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

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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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18 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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22 views

Householder reflector sign error

I am studying Householder reflectors from Trefethen and Bau but am having trouble creating a simple example for it. I am given the equation for the vector v that the Householder reflector H is based ...
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19 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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12 views

SOR and Conjugate Gradient Method

Let $H_n=[H_{ij}]\in\mathbb{R}^{n\times n}$ be Hilbert Matrix, define $h_{ij}=\frac{1}{i+j-1}$ and $x=\left(1\quad 1\quad\cdots\quad 1\right)\in\mathbb{R}^{n\times n}$ such that $b_n=H_nx$. Use SOR ...
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21 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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13 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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32 views

Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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41 views

Saddle point problem (KKT) with block-diagonal matrix

Consider the following saddle point problem originating from an interior-point method algorithm: $$ \begin{bmatrix}\mathbf{H} & \mathbf{A}^{T}\\ \mathbf{A} & \mathbf{0} ...
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34 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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21 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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40 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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30 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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20 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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40 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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28 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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63 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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66 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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19 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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24 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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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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41 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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50 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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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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50 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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27 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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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, ...
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42 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 ...
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30 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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44 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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44 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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27 views

Orthogonal polynomials induction proof

I tried writing this all out but cannot seem to get anything sensible. Basically I want to prove that assuming w(x) is the weight function of a Gram Schmidt orthogonalization process and w is an ...
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19 views

Maximal angle between kernel of rows of a matrix

Consider a matrix with 2 columns $$ \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \\ a_3 & b_3 \\ \vdots & \vdots \end{pmatrix} . $$ To each row $(a_i \;\;\; b_i)$, one draws the kernel ...
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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 ...
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57 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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53 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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44 views

Minimizing a vector constrained to a set

Sorry if this is wordy or over-complicated, I'm not sure how to isolate the problem any more than I have below without losing important context: I'm trying to implement a coordinate block descent ...
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47 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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38 views

cubic B-spline interpolation function

I read that the B-spline basis functions are the follows: $B_0(x)=(1-x)^3/6$ $B_1(x)=(3x^3-6x^2+4)/6$ $B_2(x)=(-3x^3+3x^2+3x+1)/6$ $B_3(x)=x^3/6$ The cubic b-spline interpolation function it ...
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66 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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64 views

Numerical Linear Algebra- Proof for the Backward stability of inner Product?

I know that inner product is backward stable through various sources but to prove this statement is which I'm not aware of...if someone could help me with that that would be great. ...
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24 views

Iteratively solve linear equations with rank-1 updates on LHS and RHS

What is the best way to iteratively solve updating equations of the form $$ Ax=b $$ $$ (A+c_1v_1^\intercal)x_1=b+ \alpha_1 d_1 $$ $$ (A+c_1v_1^\intercal+c_2v_2^\intercal)x_2=b+\alpha_1d_1+\alpha_2d_2 ...
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58 views

Rank and solvability of a matrix

I am working with linear complementarity problems (LCPs) which look for a solution $\mathbf{x} \in \mathbb{R}^{n}$ in the form $$ \begin{matrix} \mathbf{x} & \geq & 0 \\ ...
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20 views

need to determine weights so that quadrature formula holds

Let $l$ be an interval on the real axis, $t_1,...,t_n$ be distinct $n$ points, then there exists n numbers $m_1,...,m_n$ such that the quadrature formula, $\int_l p(t)dt = m_1p(t_1) + ... + ...
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56 views

Expressing rank condition of a matrix in terms of its elements

Let $x \in \mathbb{R}^{n}$, define $X = xx^{T}$. I have an optimization problem with some linear constraints and few quadratic constraints, and I have to solve for $x$. Using $X$ as the unknown ...
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10 views

Predict values of some numerical vectors by using other numerical vectors with all these vectors in the same vector set

I need to solve a problem about predicting values of some numerical vectors by using other numerical vectors with all these vectors in the same vector set, which is generated by one or more black box ...
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19 views

How to go from linear “equality-constrained” least squares (LSE) to linear “less-equality-constrained” LSE

I am trying to figure out how to pass from one problem to other. The linear equality-constrained least squares problem can be solved using a generalized RQ factorization (lapack solves this using ...
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26 views

Matrix conditioning with one degree of freedom

Given a not so well conditioned, NxK, N>>K matrix A with a certain structure. I have just one degree of freedom: I can multiply each row with a different factor. In formula: $$ \mathbf{B} = ...
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15 views

Singularity check for Homographies

I know that the standard singularity check for a matrix represented in some finite-precision format (IEEE-754 or whatnot) is "the matrix is singular if the reciprocal of the condition number of the ...
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17 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 ...