# Tagged Questions

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

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### What is the operation count for QR factorization using Householder transformations?

I have a hard time finding the operation count of QR factorization when using Householder transformations. The answer is $2mn^2 - \frac{2n^3}{3}$, but have no clue on how to get this count following ...
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### Numerical Method Sample Question via Truncation error Methods?

I have one multiple choice question: Approximation of integration $\int_0^{0.1} e^{x^2}dx$ by using simple formula of following options has lower Truncation error: Choice Part: $a)$ ...
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### Leading eigenvalues of large sparse unsymmetric matrix

I have a matrix $R$ which is sparse and all eigenvalues are -ve with a zero eigenvalue. Size of R is more than $10^6 \times 10^6$. But I need to calculate only few large (by value not by magnitude) ...
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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 ...
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### statistical comparison, 3 groups, multiple columns

I am using R for some statistical analysis. I have a dataset listing number of deaths by eu regions. the dataset is annual and is for 2000-2008. I divided this data into 4 subgroups according to ...
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### Is the condition number of unitary matrix always equal to 1?

I know that the 2-norm condition number $\kappa (\textbf U)={||\textbf U||_2}{||\textbf U^{-1}||_2}$ of a unitary matrix $\textbf U$ is always equal to 1. Is this true for all induced matrix norms, i....
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### Iterative methods for solving a linear equation system

There are several methods known for solving a linear equation system Ax = b (like Jacobi or Gauss-Seidel) by iterating $x_{n+1}=Mx_n+c$ with a matrix M, for which some norm is smaller than 1. But ...
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### Parallel Algorithms for SVD

I just have completed a preliminar theoretical study of the important SVD decomposition. Now, I'm moving to numerical calculation of SVD. I would like to learn directly a parallel algorithm to ...
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### Divergence of an iterative map variant.

The problem at hand. Let $M$ be a matrix from $R^p$ to $R^p$ with $\rho(M)<1$. Let $(b_{n})$ be a divergent. Show that the sequence $(x_n)$ is divergent, where $x_n=Mx_{n-1}+b_{n-1}$. Not really ...
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### Existence of a fixed point for a linear stationary iterative method

Suppose you are attempting to solve $Ax = b$ using linear stationary iteration method defined by $$x_k = G x_{k-1} + f$$ that is consistent with $Ax = b$, i.e., for which $f = (I - G)A^{-1}b$. Suppose ...
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### Improving my QZ-Algorithm (Include Shifts)

I Need to to solve an generalized Eigenvalue Problem and compare two Methods (QR and QZ) concerning their convergence rate and execution time. I started with the Basic QR-Algorithm, implemented in ...
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### One iteration of forward Gauss-Seidel followed by one iteration of backward Gauss-Seidel

Let $A = D - L - U\in\mathbb{R}^{n\times n}$ be a nonsingular matrix, where $-L$ is the matrix of strictly lower triangular elements and $-U$ is the matrix of strictly upper triangular elements. ...
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### How do I find transformation matrix with respect to given basis in the domain and/or the codomain, given the transformation in the standard basis?

I´m being given a linear transformation, for which I can find the standard basis in the domain and codomain; but then, the book ask to find the associated matrix related to a new basis for the domain.....
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### Efficient way to check if a large matrix is positive definite.

Suppose I have a large $n\times{}n$ matrix with $n>1000$ say. I would like to find the quickest way to check if it is positive definite. My matrices are sparse so at the moment I am using sparse ...
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### Eigensolver for Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
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### Following number is divisible by [closed]

If $n = 2009$ , then $N = 2009^n -1982^n -1972^n + 1945^n$ is not divisible by 659 1977 1998 2009
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### How to find the Householder transformation?

Assume $x=(1,0,4,6,3,4)^T$. Find a Householder transformation and a positive number $\alpha$ such that $Hx=(1,\alpha,4,6,0,0)^T$. I'm sorry that I don't know how to start with this problem. A ...
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### Solve the closed form solution for argmax of $x^Ty - x^T\ln(x)$

Let $y \in \mathbb{R}^n, \ln(x) = \begin{bmatrix} \ln(x_1) \\ \vdots \\ \ln(x_n) \end{bmatrix} \in \mathbb{R}^n$ Show that $$x^* = \text{argmax}_{x \in \mathcal{D}} \quad x^Ty - x^T\ln(x)$$ ...
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### Understanding power method for finding dominant eigenvalues

The power method aims to find the eigenvalue with the largest magnitude. Does magnitude still have the same meaning in this context? If so, can't we tell from the outset which eigenvalue is the ...