Hi! Is there any specific module in Julia for performing linear algebra operations like Gram-Schmidt orthogonalization for a given set of vectors?

[image] pafloxy: Specifically how exactly do I obtain the orthonormalized vectors from the given set of vectors? Do using LinearAlgebra to load the LinearAlgebra module, then do Q = Matrix(qr(A).Q) where A is a matrix whose columns are the vectors you want to orthonormalize. This yields a ma…

qr in the LinearAlgebra standard library does this. (Technically, Gram–Schmidt is just one possible algorithm for QR factorization ; in practice linear-algebra libraries tend to use different algorithms instead.)

Hi! thanks for the reply, however, could you be a bit more elaborate? Specifically how exactly do I obtain the orthonormalized vectors from the given set of vectors? P.S ~ I knew nothing about QR decomposition to this point.

Thanks for clarifying! I’m new to Julia and didn’t knew this stuff before now

No problem. This is not specific to Julia, but a lot of introductory linear-algebra courses don’t emphasize the connection of Gram–Schmidt to QR factorization. Often, introductory linear algebra classes tend to emphasize simple (perhaps simplistic) algorithms for hand calculation, whereas practica…

here is an implementation : Gram–Schmidt Orthogonalization with Julia - Julia Community 🟣

[image] VinodV: here is an implementation : Gram–Schmidt Orthogonalization with Julia - Julia Community :purple_circle: Note that this is “classical” Gram Schmidt (CGS), which is is numerically unstable (unless you do it twice ). There is a simple variant called “modified Gram Schmidt” (MGS)…

There is also a fast orthonormalization via the Cholesky decomposition. Though one must be a bit careful, it’s not as numerically stable as the QR method. using LinearAlgebra A = rand(5,3) # column vectors of length 5 U = cholesky(A'A).U B = A*inv(U) # orthonormal column vectors spanning the sam…

[image] sgaure: There is also a fast orthonormalization via the Cholesky decomposition. Note that this is not faster in the complexity sense — all of these algorithms are \Theta(mn^2) for an m \times n matrix A. But it is probably faster in actual time (i.e. a better “constant coefficient” …

I don’t think there is much written about the stability of the Cholesky approach, perhaps because it isn’t taken very seriously as a useful algorithm. There is a throw-away comment in a fairly well cited technical report suggesting that it is better than classical Gram-Schmidt without reorthogonali…

Gram-Schmidt decomposition

New to Julia

ctkelley April 11, 2024, 4:16pm 9

This has come up before. You might look at this and this

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Gram-Schmidt decomposition

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