MA2264 - Numerical Methods - Question Bank ~ Online Virtual-Tutor

NUMERICAL METHODS Question Bank

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Unit I : Solution of equations and eigen value problems

Part A

1. If g(x) is continuous in [a,b] then under what condition the iterative method x = g(x) has unique solution in [a,b].

2. Find inverse of A =

by Gauss – Jordan method.

3. Why Gauss Seidel iteration is a method of successive corrections.

4. Compare Gauss Jacobi and Gauss Siedel methods for solving linear system of the form AX = B.

5. State the conditions for convergence of Gauss Siedel method for solving a system of equations.

6. Compare Gaussian elimination method and Gauss-Jordan method.

7. What type of eigen value can be obtained using power method.

8. Find the dominant eigen value of A =

by power method.

9. How is the numerically smallest eigen value of A obtained.

10. State two difference between direct and iterative methods for solving system of equations.

Part B

Find all the eigen values of the matrix by power method (Apply only 3 iterations).

Use Newton’s backward difference formula to construct an interpolating polynomial of degree 3 for the data:

f( - 0.75) = - 0.0718125, f( - 0.5) = - 0.02475, f( - 0.25) = - 0.3349375 and f(0) = 1.101. Hence find f (-

6. Solve the system of equations using Gauss Seidel iterative methods.

20x – y – 2z = 17, 3x + 20y – z = -18, 2x – 3y +20z = 25.

7.Find the largest eigen values and its corresponding vector of the matrix

by power method.

8. Using Gauss- Jordan obtain the inverse of the matrix

Using Gauss Seidel method solve the system of equations starting with the values x = 1 , y = -2 and z = 3,

x + 3y + 5z = 173.61, x – 27y + 2z = 71.31, 41x – 2y + 3z = 65.46

Solve the following equations by Jacobi’s iteration method

x + y + z = 9, 2x – 3y + 4z = 13, 3x + 4y + 5z = 40.

Unit II : Interpolation and Approxiamtion

Part A

Construct a linear interpolating polynomial given the points (x₀,y₀) and (x₁,y₁).
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

3. Obtain the divided difference table for the following data.

X : -1 0 2 3

Y : -8 3 1 12

4. Find the polynomial which takes the following values.

X : 0 1 2

Y : 1 2 1

5. Define forward, backward, central differences and divided differences.

6. Evaluate

(1-x) (1-2x) (1-3x)--------(1-10x), by taking h=1.

7. Show that the divided difference operator

is linear.

8. State the order of convergence of cubic spline.

9. What are the natural or free conditions in cubic spline.

10. Find the cubic spline for the following data

X : 0 2 4 6

Y : 1 9 21 41

11. State the properties of divided differences.

12. Show that

13. Find the divided differences of f(x) = x³ + x + 2 for the arguments 1,3,6,11.

14. State Newton’s forward and backward interpolating formula.

15. Using Lagranges find y at x = 2 for the following

X : 0 1 3 4 5

Y : 0 1 81 256 625

Part B

1. Using Lagranges interpolation formula find y(10) given that y(5) = 12, y(6) = 13,

y(9) = 14 and y(11) = 16.

2. Find the missing term in the following table

x : 0 1 2 3 4

y : 1 3 9 - 81

3. From the data given below find the number of students whose weight is between

60 to 70.

Wt (x) : 0-40 40-60 60-80 80-100 100-120

No of

students : 250 120 100 70 50

4. From the following table find y(1.5) and y’(1) using cubic spline.

X : 1 2 3

Y : -8 -1 18

5. Given sin 45⁰ = 0.7071, sin 50⁰= 0.7660, sin 55⁰ = 0.8192, sin 60⁰ = 0.8660, find

sin 52⁰ using Newton’s forward interpolating formula.

6. Given log ₁₀ 654 = 2.8156, log ₁₀ 658 = 2.8182, log ₁₀ 659 = 2.8189, log ₁₀ 661 =

2.8202, find using Lagrange’s formula the value of log ₁₀ 656.

7. Fit a Lagrangian interpolating polynomial y = f(x) and find f(5)

x : 1 3 4 6

y : -3 0 30 132

8. Find y(12) using Newton’ forward interpolation formula given

x : 10 20 30 40 50

y : 46 66 81 93 101

9. Obtain the root of f(x) = 0 by Lagrange’s inverse interpolation given that f(30) = -30,

f(34) = -13, f(38) = 3, f(42) = 18.

10. Fit a natural cubic spline for the following data

x : 0 1 2 3

y : 1 4 0 -2

11. Derive Newton’s divided difference formula.

12. The following data are taken from the steam table:

Temp⁰c : 140 150 160 170 180

Pressure : 3.685 4.854 6.502 8.076 10.225

Find the pressure at temperature t = 142⁰ and at t = 175⁰

13. Find the sixth term of the sequence 8,12,19,29,42.

14. From the following table of half yearly premium for policies maturing at different ages, estimate the premium for policies maturing at the age of 46.

Age x : 45 50 55 60 65

Premium y : 114.84 96.16 83.32 74.48 68.48

15. Form the divided difference table for the following data

x : -2 0 3 5 7 8

y : -792 108 -72 48 -144 -252

Unit III - Differentiation and Integration

Part A

1. What the errors in Trapezoidal and Simpson’s rule.

2. Write Simpson’s 3/8 rule assuming 3n intervals.

3. Evaluate

using Gaussian quadrature with two points.

4. In Numerical integration what should be the number of intervals to apply Trapezoidal, Simpson’s 1/3 and Simpson’s 3/8.

5. Evaluate

using Gaussian three point quadrature formula.

6. State two point Gaussian quadratue formula to evaluate

7. Using Newton backward difference write the formula for first and second order derivatives at the end value x = x₀upto fourth order.

8. Write down the expression for

and

at x = x₀ using Newtons forward difference formula.

9. State Simpson’s 1/3 and Simpson’s 3/8 formula.

10. Using trapezoidal rule evaluate

by dividing into six equal parts.

Part B

1. Using Newton’s backward difference formula construct an interpolating polynomial

of degree three and hence find f(-1/3) given f(-0.75) = - 0.07181250, f(-0.5) =

- 0.024750, f(-0.25) = 0.33493750, f(0) = 1.10100.

2. Evaluate

by Simpson’s 1/3 rule with

= 0.5 where 0<x,y<1.

3. Evaluate I =

by using Trapezoidal rule, rule taking h= 0.5 and h=0.25. Hence the value of the above integration by Romberg’s method.

4. From the following data find y’(6)

X : 0 2 3 4 7 9

Y: 4 26 58 112 466 922

5. Evaluate

numerically with h= 0.2 along x-direction and k = 0.25 along y direction.

6. Find the value of sec (31) from the following data

: 31 32 33 34

Tan

: 0.6008 0.6249 0.6494 0.6745

7. Find the value of x for which f(x) is maxima in the range of x given the following table, find also maximum value of f(x).

X: 9 10 11 12 13 14

Y : 1330 1340 1320 1250 1120 930

8. The following data gives the velocity of a particle for 20 seconds at an interval of five seconds. Find initial acceleration using the data given below

Time(secs) : 0 5 10 15 20

Velocity(m/sec): 0 3 14 69 228

9. Evaluate

using Gaussian quadrature with 3 points.

10. For a given data find

and

at x = 1.1

X : 1.0 1.1 1.2 1.3 1.4 1.5 1.6

Y: 7.989 8.403 8.781 9.129 9.451 9.750 10.031

UNIT – IV : INITIAL VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

PART – A

By Taylor series, find y(1.1) given = x + y, y(1) = 0.
Find the Taylor series upto x³term satisfying .
Using Taylor series method find y at x = 0.1 if .
State Adams – Bashforth predictor and corrector formula.
What is the condition to apply Adams – Bashforth method ?
Using modified Euler’s method, find if .
Write down the formula to solve 2^nd order differential equation using Runge-Kutta method of 4^th order.
In the derivation of fourth order Runge-Kutta formula, why is it called fourth order.
Compare R.K. method and Predictor methods for the solution of Initial value problems.
Using Euler’s method find the solution of the IVP at taking .

PART-B

The differential equation is satisfied by.Compute the value of y(0.8) by Milne’s predictor - corrector formula.
By means of Taylor’s series expension, find y at x = 0.1,and x = 0.2 correct to three decimals places, given , y(0) = 0.
Given find the value of y(0.1) by using R.K.method of fourth order.
Using Taylor;s series method find y at x = 0.1, if , y(0)=1.
Given , y(1) = 1, y(1.1) = 1.233, y(1.2) = 1.548, y(1.3)=1.979, evaluate y(1.4) by Adam’s- Bashforth method.
Using Runge-Kutta method of 4^th order, solve with y(0)=1 at x=0.2.
Using Milne’s method to find y(1.4) given that given that .
Given find by Milne’s predictor-corrector method taking h = 0.2.
Using R.K.Method of order 4, find y for x = 0.1, 0.2, 0.3 given that also find the solution at x = 0.4 using Milne’s method.
Solve , y(0) = 1.

Find y(0.1) and y(0.2) by R.K.Method of order 4.

Find y(0.3) by Euler’s method.

Find y(0.4) by Milne’s predictor-corrector method.

Solve subject to using fourth order Runge-Kutta Method.

Find

and

. Using step size

Using 4^th order RK Method compute y for x = 0.1 given given y(0) = 1 taking h=0.1.
Determine the value of y(0.4) using Milne’s method given , use Taylors series to get the value of y at x = 0.1, Euler’s method for y at x = 0.2 and RK 4^th order method for y at x=0.3.
Consider the IVP

(i) Using the modified Euler method, find y(0.2).

(ii) Using R.K.Method of order 4, find y(0.4) and y(0.6).

(iii) Using Adam- Bashforth predictor corrector method, find y(0.8).

Consider the second order IVP with y(0) = -0.4 and y’(0)=-0.6.

(i) Using Taylor series approximation, find y(0.1).

(ii) Using R.K.Method of order 4, find y(0.2).

UNIT-5

PART-A

Define the local truncation error.
Write down the standard five point formula used in solving laplace equation U+ U= 0 at the point ().
Derive Crank-Niclson scheme.
State Bender Schmidt’s explicit formula for solving heat flow equations