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

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0answers
16 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 $\mathbf A_1=\mathbf A+\eta\mathbf I_n$ ...
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1answer
22 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 ...
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1answer
23 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
21 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
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2answers
60 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 ...
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1answer
24 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 ...
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0answers
14 views

Complexity of eigensolvers for sparse matrices

What is the usual complexity of the common (iterative) eigensolvers (Arnoldi, Lanczos, ...) for a very sparse (tridiagonal) symmetric $n\times n$-matrix? Can one eigenpair be computed in ...
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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,\; ...
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0answers
15 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 ...
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3answers
46 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 ...
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0answers
28 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
30 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
18 views

About reduction to Hessenberg matrix

I've read somewhere that Hessenberg decomposition is not unique unless the first column of $Q$ is given. i.e $Q^TAQ=H$ Then I read the algorithm of Arnoldi iteration and I found an amazing fact: ...
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0answers
24 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
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1answer
44 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 ...
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2answers
36 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
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1answer
39 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 ...
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0answers
37 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
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0answers
35 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
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1answer
68 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
22 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 ...
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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, ...
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1answer
34 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} ...
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1answer
28 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
57 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} ...
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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 ...
0
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0answers
29 views

condition number of orthogonal matrix

Let $A\in M_n(\mathbb R)$ be an orthogonal matrix. Then: $cond (A) =1$. I tryed to use facts about the eigenvalues but is did not help. In 2-norm it is easy to prove it since $||A||_2 = \sqrt{\rho ...
0
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0answers
10 views

Spectral norm of a Hadamard product

Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\circ$ ...
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2answers
20 views

Finding a rotation matrix

I am looking for a rotation matrix such that $$ \operatorname{rot} \cdot \begin{pmatrix} -1 & 0 & 0 \\ 1 & -1 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix} = ...
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1answer
24 views

Efficient Algorithm for Iteratively Reweighted Least Squares Problem

I'm interested in solving a weighted least squares problem of the form $X^T W X \beta = X^T W Y$ where $W$ is a diagonal, positive definite matrix, $X \in R^{m \times n}$, $Y \in R^{m \times 1}$ and ...
3
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2answers
52 views

Power iteration

If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration Especially, I often see that we ...
1
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1answer
27 views

Lipschitz continuity for generalized inverse matrix

Suppose $A$ and $B$ are full-rank and well-conditioned. Is Lipschitz continuity held for generalized inverse? $$\|A^+ - B^+\| \le \omega \|A-B\|,$$ for some $\omega > 0$, where the norm could ...
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1answer
14 views

LD$t(L)$ factorization and eigenvalues

A positive definite matrix $A$ can be factored in to $LDt(L)$form. Is the statement the eigenvalues of $A$ are the diagonals of $D$ true? If so , how to prove it?
3
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0answers
47 views

Does anyone know any reference for this matrix?

For $n \geq 4$, $A$ is $(n-1) \times (n-1)$ tridiagonal block matrix $$A = n^2 \begin{bmatrix}B & -I & 0 & \cdots & \\-I & B & -I & 0 & \\ 0 & -I & B & -I ...
2
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2answers
42 views

Solving $a_1x_1 + \cdots +a_nx_n = b$

I'm glad to ask my first question on the maths site! So here we go. I'm trying to set up prices right now and here is my problem : I know that my customer has a certain amount of money available. ...
2
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1answer
28 views

Can an iterative method converge for some initial approximation?

Studying iterative methods for solving(or approximating) linear equation systems, I came accross the following theorem$^1$: Let the following be an iterative method: $$x^{(0)},\qquad known\\ ...
1
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1answer
41 views

Inverse of constant matrix plus diagonal matrix

Is there an efficient way to calculate the inverse of an NxN diagonal matrix plus a constant term? I am looking at N of around 40000. $\left[\begin{array}{cccc} a & b & \cdots & b\\ b ...
2
votes
1answer
65 views

Linear equation: $(A^\top A+B^\top B + D)x=c$ where $A,B$ are structured sparse and $D$ is diagonal.

Updated: the goal is to solve $(A^\top A+B^\top B + D)x=c$. Maybe it is not necessary to compute $(A^\top A+B^\top B + D)^{-1}$. Denote $e=(1,1,\ldots,1)^\top\in\mathbb{R}^n$ and ...
1
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1answer
39 views

Consistency of the system $AX=b$

In the concept of consistency (compatibility) of a matrix, when $b \in \mathbb R (A)$ we know for sure that the system is consistent or there may exists no solution for it? But when $b \notin \mathbb ...
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2answers
26 views

Formatting Linear equation

How do I craft a linear equation so that it is in the form of $ax + bx + c = 0$ where $a^2 + b^2 = 1$ if I have two points? I know how to get it into the form $ax + bx + c = 0$ but I can't figure out ...
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0answers
18 views

Check feasibility of a system of integer linear equations

I'm currently working on a very large integer linear programme which cannot be solved within any reasonable time. The initial set of linear equations S={Ax<=b) is feasible. I want to add more ...
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0answers
36 views

Effective computation of matrix commutator

Is there a faster way to compute the commutator of large (at least one of them sparse) matrices $[A,B]$ then to compute $AB$ ,$BA$ and subtract them?
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0answers
28 views

Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
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0answers
58 views

Newton's method for multidimensional functions

Can Newton's method be used to find the root of a function f : $\mathbb{R}^n\to\mathbb{R}^m$. Can anyone provide a proof for this? (I have checked the method of solving system of equations with ...
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1answer
50 views

Efficient method for determining to the most positive eigenvalue of a matrix

I am trying to implement an algorithm that requires knowing the largest $\textbf{positive}$ eigenvalue of a $\textbf{real symmetric, non-sparse}$ matrix and the corresponding eigenvector. The actual ...
0
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0answers
27 views

What is the largest (dense, real, symmetric) random matrix I can diagonalize on a computer?

I have read that 10.000x10.000 is no problem for LAPACK or similar routines. I would like to know if N=20.000 or 40.000 is possible. EDIT: I don't know if it is relevant, but the matrix is positive ...
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0answers
21 views

Strange Convergence of SOR/Gauss-Seidel

I am having trouble with the convergence of my Gauss-Seidel/SOR method. The matrix $A$ in $Ax=y$ seems to be positive-(semi)definite. Its eigenvalues are: However, the method (SOR) improves the cost ...
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0answers
19 views

Change in Singular Value Decomposition of a matrix on addition of a single row

Given that I know the svd decomposition of a matrix, is there any way to compute the svd decomposition of the matrix obtained by adding a single row to the original matrix? Is there any relation ...
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0answers
21 views

If symmetric matrix in a least-square deconvolution problem positive definite?

I want to apply Gauss-Seidel method in a least square deconvolution problem. The convolution of two vectors is written in: $h * x = z$. $$z(n) = \sum_{i=0}^{N-1}h(i)x(n-i)$$ It is a linear transform ...