# Numerical Method - Unit Wise Questions

Unit 1: Solution of Nonlinear Equations
25 Questions

1. How can Horner's rule be used to evaluate the f(x) and f(x) of a polynomial at given point? Explain. Write an algorithm and program to calculate a real root of a polynomial using Horner's rule.

10 marks | Asked in 2078

1. Define the fixed-point iteration method. Given the function f(x) = x2 − 2x − 3 = 0, rearrange the function in such a way that the iteration method converses to its roots. (2+3+3)

8 marks | Asked in 2066

1. Define the types of errors in numerical calculations. Derive the formula for secant method and illustrate the method by figure. (4+4)

8 marks | Asked in 2068

1. Derive the formula to solve nonlinear equation using secant method. Using your formula estimate a real root of following nonlinear equation using secant method correct up to two decimal places x2+ ln x = 3. (3+5)

8 marks | Asked in 2069

1. What is bracketing and non-bracketing method? Explain with the help of example. Estimate a real root of following nonlinear equation using bisection method correct up to two significant figures(3 + 5)

x2sinx + e-x = 3

8 marks | Asked in 2070

1. How is the bisection method convergent to a root of an equation? Apply the bisection method to find a root of the equation (3 + 5)

8 marks | Asked in 2071

1. What are the sources of errors? Discuss various types of errors. Find the roots of the equation x2 + 5.6x − 10= 0 by trial and error method up to 4 significant digits. (1+3+4)

8 marks | Asked in 2072

1. Discuss methods of Half Interval and Newton’s for solving the nonlinear equation f(x) = 0. Illustrate the methods by figure and compare them stating their advantages and disadvantages. (8)

8 marks | Asked in 2067

1. Explain the idea of the secant method to estimate the root of any equation. Using the secant method, estimate the root of the equation x2 - 4x - 10 = 0 with the initial estimates of x1 = 4 and x2 = 2. Do these points bracket a root?(3 + 4 + 1)

8 marks | Asked in 2073

1. How can you use bisection method for the solution of nonlinear equations? Discuss with suitable example.(8)

8 marks | Asked in 2074

1. What is error? Discuss various types of errors. Estimate a real root of the following nonlinear equation using bisection method or Newton's method correct up to two decimal places sin x - x2 - x + 3 = 0 (3 + 5)

8 marks | Asked in 2075( Old Course)

1. What is non-linear equation? Derive the required expression to calculate the root of non-linear equation using secant method. Using this expression find a root of following equation.

X2 + cos(x) - e -x -  2 = 0

10 marks | Asked in 2075(New Course)

2. Describe Newton’s method and its convergence. Find the root of equation f(x) = ex− 4x2 = 0      using Newton method up to 5 decimal places. (4+4)

8 marks | Asked in 2072

1. Derive the formula for Newton Raphson Method. Solve the equation x2 + 4x - 9 = 0 using Newton Raphson method. Assume error precision is 0.01. Discuss drawbacks of the Newton Raphson method.

10 marks | Asked in 2077

4. How the half-interval method can estimate a root of non-linear equation? Find a real root of following equation using half-interval method correct up to two decimal places.

x2 - e-x - x = 1

5 marks | Asked in 2078

4. Write a program to solve a non linear equation using bisection method.

5 marks | Asked in Model Question |

5. Find the roots of the following equations using Newton’s method.

log x – cos x = 0

5 marks | Asked in Model Question |

5. Calculate a real root of the given equation using fixed point iteration correct up to 3 significant figures.

2x3 - 2x = 5

5 marks | Asked in 2078

7. Write an algorithm and C program for the secant method to find the roots of non-linear equation.(4 + 8)

OR

Write an algorithm and a C program for the Sinpson's 1/3 rule to integrate a given function.(4 + 8)

12 marks | Asked in 2073

7. Write an algorithm and program to compute all roots of polynomial using secant method.

12 marks | Asked in 2074

4. Define the terms true error and relative error? Use Horner' method to evaluate polynomial 2x3 - 3x2 + 5x - 2 at x = 3 and write down its algorithm.

5 marks | Asked in 2077

7. Write an algorithm to solve non-linear equation using secant method. On the basis of your algorithm write a C-program that reads two initial guess from keyboard and displays the following information if the solution is obtained:(5 + 7)

a) Calculated root of the equation

b) Required number of iterations

c) Functional value at calculated root

If solution is not obtained within 200 iterations your program should be terminated by displaying the following message:

NO SOLUTION WITHIN 200 ITERATIONS

TRY AGAIN WITH NEW INITIAL VALUES

12 marks | Asked in 2075( Old Course)

4. Calculate a real negative root of following equation using Newton's method for polynomial.

x4 + 2x3 + 3x2 + 4x = 5

5 marks | Asked in 2075(New Course)

7. Write an algorithm and C-program code to solve non-linear equation using Newton’s method. Your program should read an initial guess from keyboard and display the followings if the solution is obtained: (5+7)

•  Estimated root of the equation
•  Functional value at calculated root
•  Required number of iterations
12 marks | Asked in 2069

7. Write an algorithm and a C-program for the fixed point iteration method to find the roots of non-linear equation. (4+8)

OR

Write an algorithm and a C-program for the Lagrange’s interpolation to approximate the functional value at any given x from given n data. (4+8)

12 marks | Asked in 2071

Unit 2: Interpolation and Regression
25 Questions

2. Write an algorithm and a program to compute the interpolation value at a specific point, given a set of data points, using Lagrange interpolation method.

10 marks | Asked in Model Question |

2. What do you mean by interpolation problem? Define divided difference table and construct the table from the following data set. (2+2+4)

OR

Find the least squares line that fits the following data.

What do you mean by linear least square approximation?

8 marks | Asked in 2066

2. Define the linear least squares approximations. Give the data set (xi, yi) as (20.5, 765), (32.7, 826), (51.0, 873), (73.2, 942), (95.7, 1032) find the linear least square to fit given data. (2+6)

8 marks | Asked in 2068

2. Estimate f(3) from the following data using Cubic Spline interpolation.

OR

Find the best fitting quadratic polynomial from following data using least square approximation.

8 marks | Asked in 2069

2. Define interpolation. Find the Lagrange interpolation polynomial to fit the following data. Estimate the value (1 + 6 + 1)

8 marks | Asked in 2071

2. Derive the equation for Lagrange’s interpolating polynomial and find the value of f(x) at x = 1 for the following: (4+4)

8 marks | Asked in 2067

2.  Define interpolation. Find the functional value at x=3.6 from the following data using forward difference table:

8 marks | Asked in 2070

2. Given the data

Calculate f(1.35) using Newton's interpolating of order 1 through 3. Choose base points to attain good accuracy. Comment on the accuracy of results on the order of polynomial.(5 + 3)

8 marks | Asked in 2073

2. Define interpolation. Find the Lagrange interpolation polynomial to fit the following data:

Use the polynomial to estimate the value of e15.            (2 + 6)

8 marks | Asked in 2074

2. How polynomial interpolation differs with Cubic Spline interpolation? Explain. Find the best fit curve of quadratic polynomial using least square approximation from following data.

8 marks | Asked in 2075( Old Course)

2. How interpolation differs from regression? Write down algorithm and program for Lagrange interpolation.

10 marks | Asked in 2077

3. What do you mean by interpolation and approximation? Use Lagrange interpolation to estimate the value of f(0.6) from the following table of values. (2+6)

8 marks | Asked in 2072

4. Using Newton’s divided difference interpolating polynomial estimate the value of f(x) at x = 2.25 for the function defined as

8 marks | Asked in 2072

5. Write an algorithm and computer program to fit a curve y = ax2 + bx + c for given sets of (xi, yi, g. 0 = 1, ..., x) values by least square method. (4+8)
12 marks | Asked in 2066

6. Estimate the value of ln(3.5) using Newton’s backward difference formula, given the following data

5 marks | Asked in Model Question |

6. What is Newton's interpolation? Obtain the divided difference table from the following data set and estimate the f(x) at x=2 and x=5.

5 marks | Asked in 2078

7. Write an algorithm and C-program to approximate the functional value at any given x from given n no. of data using Lagrange’s interpolation. (5+7)

12 marks | Asked in 2070

7. Fit a straight line to the following set of data.

5 marks | Asked in Model Question |

5. Construct Newton's forward difference table for the given data points and approximate the value of f(x) at x = 15.

5 marks | Asked in 2077

5. What is least squares approximation of fitting a function? How does it differ with polynomial interpolation? Explain with suitable example.

5 marks | Asked in 2075(New Course)

7. What is linear regression? Fit the linear function to the following data.

5 marks | Asked in 2078

8. What are the problems with polynomial interpolation for large number of data set? How such problems are addressed? Explain with example.

5 marks | Asked in 2078

6. Fit the curve y = ae bx through the following data points.

5 marks | Asked in 2077

6. Find the lowest degree polynomial, which passes through the following points:

Using this polynomial estimate f(x) at x = 0

5 marks | Asked in 2075(New Course)

7. Fit function of type y = a + bx for the following points using least square method.

5 marks | Asked in 2075(New Course)

Unit 3: Numerical Differentiation and Integration
16 Questions

3. Derive the composite formula for the trapezoidal rule with its geometrical figure. Evaluate dx using this rule with n=5, upto 6 decimal places.(4 + 4)

8 marks | Asked in 2066

3. Write Newton-cotes integration formulas in basic form for x = 1, 2, 3 and give their composite rules. Evaluate using the Gaussian integration three point formula. (4+4)

8 marks | Asked in 2067

3. Evaluate  using trapezoidal rule with n=10. Also evaluate the same integral using Grossion 3 point formula and compare the result. (4+4)

8 marks | Asked in 2068

3. a) For the function   estimate f’(6.3) and f”(6.3) [take h = 0.01]         (4)

b) Evaluate  using Gaussian integration 3 point formula.         (4)

8 marks | Asked in 2069

3. Derive Simpson’s 1/3 rule to evaluate numerical integration. Using this formula evaluate (4 + 4)

8 marks | Asked in 2070

3. Derive Simpson’s 1/3 rule to evaluate numerical integration. Using this formula evaluate (4 + 4)

8 marks | Asked in 2071

3. How do you find the derivative if the function values are given in a tabulated form? The distance traveled by a vehicle at the intervals of 2 minutes are given as follows. Evaluate the velocity and acceleration of the

Vehicle at time T = 5, 10, 13.

8 marks | Asked in 2073

3. Derive trapezoidal rule to evaluate numerical integration. Use this formula to evaluate the integral

(2 + 6)

8 marks | Asked in 2074

3. (a) For the function estimate f'(2.1) and f''(2.7) [take h = 0.05] (3)

(b) Evaluate using trapezoidal rule taking h = 0.1 and h = 0.2. Also improve your result using Romberg integration.(4)

8 marks | Asked in 2075( Old Course)

3. Derive Composite Simpson’s 1/3 Rule for numerical integration. How does it improve the accuracy of integration?

10 marks | Asked in Model Question |

8. Estimate the first derivative of f(x)=ln x at x=1 using the second order central difference formula.

5 marks | Asked in Model Question |

9. Evaluate the following integration using Romberg integration.

5 marks | Asked in 2078

8. Calculate the integral value of the function given below from x = 1.8 to x = 3.4 using Simpson's 1/3 rule.

5 marks | Asked in 2075(New Course)

8. Write down algorithm and program for the differentiating continuous function using three point formula.

5 marks | Asked in 2077

9. Evaluate the following integration using Romberg integration.

5 marks | Asked in 2075(New Course)

9. How Simpson's 1/3 rule differs from trapezoidal rule? Derive the formula for Simpson's 1/3 rule.

5 marks | Asked in 2077

Unit 4: Solving System of Linear Equations
20 Questions

1. Compare Gauss Elimination method and Gauss Jordan method of solving simultaneous equation. Use Gauss Elimination to solve the following system of equation and also write its algorithm.

2x+3y+4z=5

3x+4y+5z=6

4x+5y+6z=7

10 marks | Asked in Model Question |

2. What is matrix factorization? How can it be used to solve system of linear equations? Factorize the given matrix A and solve the system of equations Ax=b for given b using L and U matrices.

and

10 marks | Asked in 2078

2. What is matrix factorization? Factorize the given matrix A into LU using Dolittle algorithm and also solve Ax = b for given b using L and U matrices.

10 marks | Asked in 2075(New Course)

4. Solve the following algebraic system of linear equations by Gauss-Jordan algorithm. (8)

8 marks | Asked in 2067

4. What is pivoting? Why is it necessary? Explain. Solve the following set of equations using Gauss elimination or Gauss Seidel method.(3 + 5)

x1 + 10x2 +x3 = 24

10x1 + x2 + x3 = 15

x1 + x2 + 10x3 = 33

8 marks | Asked in 2070

4. What do you mean by ill-conditioned systems? Solve the following system using Dolittle LU decomposition method.(2 + 6)

3x1 + 2x2 + x3 = 24

2x1 + 3x2 + 2x3 = 14

x1 + 2x2 + 3x3 = 14

8 marks | Asked in 2071

4. Solve the following system of linear equations using Gauss-elimination method (use partial pivoting if necessary);(8)

2x2 + x4 = 0
2x1 + 2x2 + 3x3 + 2x4 = -2
4x1 – 3x2 + x4 = -7
6x1 + x2 – 6x3 – 5x4 = 6

OR

What do you mean by eigen -value eigen- vector problems? Find the largest eigen value correct to two significant digits and corresponding eigen vectors of the following matrix using power method.(2 + 6)

8 marks | Asked in 2068

4. What do you mean by ill-conditioned systems? Solve the following system using Dolittle LU decomposition method.

3x1 + 2x2 + x3 = 10

2x2 + 3x2 + 2x3 = 14

x1 + 2x2 + 3x3 = 14

8 marks | Asked in 2073

4. Discuss Gauss-Jordan method for solving a system of linear equations. Solve the system

2x1 + 4x2 - 6x3 = -8

x1 + 3x2 +x3 = 10

2x1 - 4x2 - 2x3 = -12

Using Gauss-Jordan method.

8 marks | Asked in 2074

4. Decompose the given matrix [A] into LU form. Then using decomposed matrices L and U solve the system of linear equation  Ax = b for given [b].(4 + 4)

OR

How can you calculate the inverse of a matrix? Explain. Solve the system of linear equations Ax = b for given [A] and [b] using Gauss elimination method. (use partial pivoting) (3 + 5)

8 marks | Asked in 2075( Old Course)

4. Solve the following system of algebraic linear equations using Jacobi or Gauss-Seidel iterative (8)

6x1 – 2x2 + x3 = 11
-2x1 + 7x2 + 2x3 = 5
X1 + 2x2 – 5x3 = -1
8 marks | Asked in 2066

3. 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 marks | Asked in 2077

5. Write an algorithm and program to solve system of linear equations using Gauss- Jordan method. (4+8)

12 marks | Asked in 2068

4. Solve the following set of equation using Gauss elimination or Gauss Jordan method.         (8)

3x1 + 5x2 - 3x3 + x4 = 16

2x1 + x2 + x3 + 4x4 = 9

3x1 – 4x2 – x4 = 1

2x1 + x2 – 3x3 + 9x4 = 5

8 marks | Asked in 2069

5. Write algorithm for Gauss- Seidel method for solving the system of linear equations. Also solve the following system of linear equations using that method. (4+4)

10x1 + x2 + x3 = 12

x1 + 10x2 - x3 = 10

x1 - 2x2 + 10x = 9

8 marks | Asked in 2072

5. Write an algorithm and program to solve system of linear equations using Gauss-Seidel iterative method. (4+8)

12 marks | Asked in 2067

9. Compare and contrast between Jacobi iterative method and Gauss Seidal method.

5 marks | Asked in Model Question |

7. Discuss the Doolittle LU decomposition method for matrix factorization.

5 marks | Asked in 2077

10. Solve the following set of linear equations using Gauss Jordan method.

x2 + 2x3 + 3x4 = 9

7x1 + 6x2 + 5x3 + 4x4 = 33

8x1 + 9x2 + x4 = 27

2x1 + 5x2 + 4x3 + 3x4 = 23

5 marks | Asked in 2078

10. Solve the following set of equations using Gauss Seidel method.

x + 2y + 3z = 4

6x - 4y + 5z = 10

5x + 2y + 2z = 25

5 marks | Asked in 2075(New Course)

Unit 5: Solution of Ordinary Differential Equations
18 Questions

3. What is higher order differential equation? How can you solve the higher order differential equation? Explain. Solve the following differential equation for , taking h=0.25.

with y(1) = 1 and y'(1) = 2

10 marks | Asked in 2078

3. What is initial value problem and boundary value problem? Write an algorithm and program to solve the boundary value problem using shooting method.

10 marks | Asked in 2075(New Course)

5. (a) How can you solve higher order differential equation? Explain.        (3)

(b) Solve the following differential equation within 1 ≤ x ≤ 2 using Runge-Kutta 4th order method.        (5)

8 marks | Asked in 2075( Old Course)

5. How can you solve higher order differential equation? Explain. Solve the following differential within 0 ≤ x ≤ 1 using Heun’s method.(3 + 5)

8 marks | Asked in 2069

5. Compare Euler’s method with Heun’s method for solving differential equation. Obtain y(1.5) from given differential equation using Runge-Kutta 4th order method.(4 + 4)

OR

Solve the following boundary value problem using shooting method.(8)

8 marks | Asked in 2070

5. Solve the following boundary value problem using shooting method.(8)

8 marks | Asked in 2071

5. Obtain y(1.5) to the following differential equation using Runge-Kutta 4th order method.        (8)

taking h = 0.25

8 marks | Asked in 2073

5. How can you use Taylor series method to find the solution of ordinary differential equation? Use the Taylor method to solve the equation (5 + 3) y' = x2 + yfor x = 0.25 and x = 0.5 given y(0) = 1

OR

Define boundary value problem. Use shooting method to solve the equation in the interval(1, 2).

8 marks | Asked in 2074

6. Explain the Picard’s proves of successive approximation. Obtain a solution upto the fifth approximation of the equation such that y = 1 when x = 0 using Picard’s process of successive approximations . (2+6)

8 marks | Asked in 2067

6. Apply Runge-Kutta method of second order and fourth order to find an approximate value of y when x = 0.2 given that (8)

8 marks | Asked in 2068

7. Write an algorithm and program for computer to obtain the solution of differential equation using Runge-Kutta Method. {5+7)

12 marks | Asked in 2072

7. Define ordinary differential equation of the first order. What do you mean by initial value problem? Find by Taylor’s series method, the values of y at x = 0.1 and x = 0.2 to find places of decimal form

(2 + 6)

8 marks | Asked in 2066

10. Explain about boundary value problem with example? Differentiate it with initial value problem

5 marks | Asked in Model Question |

11. Use the Heun's method to estimate y(0.4) when

y’(x)=x2 +y2 with y(0)=0. Assume h=0.2

5 marks | Asked in Model Question |

11. Solve the following differential equation for , taking h = 0.25 using Heun's method.

y'(x) + x2y = 3x, with y(1) = 1

5 marks | Asked in 2078

10. Appropriate the solution of y' = 2x + y , y(0) = 1 using Eulers method with step size 0.1. Approximate the value of y(0.4).

5 marks | Asked in 2077

11. From the following differential equation estimate y(1) using RK 4th order method.

5 marks | Asked in 2075(New Course)

12. How boundary value problems differs from initial value problems? Discuss shooting method for solving boundary value problem.

5 marks | Asked in 2077

Unit 6: Solution of Partial Differential Equations
15 Questions

6. Derive a difference equation to represent a Poison’s equation. Solve the Poison’s equation ∇2 f = 2x2y2 over the domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1.(3 + 5)

8 marks | Asked in 2066

6. Write the finite difference formula for solving Poisson’s equation. Hence solve the Poisson’s equation∇2f = 2x2y2 over the domain 0 ≤ x ≤ 3 and 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1.        (1 + 7)

8 marks | Asked in 2071

6. What do you understand by the partial differential equation? Illustrate it with practical example and derive difference equation. (8)

OR

Find the solution of following differential equations using Taylor series method.

8 marks | Asked in 2072

6. What do you understand by the partial differential equation? Illustrate it with practical example and derive difference equation. (8)

OR

Find the solution of following differential equations using Taylor series method.

8 marks | Asked in 2072

6. Write the finite difference formula for solving Poisson's equation. Hence solve the Poisson equationover the domain 0 ≤ x ≤ 1.5 and 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 0.5        (1 + 7)

8 marks | Asked in 2073

6. Define partial differential equation. Discuss Laplace's equation along with its derivation.(2 + 6)

8 marks | Asked in 2074

6. (a) Derive a difference equation to solve Laplace equation.(3)

(b) The steady-state two-dimensional heat-flow in a metal plate is defined by . Given the boundary conditions as shown in figure below. find the temperatures at interior points A, B, C, and D.(5)

8 marks | Asked in 2075( Old Course)

6. a) How can you obtain numerical solution of a partial differential equation? Explain.(3)

b) The steady-state two-dimensional heat-flow in a metal plate is defined by Given the boundary conditions as shown in figure below, find the temperature at interior points T1, T2, T3 and T4. (5)

8 marks | Asked in 2069

6. Solve the equation over the square domain 0 ≤ x ≤ 1.5 and 0 ≤ y ≤ 1.5 with f=0 on the boundary [Take h = 0.5]. (8)

8 marks | Asked in 2070

7. Define a difference equation to represent a Laplace’s equation. Solve the following Laplace equation

For the rectangular plate given as:(3 + 5)

OR

Derive a difference equation to represent a Poison’s equation. Solve the Poison’s equation ∇2 f = 2x2yOver the domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1. (3+5)

8 marks | Asked in 2067

7. How can you solve Laplace’s equation? Explain. The steady-state two dimensional heat flow in a metal plate is defined by

OR

A steel plate of size 30 x 30cm is given. Two adjacent sides are placed at 100°C and other side held at 0°C . Find the temperature at interior points, assuming the grid size of 10 x 10cm. (3+5)

8 marks | Asked in 2068

12. Solve the Poisson’s equation ∇ 2 f=2x2 y 2 over the square domain 0 <= x <= 3 and 0 <= y <= 3 with f=0 on the boundary and h=1.

5 marks | Asked in Model Question |

12. Consider a metallic plate of size 90 cm by 90 cm. The two adjacent sides of the plate are maintained at temperature of 1000C and remaining two adjacent sides are held at 2000C. Calculate the steady state temperature at interior points assuming a grid size of 30 cm by 30 cm.

5 marks | Asked in 2078

11. A plate of dimension 18cm x 18cm is subjected to temperatures as follows: left side at 1000c, right side at 2000C. Upper part at 500 C, and lower at 1500C. If square grid length of 6cm x 6cm is assumed, what will be the temperature at the interior nodes?

5 marks | Asked in 2077

12. Solve the Poison's equation over the square domain 0 ≤ x ≤ 1.5, 0 ≤ y ≤ 1.5 with f = 0 on the boundary and h = 0.5.

5 marks | Asked in 2075(New Course)