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

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3
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1answer
64 views

Relationship between the solution to $Ax=b$ and $(A+I)x=b$

I have have a symmetric, tridiagonal, Toeplitz matrix $A$, where $A_{11} = -\frac{1}{2}$ and $A_{21} = 1$, and I need to solve the system $$ (A+I)x=b, $$ numerically where $b$ does not necessarily ...
3
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1answer
73 views

Necessary and Sufficient conditions for convergence of matrix iterations

I need some help figuring out how to go about the iteration part of the problem...I don't really know where to start. If someone can please help take me through it that would be greatly appreciated. ...
3
votes
1answer
144 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
3
votes
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) \\ ...
3
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1answer
90 views

Understanding higher order SVD

Can someone explain the singular value decomposition of a tensor (maybe a 3 dimensional matrix) with an example? It is intuitively difficult to the get the meaning from just the formulas. On a ...
3
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1answer
192 views

Computational cost, power method and page rank

When solving the PageRank problem for $n$ web pages, it is necessary to find a solution of the eigenvector equation $$(fM)*p = p,$$ where $$fM = dM + (1 - d)Z$$ $$Z =\frac{1}{n}*ee^T$$ $$e =[1, 1, ...
3
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1answer
241 views

How to Store a Banded Matrix by Diagonal

I'm taking a graduate level independent study course this semester in Matrix Computations. I'm not getting much support from the professor, so am turning to the excellent StackExchange community for ...
0
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0answers
24 views

Relative Error with Respect to Frobenius Norm

I'm look at this tiny book called "Deblurring Images: Matrices, Spectra, and Filtering" by Hansen, Nagy, O'Leary. This is a self study, but I believe my question is broad enough so that it can be of ...
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18 views

Dimensionality Reduction

Let $X\in\mathbb{R}^{100\times 100}$ matrix and let its eigenvector and eigenvalues be $X_{vec}$ and $X_{val}$ respectively. If the rank of $X$ is $5$, then is it possible to approximate $X$ with ...
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44 views

Condition number perturbation

I have a matrix of the form $\tilde{H} = H + i A A^\dagger$. It is known that $H$ is hermitian and that $\tilde{H}$ is invertible and $A A^\dagger$ has a kernel of dimension $\geq 1$. I want to study ...
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34 views

Computing a few eigenvalues of large sparse nonsymmetric matrix without LU factorisation

I'm trying to find a few targeted eigenvalues of a large sparse (N=1e6,nnz=4e6) non-symmetric matrix. Currently I'm using MATLAB's 'eigs' function with the 'sigma' option and this uses the Shifted ...
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0answers
31 views

Recasting Ax=b to use SOR method

Just need some guidance as to how to recast the matrix equation equation $Ax = b$ so that I can produce an iterative matrix to perform Succesive Over Relaxation on. These matrices are $n x n$ I ...
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0answers
17 views

Properties of specific case of the gradient descent method

Consider the gradient descent method for the system $Ax=b$ where, $A=\begin{pmatrix}1 & 0 \\0 & a\end{pmatrix}, b=\begin{pmatrix}0 \\ 0\end{pmatrix}$ and the initial vector ...
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0answers
37 views

matrix diagonalization without eigen decomposition, what other ways available?

I have a matrix, $A$ (it may be symmetric or asymmetric). I need to have a diagonal matrix without eigenvalue decomposition, please suggest what others ways are possible? Any new idea would be much ...
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0answers
29 views

computing leftmost eigenpair of positive-definite matrix

Let $A$ be an $n\times n$ real symmetric positive-definite matrix. Assume that $n$ is large and that $A$ is dense (i.e. it is not sparse). Question: What is the state-of-the-art algorithmically for ...
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46 views

Test for powers method

I have been told that for a normal matrix $A$, the powers method (i.e. computing the succession of Rayleigh quotients for a succession of vectors $z_k=A\cdot z_{k-1}$) can use the following stop ...
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39 views

Are there any sure-fire methods for correctly arranging matricies for Gaussian Elimination?

I am attempting to make a Gaussian Elimination solver for systems of linear equations that contain less than 100 equations. I have roughed out a method for creating and filling in the diagonal of a ...
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31 views

Stuck on condition number derivation of the perturbed equation $(A + \Delta)\tilde{x} = b + \delta_b$

I've almost got what I want. We start with $Ax = b $ and $(A + \Delta)\tilde{x} = b + \delta_b$. What I have then is \begin{align*} \tilde{x} - x &= -A^{-1}\Delta\tilde{x} + A^{-1}\delta_b \\ ...
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47 views

Integration of ODE equation in Matlab / Octave

I have a system of 8 ODE's where the initial conditions are in matrix form. $\frac{dT}{dS} = H T$ where T at the initial state is the identity matrix. $T(a) = I$ H is a constant 8x8 matrix T is ...
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69 views

Is the Hessenberg form of a matrix unique?

I have to calculate the Hessenberg form of an matrix using householder reflectors. For real Matrices I get the same result as the 'hess()' function in Matlab, but for Complex Matrices I keep getting ...
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0answers
79 views

Alternative to the Gram-Schmidt Procedure for Orthogonalization

I was wondering if there is an alternative to the Gram-Schmidt procedure, which instead of being a successive orthogonalization scheme, would be non-successive (simultaneous)? In other words, is there ...
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0answers
102 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 ...
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33 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 ...
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43 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
36 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 ...
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0answers
47 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
76 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. ...
0
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0answers
106 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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0answers
36 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 ...
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40 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, ...
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0answers
47 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
41 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
55 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
71 views

How to find a transformation matrix T?

(1.) Suppose there is matrix C of dimension "p by n" where p is less then n i.e. p I want to know is there any particular way exist to find transformation matrix T of dimension "n by n" such that ...
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28 views

Not enough memory for GMRES

After realizing that Gauss-Seidel is extremely slow for my simulation, i wanted to try GMRES and luckily found the C++ code here without diving into the theory. The size of the matrix in my case is ...
0
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0answers
86 views

Francis Algorithm (Implicit QR Algorithm)

In Numerical Analysis, we are touching upon QR and Francis Algorithm. I understand that for Francis's Algorithm, we reduce the matrix to its upper Hessenberg form using Householder transform. What I ...
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0answers
75 views

Upper Hessenberg Form

I am given a matrix. I would like to reduce it to its upper Hessenberg Form. We are discussing eigenvalue computations in Numerical Analysis and the textbook just gives the algorithm for it without an ...
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0answers
54 views

QR Algorithm with Shifts Question

Why must QR Algorithm with Shifts make no progress when applied to this n x n matrix? (attached as image). Also, if a matrix A is orthogonal in a QR factorization, will R be tridiagonal? How would ...
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0answers
72 views

convergence for symmetric, positive semi-definite operator

Assume $u$ is a vector in the Euclidean space $\mathbb{R}^N$, $||u||=\sqrt{\langle u, u\rangle}$, where $\langle u, v\rangle = \sum_{i=1}^N u_i v_i$. I have that $||u^{k+1}-u||\leq ||I - c ...
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0answers
22 views

Is any method which allows segmentation of long diagonalizing procedures?

This is a question for a smarter way of numerical computation. When I diagonalize a certain type of Vandermonde-matrices in Pari/GP ("mateigen(M)"), for instance of size 16x16 then this can be ...
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0answers
30 views

only calculate diagonal of cholesky decomposition

I have a massive matrix $A$ that I can't hold entirely in memory, but it is possible to easily calculate individual entries ($A(i,j)$). I'm only interested in calculating the diagonal entries of the ...
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0answers
198 views

when fixed Point Iteration does not converge?

I want to solve a nonlinear system with the fixed point iteration method. I have initial condition,and the answer is known. By using this method the answer converges very slowly about 1000 iteration ...
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0answers
68 views

How to solve a divergent linear system using iterative methods?

I have a matrix A which is symmetric and non-diagonal dominant. I tried to use Jacobi/Gauss-Seidel/SOR to solve it but it diverges. Is there any mechanism to condition the matrix for convergence ...
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204 views

Symmetric Tridiagonal QR Algorithm

I have a question regarding QR algorithm. Suppose we are being given a symmetric tridiagonal matrix A (4X4) and perform QR factorization on A: A=QR. Then we define A':=RQ. A' still possesses the ...
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0answers
62 views

Minimizing an expression with linear constraints

Given a system of under-constrained (i.e. infinite solutions) linear equations (all values will be integers, all coefficients will be 0, 1, or -1), I want to pick values for the variables to minimize ...
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0answers
50 views

Is there any risk to transform to $(B^{T} \otimes A)\operatorname{vec}(X)=\operatorname{vec}(C) $ for solving $AXB=C$ for X

To solve the equation $AXB=C$ for X, we can use the property of vec operator and kronecker product to transform to $(B^{T}\otimes A)\operatorname{vec}(X)=\operatorname{vec}(C)$, where ...
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0answers
114 views

Can antiunitary symmetry be used to calculate determinant of a matrix

Suppose I have some $N \times N$ complex matrix $A$, that commutes with some antiunitary operator $U$ that satisfies $U^2 =-1$. It can be shown that $\det(A)\ge 0$ , because for every eigenvector ...
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161 views

Divide and conquer possible on linear equation systems?

Suppose a 4-connected regular grid $$\mathcal{G}=(\mathcal{E},\mathcal{V}),$$ where $\mathcal{E}$ and $\mathcal{V}$ denote the set of edges and vertices of that grid, respectively. Given this ...
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345 views

Can this non-linear optimisation problem be converted to a linear?

I have to minimize the function: $F(x)$ $F(x) = \sum_{i=1}^{M}||x_{i+1} - x_i - K(\frac{x_{i+1} + x_i}{2})||^2 + ||x_1-c_1||^2 + ||x_N-c_2||^2$ , where $x$ is a vector of $N$ scalars, $c$ are ...
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0answers
24 views

Fast method for getting solution for underdetermined equation system

What is a fast and stable method for getting a solution for an underdetermined equation system which could be applied by a computer?