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Suppose I have a linear model depending on 2 sets of parameters $a,b$ $$ z(t) = \sum_i a_i \Phi_i(t) + \sum_j b_j \Psi_j(t) $$ Now suppose my data vector $z$ can be naturally divided into 2 sets: $x,y …

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 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_ …

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 …

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 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 $ …

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 …

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 …

I am solving nonlinear least squares problems with the normal equations approach, so on each iteration, I need to solve: $$ J^T J \delta = -J^T f $$ for the step $\delta$, where $J$ is a large (millio …

Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$: $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ This can be done with …

Consider the least squares problem, $$ \min_{\mathbf{a},\mathbf{b}} || \mathbf{f}(\mathbf{a},\mathbf{b})||^2 $$ where $\mathbf{a},\mathbf{b}$ represent the unknown parameters to be found. In my proble …

Can someone recommend the best way to solve a least squares fitting problem with B-splines, with additional equality constraints? I want to solve: $$ \min_x || b - A x ||^2, \textrm{subject to: } C x …

This is a continuation of the question asked here. I want to solve numerous least squares systems of the form $$ D_i A x \approx D_i b $$ where $D_i$ are $m \times m$ diagonal matrices with positive e …