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
1
vote
0 answers

Small perturbation of $A$ in normal equation

I'm studying the least squares problem. My professor told in class that small perturbation in $A$ does not mean small perturbation in $A^TA$, and so the relative perturbation of $x$ will depend on $\kappa^2(A),$ where $\kappa(A)$ is the condition…
dxdydz
  • 1,371
1
vote
1 answer

Algorithm to minimize the 2-norm between two matrices

What algorithm can you use to minimize the distance between two matrices? For example: $$\min_{X \in \Gamma} \| A - X\|_2$$ $\Gamma$ all m $\times$ m rank $k$ matrices. How can you think of this geometrically? If for least squares we minimize $Ax-b$…
Math Stout
  • 11
1
vote
0 answers

Tridiagonal sparse matrix - linear equation

I have to following linear system to solve : $Ax=e_1$ where $A$ is a sparse tridiagonal matrix with the main diagonal terms $a_{ii}$ being all different, and the off-diagonal terms being each others complex conjugate ($\bar{b_{ij}} = b_{ji}$). $e_1$…
Nigu
  • 111
1
vote
1 answer

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 modulus as $\rho(A)$. Can the power method give…
Eli4ph
  • 482
1
vote
0 answers

$u_{i,j}=\frac{u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}}{4}$

Let your living room be the unit square. We denote the coordinates in the square by x and y. The dots are at locations $(x_i,y_j)$ (see the image here: https://i.stack.imgur.com/cPUTU.jpg). We denote the temperature in $(x_i ,y_j)$ by $u_{i,j}$. The…
dxdydz
  • 1,371
1
vote
1 answer

Transcendental function implementation using linear algebra techniques

I'm sure this has been answered, but I am not having much luck finding it. Is there a way to implement transcendental functions ($\sin(x)$, $\cos(x)$, $e^{x}$, $\ln(x)$, etc...) using techniques from linear algebra? My current code uses a…
Daniel Rudy
  • 216
1
vote
1 answer

Large and ill-conditioned quadratic convex problem

I need to solve a convex quadratic problem numerically: $\min f(x) = \frac{1}{2} x^\top A x - b^\top x$, where $A$ is a very large and ill-conditioned semi positive definite matrix. Typical conjugate gradient method doesn't work well. SGD is too…
Joe Lacey
  • 23
1
vote
2 answers

In random matrix decomposition, how does a semi-orthogonal matrix capture the range of an input matrix?

I am reading this paper on probabilistic algorithms for matrix decomposition. I don't understand Section 1.2. Given a matrix $A \in \mathbb{R}^{m \times n}$, we want an "approximate basis for the range of $A$", $Q \in \mathbb{R}^{r \times n}$ such…
jds
  • 2,274
  • 3
  • 24
  • 35
1
vote
0 answers

Numerical linear algebra spectral norm limit.

Let $A \in \mathbb{R}^{m \times n}$ be of full rank. Consider $X_{k+1}=(2k-X_{k}A)X_{k}$, $X_0 = \alpha A^{T}$. Let $E_k = I-X_kA$, Deduce that if $||E_{0} ||_{2}<1$, then $lim_{k \rightarrow \infty}E_k=0$, I'm thinking it's something like…
simplicity
  • 3,694
1
vote
1 answer

Formulate Newtons method for system of equations.

Some background theory: By Taylor's theorem we know that an approximation of $f(x)$ with two terms around an approximation $x_k\in \mathbb{R}^n$ is given by $f(x)\approx f(x_k)+J(x_k)(x-x_k),$ where $J$ is the jacobian. So a linear model of our…
Parseval
  • 6,413
1
vote
0 answers

Numerical Linear Algebra problem (QR factorization with column pivoting)

For matrices that might be rank deficient it is common to incorporate pivoting in Householder QR factorization of A $\in$ $\Re^{mxn}$ (m $\geq$ n). Let $A^{(k)}$ denote the matrix at the start of the kth stage of the reduction (thus $A^{(1)}$ = A)…
user50017
  • 71
1
vote
0 answers

QR decomposition Eigenvalues Eigenvectors

i am interested in finding eigenvalues and eigenvectors for any real matrix - large and not symmetric. i discovered that QR-algoritm is the best one. I have questions: 1) if A-matrix (real and not symmetric) is given, then $Q$-matrix can be found…
R.Goldov
  • 41
1
vote
2 answers

How to prove whether a function is linear or affine?

so i am having hard time understanding the idea of coming up with random n-vectors to disprove the superposition equality $f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)$. or to prove that a function is linear or affine??? can someone explain to…
ASROMA
  • 579
1
vote
0 answers

Finding a row permutation that makes a matrix more "blocks-like"

Disclaimer: what follows arise in a context from Computer Science, but it seems to me that my questions were more likely to be solved from mathematicians than from computer scientists. Let suppose the following scenario, pointing out that everything…
Immanuel Weihnachten
  • 1,876
1
vote
0 answers

Spectral radius for Gauss-Siedel method convergence

Given a tridiagonal $(n-1)\times (n-1)$ matrix $A=n^2\begin{bmatrix} -2&1&0&0&0&\dots&0 \\ 1&-2&1&0&0&\dots& 0\\ 0&1&-2&1&0&\dots& 0\\ 0&0&1&-2&1&\dots& 0\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots \\ 0&0&0&0&\dots&1&-2\end{bmatrix}$ its…
Ilia
  • 489
1 2 3
…
11 12