Matrix numeric inversion, very bad conditioned

I have a matrix

A= [-1.14420182157714e-05	3.97106112979063e-09	1.71528761347565e-09	3304.87922280846	-6.43946234919730;
    3.97106112979063e-09	-7.42484390057452e-13	-5.95484638418163e-13	-0.618994914206821	5.75841022133480e-05;
    1.71528761347565e-09	-5.95484638418163e-13	-2.57140914558130e-13	-0.495586332063353	0.000965957682109416;
    3304.87922280846	-0.618994914206821	-0.495586332063353	-516040787150.579	51574737.9623166;
    -6.43946234919730	5.75841022133480e-05	0.000965957682109416	51574737.9623166	3814318.88267676];

and i want to calculate the inverse matrix. This matrix is very bad contitioned.

Normal inv oder pinv function doesnt give a stable solution. Which functions can use?

Thank you!

Answer 1

[Bhanu Prakash]

Hi Martin,

I understand that you want to calculate the inverse of the matrix ‘A’.

For a square matrix to be invertible, it should be a full row rank that means the rank of the matrix is equal to the size of the matrix.

But the rank of the given matrix ‘A’ is 2, which is not equal to its size (8).

>> rank(A) 
ans = 
2 

To get the inverse of the matrix, which corresponds to a manual calculation, you can use the ‘vpa’ function from the Symbolic Math Toolbox.

For more information on the above-mentioned functions, kindly refer to the following documentation:

For ‘vpa’ function:

https://in.mathworks.com/help/symbolic/vpa.html

For ‘rank’ function:

https://in.mathworks.com/help/symbolic/rank.html

[Martin]

Thank you!

Is it possible that the rank calculation is also a numerical problem?

Perhaps I will briefly explain how I arrived at this matrix.

I have an analytical model. Into this 5 parameters go.

The function which describes the model I call now times as Z

and the 5 parameters are called: u,v,w,x,y

I calculate my matrix now in such a way:

A = [dZ/du * dZ/du dZ/du * dZ/dv dZ/du * dZ/dw dZ/du * dZ/dx dZ/du * dZ/dy;

dZ/dv * dZ/du dZ/dv * dZ/dv dZ/dv * dZ/dw dZ/dv * dZ/dx dZ/dv * dZ/dy;

dZ/dw * dZ/du dZ/dw * dZ/dv dZ/dw * dZ/dw dZ/dw * dZ/dx dZ/dw * dZ/dy;

.....]

In the main diagonal, the derivatives are the parameter times the derivative after the parameter.

In the secondary diagonals the derivatives are crosswise. Thus this symmetrical matrix results.

In my opinion it should have theoretically full rank.

[Torsten]

I guess that your model parameters depend on each other.

In the simplest case, imagine you want to determine two parameters a and b to fit a function of the form

y = (a+b)*x

Now this is a model with two parameters, but you can fit only one, namely (a+b).

So the model can be reduced to

y = c*x

with only one parameter c to be determined.

If you compute the Jacobian of the "overdetermined" model in a and b, you will see that it is singular.

[Askic V]

I would also investigate if I can use SVD.

Please have a look at this material:

https://pillowlab.princeton.edu/teaching/mathtools21fall/notes/notes_SVD.pdf

Answer 2

[Walter Roberson]

Your array is sensitive enough that it matters that your text entries do not represent the full double precision values stored in the variables. When you use format long g to display a variable, it displays 15 significant digits, not the 16 (sometimes 17!) needed to fully resolve a decimal number to binary. For example the decimal value -0.495586332063353 (row 4 column 3) is displayed the same for -0.495586332063353 .* (1+(-4:4)*eps)

format long g
A= [-1.14420182157714e-05	3.97106112979063e-09	1.71528761347565e-09	3304.87922280846	-6.43946234919730;
    3.97106112979063e-09	-7.42484390057452e-13	-5.95484638418163e-13	-0.618994914206821	5.75841022133480e-05;
    1.71528761347565e-09	-5.95484638418163e-13	-2.57140914558130e-13	-0.495586332063353	0.000965957682109416;
    3304.87922280846	-0.618994914206821	-0.495586332063353	-516040787150.579	51574737.9623166;
    -6.43946234919730	5.75841022133480e-05	0.000965957682109416	51574737.9623166	3814318.88267676];

Ainv = inv(sym(A))

Ainv = 

Ainvd = double(Ainv)

Ainvd = 5×5

1.0e+00 * 1.33590335999762e+17 3.77238579649947e+19 8.9047199885406e+20 -44867666.7136091 61194294.0940845 3.77238579649947e+19 1.10246067006822e+22 2.51456285559039e+23 -13116744146.7408 17534675738.3868 8.9047199885406e+20 2.51456285559039e+23 5.93561184534913e+24 -299074847846.53 407902985402.706 -44867666.7136091 -13116744146.7408 -299074847846.53 0.0156059553946884 -0.0208584605760802 61194294.0940845 17534675738.3868 407902985402.706 -0.0208584605760802 0.0281524909870653

Ainv * A

ans = 

Ainvd * A

ans = 5×5

1.0e+00 * 1.00015413761139 2.33499122259673e-08 1.96378096006811e-08 -24301.5 37.96875 0.001556396484375 0.999996757716872 -1.00880861282349e-05 -4008704 -4328 0.34765625 3.042072057724e-05 1.00002479553223 82333696 134912 3.19189119579733e-15 6.62252709810775e-18 7.77237432400546e-18 1.00001826742664 -2.25118128582835e-08 4.73787675758786e-14 -4.00288712631069e-17 -1.96477762445108e-17 -1.6792444512248e-05 1.00000004631875

[Martin]

Thank you very much!