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2

The Pi function is an extension of the factorial over the reals (or even complex numbers). For integers n, Π(n) = n!, but to get a definition over the reals we define it using an integral:

Pi(z) = integral t from 0 to infinity e^-t t^z dt

In this challenge we will invert the Π function.

Given a real number z ≥ 1, find positive x such that Π(x) = z. Your answer must be accurate for at least 5 decimal digits.

Examples:

120 -> 5.0000
10 -> 3.39008
3.14 -> 2.44815
2017 -> 6.53847
1.5 -> 1.66277
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2  
Note that more often people use the Gamma (Γ) function. Π(x) = Γ(x+1). But IMO Γ is a shifted abomination, and Π is the true extension of the factorial. – orlp 12 hours ago
    
Wellp, that series expansion is enough to scare me... i.imgur.com/ttgzDSJ.gif – carusocomputing 12 hours ago
1  
@LuisMendo You may assume z ≥ 1, I edited the question. – orlp 12 hours ago
1  
All of the examples you give have other solutions as well, for example 120 -> -0.991706. This is because Π(x) goes to infinity as x goes to -1 from the right. Perhaps you mean to insist that x>0 as well. – Greg Martin 12 hours ago
    
@GregMartin Added as well. – orlp 12 hours ago
up vote 6 down vote

Mathematica, 17 15 27 bytes

FindInstance[#==x!&&x>0,x]&

Output looks like {{x -> n}}, where n is the solution, which may not be allowed.

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up vote 5 down vote

MATL, 13 bytes

1`1e-5+tQYgG<

This uses linear seach in steps of 1e-5 starting at 1. So it is terribly slow, and times out in the online compiler.

To test it, the following link replaces the 1e-5 accuracy requirement by 1e-2. Try it online!

Explanation

1        % Push 1 (initial value)
`        % Do...while
  1e-5   %   Push 1e-5
  +      %   Add
  t      %   Duplicate
  QYg    %   Pi function (increase by 1, apply gamma function)
  G<     %   Is it less than the input? If so: next iteration
         % End (implicit)
         % Display (implicit)
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up vote 3 down vote

Pyth, 4 bytes

.I.!

A program that takes input of a number and prints the result.

Test suite

How it works

.I.!    Program. Input: Q
.I.!GQ  Implicit variable fill
.I      Find x such that:
  .!G    gamma(x+1)
     Q   == Q
        Implicitly print
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up vote 2 down vote

MATLAB, 59 bytes

@(x)fminsearch(@(t)(gamma(t+1)-x)^2,1,optimset('TolF',eps))

This is an anonymous function that finds the minimizer of the squared difference betweeen the Pi function and its input, starting at 1, with very small tolerance (given by eps) to achieve the desired precision.

Test cases (run on Matlab R2015b):

>> @(x)fminsearch(@(t)(gamma(t+1)-x)^2,1,optimset('TolF',eps))
ans = 
    @(x)fminsearch(@(t)(gamma(t+1)-x)^2,1,optimset('TolF',eps))
>> f = ans; format long; f(120), f(10), f(3.14), f(2017)
ans =
   5.000000000000008
ans =
   3.390077650547032
ans =
   2.448151165246967
ans =
   6.538472664321318

You could try it online in Octave, but unfortunately some of the results lack the required precision.

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up vote 0 down vote

GeoGebra, 25 bytes

NSolve[Gamma(x+1)=A1,x=1]

Entered in the CAS input, and expects input of a number in spreadsheet cell A1. Returns a one-element array of the form {x = <result>}.

Here is a gif of the execution:

Execution of progrma

How it works

NumericallySolve the following equation: Gamma(x+1)=A1, with starting value x=1.

share|improve this answer
    
Is it guaranteed to return a positive number, and does it work for 1.5, which has broken several answers? – Pavel 4 hours ago

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