# Solve Linear System with a Ill Conditioned matrix.

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Tiago Carvalho on 25 Aug 2022
Commented: Tiago Carvalho on 26 Aug 2022
Hello,
I am working on my Master Thesis that consists in programming a 3D Multibody Simulator that solves kinematic and foward dynamic analysis, as a result I have matrices that are usually ill conditioned. In one of my lines of code I am trying to solve a linear system that follows the equation of Mq''=g, where M is an Inertia Matrix and g is a vector of forces and q'' is the accelerations that I use to start an iterative process.
From this iterative process I am able to calculate position and velocity of all bodies and the results I have are plaged with numerical instability and as a result I suspect that the culprit is this equation. Therefore my question is the following: Is there anything that I can do to improve the condition of my mass matrix? I have tried the following:
1. Use Pinv, which uses SVD, but the results are not satisfactory;
2. Tried to use Decomposition to once again see if the results were satisfatory but unsuccessfully;
3. Tried to use lsqminnorm with a high tolerance factor, where I achieved my best results.
4. Eliminated the first 7 rows and columns from the matrix and the 7 rows of the vetor which correspond to a body without movement or mas (ground) to try and improve the conditions unsucessfully ( this simplification would not change my results);
Other than this I have been reading all sort of articles and answers in this community, mainly answers from John D'Errico user, but I haven't found anything that helped me. In annex I put the example of one Inertial matrix and a vetor of forces that I use in one of my models.
EDIT: I also have been trying to use the regtools toolbox, mainly focusing on using Tikhonov regularization but currently I am having trouble in choosing an adequate method to choose the regularization parameter.
Regards,
Tiago

Ivo Houtzager on 25 Aug 2022
Edited: Ivo Houtzager on 25 Aug 2022
Please have look at the following regress function, see PBSID-Toolbox/regress.m at master · jwvanwingerden/PBSID-Toolbox · GitHub. This function uses the reglcurve and reggcv fucntions of the regtools toolbox to find the the regularization parameter using L-Curve or Generalized Cross Validation for the Tikhonov regularization or the Truncated SVD method.
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Ivo Houtzager on 25 Aug 2022
Edited: Ivo Houtzager on 25 Aug 2022
It solves the linear problem Y=X*P where X is unknown and Y and P are given, and Y can be vector as well. Seems you want to solve g=M*g2 for unknown g2 and g and M are given, If you transpose to g'=g2'*M', than the inputs are Y = g'; and P = M'; and output is g2 = X';. Thus function becomes something like
g2 = regress(g',M','TIKH','LCURVE')';
where you can select also select different method options, like 'TSVD' and/or 'GCV'.
Tiago Carvalho on 26 Aug 2022
Hello,
Thank you for your help. I have been experimenting with the toolbox that you mentioned I was able to obtain results that unfortunetly are only similar to the ones I achieved with the lsqminnorm. These results however, are not close to what I achieved with a standard commercial program.
MATLAB:
Position
Acceleration:
However, I don't think anyone else has any suggestion on how to solve this issue, and as a result I will wait more 24 hours and accept the answer, since it is very useful for regularization problems!
Regards,
Tiago