference problem.
3) Cubic Spline interpolation problem.
4) Newton’s forward and backward interpolation problem.
Unit-3
2marks:
1) Find the error in the derivative of f(x)=cos x by computing directly and using the approximation f(x)= at x=0.8 chossing h=0.01.
2) What are the errors in trapezoidal rule of numerical integration ?
3) Using trapezoidal rule evaluate by dividing the range into 6 equal parts.
4) What is the order of error in trapezoidal rule ?
5) What is the Geometric interpretation of trapezoidal rule ?
6) State trapezoidal rule to evaluate
7) What are the errors in Simpson’s rules of numerical integration ?
8) In order to evaluate by Simpson’s rule as well as by Simpson’s rule ,what is the restriction on the number of intervals ?
9) Using Simpson’s rule,find given that ,.
10) When does Simpson’s rule give exact result ?
11) State Simpson’s three-eighth’s rule.
12) State Simpson’s one-third rule.
13) State three point Gaussian quadrature formula to evaluate
14) Write the formula for evaluating double integrals using Trapezoidal method.
15) Write the formula for evaluating double integrals using Simpson’s method.
8marks:
1) Forward and Backward difference problems to compute the derivatives
2) Simpson’s 1/3 and 3/8 rule.
3) Trapezoidal rule and Evaluation of Double Integrals using Trapezoidal & Simpson’s rule.
4) Romberg’s Method.
5) Gauss two point and three point problems
Unit-4
2marks:
1) By Taylor’s series method ,find y(1.1) given ,y(1)=0.
2) Compute y(0.1) by Taylor’s series method to three places decimals given that ,y(0)=1.
3) Write the merits and demerits of Taylor’s series method of solution.
4) Using modified Euler’s method ,find y(0.1) if ,y(0)=1.
5) State modified Euler’s algorithm to solve .
6) Solve to find y(0.01) using Euler’s method.
7) Write down the formula to solve 2nd order differential equation using Runge-Kutta method of 4th order.
8) Compare Taylor series and Runge-Kutta method.
9) What are the advantages of Runge-Kutta method over taylor method ?
10) Give the formula for second order Runge-Kutta method .
11) Using Runge-Kutta method of second order find y(0.1),when )=1.
12) Write Milner’s Predictor –corrector formula.
13) What is the condition to apply Adams-Bashforth method ?
14) State Adams-Bashforth Predictor – corrector formula to solve numerically stating the assumptions.
15) Explain the meaning of explicit and implicit methods in numerical calculations.
8marks:
1) Taylor series method
2) Improved Euler method and Modified Euler’s method.
3) Runge-Kutta method.
4) Milne’s Predictor- Corrector method
5) Adam’s Bashforth Predictor-Corrector Method.
Unit-5
2marks:
1) What is the error for solving Laplace and Poisson’s equations by finite difference method ?
2) Obtain the finite difference scheme for the difference equation 2
3) State finite difference scheme of
4) Define a difference quotient.
5) State finite difference form of
6) Classify the equation
7) Write down the Bender-Schmidt recurrence relation for one dimensional heat equation.
8) Write the Bender-Schmidt recurrence relation for .
9) Write an explicit formula to solve numerically.
10) Write the Crank-Nicholson difference scheme to solve with u(0,t)= and u(x,o)=f(x).
11) Write diagonal five- point formula to solve the laplace equation .
12) state standard five- point finite difference formula for solving .
13) Write the five- point formula to solve .
14) State the difference equation that approximate elliptic equation.
15) Write the difference scheme for solving the Laplace’s equation.
8marks:
1) Problems based on finite difference method
2) Problems on One Dimensional and Two Dimensional Heat E quations.
3) Problems on One Dimensional Wave equation.