# MA 214 - Introduction to Numerical Analysis

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```					                MA 214 - Introduction to Numerical Analysis
Mid-Semester Examination (Spring 2009 - 2010)
Department of Mathematics, IIT Bombay
Max. Marks: 30             •          Time: 2:00 to 4:00 PM               •         Date: 18/02/2010

Instructions:

1. Write your Name and Roll Number clearly on your answer book as well as every supplement you may
use. A penalty of 2 marks will be imposed in case any one of these is not written.
2. Number the pages of your answer book and make a question-page index on the front page. A penalty
of 1 mark will be imposed in case the index is incomplete.
3. The answer to each question should start on a new page. If the answer for a question is split into two
parts and written in two diﬀerent places, the ﬁrst part alone will be corrected.
4. Students are not allowed to have mobile phones with them during exam hours. If any student is found
with mobile phone in the exam hall, they will not be allowed to take the examination.
5. Only scientiﬁc calculators are allowed. Any kind of programing device is not allowed.
6. Formulas used in the solution need not be proved but needs to be stated clearly. Partial marks
may be awarded for the statement.
7. The question paper contains 9 questions. Answer all the questions.

1. If the relative error of ﬂ(x) is �, then show that
|�| ≤ β −n+1

for chopping, where β is the radix and n is the number of digits in the machine ap-
proximated number.                                                       (3 marks)

2. Show that the process of evaluating the function
√         √
f (x) = x + 1 − x
is unstable for a suﬃciently large value of x. Give an equivalent expression for this
function which leads to a stable evaluation.                             (4 marks)

3. Write an algorithm for calculating the divided diﬀerence of order n for a function f (x).
(3 marks)

4. Let f (x) be a real-valued function deﬁned on I = (a, b) and n times diﬀerentiable in
(a, b). If x0 , x1 , · · · , xn are n + 1 distinct points in (a, b), then show that there exists
ξ ∈ (a, b) such that
f (n) (ξ)
f [x0 , · · · , xn ] =           .
n!
(4 marks)

5. Obtain the error formula for the ﬁrst derivative of a function f (x) using polynomial
interpolation with the assumption that this function is suﬃciently continuously diﬀer-
entiable.                                                                  (3 marks)
6. Show that the formula
f (x) − 2f (x − h) + f (x − 2h)
D (2) (x) =
h2
approximates the second derivative of the function f (x). Find the order of accuracy of
this formula.                                                               (3 marks)

7. Derive Simpson’s rule for integration and obtain the degree of precision. (4 marks)

8. Obtain the error formula for the composite trapezoidal rule. Further, ﬁnd the number
of subintervals and the step size h so that the error for the composite trapezoidal rule
�7
is less than 5 × 10−9 for approximating the integral 2 dx/x.                (3 marks)

9. Derive two point Gaussian quadrature formula for evaluating the integral
b
I(f ) =           f (x)dx.
a

Give the geometrical interpretation for the formula.                         (3 marks)

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