1
LECTURE 1.2
Dr. Vyas
Overview: (Covers pages 28 – 36)
•
Loss of significance
•
Order of approximation and BigO algebra
•
Propagation of error
•
Examples
I. Loss of Significance
1.
The loss of significance implies the loss of significant digits as a result of
arithmetic subtraction.
2.
For example, consider a quantity
x
with nine significant digits,
1.23456789
x
=
;
likewise consider another quantity
y
with nine significant digits,
1.23456742
y
=
. The subtraction of
y
from
x
yields another quantity, say
z
,
where
0.00000047
z
=
with only two significant digits. Thus, seven significant
digits were lost as a result of
subtractive cancellation
(aka loss of significance).
3.
Subtractive cancellation combined with rounding process can result in loss of
accuracy of the computed result. This is demonstrated by way of example.
Example 1:
Evaluate,
2
( )
1
f x
x
x
=
+ 
for
123.456
x
=
using six significant digits and
rounding. Show that
2
1
( )
1
g x
x
x
=
+ +
is an equivalent expression and avoids the error
resulting from the subtractive cancellation.
Solution:
Note that six significant digits and rounding means that we must round off at
the sixth significant digit at each arithmetic operation. The process is illustrated via
Matlab.
>> format long
>> x = 123.456;
(semicolon suppresses output to the command window)
>> x^2
ans =
1.52413
8393600000e+004
>> % now we round off at the sixth significant digit. Therefore,
>> % a new variable is introduced to avoid confusion of variable assignments.
>> p = 15241.4;
>> s = p + 1
(
2
1
x
+
)
s =
1.524240000000000e+004
>> % Note that the number of significant digits have not changed
>> sqrt(s)
2
1
x
+
ans =
1.234601150169560e+002
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>> % Round off again at sixth significant digit and introduce another variable.
>> t = 123.46
2
(
1)
t
rounded
x
=
+
t =
1.234600000000000e+002
>> f = t – x
2
(
1)
f
rounded
x
x
=
+

f =
0.003999999999991
:
(will defer rounding unless necessary)
2
2
2
2
2
2
2
2
1
1
1
1
( )
1
1
1
1
1
x
x
x
x
x
x
f x
x
x
x
x
x
x
x
x
+ 
+ +
+ 
=
+ 
=
×
=
=
+ +
+ +
+ +
>> % Now we evaluate g the same way. We can directly use "t" in formula for g
>> g = 1/(t + x)
g =
0.004049960310389 (in former versions, it appeared as 0.00404996031039)
>> % Now let us see the "true" value obtained by carrying all the decimal places
>> F = sqrt(x^2 + 1)  x
F =
0.00404995
949094
The fundamental idea to avoid subtractive cancellation is to use an equivalent
alternative formulation that avoids the subtraction arithmetic using methods of algebra,
trigonometry or calculus. The application of these methods is left for students to learn via
homework
(
example and exercise problems
)
.
Please refer to the Appendix to learn of another application of the loss of
significance formulation.
:
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 Spring '08
 Vyas
 Math, Algebra, Numerical Analysis, Approximation, Taylor Series, Observational error, Significance arithmetic

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