MaplePrimes - answers and comments on Question, How to solve a matricial equation (in order to coefficients) in maple?

By SolveEquations

Markiyan Hirnyk — Mon, 26 Nov 2012 16:20:57 Z

It can be done as follows.

Download SolveEquations.mw

Comparison

Markiyan Hirnyk — Mon, 26 Nov 2012 17:16:26 Z

Download comparison.mw

Minimize

acer — Tue, 27 Nov 2012 09:40:55 Z

restart:

A := Matrix(3, 3, [5*a-4, 5*sqrt(a)*sqrt(b)-5, 7*sqrt(a)*sqrt(c)-6,
                   8*sqrt(a)*sqrt(b)-5, 8*b-2, 8*sqrt(b)*sqrt(c)-9,
                   4*sqrt(a)*sqrt(c)-1, 6*sqrt(b)*sqrt(c)-4, 9*c-5]):

eqs := Equate(A, Matrix(3)):

Optimization:-Minimize( add((rhs-lhs)(e)^2,e in eqs) );

    [31.1856590880747149, [a = 0.8107833782774988, 
                           b = 0.5717485896565585,
                           c = 0.7115891789684132] ]

or, more simply,

Optimization:-Minimize( add(e^2, e in A) );

    [31.1856590880747149, [a = 0.8107833782774988,
                           b = 0.5717485896565585,
                           c = 0.7115891789684132] ]

The main problem with using Optimization:-Minimize is that it's a local minimizer. If it returns something very close to zero as the optimal objective then great, one has shown that the Matrix A can be zero (up to floating-point approximation).

But if it returns a locally optimal objective value greater than zero then nothing can be concluded from that alone about whether A could become all zero, (in general, unless we can show certain things about the convexity of the problem). I mention this for other readers, of course, as I am sure that Markiyan is aware of the one-sided benefit of using a local minimizer in such a context.

The same output is obtained

Markiyan Hirnyk — Tue, 27 Nov 2012 16:57:01 Z

@acer with the global minimizer:

yes

acer — Tue, 27 Nov 2012 18:39:51 Z

@Markiyan Hirnyk Yes, I had seen that. The points I was trying to make were that 1) Optimization:-Minimize can obtain a result (you had a little usage difficulty, that I wanted to quickly clear up), but that 2) use of a global method is preferred for the given task.