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How to create 31-tuples with nonnegative entries that sum to 1

asked 2019-11-05 11:54:26 -0600

anonymous user

Anonymous

updated 2019-11-05 12:25:20 -0600

Hello. I would like to know how to create all 31-tuples (sequence of 31 numbers) whose entries are real, nonnegative, and sum up to one. One such kind of tuple would be one where exactly one of the 31 entries is 1 and all of the other remaining 30 entries are 0.

How could I ask Sage to do this?

Thank you.

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Comments

the "numbers" are in which ring?

heluani gravatar imageheluani ( 2019-11-05 12:16:53 -0600 )edit

My apologies, I just edited my post. The entries are real and nonnegative.

merluza gravatar imagemerluza ( 2019-11-05 12:25:39 -0600 )edit
1

There are infinitely many such tuples if you do not pose additional constraints.As an example, note that $(1/n, 1-1/n,0,0, \dots, 0)$ is such a tuple for every positive integer $n$.

joakim_uhlin gravatar imagejoakim_uhlin ( 2019-11-05 13:04:48 -0600 )edit

You could pick a random number x between 0 and 1 for the first slot, then a random number between 0 and 1-x for the second slot, etc.

John Palmieri gravatar imageJohn Palmieri ( 2019-11-05 17:17:02 -0600 )edit

2 answers

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answered 2019-11-06 04:42:33 -0600

tmonteil gravatar image

If you want to consider all the points satisfying some properties, i guess the best way it to consider those points as forming a set. In your case, the set of points is a polytope. By convenience, you can construct it from a linear program as follows:

sage: p = MixedIntegerLinearProgram()
sage: x = p.new_variable(real=True, nonnegative=True)
sage: p.add_constraint(sum(x[i] for i in range(31)) == 1)
sage: P = p.polyhedron() ; P
A 30-dimensional polyhedron in RDF^31 defined as the convex hull of 31 vertices

Then you can do things like:

sage: P.contains([1/2,1/2,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
True
sage: P.contains([1/2,1/3,0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0])
False

sage: P.random_integral_point()
(0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)

sage: P.integral_points_count()
31

sage: P.dim()
30

sage: P.is_simplex()
True

sage: P.vertices_list()
...
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Comments

Clapclapclapclapclap ! Very nice (much more general than my ad hoc solution...).

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2019-11-06 04:45:55 -0600 )edit
0

answered 2019-11-05 22:12:10 -0600

Emmanuel Charpentier gravatar image

As already pointed out by commenters, there are infinitely many such tuples. One way to generate them is:

def R31S1():
    A=[random() for u in range(31)]
    S=sum(A)
    return tuple([u/S for u in A])

Let's check:

sage: T=R31S1()
sage: type(T)
<class 'tuple'>
sage: len(T)
31
sage: sum(T)
1.0
sage: all([u>=0 and u<1 for u in T])
True
sage: T
(0.006089037356475367,
 0.02653861505103852,
 0.048525911647988376,
 0.030844994609170905,
 0.02700793870208519,
 0.07872299394844966,
 0.015087636977883111,
 0.06309180347147934,
 0.05423637336235927,
 0.052700158885111356,
 0.06602701660767711,
 0.0363743838933233,
 0.013134222937337785,
 0.04594966379579753,
 0.025102035013684295,
 0.04519431764115383,
 0.002292718973017283,
 0.012639236364503856,
 0.030852444551295627,
 0.07184775173604117,
 0.04969543126517173,
 0.03573006205560776,
 0.01675001424706971,
 0.0061099530976978155,
 0.04987299807905701,
 0.015149928406498115,
 0.0027994609852037985,
 0.013127271433974403,
 0.0011150285848061565,
 0.05054156008030657,
 0.00684903623873419)

All of this is basic Python, nothing Sage-specific...

HTH,

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Comments

2

Note that this random generation is not uniform on the simplex, as it comes from the projection of an hypercube (think of a square projected on its diagonal).

tmonteil gravatar imagetmonteil ( 2019-11-06 04:44:07 -0600 )edit

Indeed. But the uniformity wasn't specified in the problem statement... and i'm lazy :-). Generating a simplex q=with uniform density isn't that simple...

Emmanuel Charpentier gravatar imageEmmanuel Charpentier ( 2019-11-07 13:38:32 -0600 )edit

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Asked: 2019-11-05 11:54:26 -0600

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Last updated: Nov 06