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matrixvectorlinear-algebraequalityscilab

Why this equality returns False on Scilab?

I am learning Scilab for educational purposes.

When dealing with Gaussian Elimination we have that Ax=b. "A" is the matrix with the coefficients from the linear equations, "x" is the vector with the variables and "b" is the vector with the "solutions" from the linear system.

Let's suppose A is:

    10.  - 7.      0.  
  - 3.     2.09    6.  
    5.   - 1.      5.

And b is:

7.    
3.91  
6.

Using the Gaussian elimination inside Scilab, we get the values of x as:

  - 3.109D-14  
  - 1.         
    1.

Obs: - 3.109D-14 is a number close to zero.

When I do the matrix multiplication as A*x, I get a 3x1 vector with the same components as vector "b":

enter image description here

Finally, my doubt is: if A*x has the same values as "b", why the equality bellow returns False in Scilab?

enter image description here

Solution

  • It is probably due to the rounding error, and the inherent limitations of the floating point math, as indicated by Normal Human in his comment.

    Another thing to remember: the Scilab console outputs rounded values, so internally there are usually more decimal places that are not shown, unless asked explicitly, e.g. mprintf("x= %.30f", x(1));

    To circumvent this problem, you can try to clean your array, that is, round the very small enties to 0. Try to see the result of:

    A*clean(x)==b