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# Questions tagged [numerical-linear-algebra]

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

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### Solving Crank-Nicolson type nonlinear vector equation

Given is the following (Crank-Nicolson discretized) equation, \mathbf{v}_{n+1}+\alpha\Big[\mathbf{A}_{n+1}\,\mathbf{v}_{n+1}-\mathbf{v}_{n+1}\left(\mathbf{v}_{n+1}^*\,\mathbf{A}_{n+1}\...
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### Solving a linear system in matrix notation

I'm trying to implement the paper "Conformation Constraints for Efficient Viscoelastic Fluid Simulation": https://www.gmrv.es/Publications/2017/BGAO17/SA2017_BGAO.pdf In implementing eqn. 6, they ...
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### Reducing a Least sqare with equality and inequality constraints problem.

First, i know that there are a lot of topic about thoose kind of questions, but i thought about a reducing approach that i did not found in the litterature and i wander if the following steps are OK ...
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### importance of Krylov subspace

Can someone please explain to me why we are using Krylov subspaces for the CG-Method, GMRES-Method and Arnoldi. Unfortunately I do not see where the advantages are und why Krylov subspaces are so ...
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### Influence of Condition Number on Arnoldi Method

I am currently working with finite elements. Goal is to find the first n eigenvalues and eigenvectors of a generalized eigenvalue problem. After some research I found the Arpack package (I am using ...
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### Can the power method give the spectral radius of a nonnegative asymmetric matrix?

I have a large sparse nonnegative asymmetric matrix $A$. Since the matrix $A$ is nonnegative, its spectral radius $\rho(A)$ is an eigenvalue of it. But $A$ may have other eigenvalues being the same ...
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### Can we anything say about $y^T y$ if we know $X^T X$ and $X^T y$

Let $y \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times p}$, where $n > p.$ We don't know the matrix X, but assume we do know $X^T X$, and make any necessary assumptions about its rank. Assume ...
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### What is the complexity order of Kalman rank condition?

Does computing the Kalman rank condition of an integer matrix have complexity polynomial in the size of the input? if yes what is the order of complexity? For a discrete-time linear state-space ...
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### Adding identity to invert matrix

I'm looking into algorithm implementation which is essentially linear regression: $$\|Ax - b\| \rightarrow \min$$ Matrix A and vector b are estimated using data, then we do $A^{-1}b$ to find x. But ...
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### Eigenvalues of sum of two matrices

I have a family of matrices depending on a parameter $\mu$ of the form: $$A(\mu)=A_0+D(\mu),$$ where $D(\mu)$ is a diagonal matrix and $A_0$ does not depend on $\mu.$ The diagonal of $D(\mu)$ is a ...
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### Inverse of a large sparse matrix in Matlab

Background: Let $\Omega$ be the state space of an absorbing Markov chain with $\Omega_a$ being the set of absorbing states, and its complement $\Omega_a^c$, being the set of transient states. The ...
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### Computing/certifying bounds for eigenvalue computations for big sparse 0-1 matrices

I am finishing my PhD, and at some point I give some bounds on let's say a constant $\lambda$. This constant is the maximum eigenvalue of a matrix $M$ with only zeroes and ones (so the Perron-...
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### Positive semidefinite inequality

Let $\mathbf{A}\in\mathbb{C}^{m\times m}$, and $\mathbf{B}\in\mathbb{H}^{m\times m}$ be an $m$-dimension Hermitian matrix, solve $\theta$ that satisfies the condition \begin{equation*} e^{\jmath\theta}...
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We know that doing a full svd for an $n\times n$ real matrix is $\mathcal{O}(n^3)$. What about just computing the leading singular vector, say using the Lanczos algorithm? It's certainly better than $\... 0answers 37 views ### Accurate method for computing the inverse of a matrix I am looking for an algorithm to compute the inverse of a matrix. I have tried some algorithms that by increasing the dimension of the matrix give a bad result. Mathematica's code works properly even ... 0answers 26 views ### Why is the minimun error bounded in Bauer-Fike's Theorem of eigenvalue computation conditioning instead of the maximum one? This is Bauer-Fike's Theorem Let the matrix$ A$be diagonalizable and let$ S $invertible, such that$S^{-1}AS=D=diag(\lambda_1,\lambda_2,...\lambda_n)$Let$\widetilde{A}$be a perturbation of ... 1answer 47 views ### Numerical stability: cannot unitarily diagonalize normal matrices I am trying to setup small numerical experiments to see if a unitary matrix can be unitarily diagonalized thanks to the spectral theorem: https://en.wikipedia.org/wiki/Spectral_theorem (see normal ... 2answers 28 views ### Critical Points of Vector Function Given a matrix$A$that is$n \times d$and a$n$-dimensional vector$w$, define the vector function$f$as:$f(v) = \frac{w^TAv}{||Av||^2}$where$v$is a$d$-dimensional vector. How can I find all ... 0answers 17 views ### Column pivoting criteria: column with maximum value or column with maximum norm I understand column pivoting is a procedure for attaining numerical stability(correct me if I am wrong). I am studying QR factorization now. I have encountered couple of ways of pivoting columns, ... 0answers 19 views ### Optimizing a system of equations to minimize purchasing cost I am new to this (posting on stack exchange and higher level math in general) so please correct me where I am in error. I am trying to minimize the cost to my company for purchasing a certain product. ... 1answer 44 views ### QR Algorithm and tridiagonal matrix Consider$A: n \times n$a symmetric matrix. a) Explain the main steps to perform on Orthogonal transformations so as to to obtain a matrix$T$, tridiagonal, orthogonally similar to$A$. b) ... 2answers 44 views ### Exercise about eigenvalue Consider$A: n \times n$, Hermitian matrix, and the ordering of eigenvalues:$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$. Prove that:$\lambda_1 = \max (x^{H}Ax)$and$\lambda_n = \min (...
I am trying to solve the following system of $n^2$ equations for $X$ $c_i^\top (A_{i,i} B + X)^{-1}(A_{j,j} B + X)^{-1} c_j = I_{i,j}, \forall i,j\in\{1,\dots,n\}$ where $c_i$ is an $n$-dimensional ...
### How to solve $Ax=b$ wihout inverting $A$?
I'm going to solve this equation: $$Ax=b$$ Onto an embedded system with using C-programing language. It need to be fast as possible. Assume that $A$ is not square. One way to solve this is to use: \$...