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

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11 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: ...
0
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0answers
22 views

Proof of theorem about iterative methods

How do I prove that if $A$ is a tridiagonal (or block tridiagonal) matrix then the corresponding $P_J$ and $P_G$ iteration matrices for the Jacobi and Gauss-Seidel methods satisfy that if $\lambda$ is ...
0
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0answers
26 views

interpolating method [on hold]

Let the function $$f(x)=\ln\left(\frac{1}{|x-1|}\right)$$ be approximated by an interpolating polynomial of degree 15 with the nodes equally spaced in the interval $[-1, 0]$. What bound can be ...
3
votes
1answer
32 views

3-sigma Ellipse, why axis length scales with square root of eigenvalues of covariance-matrix

This is my first post on math.stackexchange and i am not a mathematician, but i took some undergrad math courses and some grad mathematical modelling courses, so i come with a basic understanding of ...
1
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1answer
11 views

Condition number of a matrix bounded from below and above?

Is condition number of an invertible matrix bounded from below? And is condition number always bounded from above for an invertible matrix?
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1answer
19 views

Prove that Frobenius matrix norm is compatible with the vector norm

Show that, the Frobenius matrix norm $||.||_F$ is compatible or consistent with a vector norm $||.||_2$ , that is, $||Ax||_2 \leq ||A||_F ||x||_2, \forall x \in \mathbb{R}^n$. Where $||A||_F = \sqrt{ ...
0
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1answer
28 views

Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
-1
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0answers
41 views
+50

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
-1
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1answer
51 views

l dont know ho w to do this question [closed]

Help me in my homework question
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0answers
11 views

How is this a substitution? Linear algebra transformation matrix misunderstanding

I found the following matrix equation in '3D Surveillance System Using Multiple Cameras', (authors: Ajay Kumar Mishra, Bingbing Ni, Stefan Winkler, Ashraf Kassima) (link here): I don't follow the ...
-2
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0answers
35 views

Please help me with my home work question [closed]

Please help me with my home work question
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0answers
10 views

find the mean of a collection of the solutions of given linear systems

Suppose one has a group of $k\times k$ square matrices $A_i, i = 1,2..n$, another $k$-vector $b$ is also given. I want to exam the mean of the group of vectors $x_i$ over $i = 1,2...n$ where $A_i * ...
0
votes
1answer
61 views

Finding the smallest max eigenvalues for related matrices?

While messing around with a spectral approach to a graph coloring question, I happened upon a type of problem I hadn't seen before. Suppose you have two symmetric $n$ x $n$ matrices in the form ...
0
votes
1answer
44 views

Unique least square solutions

There is a theorem in my book that states: If $A$ is $m\times n$, then the equation $Ax = b$ has a unique least square solution for each $b$ in $\mathbb{R}^m$. But can we find a counter-example to ...
4
votes
1answer
29 views

Implementing trig functions for dual numbers

I'm curious, how do common trig functions get implemented for dual numbers? One way would be to use the power series definition, but that seems inefficient
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0answers
20 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
0
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0answers
12 views

Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute ...
0
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0answers
14 views

Nearest points / residuals on a total least squares parabola

Consider fitting a parabola $y = a + bx + cx^2$ to 2d data $X_i, Y_i$ with noise in both X and Y, using the the singular value decomposition as in Total_least_squares (TLS): $\qquad X = [ 1\ \ Xdata\ ...
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0answers
11 views

Transformation between Ideal and Warped Surface

I work on manufacturing metal panels with holes drilled in them. Suppose I have an ideal 3D surface from CAD. I want to compare it to the actual part using reference points to compare between the two. ...
0
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0answers
2 views

How can I apply a median filter directly to a time-varying rotation matrix?

I need MatLab script which would take a series of rotation matrices (referring to an actual physical object's orientation) and apply median filter to it to eliminate speckle noise from it. The way ...
2
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0answers
64 views

Numerically approximate the maximum of an element of a vector after a series of matrix multiplications.

Where S is a sigmoidal function, A_i is a matrix, and x is an input vector, and ...
0
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0answers
40 views

Fastest Algorithms for Determining the Nullity of a Matrix

How exactly does one go about determining the Nullity of a Matrix quicker than simply running Gaussian Elimination on the matrix itself? To be perfectly honest I can't think of a method that doesn't ...
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0answers
32 views

Complexity of the power method

I'd like to find out what the complexity of the power method is depending on the size of the matrix $A \in \mathbb{R}^{n\times n}$ given that the algorithm runs until a certain stop criterion. I.e. ...
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0answers
27 views

resources about sparse global constrainted optimization

Please recommend a good resources (books/articles/software) about sparse global constrained optimization?
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1answer
44 views

Solving 2x2 diagonally dominant matrix systems (non-symmetric)

I have a linear system of the form $Ax=b$ where $A\in \mathbb{R}^{2\times2}, b\in \mathbb{R}^{2\times1}$. A is diagonally dominant and non-symmetric. This is a "kernel" that I am using to solve a ...
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0answers
28 views

Numerically stable way to compute $\text{Trace}[\mathbf A\mathbf A_1^{-1}\mathbf B\mathbf B_1^{-1}]$

I have two dense (column-major) PSD matrices $\mathbf A$ and $\mathbf B$, $\mathbf A,\mathbf B\in\mathbb R^{n \times n}$ ($n$ is usually $\sim 1000$) and also $\mathbf A_1=\mathbf A+\eta\mathbf I_n$ ...
0
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1answer
34 views

Application of Conjugate Gradient Method to non-symmetric matrices

I am currently working on a problem in which I am using the Conjugate Gradient method to solve for the steady state solution of a continuous time Markov chain. I am applying the algorithm found in ...
0
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1answer
26 views

Finding SVD of a Matrix

If $a_{1},a_{2} \in \mathcal{R}^{2},$ $\ \|a_{1}\|_{2} = \|a_{2}\|_{2} = K$, and the angle $\theta$ between $a_{1}$ and $a_{2}$ is between $0$ and $\pi/2$, we want to compute the SVD of the matrix $A ...
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0answers
27 views

How to find the rank of a toeplitz matrix?

Is there any trick to compute or estimate the rank of a toeplitz matrix ? Or is this still unknown for a general toeplitz matrix ?
4
votes
2answers
68 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
0
votes
2answers
43 views

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 ...
0
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1answer
49 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
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0answers
17 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
2
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3answers
56 views

How to reverse matrix vector multiplication?

I'm using the simple matrix x vector multiplication below to calculate result. And now I wonder how can I calculate ...
1
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0answers
32 views

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 ...
2
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1answer
33 views

A generalization of GMRES

In oder to solve $Ax=b$, GMRES method finds $x_n$ in the $k$-th Krylov subspace i.e.: $$K_n=span\{b,Ab,...,A^{n-1}\}$$ and we have the condition: minimize $\|r_n\|_2$, which $r_n=b-Ax_n$ Now we ...
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0answers
26 views

Solving tridiagonal matrices where the top left element is zero

If I have a matrix like this: $$ \left[\begin{array}{rrrrrrrrr|r} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 & 0 & ...
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0answers
15 views

Finite difference method for elliptic pde stability condition

I am trying to implement a finite difference scheme for the elliptic pde $\nabla(a(x_1,x_2)\nabla\phi(x_1,x_2))=0$ where $(x_1,x_2) \in (0,l_1)\times(0,l_2)$ with $l_1 \neq l_2$ necessarily. I know ...
2
votes
1answer
47 views

Positive definite martix

I understand the majority of this solution, it's just I don't understand why I have to use both $\epsilon_1 $ and $\epsilon_2 $ rather than just $\epsilon$. I understand that i'm working with ...
0
votes
2answers
43 views

Least Squares Solution Confusion

Say if I have an overdetermined system $A\vec x=\vec b$, I can use the normal equations $\implies$ $A^TA\vec x=A^T\vec b$. If I solve for $\vec x$ I will get a "solution" with an error. It says in ...
2
votes
1answer
41 views

QR transformation with Householder transformation

It's a task i do to understand minimizing the error including the QR transformation with the help of Householder transformation. I think i really do something wrong but i dont get it running i hope ...
1
vote
0answers
40 views

Numerical rounding errors in intersection code

I hope this question is in the right place, as it is as much about programming as it is about math. I'm trying to find the intersection between a circle and a line. My implementation of the algorithm ...
0
votes
0answers
36 views

Efficient Factorization for Family of Matrix Equations

I am looking for an efficient solution to the following problem $$ A+\lambda I = b \tag{1} $$ where $A\in\mathbf{S}^{n}$ is a symmetric matrix with nonzero eigenvalues, $b\in\mathbf{R}^n$ is fixed, ...
3
votes
1answer
76 views

I want to study Numerical linear algebra [closed]

Would you like to recommend a book to me? the proof is explicit and easy to understand is preferred.
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0answers
23 views

Gaussian elimination vs. Jacobi iteration

How can I determine which of the matrix solver is faster for a given set of equations: Gaussian elimination or Jacobi iteration? In case, I have a banded matrix, is it advisable to use LU ...
1
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0answers
27 views

When is the LU decomposition unique?

I want to find out when a matrix decomposition $A = LU $ (L lower and U upper matrix) is unique? Clearly, if $A$ is not invertible, there is no chance that this decomposition is unique. Hence, ...
0
votes
1answer
36 views

Simplying linear equation to get quartic in q using Maple and then using Descarte’s rule of sign

Using the maple I am trying to get quardic in q from this big linear equation. Then use Descarte’s rule of signs to determine the number of positive roots. \begin{equation} ...
0
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1answer
34 views

Solve for a matrix in a linear equation

This is probably a really basic question, but we are stuck and the usual keyword lead to the normal "Solving linear equations with matrices"/Gaussian-elimination pages ... I have an equation $A X B + ...
2
votes
2answers
59 views

Determinant of an ill conditioned matrix

I have the following ill conditioned matrix. I want to find its determinant. How is it possible to calculate it without much error \begin{equation} \left[\begin{array}{cccccc} ...
0
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0answers
33 views

System of linrar equations and condition number

The relative error of the solution of a system of linear equation $Ax=b$, for any natural norm $\|\cdot\|$ is bounded by $$ \frac{1}{\| A\| \|A^{-1} \|} \frac{\|r\|}{\|b\|} \le \frac{\|e\|}{\|x\|} \le ...