Page 221 - Engineering Mathematics Workbook_Final

Numerical Methods

[GATE 2018 (ME-Morning Session)] (a) x converges to 2 with rate of
n
65. Consider the iteration function for convergence 1.
f ( ) x
Newton’s method ( ) x = x − (b) x converges to 2 with rate of
g
n
' f ( ) x convergence 2.
and its application to find (c) The given iteration is the fixed
(approximate) square root of 2, point iteration for ( ) x = x − 2 2.
f
2
starting with x = . Consider the
0
first and the second iterates x and (d) The given iteration is the
1
2
f
x , respectively; then Newton’s method for ( ) x = x − 2.
2
68. The following numerical integration

(a) 1.5 x  1 2 formula is exact for all polynomials
of degree less than or equal to 3
(b) 1.5  x  1 2
(a) Trapezoidal rule

(c) x  1 1.5; x  2 1.5 1
(b) Simpson’s rd rule
(d) x = 1 1.5; x  2 1 3

3
66. Let f be a continuous map from the (c) Simpson’s th rule
interval [0,1] into itself and consider 8

the iteration x n+ 1 = f x n (d) Gauss-Legendre 2 point formula
( ). Which of
the following maps will yield a fixed 1  2 

point for f? 69. The iteration x n+ =   x +   ,
1
2   n x n  
f
0
(a) ( ) x = x 2 / 4 n  0 for a given x  is an
0
instance of
(b) ( ) x = x 2 /8
f
(a) fixed point iteration for
f
(c) ( ) x = x 2 /16 f ( ) x = x − 2 2
(b) Newton’s method for
(d) ( ) x = x 2 /32 f ( ) x = x − 2 2
f

67. Consider the iteration (c) fixed point iteration for

2
1   2   f ( ) x = x + 2
0
x =  x +  , n  for a 2x
n+
1
2   n x n  
0
given x  . Then (d) Newton’s method for
0
f ( ) x = x + 2 2

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