How to compute the sum of exponentials of a list of numbers, if we know the summation of the numbers?

Question

Imagine we know the summation of a list of numbers (that is computed by dynamic programming):

a+b+c+d+...

How can we compute the sum of exponentials of these numbers without re-iterating over all numbers:

Math.exp(a)+Math.exp(b)+Math.exp(c)+...

Just a side note, if needed: The numbers are the probabilities of the paths in a lattice structure. The summation of the probabilities of all paths is computed by dynamic programming. However, I need the exponential sum without re-iterating over the lattice structure.

It is in Java. If there is a solution in numpy, I can try to convert to Java. . — malaguena, Aug 17 '17 at 09:33
Like "we know the sum of the numbers, but not the numbers themself" ? ^^ explain better — azro, Aug 17 '17 at 09:35
Do you literally have a list, ie a `List` or do you have an array `int[]`? — Bohemian, Aug 17 '17 at 09:38
The numbers are the probabilities of the paths in a lattice structure. The summation of the probabilities of all paths is computed by dynamic programming. However, I need the exponential sum without re-iterating over the lattice structure. — malaguena, Aug 17 '17 at 09:39
Are you asking how to compute the sum of the exponentials of `a, b, c, ...` working from *only the sum* `a+b+c+...` ? Because that is clearly impossible — khelwood, Aug 17 '17 at 09:40
Now i'm understanding, for example you have a result of `10` and you want to calculate which `Math.exp(nums)` can give that result by summing them, am i right? — Marco Salerno, Aug 17 '17 at 09:42
@malaguena Then you're asking for something that is mathematically impossible. — khelwood, Aug 17 '17 at 09:44
Ok, then I need to find another solution to compute the sum of exponentials within the dynamic programming. Thanks. — malaguena, Aug 17 '17 at 09:45
Well maybe it isn't mathematically impossible but it needs a lot of computating power, by simply trying every possibility — Marco Salerno, Aug 17 '17 at 09:51
@MarcoSalerno So you think that if I tell you "I have 2 numbers that add up to 10", with a lot of computing power, you would be able to tell me what the sum of the exponents is? — khelwood, Aug 17 '17 at 10:59

score 1 · Answer 1 · answered Aug 17 '17 at 09:41

1

Use a stream:

int[] numbers;  // given some numbers
double expSum = IntStream.of(numbers).mapToDouble(Math::exp).sum();

answered Aug 17 '17 at 09:41

Bohemian

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The problem is that OP doesn't have the numbers. The desired solution has to magically guess the individual numbers from a given sum... – f1sh Aug 17 '17 at 09:49
This is not possible in dynamic programming. The result is the summation of a list of numbers but without having the numbers separately. All partial solutions are kept in the lattice structure. – malaguena Aug 17 '17 at 09:51

How to compute the sum of exponentials of a list of numbers, if we know the summation of the numbers?

1 Answers1