Numerical Method - Unit Wise 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.

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)

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)

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)

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

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)

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

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)

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)

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)

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

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

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

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.

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

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

     log x – cos x = 0

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

        2x³ - 2x = 5

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

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.

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

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

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)

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)

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

x⁴ + 2x³ + 3x² + 4x = 5

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.

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?

2. Define the linear least squares approximations. Give the data set (x_i, y_i) 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)

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.

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

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

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

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)

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

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

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.

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

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)

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

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

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.

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)

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

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

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

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

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

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

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

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

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

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

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.

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)

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)

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)

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

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)

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

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

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

                                                (2 + 6)

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

9. Evaluate the following integration using Romberg integration.

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

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

9. Evaluate the following integration using Romberg integration.

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

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

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 

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.

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)

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

        x + 2y + 3z = 4

        6x - 4y + 5z = 10

        5x + 2y + 2z = 25

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

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

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

    taking h = 0.25 

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

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 4^th order method.        (5)

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

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)

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 + y² for 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).

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)

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)

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

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

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

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

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

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

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).

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

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

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)

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)

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)

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.

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

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)

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)

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)

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.

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 = 2x²y² Over the domain 0 ≤ x ≤ 3, 0 ≤ y ≤ 3 with f = 0 on the boundary and h = 1. (3+5)

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)

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.

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.

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?

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.