Survey for Special Math Problem Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 06-02-2012, 05:41 PM 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. Thanks! Namir Edited: 2 June 2012, 5:42 p.m. Luiz C. Vieira (Brazil) Senior Member Posts: 591 Threads: 16 Joined: Feb 2012 06-02-2012, 08:32 PM 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. Cheers. Luiz (Brazil) Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 06-03-2012, 12:15 AM Luiz, 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? namir Luiz C. Vieira (Brazil) Senior Member Posts: 591 Threads: 16 Joined: Feb 2012 06-03-2012, 01:37 AM Quote: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) Les Wright Posting Freak Posts: 1,368 Threads: 212 Joined: Dec 2006 06-03-2012, 02:40 PM 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. Les Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 06-03-2012, 05:55 PM Les, 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." Les Wright Posting Freak Posts: 1,368 Threads: 212 Joined: Dec 2006 06-03-2012, 07:31 PM 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? Les Namir Posting Freak Posts: 2,247 Threads: 200 Joined: Jun 2005 06-03-2012, 09:46 PM 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. Namir « Next Oldest | Next Newest »

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