# 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?

IÂ´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 $$x^* = \text{argmax}_{x \in \mathcal{D}} \quad x^Ty - x^T\ln(x)$$ ...