mersenneforum.org  

Go Back   mersenneforum.org > Great Internet Mersenne Prime Search > Math

Reply
 
Thread Tools
Old 2011-04-13, 17:31   #1
Raman
Noodles
 
Raman's Avatar
 
"Mr. Tuch"
Dec 2007
Chennai, India

100111010012 Posts
Default Inverse of functions

That website http://wims.unice.fr contains a plenty of useful online calculators,
really a lot, but I couldn't find out a place where inverse of functions is being sought.

For example take up with 1. e^x + sinh(x)
Can put with sinh(x) = \frac{e^x - e^{-x}}{2} and then solve with that quadratic for e^x.
The inverse is being given by ln (\frac{x\pm\sqrt{x^2+3}}{3})

2. cosh(x) + sinh(x)
The inverse of this function is given by ln\ x
just simply this case goes away within that way.
3. cos(x)+sin(x)
Writing with this function as \sqrt{2}\ sin(x+\frac{\pi}{4})
The inverse can be given by sin^{-1}\frac{x}{\sqrt 2}-\frac{\pi}{4}

OK, then what will be the inverses of these following functions?
It seems that they cannot be solved by using all those elementary mathematical functions at all!
(4) x^x
(5) x^{\frac{1}{x}}
(6) x+sin(x)
(7) x+tan(x)
(8) x+e^x
(9) e^x+sin(x)
(10) sin(x)+sinh(x)

Thanks for your help, if possible within any case

Disclaimer:
I had faced a problem which asked me to give an algorithm to check out if N is a perfect power, xy where that value of y is ≥ 2. I told that it can be done by using checks of \sqrt{x}, log_2(x), ^3\sqrt{x}, log_3(x), ^4\sqrt{x}, log_4(x), ... alternately, what value to check upto for base? Till N = zz, for some value of z. Thus, how to write up with that value of z as a function of N? How does that way work out rather...

Last fiddled with by Raman on 2011-04-13 at 17:56
Raman is offline   Reply With Quote
Old 2011-04-13, 19:17   #2
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

3·52·79 Posts
Default

Quote:
Originally Posted by Raman View Post
That website http://wims.unice.fr contains a plenty of useful online calculators,
really a lot, but I couldn't find out a place where inverse of functions is being sought.
The problem is hard and in general cannot be done symbolically.

Quote:
Originally Posted by Raman View Post
I had faced a problem which asked me to give an algorithm to check out if N is a perfect power, xy where that value of y is ≥ 2. I told that it can be done by using checks of \sqrt{x}, log_2(x), ^3\sqrt{x}, log_3(x), ^4\sqrt{x}, log_4(x), ... alternately, what value to check upto for base? Till N = zz, for some value of z. Thus, how to write up with that value of z as a function of N? How does that way work out rather...
For that one you'll need a special function like Lambert's W, or numerical techniques. I think it's exp(W(log N)). W(x) is about log x - log log x, so this is roughly exp(log log N - log log log N) = log N / log log N, where log is the natural log.

Last fiddled with by CRGreathouse on 2011-04-13 at 19:20
CRGreathouse is offline   Reply With Quote
Old 2011-04-13, 19:58   #3
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by CRGreathouse View Post
The problem is hard and in general cannot be done symbolically.



For that one you'll need a special function like Lambert's W, or numerical techniques. I think it's exp(W(log N)). W(x) is about log x - log log x, so this is roughly exp(log log N - log log log N) = log N / log log N, where log is the natural log.
if he didn't want work I'd say use PARI's ispower() with a loop through base values.

Last fiddled with by science_man_88 on 2011-04-13 at 19:58
science_man_88 is offline   Reply With Quote
Old 2011-04-13, 23:05   #4
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

134458 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
if he didn't want work I'd say use PARI's ispower() with a loop through base values.
With ispower() there's no need to loop at all -- ispower(N) does the right thing. ispower(N, b) is only when you want to check for a particular base and no others.

You should look at the code for it some time, it's fascinating. They check for 2nd, 3rd, and 5th powers separately, then move on to general testing for larger bases. Some amount of trial division is also done so you don't have to check quite as high, IIRC.

Incidentally, Bernstein has a nice paper about doing this test efficiently, in almost-linear time. Pretty impressive stuff.

Last fiddled with by CRGreathouse on 2011-04-13 at 23:06
CRGreathouse is offline   Reply With Quote
Old 2011-04-13, 23:22   #5
science_man_88
 
science_man_88's Avatar
 
"Forget I exist"
Jul 2009
Dumbassville

26·131 Posts
Default

Quote:
Originally Posted by CRGreathouse View Post
With ispower() there's no need to loop at all -- ispower(N) does the right thing. ispower(N, b) is only when you want to check for a particular base and no others.

You should look at the code for it some time, it's fascinating. They check for 2nd, 3rd, and 5th powers separately, then move on to general testing for larger bases. Some amount of trial division is also done so you don't have to check quite as high, IIRC.

Incidentally, Bernstein has a nice paper about doing this test efficiently, in almost-linear time. Pretty impressive stuff.
Bernstein ? I don't have a math degree and have never read the name to my knowledge.
science_man_88 is offline   Reply With Quote
Old 2011-04-13, 23:29   #6
CRGreathouse
 
CRGreathouse's Avatar
 
Aug 2006

3·52·79 Posts
Default

Quote:
Originally Posted by science_man_88 View Post
Bernstein ? I don't have a math degree and have never read the name to my knowledge.
Read and be enlightened:
http://cr.yp.to/

In particular, his math stuff is mostly in one of these:
http://cr.yp.to/ntheory.html
http://cr.yp.to/factorization.html
http://cr.yp.to/arith.html
CRGreathouse is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Inverse of Smoothness Probability paul0 Math 6 2017-07-25 16:41
mi64: inverse does not exist wreck NFS@Home 1 2016-05-08 15:44
Lurid Obsession with Mod Inverse only_human Miscellaneous Math 26 2012-08-10 02:47
Electricity, GIMPS, and inverse relationships stars10250 Soap Box 31 2012-04-10 15:41
Inverse Laplace Transform flouran Math 1 2010-01-18 23:48

All times are UTC. The time now is 07:51.

Fri Sep 25 07:51:01 UTC 2020 up 15 days, 5:01, 0 users, load averages: 1.63, 1.44, 1.36

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.