# Questions tagged [least-squares]

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### Newton-based methods in optimization vs. solving systems of nonlinear equations

I asked for clarification about a recent question about minpack, and got the following comment: Any system of equations is equivalent to an optimization problem, which is why Newton-based methods ...
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### Solving a least squares problem with linear constraints in Python

I need to solve \begin{alignat}{1} & \min_{x}\|Ax - b\|^2_{2}, \\ \mathrm{s.t.} & \quad\sum_{i}x_{i} = 1, \\ & \quad x_{i} \geq 0, \quad \forall{i}. \end{alignat} I think it is a ...
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### Purely rotational least squares match

Could anyone recommend a method for the following least-squares problem: find $R \in \mathbb{R}^{3 \times 3}$ that minimizes: $\sum\limits_{i=0}^N (Rx_i - b_i)^2 \rightarrow \min$, where $R$ is a ...
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### Least squares approximation question

I am taking a course on scientific computation, and we just went over least squares approximation. My question is specifically about approximating using polynomials. I understand that if you have n+1 ...
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### Fitting one set of points to another by a rigid motion

I'm not really sure how to explain this problem clearly, so please bear with me. I have a basis of 3 orthonormal unit vectors and a position, a standard 4x4 transform matrix in computer graphics. ...
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### Least Squares and Fourier Series

I have a little bit of problem figuring out the relation between Fourier series and Least Squares. As far as I understand, LS is a way of minimizing the quadratic error between a measured value $y_i$ ...
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### “Geometry of ill-conditioning” for least-squares problems

It is an idea that dates back to Demmel, 1987 that the condition number of a problem is often related to the distance to the closest ill-posed problems. In Section 3 of the above paper, the author ...
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### Least Squares: Numerically, is solving normal equations okay for nice matrices?

I have to solve a least squares problem: $$x=\arg \min\|Ax-b\|$$ where $A$ is a $m\times n$ matrix, $m>n$, and $b\in\mathbb{R}^m$. I always thought that doing this via QR factorization is ...
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### How to determine whether two cylinders intersect or not?

Considering any two cylinders, defined as: the center of their bottoms $A_i$, the radius of their bottom $R_i$, the unit vector $W_i$ of their axis direction, and the length $L_i$ of the cylinders, ...
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### Large scale triangular least squares

I have to solve the following least squares problem: $$\| \left[ \begin{smallmatrix} \mathbf{L} \\ \mathbf{I} \end{smallmatrix} \right]\mathbf{x} - \mathbf{b} \|_2^2$$ ...
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### Pseudoinverse of perturbed matrix

How does the pseudo inverse of a full column rank matrix change if I rescale a single row? In more detail the problem is the following: We have a fixed matrix $V$ with linear independent columns and ...
142 views

### Solve ill-posed linear system without transposing matrices?

I am attempting to use an iterative solver to solve $p$ in $$Jp = -r$$ where $J$ is an $m\times m$ matrix ($m$ is in the order of $10^5$ and never explicitly stored). $J$ is a dense matrix ...
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### Nonlinear least-squares solvers vs. generic minimization

A nonlinear least-squares problem with $F:\mathbb{R}^m\to\mathbb{R}^n$, $$F(x) \to \min_x \quad (\text{in the least-squares sense})$$ really means minimizing $$\frac{1}{2} \|F(x)\|^2 \to \min_x.$$ ...
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### Best solver/preconditioner for least-squares finite element method

I have seen a lot of literature, lecture videos, etc. on solvers/preconditioners for non-symmetric and/or indefinite systems. However, now I want to solve the mixed poisson/Darcy equation using the ...
578 views

### Complex least-squares problem

Having a matrix $\mathbf{A} \in \mathcal{C}^{m\times n}$ I solve following least-squares problem $$Re(\mathbf{A}^H \mathbf{A})x=Re(\mathbf{A}^H\mathbf{b}).$$ If the matrix $\mathbf{A}$ was a real ...
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### Factorization for reweighted least squares

I am solving a problem using an iteratively-reweighted least squares method: http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares Essentially this requires solving a number of least-...
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### Non-negative least squares with very small numbers

(I have asked this question on StackOverflow previously but it has been pointed to me that CSSE or MSE could be more appropriate) I have to solve a constrained optimization problem of the following ...
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### Nonlinear least squares optimized Jacobian calculation

I have a nonlinear least squares problem, in which I am trying to minimize residuals which can be divided into four classes: $$\min_x ||\epsilon(x)||^2 + ||\xi(x)||^2 + ||\delta(x)||^2 + ||s(x)||^2$$...
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### Using MINPACK for curve fitting: implementation?

I need to implement a non-linear fitting algorithm in Fortran and chose to use MINPACK's flavor of the Levenberg-Marquardt algorithm as a basis for the least-squares stuff. However, I seem to ...
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### Cholesky decomposition vs LDL decomposition

In different books and on Wikipedia, you can see mentions of Cholesky decomposition and only sometimes of LDL decomposition. As far as I understand, LDL decomposition can be applied to a broader ...
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### Solving multiple least-square problems with the same constraints

The following least-square problem can be solved efficiently (e.g. using matlab's lsqlin): $$\vec{x}^*=\arg\min_\vec{x} ||C\vec{x}-\vec{t}||^2\,\ s.t.\ Ax \le \vec{b}$$ where the parameters of the ...
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### indirect method for least-squares with inequality constraints

I aim to find $x \in \mathbb{R}^n$ that $\min_x |D \cdot F \cdot x|^2$ subject to $x_i = X_i$ and $x_j \geq X_j$ , $i \in I, j \in J$ and I and J partition ${1\cdots N}$ into two sets. it is ...
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### How do you formulate the linear least-squares method for radiometric calibration?

In Debevec and Malik (mentioned similarly in Forsyth and Ponce's Computer Vision: A Modern Approach) they highlight a method of solving the camera response function using linear least-squares. We ...
For a given set of data points $$\{(x_i, y_i, z_i)\}$$ there exists some $$f_{ABC}(x,y)=Ax^2+Bxy+Cy^2$$ that minimizes $$\sum_i(f_{ABC}(x_i,y_i)-z_i)^2$$ $A$, $B$, and $C$ can be found quickly ...