Assume I want to solve Ax = b for a bunch of different b's, but keeping A fixed. Furthermore, assume A is a tall matrix. The question is, what are the pros and cons of precomputing the pseudoinverse of A and precomputing the QR factorization of A? In the documentation of LinearSolve.jl , for example…

If you have enough right-hand sides, computing and multiplying by the pseudoinverse will be faster. However, you might want to think about how many right-hand sides you have and benchmark your computation of F and invA. Computing the pseudoinverse is slower than computing F. So you might or might…

[image] marcsgil: F = qr(A) If you’re going to use QR for this, I would use qr(A, ColumnNorm()) to use column-pivoted QR, in case A happens to be badly conditioned (nearly dependent columns). That’s what A \ b uses by default. [image] mstewart: Also computing, the pseudoinverse is not b…

[image] stevengj: [image] mstewart: Also computing, the pseudoinverse is not backward stable. Really? Yes. I believe that as far as it goes, that specific statement is correct. But you are right that I was neglecting the truncation in my example. I think it is harder to show backwar…

[image] mstewart: The fact that explicit (pseudo)inverses are not backward stable is something that applies even in the special case of a square invertible matrix. Nevertheless, solving Ax = b by inv(A) * b usually has good forwards accuracy (comparable to using LU factors) and is backwards s…

Agreed. But I think it’s worth noting that even with good forward accuracy, you do typically have larger relative residuals with the inverse when A is ill conditioned and \|x\| \ll \|A^{-1}\| \|b\|. Whether forward accuracy is sufficient even with larger residuals depends on what you are doing wit…

[image] mstewart: I still like to warn people away from explicit inversion unless they really want to worry about the details. Agreed. Explicit matrix inversion (or pseudo-inversion) is only a good idea under relatively rare circumstances, even just from a performance perspective .

Pseudoinverse x Factorization in a repeated linear solve: when to use which?

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stevengj October 6, 2024, 9:41pm 7

Agreed. Explicit matrix inversion (or pseudo-inversion) is only a good idea under relatively rare circumstances, even just from a performance perspective.

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Pseudoinverse x Factorization in a repeated linear solve: when to use which?

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