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3

Most people use (algebraic as well as geometric) multigrid as preconditioners these days. It's an empirical observation that that leads to faster convergence in terms of iterations, given that in a typical CG iteration, applying a multigrid preconditioner takes up the vast majority of the effort compared to all other operations. In other words, you get the ...

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in general there is no formula for updating the cholesky factorization of the covariance matrix when the hyperparameter is changed. There are other techniques for speeding up the computations though. For instance the method SKI (http://proceedings.mlr.press/v37/wilson15.pdf) can be used to compute matrix-vector products with the covariance matrix. Lanczos ...

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Fortunately, LAPACK provides routines to deal with the $\mathbf Q$ factor from the $\mathbf A = \mathbf Q \mathbf R$ decomposition, [dgeqrf]. To find the projection of an arbitrary $\mathbf B$ onto the space orthogonal to $\mathrm {range}(\mathbf A)$, you want to form $\mathbf C = \left(\mathbf I - \mathbf Q \mathbf Q^T\right) \mathbf B$. Here are two ...

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Basic linear algebra states that $\det(I-S)$ must be non-zero so that a solution to your linear system exists. On the other hand, if your determinant is (numerically) zero, the basis vectors in your coefficient matrix $(I-S)$ do not span the whole vector space, so that you can't construct a general right-hand side $b$. Think of this picture: In the example, ...

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