Questions tagged [numerical-linear-algebra]

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

868 questions
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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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Conditioning of the linear systems in the inverse or Rayleigh quotient iteration algorithms

I'm working through the book Numerical Linear Algebra by Trefethen and Bau. In Lecture 27 (and exercise 27.5), the following claim is made about the inverse iteration algorithm: Let $A$ be a real, ...
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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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Numerically robust 2x2 determinant?

How can the determinant of a 2x2 matrix $$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = a d - b c$$ be computed in floating point without suffering unnecessary catastrophic cancellation? ...
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Maximizing the pairwise Frobenuis distance between M othrogonal matrices

I want to maximize the pairwise Frobenius distance between $M$ orthogonal matrices. That is, I'm looking for $Q_{i}, i = 1, 2, ... M$ such that \begin{equation*} \begin{aligned} & \underset{ 1 \...
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IEEE 754 as a mathematical space

Integer operations in computers (i.e. 32-bit integers) probably can be represented best by modular arithmetic (because of integer overflows/underflows). What about IEEE 754 floating point arithmetic? ...
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Find the eigenvector with maximum overlap

Given a large symmetric matrix $A$, there are methods to find the largest or smaller eigenvalue, or the eigenvalue closest to some initial value. Is there any method to find the normalized ...
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Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum\limits_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on ...
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Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$\Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}.$$ Applying the Woodbury matrix identity ...
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What is the significance of the matrix in the LAPACK logo?

This is the LAPACK linear algebra library logo: What is the significance of this matrix?
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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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A new way to generate Triangular Numbers…interesting?

An open ended question... In my work in aerodynamics, lifting line theory specifically, there is a matrix $Q$ that relates vortex strength to wing downwash. While fiddling with the problem and ...
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lower bound on the norm of Hadamard product

Suppose $A \in \mathbb{R}^{n \times n}$ is a nonsingular matrix, $D_1, D_2 \in \mathbb{R}^{n \times n}$ are both diagonal matrices with decreasing positive real values on the diagonals. Can you ...
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Efficient way to rigorously learn AI prerequisites

Question: My formal goal is to be able to rigorously understand the mathematical basis for modern statistical learning methods (ML, deep learning). I am told by math people that this involves: linear ...