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Complex set of linear equations
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I'm trying to find $beta$'s which solve the following problem:
$sum Vi beta i = V$ Or that at least minimize ($sum Vi beta i - V)$
where $Vi$ are vectors.
Additionally, there are a few other constraints/properties for these numbers.
$sum beta i =1 $ and $beta i gt 0$
Also, the sum of the vector components is also 1.
Another formulation is imagine you have a population of Things. For each one of these, you can apply some random input (from a set of 6 inputs, not necessarily equally likely), and you get some output that depends solely on the input, given the Thing. I need to find the best distribution of inputs so that, if I apply them to the entire population, my output match some control totals that are pre-established. In this case though, I don't need to exactly match everything, but find the solution which gives the closest (or close enough) set of outputs to my controls.
I'm not really sure where to start.
I've tried setting it up as a matrix equation, by collapsing some of my vector categories to get a square matrix, and inverting it, but I get a lot of negative $beta$'s.
Are there numerical methods which try to solve this sort of thing?
Thanks!
systems-of-equations linear-programming matrix-equations
asked Aug 3 at 19:29
Ilya
6
add a comment |Â
up vote
0
down vote
favorite
I'm trying to find $beta$'s which solve the following problem:
$sum Vi beta i = V$ Or that at least minimize ($sum Vi beta i - V)$
where $Vi$ are vectors.
Additionally, there are a few other constraints/properties for these numbers.
$sum beta i =1 $ and $beta i gt 0$
Also, the sum of the vector components is also 1.
Another formulation is imagine you have a population of Things. For each one of these, you can apply some random input (from a set of 6 inputs, not necessarily equally likely), and you get some output that depends solely on the input, given the Thing. I need to find the best distribution of inputs so that, if I apply them to the entire population, my output match some control totals that are pre-established. In this case though, I don't need to exactly match everything, but find the solution which gives the closest (or close enough) set of outputs to my controls.
I'm not really sure where to start.
I've tried setting it up as a matrix equation, by collapsing some of my vector categories to get a square matrix, and inverting it, but I get a lot of negative $beta$'s.
Are there numerical methods which try to solve this sort of thing?
Thanks!
systems-of-equations linear-programming matrix-equations
asked Aug 3 at 19:29
Ilya
6
You are asking, in effect, how to solve a linear system over the field of complex numbers. The standards methods (e.g. Gauss-Jordan elimination) will do that, and if the linear system is inconsistent (has no solutions) then one can ask about least squares methods (to minimize the residuals $(sum V_i beta_i) - V$).
– hardmath
Aug 3 at 19:35
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to find $beta$'s which solve the following problem:
$sum Vi beta i = V$ Or that at least minimize ($sum Vi beta i - V)$
where $Vi$ are vectors.
Additionally, there are a few other constraints/properties for these numbers.
$sum beta i =1 $ and $beta i gt 0$
Also, the sum of the vector components is also 1.
Another formulation is imagine you have a population of Things. For each one of these, you can apply some random input (from a set of 6 inputs, not necessarily equally likely), and you get some output that depends solely on the input, given the Thing. I need to find the best distribution of inputs so that, if I apply them to the entire population, my output match some control totals that are pre-established. In this case though, I don't need to exactly match everything, but find the solution which gives the closest (or close enough) set of outputs to my controls.
I'm not really sure where to start.
I've tried setting it up as a matrix equation, by collapsing some of my vector categories to get a square matrix, and inverting it, but I get a lot of negative $beta$'s.
Are there numerical methods which try to solve this sort of thing?
Thanks!
systems-of-equations linear-programming matrix-equations
asked Aug 3 at 19:29
Ilya
6
I'm trying to find $beta$'s which solve the following problem:
$sum Vi beta i = V$ Or that at least minimize ($sum Vi beta i - V)$
where $Vi$ are vectors.
Additionally, there are a few other constraints/properties for these numbers.
$sum beta i =1 $ and $beta i gt 0$
Also, the sum of the vector components is also 1.
Another formulation is imagine you have a population of Things. For each one of these, you can apply some random input (from a set of 6 inputs, not necessarily equally likely), and you get some output that depends solely on the input, given the Thing. I need to find the best distribution of inputs so that, if I apply them to the entire population, my output match some control totals that are pre-established. In this case though, I don't need to exactly match everything, but find the solution which gives the closest (or close enough) set of outputs to my controls.
I'm not really sure where to start.
I've tried setting it up as a matrix equation, by collapsing some of my vector categories to get a square matrix, and inverting it, but I get a lot of negative $beta$'s.
Are there numerical methods which try to solve this sort of thing?
Thanks!
systems-of-equations linear-programming matrix-equations
asked Aug 3 at 19:29
Ilya
6
asked Aug 3 at 19:29
Ilya
6
asked Aug 3 at 19:29
Ilya
6
asked Aug 3 at 19:29
Ilya
6
6
You are asking, in effect, how to solve a linear system over the field of complex numbers. The standards methods (e.g. Gauss-Jordan elimination) will do that, and if the linear system is inconsistent (has no solutions) then one can ask about least squares methods (to minimize the residuals $(sum V_i beta_i) - V$).
– hardmath
Aug 3 at 19:35
add a comment |Â
You are asking, in effect, how to solve a linear system over the field of complex numbers. The standards methods (e.g. Gauss-Jordan elimination) will do that, and if the linear system is inconsistent (has no solutions) then one can ask about least squares methods (to minimize the residuals $(sum V_i beta_i) - V$).
– hardmath
Aug 3 at 19:35
You are asking, in effect, how to solve a linear system over the field of complex numbers. The standards methods (e.g. Gauss-Jordan elimination) will do that, and if the linear system is inconsistent (has no solutions) then one can ask about least squares methods (to minimize the residuals $(sum V_i beta_i) - V$).
– hardmath
Aug 3 at 19:35
You are asking, in effect, how to solve a linear system over the field of complex numbers. The standards methods (e.g. Gauss-Jordan elimination) will do that, and if the linear system is inconsistent (has no solutions) then one can ask about least squares methods (to minimize the residuals $(sum V_i beta_i) - V$).
– hardmath
Aug 3 at 19:35
add a comment |Â
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You are asking, in effect, how to solve a linear system over the field of complex numbers. The standards methods (e.g. Gauss-Jordan elimination) will do that, and if the linear system is inconsistent (has no solutions) then one can ask about least squares methods (to minimize the residuals $(sum V_i beta_i) - V$).
– hardmath
Aug 3 at 19:35