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cdeagle
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Joined: 22/06/2014
Location: United States
Posts: 261

Posted: 08:26am 21 Jun 2015

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Here's a short program that determines the machine epsilon of the Micromite running version 4.6B. Machine epsilon can be defined several ways;

(1) that number, when added to one, is still one (according to the computer).

(2) the distance from 1.0 to the next largest floating point number.

Machine epsilon is important in numerical analysis because it can be used to "control" the number of iterations (convergence) of an algorithm. The square root of epsilon is often used for the convergence criterion of an iterative number method. It's also an indication or guideline when looking for a "reasonable" answer without spending lots of CPU cycles to gain very little in the final result.

' determine Micromite machine epsilon

x = 1.0

DO WHILE ((1.0 + x) > 1.0)

x = x/2.0

PRINT "x = ", x

LOOP

PRINT " "
PRINT "machine epsilon = ", x

END

Here are the results.

RUN
x = 0.5
x = 0.25
x = 0.125
x = 0.0625
x = 0.03125
x = 0.015625
x = 0.0078125
x = 0.00390625
x = 0.00195312
x = 0.000976563
x = 0.000488281
x = 0.000244141
x = 0.00012207
x = 6.10352e-05
x = 3.05176e-05
x = 1.52588e-05
x = 7.62939e-06
x = 3.81470e-06
x = 1.90735e-06
x = 9.53674e-07
x = 4.76837e-07
x = 2.38419e-07
x = 1.19209e-07
x = 5.96046e-08

machine epsilon = 5.96046e-08

srnet
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Joined: 08/08/2014
Location: United Kingdom
Posts: 164

Posted: 10:30am 21 Jun 2015

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Quote (2) the distance from 1.0 to the next largest floating point number.

How can 1 and a floating point number have distance ?

$50SAT is Silent but probably still working.
For information on LoRa visit http://www.loratracker.uk/

shoebuckle
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Joined: 21/01/2012
Location: Australia
Posts: 189

Posted: 02:02pm 21 Jun 2015

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Pure mathematics is indeed a mystery unto itself. Not of great use to us mortals but it came in handy to prove Fermat's Last Theorum!

JohnS
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Joined: 18/11/2011
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Posted: 08:29pm 21 Jun 2015

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Except this is Applied Mathematics and of real importance.

JohnEdited by JohnS 2015-06-23

vegipete

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Joined: 29/01/2013
Location: Canada
Posts: 1085

Posted: 11:03am 22 Jun 2015

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srnet said

Quote (2) the distance from 1.0 to the next largest floating point number.

How can 1 and a floating point number have distance ?

Distance as in how far apart.
Micromite floating point numbers are represented by 32 bits (is that right?) so epsilon represents the difference to the next discrete value.

1.0000000000000 is followed by
1.0000000596046 if I've got my decimal places right.
Which means for example that the number
1.0000000298000 cannot be represented by the Maximite.

I don't have a Micromite setup so I can't test if epsilon is the same if calculated differently, for example, by dividing by, say, 1.9, instead of 2.0

Visit Vegipete's *Mite Library for cool programs.

cdeagle
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Joined: 22/06/2014
Location: United States
Posts: 261

Posted: 01:21am 23 Jun 2015

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"What good are numbers?" asked the plumber. "What good is a 1 1/2 inch P-trap when I need a 1 1/4 inch P-trap?" replied the mathematician.

CORRECTION: Because of the way the DO WHILE loop executes in the source code example at the beginning of this post, the machine precision should be 1.19209e-07.

The math implementation on a micro-controller and most CPUs is a function more of the software used to calculate than the hardware. For example, the epsilon for double precision is typically 2.23...e-16. When calculated in MATLAB, the double precision epsilon is the same for both 32-bit and 64-bit versions of Windows 7. This is also true for both 32-bit and 64-bit versions of MATLAB.

Math calculations in most modern Intel CPUs is now performed in hardware using a floating point unit (FPU).

For a micro-controller example, the machine epsilon for the the EzSBC1 controller is 2.23....e-16. This 32-bit CPU uses IEEE 64-bit math in the Control BASIC firmware.

To visualize distance between numbers, image you are on the equator walking east or west. The machine epsilon is the distance you would have to walk before the computer realizes you have actually moved. Physically, this distance could be feet, meters or any other useful measure.

IMHO, there seems to be a "time constant" between mathematics and society. It may be many years between the invention of a new mathematical concept or theory and the actually use of "new math" by society.

Hope this helps.

G8JCF

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Joined: 15/05/2014
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Posts: 676

Posted: 04:46am 23 Jun 2015

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With Windows 7 x64 in VB.NET I get

Single precision machine epsilon = 1.110223E-16
Double precision machine epsilon = 1.11022302462516E-16

Module MachineEpsilon

Sub Main()

Dim x As Double
' determine Micromite machine epsilon

x = 1.0

Do While ((1.0 + x) > 1.0)

x = x / 2.0

Console.WriteLine("x = " & x)

Loop

Console.WriteLine(" ")
Console.WriteLine("machine epsilon = " & x)

End Sub

End Module

Maths underpins our understanding of the real world. Without it we'd be still praying to statues, sacrificing animals, reading tea-leaves ...

Peter

The only Konstant is Change

twofingers
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Joined: 02/06/2014
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Posted: 05:40am 23 Jun 2015

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G8JCF said ...
Maths underpins our understanding of the real world. Without it we'd be still praying to statues, sacrificing animals, reading tea-leaves ...

Peter

what is wrong about sacrificing animals?

Regards Michael

isochronic
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Joined: 21/01/2012
Location: Australia
Posts: 689

Posted: 05:37pm 25 Jun 2015

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(Please forgive typos, black text on a wjite bacground is almost invisible to me )

[EDIT]
I alaways thought the epsilon was the smallest number so ..I thought this
would be equivalent...but ..obviously not !
Why is it so different ?

]>L
PROGRAM epsilon
C
DOUBLE a
FORMAT (E10.8,/)
a = 1.0
DO WHILE ( a > 0.0 )
WRITE (6,0) a
a = a / 2.0
END DO
END
\

1.00000000e+00
5.00000000e-01
2.50000000e-01
1.25000000e-01
...
1.01184644e-320
5.05923221e-321
2.52961611e-321
1.26480805e-321
6.32404027e-322
3.16202013e-322
1.58101007e-322
7.90505033e-323
3.95252517e-323
1.97626258e-323
9.88131292e-324
4.94065646e-324

BTW for plotting data it may be worth the TeraTerm Tektronix
4010 emulation. Kind of ancient but..

ripple from plugpak ,,

and a faster AD run , looking at 9600 rs232

The slow scan used this code or very similar, the TX routines are small and straightforward

PROGRAM sdisto
C
INTEGER*2 dat(256)
CALL tekinit ( )
CALL tekrect ( )
CALL acquiredat ( dat )
CALL tekplot ( dat )
PRINT *,
C
END
C
SUBROUTINE acquiredat ( acdat )
INTEGER*4 i
DO 20 i = 1, 256
20 acdat(i) = ADX5
RETURN
END
C
C
SUBROUTINE tekinit ( )
TXINIT
RETURN
END
C
C
SUBROUTINE tekrect ( )
TXMOVE 32, 250
TXDRAW 544, 250
TXDRAW 544, 762
TXDRAW 32, 762
TXDRAW 32, 250
RETURN
END
C
SUBROUTINE tekplot ( dat )
INTEGER*4 i, j
INTEGER*2 nn, mm
nn = 32
mm = dat(1) / 2 + 250
IF ( mm > 762 ) mm = 762
TXMOVE nn, mm
DO 40 i = 2, 256
nn = nn + 2
mm = dat(i) / 2 + 250
IF ( mm > 762 ) mm = 762
TXDRAW nn, mm
40 CONTINUE
TXMOVE 32, 250
RETURN
END
\

Edited by chronic 2015-06-27

isochronic
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Joined: 21/01/2012
Location: Australia
Posts: 689

Posted: 06:42pm 26 Jun 2015

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D'OH !!
It says, "when added to one "... and doing so, I get the dp epsilon OK..
The difference being, one quantity is the smallest number that can be compared to zero whereas the other is the smallest that can be used with the precision specified...went right past itEdited by chronic 2015-06-28

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