Page 75 - Applied Numerical Methods Using MATLAB

64 MATLAB USAGE AND COMPUTATIONAL ERRORS
command window.

>>x = 2^30;x+2^-23== x,x-2^-23==x
which will give you the logical answer 1 (true) and 0 (false). Justify this
31
29
30
result based on the difference of resolution of two ranges [2 ,2 )and [2 ,
30
30
30
2 ) to which the true values of computational results (2 + 2 −23 )and (2 −
2 −23 ) belong, respectively. Note from Eq. (1.2.5) that the resolutions—that is,
the maximum quantization errors—are E = 2 E−52 = 2 −52+30 = 2 −22 and
2 −52+29 = 2 −23 , respectively. For details, refer to Fig. P1.18, which illustrates
the procedure of addition/subtraction with four mantissa bits, one hidden bit,
and one guard bit.
1.19 Resolution of Number Representation and Quantization Error
(a) What is the result of typing the following statements into the MATLAB
command window?
>>7/100*100 - 7

How do you compare the absolute value of this answer with the reso-
lution of the range to which 7 belongs?
(b) Find how many numbers are susceptible to this kind of quantization
error caused by division/multiplication by 100, among the numbers
from 1 to 31.
(c) What will be the result of running the following program? Why?

%nm1p19: Quantization Error
x = 2-2^-50;
for n = 1:2^3
x = x+2^-52; fprintf(’%20.18E\n’,x)
end

1.20 Avoiding Large Errors/Overflow/Underflow
(a) For x = 9.8 201 and y = 10.2 199 , evaluate the following two expressions
that are mathematically equivalent and tell which is better in terms of
the power of resisting the overflow.

2
(i) z = x + y 2 (P1.20.1a)

2
(ii) z = y (x/y) + 1 (P1.20.1b)
Also for x = 9.8 −201 and y = 10.2 −199 , evaluate the above two expres-
sions and tell which is better in terms of the power of resisting the
underflow.
8.5
7.4
(b) With a = c = 1 and for 100 values of b over the interval [10 ,10 ]
generated by the MATLAB command ‘logspace(7.4,8.5,100)’,