Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [numerical-linear-algebra]

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

Questions tagged with this tag can be about, but not limited to:

  1. Matrix decompositions like SVD, QR, Cholesky, etc.
  2. The solution of linear systems and least squares problems.
  3. Analysis of numerical linear algebra algorithms like condition numbers and stability analysis.
  4. Eigenvalue problems.
  5. The designs of direct or iterative methods to solve linear systems.
3541 questions
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Spectral radius and iterative method convergence

I'm trying to show that if the spectral radius of $R$, $\rho(R)\geq 1$, then there exist iterations of the form, given $\mathbf{x}_0$, $\mathbf{x}_{n+1}=R\mathbf{x}_n+\mathbf{c}$ Which do not converge. Let $(\lambda, \mathbf{x}_\lambda)$ be the…
AlmostSurely
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LD$t(L)$ factorization and eigenvalues

A positive definite matrix $A$ can be factored in to $LDt(L)$form. Is the statement the eigenvalues of $A$ are the diagonals of $D$ true? If so , how to prove it?
user133834
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Formatting Linear equation

How do I craft a linear equation so that it is in the form of $ax + bx + c = 0$ where $a^2 + b^2 = 1$ if I have two points? I know how to get it into the form $ax + bx + c = 0$ but I can't figure out the algorithm for satisfying the second…
Chancler
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Numerical Linear Agebra

how to Prove the backward stability of the inner product…
Aditya Sai Gundu
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How to make a function lie in the interval [0,1]

Is there a way to convert a function g(x) so that the result lies between [0,1]? Thank you in advance.
emy
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What is a good unstructured matrix solver?

If I were to hand you a general unstructured matrix A and a right hand side b, what would be your preferred iterative solver for solving Ax=b? Why?
Matt
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Estimated time required to apply to matrix

I'm completely lost on this question, any help would be appreciated. Suppose the application of the Gaussian Elimination algorithm on a 50 by 50 matrix is timed at 500 μ seconds. How much time do you estimate would be required to apply the algorithm…
oremor94
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function of matrix and eigen values

I want to calculate exp(A), A is matrix, with numann series. is this series depend of matrix's eigen values? for example if it's eigen values are large, is numann series useful for this function?
user115927
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only calculate diagonal of cholesky decomposition

I have a massive matrix $A$ that I can't hold entirely in memory, but it is possible to easily calculate individual entries ($A(i,j)$). I'm only interested in calculating the diagonal entries of the Cholesky factor of the matrix. Any way to do this…
not_even_wrong
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Are Similar Matrices and Unitary Property related?

Recall that 2 matrices $A, B\in R^{n,n}$ are similar if there exists a matrix $P$ such that $A=P^{-1}BP$. In this case is $P$ always orthogonal?
dresden
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Find upper Hessenberg by Householder transformation

I have a matrix that looks like this: $$ \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \epsilon & 0 & 0 & 0 \\ \end{pmatrix} $$ Compute a Householder transformation so that the matrix becomes an Upper Hessenberg. Is…
dresden
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Let $k$ and $w$ be digits and let $X$ be some positive integer with one or more digits. Using the two digits, $kw_7$ is a two digit base 7...

I couldn't fit the whole question in the title so here it is re-typed. Let $k$ and $w$ be digits and let $X$ be some positive integer with one or more digits. Using the two digits, $kw_7$ is a two-digit base 7 number and $wk_9$ is a two-digit base 9…
Adhrit Mahesh
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How do define an integral of a matrix valued function?

Let $g:\mathbb{C} \mapsto \mathbb{C}^{n \times n}$. How does one define the integral $$\int_\Gamma g(z) dz,$$ for some closed curve $\Gamma \subset \mathbb{C}$? Is it understood entry-wise or as an special case of the Bochner Integral? I'm trying to…
Duncan Idaho
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Why are Householder reflectors used in numerical linear algebra if the outer product isn't stable?

The Householder reflector $H = I - 2vv^T$ contains an outer product with $v$, which is not backwards stable. Why are Householder reflectors still used -- wouldn't that lack of stability cause an issue?
greg115
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QR decomposition using Given transformation in $O(n^2)$ operations

Let a matrix $A\in \mathbb{R}^{n\times n}$ be of the form \begin{pmatrix} \alpha_1 & \gamma_1 & \gamma_2 &... &\gamma_{n-1} \\ \beta_2 & \alpha_2 & 0 & ... & 0\\ \beta_3 & 0 & \alpha_3 & ...&0 \\ \vdots & 0 & \ddots& ...&0\\ \beta_n& &…
Aritra Roy
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