I am multiplying a matrix $A$ with vector $p$. However, the matrix $A$ isn't accurate. Some (a very small fraction) of the element's value is changed from $a_{i,j}$ to {0,$-a_{i, …

In usual formulation of conjugate gradient algorithm initial search direction is taken to be the residual (so residual and search direction spans Krylov subspace). However, in case …

I am interested in cases where Conjugate gradient works much better than GMRES method. In general, CG is preferable choice in many cases of SPD because it requires less storage an …

Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the …

It is well known that Gaussian elimination without pivoting is numerically unstable, and in practice Gaussian elimination is done with row pivoting (partial pivoting). A theorem of …

What are the standard methods of computing the rank-k truncated SVD of large dense matrices? My literature search yields results only for large sparse matrices. I assume for k sma …

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} v_{1,0}& …

I have a large $m\times n$ data matrix $A$, $m>n$, and response $m$-vector $b$. I need to calculate $E = ||Ax-b||_2$ as quickly as possible, where $x$ is the least squares solutio …

Solve the following nonlinear equations for $v$ and $w$ $Avv^TAw=\lambda_1v+\lambda_2w$ $Aww^TAv=\lambda_1w+\lambda_2v$ $v^Tw=w^Tv=0$ $v^Tv=w^Tw=1$ where $\lambda_1, \lambda_ …

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $M$. $M$ is a Laplacian matrix, and it has the f …

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality o …

Solve the following nonlinear equations for $v$ and $w$ $Avv^TAw+Bvv^TBw=\lambda_1v+\lambda_2w$ $Aww^TAv+Bww^TBv=\lambda_1w+\lambda_2v$ $v^Tw=w^Tv=0$ $v^Tv=w^Tw=1$ where $\la …

The Cholesky factorization is the classic test to check if a matrix is positive definite. In infinite precision it is also an exact test: A matrix has a Cholesky factorization iff …

It is well-known that Chebyshev polynomials are the polynomials of minimal $L_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L_\infty$ polynom …

Let $\Sigma$ be a hermitian positive definite matrix and $L$ be it's Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda …