Download Representation of Numbers

Computational Physics 1
Limits of computers




Computers are powerful but finite.
There is limited amount of space.
This has an effect on how numbers can be
represented and any calculations we carryout.
We first look at how numbers are stored and
represented on computers.
Number Representation
Representation of Numbers

Computer represent all numbers, other than
integers and some fractions with imprecision.
Representation of Numbers


Computer represent all numbers, other than
integers and some fractions with imprecision.
Numbers are stored in some approximation
which can be represented by a fixed number of
bits or bytes.
Representation of Numbers



Computer represent all numbers, other than
integers and some fractions with imprecision.
Numbers are stored in some approximation
which can be represented by a fixed number of
bits or bytes.
There are different types of “representations”
or “data types”.
Representation of Numbers

Data types vary in number of bits used (word
length) and representation - fixed point (int or
long) or floating point(float or double) format.
Fixed point Integer Representation
Fixed point Integer Representation

A common method of integer representation is
sign and magnitude rep.
Fixed point Integer Representation


A common method of integer representation is
sign and magnitude rep.
One bit is used for the sign and the remaining
bits for the magnitude.
Fixed point Integer Representation



A common method of integer representation is
sign and magnitude rep.
One bit is used for the sign and the remaining
bits for the magnitude.
Clearly there is a restriction to the numbers
which can be represented.
Fixed point Integer Representation

With 7 bits reserved for the magnitude, the
largest and smallest numbers represented are
+127 and –127.
Sign bit (+ve number)
+127
-127
=
=
0 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1
Sign bit (-ve number)
Fixed point Integer Representation

1.
2.
3.
Things to note:
Fixed point numbers are represented as
exact.
Arithmetic between fixed point numbers is
also exact provided the answer is within
range.
Division is also exact if interpreted as
producing an integer and discarding any
remainder.
Floating point Representation
Floating point Representation




In floating point representation, numbers are
represented by a sign bit s, an integer
component e, a positive integer mantissa M.
Eg of floating pt.
e E
sM  B
e-exact exponent. E –bias : fixed int and
machine dependent.
B- base, usually 2 or 16.
Floating point Representation

Most computers use the IEEE representation
where the floating point number is normalized.
Floating point Representation


1.
2.
Most computers use the IEEE representation
where the floating point number is
normalized.
The two most common IEEE rep are:
IEEE Short Real (Single Precision): 32 bits –
1 for the sign, 8 for exponent and 23 mantissa
IEEE Long Real (Double Prec): 64 bits – 1
sign bit, 11 for exp and 52 for the mantissa
Floating point Representation

Example of a single precision number.
exponent
31 30
1
sign
23 22
11111111
0
11111111111111111111111
mantissa
Floating point Representation

For example 0.5 is represented as:
0

01111111
10000000000000000000000
Where the bias is 0111 11112 =12710
Floating point Representation


By convention by floating point number is
normalized so that 31278 is normalized as
3.1278 x 104.
As a result the left most bit in the mantissa is 1
and does not have to stored. The computer
only needs to remember that there is a
phantom bit.
Precision


A consequence of the storage scheme is that
there is a limit to precision.
Consider the following example, to see how
machine precision affects calculations.
Precision

Consider the addition of two 32-bit words
(single precision): 7 + 1.0 x 10-7.
Precision


Consider the addition of two 32-bit words
(single precision): 7 + 1.0 x 10-7.
These numbers are stored as:
7= 0
10000010
11100000000000000000000
10-7 = 0
01100010
11010110101111111001010
Precision

Because the mantissa of the numbers are
different, the exponent of the smaller number is
made larger while the mantissa is made
smaller until they are same(shifting bits to the
right while inserting zeros).
Precision


Because the mantissa of the numbers are
different, the exponent of the smaller number is
made larger while the mantissa is made
smaller until they are same(shifting bits to the
right while inserting zeros).
Until:
10-7 = 0

01100010
00000000000000000000000(0001101…)
NB: The bits in the brackets are lost.
Precision



As a result when the calculation is carried out
the computer will get: 7 + 1.0 x 10-7 = 7
Effectively the 10-7 term has been ignored.
The machine precision  mcan be defined as the
maximum positive value that can be added to 1
so that its value is not changed.
Errors and Uncertainties
Living with errors
The problem



Errors and uncertainty are an unavoidable part
of computation.
Some are human error while others are
computer errors (eg. Due to precision).
We therefore look at the types of errors and
ways of reducing them.
Errors and Uncertainties

1.
2.
3.
4.
5.
There five types of errors in computation:
Mistakes
Random error
Truncation error
Roundoff error
Propagated error
Errors and Uncertainties

Mistakes: are typographical errors entered with
program or maybe running the program using
the wrong data etc.
Errors and Uncertainties

Random errors: these are caused by random
fluctuations in electronics due to for example
power surges. The likelihood is rare but there
is no control over them.
Errors and Uncertainties

Truncation or approximation errors: these
occur from simplifications of mathematics so
that the problem may be solved. For example
replace of an infinite series by a finite series.

n
x
x
 Eg: e  
n  0 n!
Errors and Uncertainties

Truncation or approximation errors: these
occur from simplifications of mathematics so
that the problem may be solved. For example
replace of an infinite series by a finite series.

n
n
N
x
x
x
 Eg: e  

 e x   x, N 
n  0 n!
n 0 n!
Errors and Uncertainties

Where  x, N  is the total absolute error.
Errors and Uncertainties




Where  x, N  is the total absolute error.
The truncation error vanishes as N is taken to
infinity.
For N much larger than x, the error is small.
If x and N are close then the truncation error
will be large.
Errors and Uncertainties


Roundoff error: since most numbers are
represented with imprecision by computers
(and general restrictions) this leads to a
number being lost.
The error as result of the roundoff or truncation
of digits is known as the roundoff error.
1 2
 Eg: 2    0.6666666  0.6666667  0.0000001  0
 3 3
Errors and Uncertainties

Propagated error: this is defined as an error in
later steps of a program due to an earlier error.
Errors and Uncertainties


Propagated error: this is defined as an error in
later steps of a program due to an earlier error.
This error is added the local error(eg. to a
roundoff error).
Errors and Uncertainties



Propagated error: this is defined as an error in
later steps of a program due to an earlier error.
This error is added the local error(eg. to a
roundoff error).
Propagated error is critical as errors may be
magnified causing results to be invalid.
Errors and Uncertainties




Propagated error: this is defined as an error in
later steps of a program due to an earlier error.
This error is added the local error(eg. to a
roundoff error).
Propagated error is critical as errors may be
magnified causing results to be invalid.
The stability of the program determines how
errors are propagated.
Stability of algorithms


For stable methods early errors die out as the
program.
However unstable programs continuously
magnify any error.
Errors and Uncertainties

In a program all or some of the types of errors
occur and will interact to some degree.
Calculating the Error


A simple way of looking at the error is as the
difference between the true value and the
actual value.
Ie:
Error (e) = True value – Approximate value
Example:

Find the truncation error for at x= e x if the first 3
terms in the expansion are retained.

Sol: Error = True value – Approx value

x 2 x3  
x2 
 1 x   ...  1 x  

 

2
!
3
!
2
!

 


x3 x 4 x5
   ...
3! 4! 5!
Calculating the Error




Three other ways of defining the error are:
Absolute error
Relative error
Percentage error
Calculation the Error

Absolute error.
ea = |True value – Approximate value|
ea  X  X   Error
Calculating the Error

Absolute error:
ea = |True value – Approximate value|
ea  X  X   Error

Relative error is defined as:
Error
er 
TrueValue
X  X

X
Calculating the Error

Percentage error is defined as:
X X
e p 100er  100
X
Examples

Suppose 1.414 is used as an approx to 2. Find
the absolute, relative and percentage errors.
Examples


Suppose 1.414 is used as an approx to 2. Find
the absolute, relative and percentage errors.
2 1.41421356
Examples


Suppose 1.414 is used as an approx to 2. Find
the absolute, relative and percentage errors.
2 1.41421356
ea  True value – Approximat e value
ea  1.41421356 -1.414
 0.00021356
(absolute error)
Examples


Suppose 1.414 is used as an approx to 2. Find
the absolute, relative and percentage errors.
2 1.41421356
Error
er 
TrueValue
er  0.00021356  0.151103
2
(relative error)
Examples


Suppose 1.414 is used as an approx to 2. Find
the absolute, relative and percentage errors.
2 1.41421356
Error
er 
TrueValue
er  0.00021356  0.151103
(relative error)
2
ep  er 100 0.151101 %
( percentage error)
Well posed(conditioned) vs ill
posed problems
Conditioning of a problem and
stability

The accuracy of a solution depending on how a
problem is stated (as well as the computer’s
accuracy).
Conditioning of a problem and
stability


The accuracy of a solution depending on how a
problem is stated (as well as the computer’s
accuracy).
Not all solutions to a problem are well posed
and hence stable.
Conditioning of a problem and
stability



The accuracy of a solution depending on how a
problem is stated (as well as the computer’s
accuracy).
Not all solutions to a problem are well posed
and hence stable.
A problem is well posed if a solution (a) exits,
(b) is unique, and (c) has a solutions that
varies continuously as its parameter vary
continuously
Conditioning of a problem and
stability

If the problem is ill-posed it should be replaced
by alternative form or another that has a
solution which is close enough.
Conditioning of a problem and
stability


If the problem is ill-posed it should be replaced
by alternative form or another that has a
solution which is close enough.
For example simplifying complicated functions
having values which are almost the same.