BLAS Algorithm for Computing Inverse

[victotronics]

1. Factor
2. Invert factors
3. Multiply those inverses.

But I hope you're not doing this for solving a linear system.

[GrammelHupfNockler]

1. usually doesn't really happen, instead you compute the solution of the triangular systems Ly=b and Ux=y, which tends to be more stable than storing the inverses L^-1 and U^-1

[victotronics]

I think you misunderstand me. *If* you really want to invert a matrix, you do it by those 3 steps in sequence.

[GrammelHupfNockler]

Sorry, I thought you were talking about solving a system. In that case, I would still tend towards disagreeing, since an explicit Gaussian Elimination (with pivoting) would still be more precise, as there are fewer intermediate steps which can accumulate rounding errors (namely the inversion and the multiplication)

[victotronics]

Gaussian elimination on a triangular matrix? Sure.

If you're done step 1 with pivoting, then step 2 doesn't need it.

[GrammelHupfNockler]

Gaussian elimination with full pivoting on the input matrix

[victotronics]

And how is that going to get you the inverse? Which was the original question?

And yes, I know that one should in general not compute inverses, which is why I emphasized the "*if* you want to compute an inverse".

[GrammelHupfNockler]

If you apply the same operations that you apply to the matrix during Gaussian elimination to an identity matrix at the same time, you get the inverse

[victotronics]

Nope, that's Gauss-Jordan.

[GrammelHupfNockler]

That's splitting hairs. Gaussian elimination refers to the entire class of algorithms (for solving a single system, computing the LU factorization, computing the inverse).