A problem with pseudo inverse

asked 2023-02-09 11:57:41 +0100

Cyrille
1409 ●35 ●77 ●140

I have defined the following function

 def pseudo_inverse(mat) :
        from scipy import linalg
         return matrix(linalg.pinv(mat))

From the eye it works correctly. But if I apply it to

M = matrix(RR,2,3,(1,2,2,4,5,6))

that is

M+=pseudo_inverse(M)
M*M+*M == M

returns False (when I can see that it is formaly true but not operationaly). I think I understand where is the problem but even in changing the ring ( with the help of the function change_ring() I cannot obtain the desired identity.

answered 2023-02-09 18:13:20 +0100

Max Alekseyev

6560 ●9 ●47 ●155

You cannot use M+ as a variable name. In fact, M+=... is interpreted as incrementing M by what is given in the right hand side. Try this out:

Mp = pseudo_inverse(M)
M * Mp * M == M

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Comments

Additionally, you should not check the equality of real numbers. You should check if their absolute difference is smaller than the error limit that you will define (for example, 10^-7). You can display M*Mp*M and M to see what I mean.

tolga ( 2023-02-10 07:08:11 +0100 )edit

To complement @tolga's remark :

RR is not $\mathbb{R}$, but a (pseudo-) ring of floating-point approximations. Example :

sage: 3.17.parent()
Real Field with 53 bits of precision

The best you can do to declare a variable as real without giving it a value is to declare it as such. Compare :

sage: sqrt(x^2).simplify()
sqrt(x^2)
sage: var("y", domain="real")
y
sage: assumptions()
[y is real]
sage: sqrt(y^2).simplify()
abs(y)

For numerical values, AA allows for exact representation of algebraic values. Compare :

sage: cos(pi/6)
1/2*sqrt(3)
sage: AA(cos(pi/6))
0.866025403784439?
sage: RR(cos(pi/6))
0.866025403784439

The ? ending the representation of the AA value denotes the "arbitrary precision".

HTH

Emmanuel Charpentier ( 2023-02-10 08:32:39 +0100 )edit

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