vpasolve: numeric instability leading to wrong solutions?

Question

Hubert Villeneuve 2016-8-11

1
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https://ww2.mathworks.cn/matlabcentral/answers/299313-vpasolve-numeric-instability-leading-to-wrong-solutions

回答： Chidvi Modala 2020-10-27

Hi,

I'm working on a real-time distance estimation program using data from successive image frames. To do so, I employ Perspective-3-Points, which boils down to solving systems of three nonlinear equations derived from the law of Cosines. I'm using vpasolve, as there are 8 sets of solutions, and while I often get the solutions I expect, I am occasionally getting erroneous values that come with the following warning:

Warning: Solution '[x1 = 50.513354258655913523642438228345, x2 = 70.834152303683331599397540128635, x3 = -3.6823353014471324108432162378347]' seems to be affected by some numeric instability. Inserting this solution into equation '(5682387108428029*x1^2)/36893488147419103232 - (2779811138476361*x1*x2)/9223372036854775808 + (1396576697819169*x2^2)/9223372036854775808 - 3837877865952747/73786976294838206464 = 0' produces the residue '0.074294366099841203751338186504608'.

I attached two sets of very similar variables, one resulting in errors and the other in expected solutions (with the right order of magnitude), to check with the code below:

clear; clc;
load('P3Pvar2.mat');
syms x1 x2 x3
F1=A*x1^2+B*x2^2-2*D(1)*x1*x2-d_bar(1)==0;
F2=B*x2^2+C*x3^2-2*D(2)*x2*x3-d_bar(2)==0;
F3=C*x3^2+A*x1^2-2*D(3)*x3*x1-d_bar(3)==0;
S = vpasolve([F1,F2,F3],[x1, x2, x3]);
sol=double([S.x1 S.x2 S.x3])