I want to calculate pi^2 to 22 decimal places

I want to calculate pi square to 22 decimal places. MATLAB can do it if we use vpa/digits but these require symbolic math toolbox. Is there any other way in MATLAB without toolbox?

Thanks

Answer 1

[Walter Roberson]

Use the opposite of "long-division". That is, use decimal multiply and add and carry.

Have vectors of decimal digits. Multiply the current "bottom" digit by the current "top" digit; take the final digit of the result and add it to the appropriate position of the result, and if there is a carry digit then add it to the digit to the left of the result (and if the addition results in a value > 9 then reduce by 10 and carry the 1.)

When it is time to move to the next digit on the left in the second value, then move the destination for the first multiplication one position to the left.

Remember that if you need to, expand the working storage to the left.

(For technical reasons it can be easier to work in reverse, storing digits left to right instead of right to left -- makes it easier to expand the destination without having to move anything else in the array.)

Answer 2

[James Tursa]

You can use this excellent FEX submission by John D'Errico:

https://www.mathworks.com/matlabcentral/fileexchange/36534-hpf-a-big-decimal-class?s_tid=srchtitle

[John D'Errico]

As James suggests, there is really no alternative to the use of HPF, unless @CanFox is willing to write their own higher precision tools. Doable, but never an easy thing, as then you need to write addition, subtractino, multiplication division, powers, yada yada yada.

Ok, there is a second alternative. There is a bigfloat tool in Java.math. I don't recall the name of the library. I even considered it, when/before I wrote HPF. That is, I could have put a MATLAB wrapper around the calls. In fact, as I recall, I did exactly that to test it out and compare to the HPF tools I wrote. Personally, I did not like some aspects of it. They made it terribly difficult to use, compared to what I was able to do in HPF.

Anyway, as I recall, I computed pi to either a million or a half million digits using HPF. (I can't remember which. And that was on my older slower CPU, fewer cores, etc.) Then I verified the digits from online sources, and all were correct. The biggest trick is which series to use. There are series that give you something like 20 digits per term, so 22 digits would take hardly any effort at all.

[Walter Roberson]

Though to be fair if the user is permitted to start with a number of digits of π then it might just come down to extended precision multiplication.

[CanFox]

This needs

• Mapping Toolbox
• Phased Array System Toolbox
• Symbolic Math Toolbox

Answer 3

[John D'Errico]

As a strong suggestion, you want to do some reading here:

https://en.wikipedia.org/wiki/Approximations_of_π

Again, you CANNOT do this using doubles, at least not without the effort of writing a higher precision arithmetic tool of your own. I think I used a Chudnovsky series wen I computed pi to all those digits. Remember that series computes the inverse of pi.

But if you are looking for a good approximation that will not even require writing a series, then the wiki page I show has an approximation valid to 30 decimal digits.

I'll write it using the symbolic toolbox, since that is easy here (and I can't easily run HPF in Answers.)

pie = log(sym(640320)^3 + 744)/sqrt(sym(163))

pie = 

vpa(pie,40)

ans = 

3.141592653589793238462643383279726619348

vpa(pi,40)

ans = 

3.141592653589793238462643383279502884197

vpa(pie - pi,40)

ans = 

0.0000000000000000000000000000002237351503804815087144226044020191064753

If you don't want to count how many digits are the same, this will do:

double(ans)

ans = 2.2374e-31

So it is indeed accurate to 30 decimal digits. Again though, you cannot use doubles there. The everything past roughly 16 digitis will be random garbage.

When you are done, and you have computed pi to a sufficient number of digits, multiply it by itself.

And if you really want a series approximation, there is one listed on that page that gives you 8 digits per term of the series. (Thanks to Ramanujan.) So 3 terms will suffice.

(Oh - I checked, and it looks like I computed pi to 500100 decimal digits using HPF.)