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Topic: Solving N-1 equations in N unknowns with initial state?
Replies: 4   Last Post: May 9, 2013 1:23 PM

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 William Elliot Posts: 2,637 Registered: 1/8/12
Re: Solving N-1 equations in N unknowns with initial state?
Posted: May 8, 2013 5:46 AM

On Tue, 7 May 2013, Don Del Grande wrote:

> I have a system of N-1 equations in N unknowns.

> If I start by assigning values to the N unknowns and then recalculate
> the values recursively:

Huh? If you assign values to the n unknowns, then the n - 1 equations
are all true or one or more is false.

> (a) Will they always reach a single solution (assuming there is a
> solution for the equations in the first place), and

> (b) If a single solution can be reached, is there a way to determine it
> in terms of the initial values without having to go through the
> recursion, and if so, what is it?

> Example:
> 3a - b - c - d = 9
> -a + 3b - c - d = 3
> -a - b + 3c - d = -3
> -a - b - c + 3d = -9

That's not three equations in four unknowns;
it's four equations in four unknowns.

> If you combine the first three equations, you get a + b + c - 3d = 9,
> which is -1 times the fourth equation, so it is effectively 3 equations
> in 4 unknowns.

What do you mean by combining?
I get a + b + c - 3d = 9 by adding the first three equations.

> If the values are initially set to a = 3, b = 1, c = -1, and d = -3, and
> the equations repeated recalculated recursively, they eventually become
> a = 9/4, b = 3/4, c = -3/4, and d = -9/4.

You can't set d. What d is, is determined by setting a,b and c.
What's a recalculation?

> Can those values be determined in a non-recursive method from the
> equations and the initial values?

Given the three equations
3a - b - c - d = 9
-a + 3b - c - d = 3
-a - b + 3c - d = -3
that is
3a - b - c = 9 + d
-a + 3b - c = 3 + d
-a - b + 3c = -3 + d
we have
2a + 2b - 2c = 12 + 2d; a + b - c = 12 + d
-2a + 2b + 4c = 2d; -a + b + 2c = d
2b + 2c = 12 + 2d; b + c = 6 + d
from which we should be able to calulate a,b,c in terms of d.
So d is the only parameter that can be set and by setting d,
a,b,c are sets.

Date Subject Author
5/7/13 Don Del Grande
5/7/13 Ken.Pledger@vuw.ac.nz
5/7/13 Don Del Grande
5/9/13 gnasher729
5/8/13 William Elliot