# Tagged Questions

Questions on the various algorithms used in linear algebra computations (matrix computations).

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### Computing the SVD factorization on C++ (using the proof of the existence of the SVD factorization)

I am doing a C++ program that computes the SVD factorization of a real matrix A without using any known library of algebra that contains the implementation. In addition, QR descomposition is not ...
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### How to solve linear system of form $(A \otimes B + C^{T}C)x = b$ when $A \otimes B$ is too large to compute?

For the given linear system: $$(A \otimes B + C^{T}C)x = b$$ where $\otimes$ is the Kronecker product, $A$ and $B$ are dense and symmetric positive-definite, and $C^{T}C$ is a sparse symmetric block ...
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### Simulating from a Multivariate Gaussian without Cholesky

I'd like to draw a sample from a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$, where $\mu$ is the mean vector (can assume it to be $\boldsymbol{0}$), and $\Sigma$ is a sparse positive ...
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### Express Lagrange polynomial in term of Cauchy matrix

Given 2n distinct real numers $s_1,s_2, \dots, s_n$ and $t_1, t_2, \dots,t_n$ define the $n \times n$ Cauchy matrix $C = C(t,s)$ by $C_{ij} = \frac{1}{t_i - s_j}$. Express the Lagrange interpolation ...
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### Lagrange multiplier for more than one constraints.

How to minimize $x^TAx$ over the set $D=(x\geq 0, x^TBx=1$ and $(I-A^\dagger A)x=0$), where $A$ is copositive matrix of order $n-1$ and $B$ is strictly copositive matrix of order $n$. If I drop the ...
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### Solver for sparse linearly-constrained non-linear least-squares

Reposted from stackoverflow on the advice of Nick Rosencrantz: Are there any algorithms or solvers for solving non-linear least-squares problems where the jacobian is known to always be sparse, and ...
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### Why is Cholesky factorization numerically stable

It's often stated (eg: in Numerical Recipes in C) that Cholesky factorization is numerically stable even without column pivoting, unlike LU decomposition, which usually need pivoting schemes. But I'...
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### What's the state of the art for computing the largest singular value of a matrix

My matrix is not sparse, and is sized 30k by 30k. Most importantly, the gap between the largest and the second largest singular values is small or even 0. ARPACK, SLEPc, Matlab, PROPACK? Which ...
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### Numerical methods to find eigenvectors with 0 eigenvalue

I'm curious if there's any numerical way of directly finding the eigenvectors with eigenvalue 0. If I didn't have to do it directly, I would probably do it like this in pseudocode: ...
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### Round-off: cross- vs. dot-products

I have three vectors e0 = numpy.array([1.0, 0.0, 0.0]) e1 = numpy.array([-0.5, 1.0e-4, 0.0]) e2 = numpy.array([0.5, 1.0e-4, 0.0]) (Edges that form a very flat ...
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### Is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay

I'm studying for a test and I'd like to know is it possible to design a strongly stable linear multistep method of order 7 which has stiff decay. I have no clue to verify the claim. Can anyone give me ...
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### LU Decomposition vs. QR Decomposition for similar problems

Suppose I want to solve the 2D Poisson equation with Neumann boundary conditions. The solution is non-unique up to an additive constant. I have previously asked a related question here for the 1D ...
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### Eigenvalues of a sum of matrices given eigenvalues of different sum

Firstly, what I want are the eigenvalues of a sum of matrices $(A + C)$. I am not asking how to express them in terms of the eigenvalues of the summands*. What I am hoping for is that there may be ...
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### Why use the logarithm of the relative error?

In my numerical analysis course, we had an assignment to use MATLAB to numerically solve the Poisson Equation $-\nabla\cdot\nabla u = 0$ in one dimension. We computed the numerical solution, plotted ...
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### Looking for matrices such that $\kappa(A) =1$

Looking clues for this problem. Find all the matrices such that $\kappa(A) = 1$ We define $\kappa(A) = \|A\|\,\|A^{-1}\|$. If I'm looking matrices such that $\kappa(A) = 1$, I was thinking in ...
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### Number of Arithmetic Operations in Gaussian-elimination/Gauss-Jordan Hybrid Method for Solving Linear Systems

I am stucked at this problem from the book Numerical Analysis 8-th Edition (Burden) (Exercise 6.1.16) : Consider the following Gaussian-elimination/Gauss-Jordan hybrid method for solving linear ...
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### Fastest way to find linearly independent columns of a matrix

Given a rectangular matrix $X$ of size $n\times m$ with $m>n$, what is the fastest way to find the linearly independent coloums. Robust methods like SVD or RRQR decompostion have complexity of ...
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