Survey for Special Math Problem

We are all familiar with the system of linear equations:

a11 x1 + a12 x2 + ... + a1n xn = b1
a21 x1 + a22 x2 + ... + a2n xn = b2

an1 x1 + an2 x2 + ... + ann xn = bn

Has anyone seen an application for the variant of the above such that:

a11 x1^2 + a12 x2 + ... + a1n xn = b1
a21 x1 + a22 x2^2 + ... + a2n xn = b2
an1 x1 + an2 x2 + ... + ann xn^2 = bn

In other words, where the i'th diagonal element has the square (or even any other power that is not equal to one) of xi instead just xi.



Edited: 2 June 2012, 5:42 p.m.


Hi, Namir.

I am curious now. You have surely imprinted the question in my mind.

Please, allow me a candid question: do you actually want to know if this sort of 'hybrid' system - not linear, though - is used in some particular application or you already know and would like to check if anyone else has also seen it?

Thanks for bringing the subject up. I confess this is new to me.


Luiz (Brazil)



I have been studying iterative linear methods (both stationary and non-stationary). The problem I am asking about is something that just popped in my mind two days. Interestingly, you can solve for the matrix coefficients a(i,j) for the 'hybrid' system using the iterative Gauss-Seidel method that is also used for linear system.

My question is, does the problem I am asking about model some application?



Interestingly, you can solve for the matrix coefficients a(i,j) for the 'hybrid' system using the iterative Gauss-Seidel method that is also used for linear system.
When dealing with electrical distribution networks and load-flow studies, back in the days I was concluding the Electrical Engineering course, I wrote a program for the HP42S to solve a load-flow system by using an alternate Accelerated Gauss-Siedel method. This was done because I decided not to use a computer and FORTRAN back then (1990). The teacher accepted my solution only because I implemented the whole iteration process without using the HP42S SOLVE, only discrete program steps. As you mentioned, Gauss-Siedel is used in this case because the load-flow studies presume a linear system.

Not too much to add, sorry...

Luiz (Brazil)


Can't say I have seen this particular nonlinear system treated as a special case, but on looking at its symmetries it would seem that general solution approaches would be a little simpler to set up.

For example, if you were to take a multidimensional Newton-Raphson approach, the Jacobian matrix that appears in the expression of that problem is simply an nxn matrix with 2*aii*xi along the diagonal and the remaining elements simply constant values of the original matrix. With this sort of redundancy you save a lot of register space when storing elements.




The iterative solution for these simple nonlinear equations can employ Gauss-Sediel which works for liner system. My question to all is, "Have you seen an application that can be modeled with such system of nonlinear equations."


I just read up on Gauss-Seidel and it is very easy to understand. Can you tell us how the algorithm is modified to accommodate your nonlinear diagonal terms?



To solve the nonlinear/hybrid system, here is the core steps expressed in Excel VBA:

For J = 1 To N
Sum = B(J)
For I = 1 To N
If I <> J Then
Sum = Sum - A(J, I) * X(I)
End If
Next I
X(J) = Sqr(Sum / A(J, J))
Next J

The core steps of the Gauss-Seidel for linear systems is:

For J = 1 To N
Sum = B(J)
For I = 1 To N
If I <> J Then
Sum = Sum - A(J, I) * X(I)
End If
Next I
X(J) = Sum / A(J, J)
Next J

Since we using the squares of the variables n the diagonal elements, calculating them iteratively would require the square root values of Sum/A(I,I).

Using the minor modification of the first code snippet allows Gauss-Seidel method to work with the nonliner/hybrid system of equations.


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