Solve 100 equations, each with a different amount of unknowns in matlab

Hi folks,

I have 100 equations which I want to solve simultaneously. There are 9 unknowns (A, B, C, D, E, F, G, H, J). Not each equation has 9 unknowns.

Below are a couple of the equations which I am trying to solve simultaneously (I'm aware there may be limitations considering not each equation has the same unknowns, but considering there is so many equations I'm hoping the computational capabilities of matlab could bypass this issue?).

PGASiteCorrectedIntraResidual(21)= 0.8965*B
PGASiteCorrectedIntraResidual(22)= 1.4350*F + 0.2272*B
PGASiteCorrectedIntraResidual(23)= 0.1902*K + 1.5069*L
PGASiteCorrectedIntraResidual(24)= 1.3683*F
PGASiteCorrectedIntraResidual(25)= 1.3260*L
PGASiteCorrectedIntraResidual(26)= 1.4388*L
PGASiteCorrectedIntraResidual(27)= 0.1428*A + 1.3456*E + 0.1208*F
PGASiteCorrectedIntraResidual(28)= 1.7080*F
PGASiteCorrectedIntraResidual(29)= 0.9675*B + 0.4395*F
PGASiteCorrectedIntraResidual(30)= 1.1125*F
PGASiteCorrectedIntraResidual(31)= 0.3608*A + 1.1628*E + 0.2967*F
PGASiteCorrectedIntraResidual(32)= 1.1762*C
PGASiteCorrectedIntraResidual(33)= 1.2025*F + 0.0642*B
PGASiteCorrectedIntraResidual(34)= 1.2082*F + 0.0638*B
PGASiteCorrectedIntraResidual(35)= 1.5749*F
PGASiteCorrectedIntraResidual(36)= 0.2690*A + 0.8727*B + 0.7128*F
PGASiteCorrectedIntraResidual(37)= 1.3713*L
PGASiteCorrectedIntraResidual(38)= 0.7236*B + 0.7163*F
PGASiteCorrectedIntraResidual(39)= 0.7961*B + 0.7752*F
PGASiteCorrectedIntraResidual(40)= 1.8229*E + 0.2857*F + 0.4807*B
PGASiteCorrectedIntraResidual(41)= 1.9081*A + 0.7745*B
PGASiteCorrectedIntraResidual(42)= 0.3616*K + 1.7008*L
PGASiteCorrectedIntraResidual(43)= 0.7649*B + 0.8394*F
PGASiteCorrectedIntraResidual(44)= 1.8750*A + 1.0423*B
PGASiteCorrectedIntraResidual(45)= 2.2159*B + 0.5612*C
PGASiteCorrectedIntraResidual(46)= 2.2250*E + 0.4393*D

Is there a function in matlab which could solve for these unknowns easily?

Thanks for any help

Answer 1

[Bjorn Gustavsson]

What you have (shown) is a linear system of equations, with 100 equations and only 9 unknowns. This is exactly what matlab is designed to solve. Since the system is rather overdetermined, 91 more equations than unknowns, it is likely that all equations cannot be solved exactly at the same time, but you should be able to get a least-square solution without problems. What you should do is to create a matrix, M, with the coefficients for each equation row-by-row. After you've done that you have a simple matrix equation:

PGAS_vector = M*ABCDEFGHI;

Where the ABCDEFGHI array is an array with your 9 unknowns, and the PGAS_vector is the column-vector with all your PGASiteCorrectedIntraResidual components. Matlab is expertly designed to solve this type of overdetermined equations:

ABCDEFGHI = M\PGAS_vector;

That way you get a least-square solution for all your 9 unknowns.

Now, if the equations you've not shown contains non-linear terms in the unknowns this is not a solution.

HTH

[BOB]

That's great thanks. To clarify, my matrix should look something like (for a small group of the 100 equations):

Where the zeros indicate the coefficient is not present for that equation equation. And how would I input an array of unknown values? I get the "Undefined function or variable" return if I try to construct it using:

ABCDEFGHIJKLO1O5O8 = [A B C D E F G H I J K L O1 O5 O8]

[Bjorn Gustavsson]

Yeah, that would be a plausible couple of rows in the matrix M. You don't need to, and shouldnt, define your unknowns-array like that. Just calculate it directly with the matrix inverse:

ABCDEFGHIJKLO1O5O8 = M\PGASite_part_array;

That will give you the values of all 9 components. From there on you extract the components you're interested in:

A = ABCDEFGHIJKLO1O5O8(1);
L = ABCDEFGHIJKLO1O5O8(9);

and so on.

[BOB]

Ok great. One last thing. The PGAS_vector dimensions should be 100 x 1? If so I'm currently getting the matrix dimensions do not match issue. I'm assuming there should only be one column in this vector though?

[Bjorn Gustavsson]

Yes, it should have as many rows as you have in M - the number of equations. You should get somewtinh like this from whos:

>> whos  
  Name           Size            Bytes  Class     Attributes

  M              100x9             168  double              
  PGAS_vector    100x1              56  double