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1
CHE338_Lecture#2
For reading
Reference Chapter Page
Chapra 3 52-77
Approximation and Errors
2
Objectives
To introduce computer numbers (binary floating-point
numbers)
To introduce round-off errors
To introduce machine epsilon
3
Numerical errors
Numerical methods generate:
approximate solutions that are close to the exact
solutions (analytical solutions).
Exact numerical errors are difficult to be computed:
a) Given input data from measurements are not
exact
b) Numerical algorithm itself generate errors such
as round-off errors.
...14159265.3
7,,
ore
4
Accuracy and Precision
Errors related to both calculations and measurements
are described:
How close is a computed or measured value
to the true value
How close is a computed or measured value
to previously computed or measured values.
A systematic deviation from the actual value.
Magnitude of scatter.
5
Accuracy and precision
Let’s say
49Km/h is exact.
Read the meter 10
times.
6
Significant digits (figures)
Number of significant figures indicates precision. Significant digits of number are those that can be used with confidence, e.g., the number of certain digits plus one estimated digit.
To present numbers with how precise your measurements or predications are
48.9
49.0
49.8
.
.
.
7
Significant digits (figures)
Number of significant figures indicates precision. Significant digits of number are those that can be used with confidence, e.g., the number of certain digits plus one estimated digit.
5.38 3
5.380
5.3800
Zeros are sometimes used to locate the decimal point not significant figures.
0.00001753 4
0.0001753
0.001753
37.
37000
8
Significant digits (figures)
Number of significant figures indicates precision. Significant digits of number are those that can be used with confidence, e.g., the number of certain digits plus one estimated digit.
5.38 x 104 3
5.380 x 104
5.3800x 104
Zeros are sometimes used to locate the decimal point not significant figures.
0.00001753
0.0001753
0.001753
9
Definition of errors
Example 3.1 (Textbook): Measuring the lengths of a
bridge and a rivet
True length of the bridge: 10,000cm
Measured length: 9999cm
True length of the rivet: 10cm
Measured length: 9cm
10
Definition of Numerical errors
To account for the magnitude of the quantities
No true solution (analytical solution) is given. To numerically
find solutions close to true values, we use approximated
approaches to represent exact mathematical operations and
physical quantities
This leads to errors in solutions.
11
Definition of Numerical errors
No true solution (analytical solution) is given. To find
solutions to close to true solutions, we use approximated
approaches to represent exact mathematical operations and
physical quantities
%100ionApproximat
error eApproximat
a
Iterative approach to find a solution, example
Newton’s method
12
Tolerance
Stopping Criterion:
)%n)-(2
s 10 (0.5Say that the results is correct
to at least n significant figures.
Specified tolerance, user defined
If the following condition is met, then
13
Example 2.1 (Example 3.2 in the textbook)
!321
32
n
xxxxe
nx
Error=approximation +error
Adding more terms in sequence leads to a
better estimate of the function.
Exponential function can be represented by the
following infinite series.
14
Example 2.1 (Example 3.2 in the textbook)
!321
32
n
xxxxe
nx
Evaluated at x=0.5 and at least three significant figures,
0.05%)
How many terms should be added ?
)%n)-(2
s 10 (0.5
15
Errors: Round-off
Representation of numbers by computer:
...14159265.3
7,,
oreNumbers then can be expressed by a finite
number of significant digits
Numbers stored in a binary format (2-base)
Fractional quantities stored in “floating point” form
16
Binary numbers
10-base numbers are mostly used for mathematical operations.
17
Binary numbers
10-base numbers are mostly used for mathematical operations.
1563=(1x103)+(5x102)+(6x101)+(3x10o)
2-base
1563=(1x210)+(1x29) )+(0x28) )+(0x27) +(0x26) +(0x25)
+(1x24) +(1x23) +(0x22) +(1x21) +(1x20)
1563=11000011011two
16-bit computer for -173
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Floating-point (machine number) for real numbers
General floating-point form:
em.bexponent
Base of the number system
used
mantissa
for a base-10 system 0.1 ≤m<1 for a base-2 system 0.5 ≤m<1
19
Floating-point (machine number) for real numbers
Four decimal digit (4 bit) for mantissa with the exponent
of -3,-2,---,4.
A finite set of numbers can be represented
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Example 2.2 (Example 3.5 in the textbook)
Let’s say: if you are only allowed to have a system:
+/- +/- 21 20
2-1
2-2
2-3
10-BASE Gap (interval)
∆x
0 1 1 1 1 0 0 0.062500 Smallest 0.01562
5
(e=-3)
0 1 1 1 1 0 1 0.078125
0 1 1 1 1 1 0 0.093750
0 1 1 1 1 1 1 0.109375 0.015625
0 1 1 0 1 0 0 0.125000 0.03125
(e=-2) 0 1 1 0 1 0 1 0.156250
0 1 1 0 1 1 0 0.187500
0 1 1 0 1 1 1 0.218750
: : : : : : : :
0 0 1 1 1 1 1 7 Largest
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Floati
Floating point representation allows both fractions and very large numbers to be expressed on the computer.
Floating point numbers take up more room. Take longer to process than integer numbers. Round-off errors are introduced because mantissa holds only a finite number of significant figures.
Floating-point (machine number) for real numbers
22
Floati Floating-point (machine number) for real numbers
Round-off errors due to a finite number of quantities
that can be presented within the limited range. Thus,
Some numbers can not precisely matched (limited
precision) referred as quantizing errors.
Example: p=3.14159265358 to be stored on a base-10 system carrying 7 significant digits. p=3.141592 chopping error ԑt=0.00000065 p=3.141593 rounded ԑt=0.00000035
23
Machine epsilon
The interval (∆x) increases as x ( the numbers in
magnitude) increases. This means the quantizing errors
are proportional to the magnitude of the numbers.
For chopping For rounding
For chopping with 2-base and t =3, then
24
Normalized Binary floating number (m2e)
Quantizing errors and round-off errs
Machine epsilon
Summary
x
x
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