Anna University Coimbatore 2010 B.E Computer Science and Engineering Two ks with Answers for Numerical Methods(unit-1) - Question Paper

TWO MARKS QUESTIONS WITH ANSWERS

UNIT-1

 

1.     State the iterative formula for Regula falsi method to solve.

Solution: The iteration formula to find a root of the equation which lies between is .

 

2.     Give an example of a) algebraic b) transcendental equation.

Solution:

a)     Algebraic equation: The equation is called Algebraic if is a polynomial.

Example: (1)

(2)

b)    Transcendental equation: The equation is called Transcendental if contains logarithmic, exponential and trigonometric functions.

Example: (1)

(2)

 

3.     Mention the methods to solve the equation which is either, algebraic or transcendental.

Solution:

a)     Bisection method

a)     Regula Falsi method

b)    Iteration method

c)     Newton Raphson method.

 

4.     What are the two types of errors involving in the numerical computation?

Solution:

(1) Round off Error

(2) Truncation Error

 

5.     Define Round off Error.

Solution: While dealing with decimal numbers, it is very inconvenient to work with all

decimal places. So we take approximates to facilitate calculation work. These

approximations lead to error in the final result, known as Round off error.

 

6.     Define Truncation Error.

Solution: The error caused by using approximate formula in computations is known as

Truncation Error.

Example: . If we are writing (approximately).

We get Truncation Error.

 

 

 

7.     What is the criterion for the convergence in Newton Raphson method?

Solution: The sequence converges to the exact value if .

i.e., if .

 

8.     Write the iterative formula of Newton Raphson method.

Solution: The iterative formula of Newton Raphson method is .

 

9.     Show that the iterative formula for finding the reciprocal of N is .

Solution: Let i.e., .

Let

W.K.T.,

.

 

10.                        Derive Newtons algorithm for finding the root of a number N.

Solution: If then is the equation to be solved.

Let .

By Newton Raphson method, if iterate then

 

11.                        What is the condition for applying the fixed point iteration method(successive approximation method) to find the real root of the equation?

(OR) If is continuous in , then under what condition the iterative method has

a unique solution in ?

Solution: Let be a root of . Let I be an interval combining the point . If

for all x in I, the sequence of approximation will converge

to the root r , provided that the initial approximation is chosen in I.

 

12.                        What is the order of convergence for fixed point iteration?

Solution: The convergence is Linear and the convergence is of order one.

 

13.                        In what form is the coefficient matrix transformed into when AX=B is solved by Gauss-Elimination method.

Solution: Upper Triangular Matrix.

 

14.                        In what form is the coefficient matrix transformed into when AX=B is solved by Gauss-Jordan method.

Solution: Diagonal Matrix.

 

15.                        State the principle used in Gauss-Jordan method.

Solution: Coefficient matrix is transformed into Diagonal matrix.

 

16.                        When Gauss Elimination method fails?

Solution: This method fails if the element in the top of the first column is zero. We can

rectify this by interchanging the rows of the matrix.

 

17.                        Write a sufficient condition for Gauss-Seidel method to converge.

Solution: The process of iteration by Gauss-Seidel method will converge if in each

equation of the system, the absolute value of the largest coefficient is greater than

the sum of the absolute values of the remaining elements in that row.

[ i.e., The Coefficient of matrix should be Diagonally dominant].

 

18.                        Give two indirect methods to solve a system of linear equations.

Solution: (1) Gauss-Jacobi method

(2) Gauss-Seidal method.

 

19.                        Give two direct methods to solve a system of linear equations.

Solution: (1) Gauss-Elimination method

(2) Gauss-Jordan method.

 

20.                                    Why Gauss-Seidal method is a better method than Jacobis iterative method?

Solution: Since the current value of the unknowns at each stage of iteration are used in

proceeding to the next stage of iteration, the convergence in Gauss-Seidal method

will be more rapid than in Gauss-Jacobi method.

 

21.                                    Solve by Gauss-Seidal method correct to four decimal places.

Answer:

 

22.                                    What do you mean by Diagonally dominant?

Solution: A matrix is diagonally dominant if the absolute value of the leading diagonal

element in each row is greater than or equal to the sum of the absolute values of

the remaining elements in that row.

 

23.                        Find the inverse of the coefficient matrix by Gauss-Jordan elimination method

Answer:

 

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

Solution: The dominant eigen value can be obtained by power method.

 

25.                        Find the dominant eigen value of by power method.

Answer: The dominant eigen value and the corresponding eigen vector

 

26.                        Determine the largest eigen value and the corresponding eigen vector of the matrix correct to two decimal places using power method.

Answer: The largest eigen value = 2 and the corresponding eigen vector

 

27.                        How to reduce the number of iterations while finding the root of an equation by Regula falsi method?

Solution: The number of iterations to get a good approximation to the real root can be

reduced, if we start with a smaller interval for the root.

 

28.                        What are the merits of Newtons method of iteration?

Solution: Newtons method is successfully used to improve the result obtained by other

methods. It is applicable to the solution of equations involving algebraical

functions as well as transcendental functions.

 

29.                        Define order of convergence.

Solution: Let be the successive approximations of the root of .

Let be the error in the root , .

If is the exact root, .

If can be found out such that where k is a positive constant for

every i, then p is called the order of convergence.

 

Note: If P = 1, the convergence is linear.

If P = 2, the convergence is quadratic.

 

30.                        Is the iteration method, a self correcting method always?

Solution: In general, iteration is a self correcting method, since the round off error is

smaller.

 

 

 

31.                        Explain power method of finding the eigenvalues of a matrix.

Solution: The power method is an iterative technique. The method may not converge very

fast. We can accelerate the convergence as well as get eigenvalues of magnitude

intermediate between the largest and smallest by shifting. The power method with

its variations is fine for small matrices. However, if a matrix has two eigenvalues

of equal magnitude, the method fails in the successive normalization factors

alternate between two numbers. The duplicated eigenvalues in this case is the

square root of the product of the alternating normalization factors. If we want all

the eigenvalues for a larger matrix, there is a better way.

 

32.                                  Write the types of Pivoting.

Solution: There are two types of Pivoting

(1) Partial Pivoting,

(2) Complete Pivoting.

 

 

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