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Old 2011-03-28, 20:29   #1
petrw1
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Default Unusual system of 11 unknowns

I have a series of equations with 11 unknowns (a-k)

On the plus side I have many equations ... more than 100.

The values of a-k are NOT integers.
However in each of the equations the resultant(? - the number after the equal sign) is reported as an integer; though is likely isn't an integer.

Furthermore, it is now knows whether this resultant is rounded or truncated.

Is it possible?, difficult? to solve for a - k?

Thanks
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Old 2011-03-28, 21:33   #2
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Quote:
Originally Posted by petrw1 View Post
I have a series of equations with 11 unknowns (a-k)

On the plus side I have many equations ... more than 100.

The values of a-k are NOT integers.
However in each of the equations the resultant(? - the number after the equal sign) is reported as an integer; though is likely isn't an integer.

Furthermore, it is now knows whether this resultant is rounded or truncated.

Is it possible?, difficult? to solve for a - k?

Thanks
This is similar (but not equal) to: http://www.mersenneforum.org/showthread.php?t=15462

however no knowing it's integer allows for approximation only as far as i know( after all the values known are approximate). care to give a list of equations ? maybe then i can help approximate.
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Old 2011-03-28, 22:50   #3
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That's a lot of equations for only 11 variables - do you have reason to believe there is a simultaneous solution?

I'd try solving it as a minimization problem - the error for each of the equations is the left side minus the right side squared. The total error is the sum of the errors. Use unconstrained minimization methods to search for the smallest error. You might get fancy with the individual error terms to account for truncations or roundoff - I'd try that as a next stage after seeing if the simple approach brought good enough results.
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Old 2011-03-31, 01:18   #4
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You could also try solving the equations 11 at a time and seeing how far apart the low-dimensional (finite number of points) solutions fell.

Of course, I'm assuming your functions are reasonably smooth...otherwise, as in the previous thread on lots of equations, we are going to need some additional information to help us understand the problem.
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Old 2011-03-31, 04:04   #5
petrw1
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It is a weighted scoring system we use in one of my organizations.

A team is scored in each of the 11 factors and each factor has a "secret" weighting. The scores are whole numbers but the weights may or may not be. Then the final answer is truncated / rounded (I don't know which one).

In all the years over 100 teams have been scored. For every team I know the scores they got on each of the 11 factors and the final score as an integer.

The scores for each factor have a low of 1 (cannot be 0) and a high somewhere between 5 and 12.

The final answer is between 1 and 10.

The curious mathematical side of me is interested in knowing if this can be solved.

If it helps / matters I am fairly certain that a>=b>=c ,... >=k
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Old 2011-03-31, 04:53   #6
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Ah. In that case I'd go with wblipp's suggestion.
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Old 2011-03-31, 04:56   #7
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Quote:
Originally Posted by petrw1 View Post
The scores for each factor have a low of 1 (cannot be 0) and a high somewhere between 5 and 12.

The final answer is between 1 and 10.

1. From the above two statements, I would conclude that the sum of the weights = 1 exactly.

2. If it is a human-generated system, odds are, the coefficients will be nice round fractions.
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Old 2011-03-31, 21:09   #8
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Quote:
Originally Posted by petrw1 View Post
It is a weighted scoring system we use in one of my organizations.

A team is scored in each of the 11 factors and each factor has a "secret" weighting. The scores are whole numbers but the weights may or may not be. Then the final answer is truncated / rounded (I don't know which one).

In all the years over 100 teams have been scored. For every team I know the scores they got on each of the 11 factors and the final score as an integer.

The scores for each factor have a low of 1 (cannot be 0) and a high somewhere between 5 and 12.

The final answer is between 1 and 10.

The curious mathematical side of me is interested in knowing if this can be solved.

If it helps / matters I am fairly certain that a>=b>=c ,... >=k
So this is a *linear* system....so solve for the latest 11 teams and see if you get the approximately the same result for the first 11 teams. Have fun!
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Old 2011-12-11, 22:04   #9
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I'm fairly positive this can easily be solved with dynamic programming. Even though the state space is a priori much too large to tackle with explicit enumeration. The number of binding constraints is so large that one can probably dynamically reduce the searching space as the execution of the algorithm goes along.
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