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If you will be adding lots of rows, then you will want to use the tall skinny QR algorithm (TSQR) of Demmel et al, 2008, https://arxiv.org/abs/0806.2159 This algorithm can be combined with the level …

Since the $f_i(\theta,\phi)$ are linear combinations of spherical harmonics, we can write $$ \mathbf{f} = F \mathbf{Y} $$ where $\mathbf{Y}$ is a vector of the orthonormalized spherical harmonics - i. …

I need to solve many least squares problems with the following matrices: $$ \pmatrix{ R \\ D_i } $$ where $R$ is upper triangular and $D_i$ is diagonal. $R$ is the same for all the problems, while $D_ …

Consider the regularized least squares problem $$ \min_x || b - A x ||^2 + \lambda^2 ||x||^2 $$ which is equivalent to $$ \min_x \left|\left| \pmatrix{b \\ 0} - \pmatrix{A \\ \lambda I} x \right|\righ …

If we assume that $D$ is nonsingular, then there is a relatively straightforward (and efficient) solution based on an $LU$ decomposition. If we write $$ \pmatrix{D & B \\ B^T & A} = \pmatrix{ L_{11} & …

I have a nonlinear least squares problem, in which I am trying to minimize residuals which can be divided into four classes: $$ \min_x ||\epsilon(x)||^2 + ||\xi(x)||^2 + ||\delta(x)||^2 + ||s(x)||^2 $ …

For a linear least squares problem, it is possible to define a resolution matrix, relating the estimated model parameters to the true model parameters. If we are solving a regularized problem, $$ \min …

I want to compute the matrix $$ A = \sum_{i=1}^N v_i v_i^T $$ where each $v_i$ is a given vector of length $2500$, so that $A$ is $2500 \times 2500$, and my $N$ is about 2 million. Rather than call DS …

Consider the Laplace equation, $$ \nabla^2 f(r,\theta,\phi) = 0 $$ in spherical coordinates. We know that the solution to this equation can be derived analytically, and is given by, $$ f(r,\theta,\phi …

Can anyone recommend a good software for solving generalized symmetric eigenvalue problems of the form, $$ A x = \lambda B x $$ where $A,B$ are symmetric and sparse, and $B$ is positive definite? I ha …

I have a set of measurements $y_i$, $1 \leq i \leq N$, and I want to model these measurements with a linear model. I have two possible models I can use, $$ y \approx A c $$ and $$ y \approx B d $$ whe …

I am fitting some B-splines to data, but the data has a "gap" region where the spline is less constrained by the data. I want to devise a regularization scheme to help prevent the spline from deviatin …