Accurate calculations

bobbym · 2012-04-23 19:31:50

Hi;

This question popped up in another thread and was answered there. Here we look at why that idea worked and the general theory.

It obvious that both square roots are going to be nearly equal and therefore that subtraction will cause much subtractive cancellation. This will cause the loss of significant digits and indeed many calculators will return 0.

The Taylor series is the main tool of numerical work. It comes to the rescue here too!

If we expand the following around 0 in terms of epsilon we get:

we can reaarange this to,

This is exactly what we need, we call x = 3 and epsilon = .000 000 000 000 000 1

We can truncate 2) to

now just plug in:

Which is quite accurate for only 2 terms of a taylor series.

anonimnystefy · 2012-04-23 20:11:55

Hi bobbym

What was wrong then whit my suggestion in the other thread?I just set x=3 immediately.

bobbym · 2012-04-23 20:13:03

Did you try it and see if it got the right answer?

anonimnystefy · 2012-04-23 20:19:44

http://www.wolframalpha.com/input/?i=x% … 0000000001

It is just a problem that Wolfram doesn't want to calculate it to more digits.

bobbym · 2012-04-23 20:22:32

Before I can comment on anything I ask, why are you using Wolfram when you have mathematica?

anonimnystefy · 2012-04-23 20:30:11

I don't know.It makes my computer slow.

Last edited by anonimnystefy (2012-04-23 20:30:27)

bobbym · 2012-04-23 20:31:27

Even when it is off?!

anonimnystefy · 2012-04-23 20:32:19

No,when it is open.I haven't deleted it!

bobbym · 2012-04-23 20:36:46

So then shut it down when you are finished with it.

First, Alpha in its desire to make an engine that understands English has ruined the mathematica language! You will never learn the correct syntax by using alpha.

Here is what you should have entered

N[x/(2sqrt(3))-x^2/(24sqrt(3)),20] for x=10^(-16)

anonimnystefy · 2012-04-23 20:42:27

I have opened Mathematica and entered:

N[x/(2*Sqrt[3]) - x^2/(24*Sqrt[3]), 20] for x = 0.0000000000000001;

but it gives me:

"Set::write: Tag Times in 2.88675*10^-17 1.*10^-16 for is Protected. >>"

bobbym · 2012-04-23 20:43:27

That is the problem! That code is for Alpha. They are not the same!

anonimnystefy · 2012-04-23 20:45:35

I even tried setting x=... before it.It doesn't give me more digits.How do I enter the code so that I get more digits?

bobbym · 2012-04-23 20:48:21

You are not listening. Mathematica and alpha are very different.

Put this in alpha:

N[x/(2sqrt(3))-x^2/(24sqrt(3)),20] for x=10^(-16)

anonimnystefy · 2012-04-23 20:57:20

Can't you show me what to do in Mathematica?I want to learn Mathematica language and functions.

bobbym · 2012-04-23 21:01:55

You use Mathematica functions for alpha too!

Mathematica allows many forms of input. This is the simplest:

N[x/(2Sqrt[3])-x^2/(24Sqrt[3]),20] /. x->10^(-16)

anonimnystefy · 2012-04-23 21:07:54

It still gives me 2.88675*10^-17

bobbym · 2012-04-23 21:12:19

Your display digits is set too low.

anonimnystefy · 2012-04-23 21:14:11

How do I change that?

bobbym · 2012-04-23 21:18:29

Try this first, Edit then Preferences then Appearance then Numbers. Tell me when you are there.

anonimnystefy · 2012-04-23 21:23:58

Actually,I found it,and I set 100,and it gives me this output:

2.8867513459481294*10^-17

bobbym · 2012-04-23 21:26:18

100 is too large. Set it to 16 and then calculate Sqrt[2.], what do you get?

anonimnystefy · 2012-04-23 21:30:50

1.414213562373095

bobbym · 2012-04-23 21:33:23

Then it is working correct. Go into preferences and get to internet connectivity. Uncheck allow mathematica to access the internet.

anonimnystefy · 2012-04-23 21:38:40

Why?

bobbym · 2012-04-23 21:42:25

Shutting off the internet to mathematica might speed up the machine greatly. If you want to communicate with alpha use your browser.

Math Is Fun Forum

#1 2012-04-23 19:31:50

Accurate calculations

#2 2012-04-23 20:11:55

Re: Accurate calculations

#3 2012-04-23 20:13:03

Re: Accurate calculations

#4 2012-04-23 20:19:44

Re: Accurate calculations

#5 2012-04-23 20:22:32

Re: Accurate calculations

#6 2012-04-23 20:30:11

Re: Accurate calculations

#7 2012-04-23 20:31:27

Re: Accurate calculations

#8 2012-04-23 20:32:19

Re: Accurate calculations

#9 2012-04-23 20:36:46

Re: Accurate calculations

#10 2012-04-23 20:42:27

Re: Accurate calculations

#11 2012-04-23 20:43:27

Re: Accurate calculations

#12 2012-04-23 20:45:35

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#13 2012-04-23 20:48:21

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#14 2012-04-23 20:57:20

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#15 2012-04-23 21:01:55

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#16 2012-04-23 21:07:54

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#17 2012-04-23 21:12:19

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#18 2012-04-23 21:14:11

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#19 2012-04-23 21:18:29

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#20 2012-04-23 21:23:58

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#21 2012-04-23 21:26:18

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#22 2012-04-23 21:30:50

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#23 2012-04-23 21:33:23

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#24 2012-04-23 21:38:40

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#25 2012-04-23 21:42:25

Re: Accurate calculations

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