Short Questions and Answers

SHORT QUESTIONS AND ANSWERS.

1. What is the order of convergence of Newton-Raphson method if the multiplicity of the root is one.[M.U. Oct.95, A.V. A/M 2005] [A.U N/D 2017 R-13]  [A.U A/M 2018 R-13]

Solution: Order of convergence of N.R. method is 2.

2. Newton Raphson method is also known as the method of …….

Solution: Tangent.

3. Derive Newton's algorithm for finding the p^th root of a number N.

Solution :

4. When should we not use Newton-Raphson method? (OR)

When does the Newton -Raphson method fail? [M.U. April, 1997]

Solution: The method should not be used in cases where the graph of the function when it crosses the X axis is nearly horizontal.

5. What is the rate of convergence in N-R method ?  [M.U. April, 1998] [A.U N/D 2009]

Solution: The rate of convergence in N-R method is of order 2.

6. What is the criterion for the convergence in Newton-Raphson method? [A.U. April, 2000, A/M 2011, M/J 2012]

[A.U N/D 2015 R13] [A.U M/J 2016 R8-10, A/M 2017 R-13] [A.U A/M 2018 R-13 SNM]

Solution: (i) f'(xo) should not be equal to zero. If f' (x₀) = 0 then initial approximation must be changed.

 (ii) For better convergence the product f (x₀)f" (x₀) should be positive.

7. Write the iterative formula of Newton-Raphson method.

[M.U. Oct. 2000] [A.U N/D 2009, CBT N/D 2010, A/M 2017 R-13]   [A.U A/M 2018 R-13 SNM]

Solution:

8. Show that the N.R. formula to find va can be expressed in the form  x_n + 1 = 1/2  [x_n + a / x_n], n = 0, 1, 2, 3, . [A.U. N/D 2003, N/D 2006] [A.U. CBT N/D 2011] [A.U N/D 2016 R13] [A.U N/D 2016 R8-10]

[A.U N/D 2019 R-17]

Solution :

9. What are the merits of Newton's method of iteration ?

Solution :

1. Newton's method is successfully used to improve the result obtained by other methods.

2. It is applicable to the solution of equations involving algebraical functions as well as transcendental functions.

10. Say true or false.

Newton's method is useful when the graph of the function when it crosses the x axis is nearly vertical.

Solution: True.

11. Say true or false.

Newton's method is useful in cases where the graph of the function when it crosses the x axis is nearly horizontal.

Solution: False.

12. State the order of convergence and convergence condition for Newton-Raphson method. [A.U. Nov./Dec. 2004] [AU M/J 2012] [A.U CBT M/J 2010] [A.U M/J 2016 R13 SNM]

Solution: The order of convergence is 2.

Condition for convergence is | f (x) f''(x) | < |ƒ' (x) |²

13. Newton-Raphson method is convergent.

(a) Linearly,

(b) Quadratically,

(c) Cubically,

(d) Biquadratically.

Solution:

 [Ans. (b) Quadratically]

14. What is the computational cost of the N-R method?

Solution: The computational cost of the method is one evaluation of f(x) and one evaluation of the derivative f'(x) per iteration.

15. State the error term for N-R method.

Solution:

16. State the criterian for the convergence in N-R method?

Solution: Newton's formula converges if

\f(x)ƒ'' (x) | < |f'(x) | ² in the interval considered.