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

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1answer
45 views

Solving Ax = b where A is composed of diagonal blocks

I would like to solve the equation $Ax=b$ where $x\in\mathbb{R}^n$ and $A$ is of the form: $$A= \begin{bmatrix} D_1 & D_2 &D_3 \\ D_2 & D_4 & D_5 \\ D_3 & D_5 & D_6 ...
1
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1answer
59 views

Checksum Invariants for Matrix Inversion via Gaussian Elimination

In general, when solving $Ax=b$, we make the $[A|b]$ matrix and doing row operations to reduce the left hand side to an identity. It's painfully annoying to find mistakes in the process. Assuming we ...
1
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1answer
22 views

Equality of iterates produced by Minres and GMres for practically symmetric matrix

My system is from time-integration of the semi-discretized Stokes equation. The time update of the variables $(v,p)$ is defined via the solution of $$ \begin{bmatrix} A & -\tau B^T \\ B & 0 ...
2
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0answers
55 views

Examples of non trivial problems in this structure.

I'm looking for examples of non trivial problems that match with the follow structure. Let the function $$g: U \times V \rightarrow \mathbb{R}$$, where $U$ and $V$ are complex vetorial spaces of ...
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2answers
80 views

Legendre Polynomial Orthogonality and Size

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$ I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and ...
0
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1answer
77 views

Householder Reflection

I am working on algorithms for SVD by first performing Householder transformation. I got my algorithm to work but I'm trying to gain a better intuition of it. My understanding is that the ...
1
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1answer
63 views

Can a tridiagonal matrix be rectangular?

My program works with tridiagonal matrices (calculates its LU decomposition) so before doing anythig, it stores the matrix in 3 vectors: the three diagonals only. So far my conclusion was, a ...
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2answers
90 views

Square Idempotent matrix: efficient algorithms for finding eigenvectors

Given a square idempotent $N \times N$ matrix $A$ with large $N$, and a priori knowledge of the rank $K$, what is the most efficient way to compute the $K$ eigenvectors corresponding to the $K$ ...
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1answer
101 views

matrix exponential and Spectral abscissa

Prove that $\lim_{t \rightarrow \infty} \|e^{tA}\| = 0$ if and only if $\alpha(A) < 0 $, where $\alpha$ is the Spectral abscissa, defined as $\max{Re(\lambda_i)}$. I tried to approach this ...
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1answer
122 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
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2answers
96 views

non-sequential sequence function

if i remember correctly (i had one workshop on numerics years ago, sorry for my lack of knowledge) there is a way to create some sort of hash function that gives you a non sequential sequence. This ...
2
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0answers
34 views

Smallest set of Liner equations, which exactly fit a set of points

I have a set of 2-d points,(it can be of any arbitrary dimension n). I want to find the minimum set of straight lines(linear equations) which exactly passes through the given 2-d points (unlike ...
4
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4answers
325 views

Finding nonnegative solutions to an underdetermined linear system

Here's the environment of my problem: I have a linear system of 4 equations in 8 unknowns (i.e. $Ax = b$, where $A$ is $4 \times 8$, $x$ is $8 \times 1$, and $b$ is $4 \times 1$, with $A$ given and ...
0
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1answer
55 views

Is there a limit for how “good” a numerical method can be?

Multiplying two matrices $A \cdot B$ of size $n \times n$ in the trivial way requires $n^3$ computations. However, more efficient algorithms such as the Strassen algorithm have a lower complexity of ...
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3answers
69 views

About matrix products $A^{T}A$ and $ AA^{T} $

I'm investigating the relationship between 2-norms and eigenvalues of $A^{T}A$ and $ AA^{T} $, in order to better understand the SVD decomposition. How can I prove that $A^{T}A$ and $ AA^{T} $ are ...
2
votes
0answers
133 views

Inverse of the sum of a symmetric and diagonal matrices

I have two square matrices, $A$ and $B$. $A$ is a block symmetric matrix with 1's along the diagonal (and therefore 1's in whole blocks along the diagonal). $B$ is diagonal, with the same block ...
0
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1answer
63 views

Complexity of sparse back substitution

What is the complexity of sparse backsubstitution $Rx = b$, given $n$, the dimensions of dense $x$ and $b$ as well as of the sparse $R$ and $nnz$, the number of nonzero entries in $R$?
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1answer
44 views

How to solve an Optimization problem with linear as well as Quadratic constraints.

I want to solve the following problem, \begin{equation} \begin{aligned} & \underset{\mathbf{x}}{\text{minimize}} & & \mathbf{x^T}\mathbf{Px} \\ & \text{subject to} & & ...
5
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2answers
706 views

Algorithm to find an orthogonal basis (orthogonal to a given vector)

Let $K$ be a given integer, with $K$ even (and "large"). Let $\mathbf{v} \in \mathbb{R}^{K \times 1}$ be a given non-zero (column) vector. Write a (possibly efficient) algorithm to construct a matrix ...
0
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1answer
43 views

Is it possible to solve a system of equations comprising FFTs?

Consider the following known matrices, A, B, C and these unknown matrices X,Y, all of which comprise values in the Real domain. Also consider $F(x)$ as the *Fast Fourier Transform function* (the ...
3
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1answer
129 views

How to solve this system of 3 equations with 3 variables?

I stumbled upon this system with constants $a_{i,j}>0$ that I want to solve for $x,y,z \in\mathbb{R}$: \begin{align} a_{2,1}y+a_{3,1}z=& x(y+z) \\ a_{1,2}x+a_{3,2}z=& y(x+z) \\ ...
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1answer
86 views

Inverse Square root of a rectangular matrix

I am trying to compute the inverse square root ($X^{-1/2}$) of a $n \times p$ matrix with $n > p$. I was wondering if we can compute it via SVD just as we do it for square diagonalizable matrices ...
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2answers
100 views

Matlab terminate code if warning occurs [closed]

I am running a code where by I want to maximise a parameter (basically the range of integration) by increasing it in steps until a warning occurs. It involves inverting a linear system Ax=b and I want ...
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1answer
45 views

A problem on numerical analysis [closed]

Let $f(x)=e^x$ be approximated by Taylor's polynomial of degree n at the point $x=\frac{1}{2}$ and on the interval $[0,1]$. If the absolute error in this approximation does not exceed $10^{-2}$ , then ...
0
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1answer
32 views

Residual norm for iterative scheme

Consider a linear system $A\vec{x} = \vec{b}$, where $A \in \mathbb{R}^{m\times{}m}$ is non-singular and positive definite. Given the following iteration scheme $\vec{x}^{(k+1)} = \vec{x}^{(k)} + ...
5
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0answers
82 views

IEEE 754 as a mathematical space

Integer operations in computers (i.e. 32-bit integers) probably can be represented best by modular arithmetic (because of integer overflows/underflows). What about IEEE 754 floating point arithmetic? ...
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1answer
76 views

Special Matrix 2-norm and F-norm Inequalities

This is a homework problem for my Numerical Linear Algebra course. It states the following: If A is an mxm nonsingular matrix, prove the following: (1)$\|A+(A^{*})^{-1}\| _{2} \ge 2$ ...
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1answer
75 views

Using the Gauss-Seidel method, will the matrix A converge

Just came back from my Numerical Analysis midterm, posting up the questions and my solutions for an estimation as to how I did. If you were to perform the Gauss-Seidel method on a matrix $A$, where ...
1
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0answers
14 views

Finding eigenvalues in a region

I have an eigenvalue problem (of a very large (n~1000000) but sparse complex system) wherein I need to determine all the eigenvalues in a certain region(a rectangle) in the positive half plane. I am ...
3
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1answer
249 views

Absolute values in linear programming

Suppose I have an objective function in my LP as follows $max$ $|x|$ Based on some googling, I have found there are two ways to convert this into a standard LP. Method 1. $|x|$ = $ x^+ + x^-$ $x ...
2
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1answer
44 views

Is it true: $||A||_2 = \min\{ ||A||_1 ,||A||_3,||A||_4,\ldots \ldots, ||A||_{\infty},\|A\|_F\} $?

While running one algorithm , I observed the following peculiar relationship (at-least to me). I am not quite sure whether it is true in general, but I could not succeeded either in producing any ...
1
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2answers
77 views

Relation between condition number and perturbed matrix

Prove that if $A\vec{x} = \vec{b}$ and $(A+\delta{}A)(\vec{x}+\delta\vec{x}) = \vec{b}$, then $\dfrac{\|\delta\vec{x}\|/\|\vec{x}+\delta\vec{x}\|}{\|\delta{}A\|/\|A\|} \le \kappa{(A)}$, where ...
1
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1answer
67 views

What does it mean for a matrix to be nearly singular?

I am currently enrolled in numerical analysis course, and a terminology I have not heard of came up; nearly singular matrix. I know that a non-singular matrix is one where the column vectors are ...
0
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1answer
116 views

Condition number vs. reconstruction error

Suppose I want to solve a simple, linear inverse problem given by $\mathbf{y} = \mathbf{A} \cdot \mathbf{c}$ where $\mathbf{A}$ is an $M \times K$ matrix and I want to solve for $\mathbf{c}$ ($M$ = ...
2
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1answer
471 views

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 ...
1
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4answers
376 views

How to tell if two matrices are equal up to a permutation

Given two real rectangular matrices A, B how can I tell if they are equal up to a permutation of their rows/column without trying all possible permutations? (This is closely related to the question I ...
2
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0answers
48 views

Find the eigenvector with maximum overlap

Given a large symmetric matrix $A$, there are methods to find the largest or smaller eigenvalue, or the eigenvalue closest to some initial value. Is there any method to find the normalized ...
0
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1answer
53 views

A problem in a Question paper on Linear Transformation

anyone please solve it . Let the linear transformation $T: F^2\to F^3$ be defined by $T(x_1,x_2)=(x_1,x_1+x_2,x_2)$ . Then the nullity of T is 0 1 2 3 Also please mention how it is solved
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0answers
46 views

Representation of Uncertainty in linear systems

I have a linear uncertain system represented by a family of models: $\dot{x}=A_ix$,$i=1,\cdots,N$ I want to represent the system as: $A_i=A_0+B\Delta_iC$ subject to the condition that $\lVert ...
1
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0answers
86 views

Algorithm to determine matrix equivalence

I'm a physicist who's not particularly good at linear algebra so please accept my apologies if this is standard textbook stuff that I'm just unaware of. I have two real rectangular matrices $A_{mxn} ...
3
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0answers
59 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...
1
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1answer
67 views

Inverse of a triangular matrix in a statistical problem

Can any one give to me idea how to solve this problem? Find the inverse of the triangular matrix T, where $ T =\left[ \begin{array}{ccc} I & J & J \\ 0 & I & J \\ 0 & 0 & I ...
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2answers
47 views

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?
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0answers
85 views

is it possible to generate a unique number given a set of N integers regardless of their permutation?

I need to efficiently compute an "id" for a set of N integers, the id needs to be unique if any of the numbers is different from some other set. At the same time the id needs to be the same if the ...
2
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2answers
173 views

QR-factorization of a tridiagonal matrix super diagonals question

I understand it is possible to QR-factorize a tridiagonal matrix A by performing Given's plane rotations: $$ J(n-1,n)J(n-2,n-1)... J(1,2) A =R$$ where $R$ is upper triangular. I have read that in ...
3
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2answers
158 views

Singular Values of Matrix as Optimization Problem

Assume that $A$ is a positive semidefinite symmetric matrix. It is known that $$\max_{||y||\leq1} \quad y^TAy$$ Has an analytical solution which is the maximum eigenvalue of $A$. This isn't hard ...
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0answers
54 views

How can solve this differential equation (third equation )?

How can I solve this differential equation? $$ \frac{dy}{dx}=\sqrt{\frac{A}{y}+\frac{B}{y^2}+\frac{C}{y^4}+\frac{D}{y^5}+\frac{1}{(\frac{1}{y}+\frac{3}{y^2})^2}} $$ where $A,B,C,D$ are constants.
0
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2answers
52 views

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 ...
2
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2answers
54 views

Solve linear equation system $A'Ax=A'Bz$

For $A$ and $B$ known matrices which are not square matrices, I have the following equation sistem i would like to solve numerically \begin{equation} A'Ax=A'Bz \end{equation} I want to know which is a ...
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2answers
138 views

Showing that $A=B+\alpha \cdot I$ is an invertible matrix

Let $B$ be a non-zero random $n\times n$ matrix generated using the matlab command $B=rand(n,n)$. I need to show that $A=B+\alpha \cdot I$ is an invertible matrix, where $\alpha=\|B\|_{\infty}$. I ...