solve overdetermined linear system

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Alexandr Fedorov
Alexandr Fedorov 2017-12-27
编辑： Alexandr Fedorov 2017-12-27
Hello. I need to solve overdetermined linear system A*x=B, where x is [a1 a2 a3 a4 a5]. I got a solution with command
inv(A'*A)*A'*B;
but the answer is not what I need, because i have a theoretical solution for this problem. Can I somehow set the initial approximation to get the solution I need?

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John D'Errico
John D'Errico 2017-12-27
First of all, DON'T USE THAT SOLUTION!!!!!!!! That is a bad idea, because it is terrible when applied to moderately ill-posed problems.
Instead, use
coef = A\b;
This will be more accurate.
You say that the answer is not what you wanted. Too bad. You cannot set an initial start point for a problem that has a unique solution.
An alternative solution is to use lsqr.
coef = lsqr(A,b);
You can set the start point for lsqr. From the help for lsqr...
X = lsqr(A,B,TOL,MAXIT,M1,M2,X0) specifies the P-by-1 initial guess. If
X0 is [] then lsqr uses the default, an all zero vector.
But as long as A is full rank, then you will get the same solution, no matter what the start point.
So, is A of less than full rank? If so, then the solution is non-unique, and there are infinitely many equally good solutions.
If you have additional information about the problem, then you need to explain what it is.
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John D'Errico
John D'Errico 2017-12-27
编辑：John D'Errico 2017-12-27
I cannot run your code though, because I do not know what input to provide.
Your .mat file does not contain A and B, which is all that we need here for the moment.
But if A has rank 2 AND it has 2 columns, then this explains why you get the same answer from both backslash and lsqr. It will be of full rank. If A had 3 or 4 columns and 30-40 rows, and still was rank 2, then backslash and lsqr can return different results, because there will be infinitely many results. So I think you are trying the case where A has 2 columns.
Alexandr Fedorov
Alexandr Fedorov 2017-12-27
编辑：Alexandr Fedorov 2017-12-27
But how i can make the rank of A is 3 or 4, when i take like 3 members in sum? I take for example 40 point, 3 members and 223 MPa of stress. Ok. I finding K1 which is mode I stress-intensity factors by solve this system
The left side of expression is stress from ANSYS and F1n is (q=n/2):
. K1=a1*sqrt(2*pi). In theory K1=Sigma*sqrt(2*pi*l), where l is lenght of fracture, Sigma is the tensile load value. So theta and r i got from picture of stress intensity from ANSYS.
This theory is described in Interference-optical Methods of Solid Mechanics by Razumovsky, Igor A.

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