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HP Forum Archive 21

 Survey for Special Math ProblemMessage #1 Posted by Namir on 2 June 2012, 5:41 p.m. 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.

 Re: Survey for Special Math ProblemMessage #2 Posted by Luiz C. Vieira (Brazil) on 2 June 2012, 8:32 p.m.,in response to message #1 by Namir 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)

 Re: Survey for Special Math ProblemMessage #3 Posted by Namir on 3 June 2012, 12:15 a.m.,in response to message #2 by Luiz C. Vieira (Brazil) 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

 Re: Survey for Special Math ProblemMessage #4 Posted by Luiz C. Vieira (Brazil) on 3 June 2012, 1:37 a.m.,in response to message #3 by Namir 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)

 Re: Survey for Special Math ProblemMessage #5 Posted by Les Wright on 3 June 2012, 2:40 p.m.,in response to message #1 by Namir 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

 Re: Survey for Special Math ProblemMessage #6 Posted by Namir on 3 June 2012, 5:55 p.m.,in response to message #5 by Les Wright 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."

 Re: Survey for Special Math ProblemMessage #7 Posted by Les Wright on 3 June 2012, 7:31 p.m.,in response to message #6 by Namir 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

 Re: Survey for Special Math ProblemMessage #8 Posted by Namir on 3 June 2012, 9:46 p.m.,in response to message #7 by Les Wright 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

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