Questions tagged [numerical-linear-algebra]
Questions on the various algorithms used in linear algebra computations (matrix computations).
3,672 questions
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Inversion of an ill-conditioned triangular matrix
Let $L$ be a lower triangular matrix where all elements on and below the diagonal are positive. The matrix is also highly ill-conditioned. I want to solve $Lx=1$. Additionally, I know that all ...
- 11
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Factorizing a matrix as a product of band matrices
Let $A$ be $n\times n$ matrix. Can $A$ be written as $A=B_1\cdots B_k$ where $B_i$ are band matrices with constant bandwidth, and $k=O(n)$? Is it always possible? Is there an efficient algorithm for ...
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Inequality of Perturbation of Linear System [duplicate]
Problem: Let $A \in \mathbb{R}^{m \times n}$ be a nonsingular matrix. Consider the system $Ax = b$. If $b$ is perturbed by $\delta b$, suppose that the solution of the original system is perturbed by $...
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$LU$ decomposition and Leading principal submatrix [duplicate]
Problem Setup: Let $A \in \mathbb{C}^{m \times m}$ be nonsingular. For each $k$ with $1 \leq k \leq m$, the upper-left $k \times k$ block $A_{1:k,1:k}$ is nonsingular. We want to show that $A$ has an ...
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Condition Number of $A$
Let $A = \begin{bmatrix}
1.0000 & 2.0000\\
1.0001 & 2.0000
\end{bmatrix}$. Suppose we wish to find $Ax = b$, where $b = (3.0000,3.0001)^T$. Instead of $x$, we obtain $x' = ...
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Nozero diagonal entries imply full rank
Let $A$ be an $m \times n$ matrix with $m \geq n$. And let $A = \hat{Q}\hat{R}$ be the reduced $QR$ factorization. Suppose that all the diagonal matrix entries $\hat{R}$ are nonzero. We want to show ...
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I need help with solving a double integral exp of a quadratic variable
I have a double integral, which I want to solve:
$\mathcal{I}_{au}=\int_0^{t'}e^{(a-c)t''}\int_0^{t''}e^{(-b+c)t'''}\text{exp}\big(-\frac{\lambda^2}{2}\left( S_2(t''')^2+S_1(t')^2+S_1(t'')^2+2S_{12}t'...
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condition number of the intersection point of two lines for infinitesimal perturbations
I am given a question that we consider two lines in $\mathbb{R}^2$ given by the equations $y=0$ and $ax+y=b$, with $a,b, \in \mathbb{R}$. We are to compute the intersection point $S$ of the two lines ...
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Calculating jacobians in complicated systems
EDIT NOTE: According to a commenter Ted Black this is in fact correct. I am just going to slightly edit the original post for a slight bit more clarity.
I am going to spare the details of the problem ...
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Computing truncated singular value decomposition (SVD) in alternative inner products
Let $\mathbf{N}\in \mathbb{R}^{n \times n}$ and $\mathbf{M} \in \mathbb{R}^{m \times m}$ be symmetric positive definite matrices. How can we efficiently compute the truncated singular value ...
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Matrix with monomial entries
I am interested in the following matrix
Here $A$ and $B$ are complex numbers.
Are matrices of this type known in the literature? Is there some "obvious" reason why they should have maximal ...
- 868
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Multigrid methods - restriction for even numbered vectors
I've implemented a multigrid V-cycle for solving 1D Poisson equation in MATLAB. It works at least for the examples I've used. There are prolongation and restriction operators that change the grid size ...
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Derivation of Conjugate Gradient Iteration
I’m currently studying Numerical Analysis for the first time and got stuck while working on a problem involving the Conjugate Gradient method.
I’ve tried to consult as many resources as possible, and ...
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Can Krylov subspace methods like Lanczos be used for generalized eigenvalue problems with defective eigenvalues?
I’m studying generalized eigenvalue problems of the form $A x = \lambda B x$, and I’m particularly interested in defective eigenvalues—those with fewer linearly independent eigenvectors than their ...
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Source and information about this algorithm for solving positive-semidefinite system
I encountered the following algorithm (Algorithm 1 in a paper from Stroubolis on generalized finite element methods) for the solution of a the system $Ax = b$ where $A$ is positive semi-definite or ...
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Where does the smoothing property of Jacobi/Gauss-Seidler methods come from?
I'm studying multigrid methods for solving linear systems. The overall goal is to understand them on a satisfactory level and then apply them to 2D Poisson's equation. I'm rather familiar with how ...
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Why can a line search make an inexact Newton method perform more poorly on a linear system of equations?
Why can a line search make an inexact Newton method perform more poorly on a linear system of equations?
I have a problem where I need to solve multiple linear systems of equations where the system ...
- 2,692
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Efficient matrix inversion over $\mathrm{GF}(2)$
Is there an efficient way to find the inverse of an invertible square matrix over $\mathrm{GF}(2)$ (the binary field)? That is, something more efficient than Gaussian elimination.
I've tried looking ...
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Best method to numerically sove a linear constant-coefficient ode
So I have this very simple problem. I want to solve for $\lambda : [0,1] \to \mathbb{R}^n$ (twice differentiable) the following ODE :
$$ \ddot{\lambda} + A\lambda = f $$
with some Neumann boudary ...
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Inverse of sum of two Kronecker products with explicit inverse available
Consider a matrix of the form
$$L = A \otimes I + J \otimes B,$$
where $I$ and $J$ denote identity matrices (of different sizes), and $A$, $B$ are square matrices whose inverses are explicitly known. ...
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SVD Frobenius norm [duplicate]
Let $A \in \mathbb{C}^{m \times n}$ be a matrix with $r$ nonzero singular values, $\sigma_1 \geq \cdots \geq \sigma_r$. Let $u_1, \ldots, u_r$ and $v_1, \ldots, v_r$ be corresponding left and right ...
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Is it possible to create this system of linear equations
Given a system of linear diophantine equations such that:
The number of equations are exactly 1 less than the number of variables.
The equations are linearly independent.
There are 3 variables in the ...
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Defining basis vectors in triclinic lattice (general case)
I have been looking at ways to be able to define basis vectors for a lattice in the most general case when the unit cell lengths $a \neq b \neq c$ and the angles $\alpha \neq \beta \neq \gamma$. What ...
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$(A+I)^{-1}$ when you already have $A^{-1}$
I'm trying to optimize some computations, and the current bottleneck, by like 20x, is a matrix inverse. (Or rather, a solve of an $Ax=y$ system, still $O(n^3)$.) I've managed to factor out and ...
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One column of the inverse of a Toeplitz matrix
Let $\bf A$ be a $n \times n$ non-singular Toeplitz matrix whose first column $\bf c$ and first row $\bf r$ are given. Let ${\bf e}_n$ be the last element of the standard basis for ${\Bbb C}^n$. I ...
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