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3

tch already gave you a good answer, but I have a suggestion for a simpler problem that you could address pretty easily and which should help you sanity check your results. Let $$M = \sum_i\lambda_iv_iv_i^\top$$ be the eigenvalue decomposition of $M$ and furthermore take the eigenvalues to be sorted in decreasing order: $\lambda_1 \ge \ldots \ge \lambda_n$. ...


6

This is a slightly modified version of my response on math.stackexchange. One standard approach to computing matrix functions times a vector $f(M)x$ or quadratic forms $x^Tf(M)x$ when $M$ is symmetric is via the Lanczos algorithm. Lanczos computes an orthonormal basis $Q_k = [q_1, \ldots, q_k]$ for Krylov subspace $\operatorname{span}(x,Ax,\ldots, A^{k-1}x)$ ...


3

For symmetric, more generally, normal, matrices there is no danger in diagonalizing a matrix to evaluate a matrix function, because all eigenvectors are perfectly conditioned. You should get answers with errors close to the machine epsilon times the condition number of the function evaluation problem. This is as accurate as you can get in floating point. ...


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