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

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70 views

$F(x)=Ax+b$ is a contraction mapping

I want to proof that $f(x)=Ax+b$ with \begin{align} A = \begin{bmatrix} 0 & -\frac{1}{8} & \frac{1}{4} \\ 0 & \frac{1}{3} & 0 \\ -\frac{1}{2} & -\frac{5}{22} & \frac{3}{4} ...
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88 views

Applications of Numerical methods

I'm in a course of Numerical Methods and part of an assignment is find an article about an application of numerical methods, explain this article and present a program (in matlab/octave) that ...
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72 views

How to condense a matrix to a vector

I'm not an experienced person in mathematics and this might either sound like a trivial question or a stupid one. However, this problem arose to me when I was writing a program. Following is my ...
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62 views

Reformulating objective function of canonical correlation analysis

Given two column vectors $X = (x_1, \dots, x_n)'$ and $Y = (y_1, \dots, y_m)'$ of random variables with finite second moments, canonical-correlation analysis seeks vectors $a$ and $b$ such that the ...
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42 views

Matrix computing

Given an $n$-vector $x$, show that floating-point computation of the Householder vector $v$ such that $P x = (I − 2vv^{T} )x = \pm\left\|x\right\|_{2}e_{1} $ gives a forward stable result $v^{\prime}$ ...
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119 views

Matrix-valued expansion in spherical harmonics

I am seeking a clever solution to the following problem. Given $$X(\theta,\phi) = exp(-iA(\theta,\phi))\; B\; exp(+iA(\theta,\phi))$$ with the square, Hermitian matrix $A$: $$A(\theta,\phi) = A_{0,0} ...
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922 views

The spectral radius of the matrix $A$ is less than or equal any natural norm

Show that the spectral radius of the matrix A is less than or equal any natural norm, i.e: $$\rho(A) \leq ||A||=\max_{||x||=1}{||Ax||}$$ where $\rho(A)=\max\{|\lambda|:\lambda \text{ is a eigenvalue ...
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104 views

Fast way to find a matrix with only $0$ and $1$ as entries full-rank or not?

I have a huge number of small Zero-One Matrices($4\times 4$, $5\times5$,$6\times6$) and I want to determine whether they are full-rank or not one by one. Gaussian elimination is a option, I want to ...
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663 views

Least Squares “analytic expression” for fitting a 2D quadratic function to measurements

I have n scattered elevation measurements: $ \{x_i,y_i,z_i\}_{i=1..n} $ that I want to fit a quadratic function to: $ z = ax^2 + by^2 + cxy + dx + ey + f$. The problem can be written as a vector ...
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90 views

Matrix in Matlab

I'd like to compute the centralizer of a subgroup $H$ of orthogonal group $O(8, R)$, so I need to solve the equation $AX=XA, BX=XB \mbox{ where } H=\langle A, B\rangle.$ The problem that I have is ...
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186 views

Matrix spectral decomposition

Let $A$ be a square matrix $(N \times N)$ and $a_{ij} \in \mathbb{R}$. Suppose A has N eigenvalues $\lambda_{1} < \lambda_{2} < ...\lambda_{n} \in \mathbb{R}$. $A$ = $R \Omega R^{-1}$ its ...
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23 views

Short-cut to a group of long sums/differences

If I have data $a,b,c,d$, and want to calculate $x=a+b-c-d$, $y=a-b-c+d$ and $z=a+b+c-d$, I can save three adds by doing $e=a-c$, $f=b-d$, then $x=e+f$,$y=e-f$, $z=a+c+f$. If I have 100 data values ...
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159 views

How to Diagonalize an Extremely Large Sparse Matrix in SLEPc/PETSc

Dear Friends, Recently I have started with learning SLEPc/PETSc, but I didn't find a way to solve my problem. I have to solve a big sparse matrix which is a two dimensional quantum ...
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113 views

Existence criteria for the LU decomposition of a tridiagonal matrix

In this link, the following result is presented without proof: Let $a, b, c$ be the lower off diagonal, diagonal, and upper off diagonal elements of a tridiagonal matrix. A pivotless LU ...
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212 views

About the Generalized singular value decomposition (GSVD).

I have studied about Singular value decomposition (SVD) and had solved few numerical examples to understand SVD. Now I am studying Generalized singular value decomposition (GSVD). I followed this ...
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38 views

Algorithm to compute similarity computation

I have a similarity transformation of matrices from the type $B = P^{-1}AP$. It is known that $A$ and $P$ are invertible matrices, but not orthogonal. Given that I have the matrices $P$ and $A$ I ...
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297 views

Multigrid Interpolation and Restriction operators

I have a question about the restriction and the interpolation operators of a Multigrid algorithm. Let those be given: The full weighting restriction stencil (in 2D): $\frac{1}{16} \left[ ...
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52 views

Preconditioning and effects on precision of solution of LSE

In my courses on numerical analysis I have been tought that the main and principle motivation for preconditioning linear systems of equations is to increase the convergence rate of iterative solvers ...
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207 views

Criterion for detecting rank-deficiency via QR decomposition?

I apologize in advance if this is an ill-posed question -- I'd appreciate advice on what pieces are missing as much as an answer. I'm solving a system like $P \approx X Y^T$, where P is a large ...
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100 views

Nearest point to a convex polytope

I am looking for fast, memory-efficient computational algorithms to solve the following problem: Minimize: $||x - x*||_2^2$, subject to constraints $A x = a, B x <= b, l <= x <= u$, where ...
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251 views

Weighted linear least squares parameter covariance

I am currently trying to figure out the parameter covariance for a weighted linear least squares problem where $$y = X\beta$$ The parameters for which my objective function is lowest are given by ...
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22 views

How to construct an optimal subspace with 3 indices.

I have a 3-dimensional array that is potentially very large and I need to do quite a lot of operations with it. Is there a systematic way to choose a subspace of a certain size, such that the norm ...
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429 views

covariance matrix eigenvalues eigenvectors

Is there a probabilistic or analytical meaning of the eigenvalues/eigenvectors of covariance matrix of multivariate normal distribution? Thank you
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44 views

Numerical linear algebra spectral norm limit.

Let $A \in \mathbb{R}^{m \times n}$ be of full rank. Consider $X_{k+1}=(2k-X_{k}A)X_{k}$, $X_0 = \alpha A^{T}$. Let $E_k = I-X_kA$, Deduce that if $||E_{0} ||_{2}<1$, then $lim_{k \rightarrow ...
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119 views

fixed point spectral radius

We have the following stationary matrix iteration $$x_{k+1} = Mx_k + c$$ where $M$ is nxn matrix and $c$ is a vector. Let $r(M)$ denote the spectral radius of $M$. Show that spectral radius ...
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213 views

Numerical Linear Algebra problem (QR factorization with column pivoting)

For matrices that might be rank deficient it is common to incorporate pivoting in Householder QR factorization of A $\in$ $\Re^{mxn}$ (m $\geq$ n). Let $A^{(k)}$ denote the matrix at the start of the ...
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129 views

Show that its a Generalized Eigenvalue problem

Show that the minimizer is obtained by a generalized eigenvalue problem. $$\alpha=\underset{1^TK\alpha=0; \ \alpha^TK^2\alpha=1}{\text {arg min}} \gamma ||f||_{K}^2+f^TLf$$ Details: $K$ ...
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176 views

Simultaneous Eigenvalue Problem

I have what I think is a simultaneous eigenvalue problem in three parameters: $$\alpha A_1x + \beta B_1x + \gamma C_1x + D_1x = 0$$ $$\alpha A_2x + \beta B_2x + \gamma C_2x + D_2x = 0$$ $$\alpha A_3x ...
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51 views

Algorithms for Performing Large Integer Matrix Operations w/ Numerical Stability

I'm looking for a library that performs matrix operations on large sparse matrices w/o sacrificing numerical stability. Matrices will be 1000+ by 1000+ and values of the matrix will be between 0 and ...
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265 views

Finding a row permutation that makes a matrix more “blocks-like”

Disclaimer: what follows arise in a context from Computer Science, but it seems to me that my questions were more likely to be solved from mathematicians than from computer scientists. Let suppose ...
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89 views

How to solve Rayleigh Quotient type problem?

How to solve Rayleigh Quotient type problem? $$\max (w+w_0)^tC(w+w_0) \text{ s.t. } w'w=1,$$ where $w_0$ is given. Thank you!
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56 views

formulas for exact values of singular values in low dimension?

Are there formulas for the singular values of a real matrix in low dimension, i.e. for a $2 \times 2$ matrix or a $2 \times 3$ matrix? Any comment is welcome.
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150 views

The fastest algorithm of computing Principal eigenvector of a non-negative-entries matrix

I am studying the QR algorithm, is it the fastest one in this situation?
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134 views

Stable and efficient projection onto subspace along another subspace

Suppose we are given the euclidean space $\mathbb R^{n+m}$ with the decompositin $\mathbb R^n = V \oplus W$, which we however do not expect to be orthogonal. Let us describe the matrix $P$ that ...
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146 views

Nonlinear least squares and polygon area

I found this paper that describes preserving the global area of a polygon given some deformation (section 5): http://www.kunzhou.net/publications/2DShape.pdf I'm trying to do something very similar. ...
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74 views

Looking for a specific paper not available electronically

not really sure that it is the right place to post, but I'll give it a go. I really would love to have a look to this technical report J.G. Lewis, Algorithms for Sparse Matrix Eigenvalue Problems, ...
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102 views

Fast simultaneous orthonormal basis computation for multiple nullspaces

Consider vectors $a_i\in R^{m\times n}$ and $B\in R^{m\times p}$, with $n +p < m$, and assume that the columns of $(A, B)$ are linearly independent. To compute an orthobasis for $\text{ker}(A)$, it ...
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402 views

Tridiagonal sparse matrix - linear equation

I have to following linear system to solve : $Ax=e_1$ where $A$ is a sparse tridiagonal matrix with the main diagonal terms $a_{ii}$ being all different, and the off-diagonal terms being each others ...
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12 views

Does the Conjugate Gradient Method provide an eigenvalue estimate?

Suppose that we apply a Krylov subspace method to the linear system $A x = b$. For example, if $A$ is symmetric positive-definite, then the Conjugate Gradient method may be used. I remember that the ...
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16 views

Enstrom and kakeya theorem

i need some numerical application of enestrom Kakeya thereom ? in fench je cherche une application du theoreme d enstrom et kakeya application numerique ou une application dans un domaine de ...
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24 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve $Ax = b$, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous ...
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11 views

choice of iterative linear system method

while implementing an unconstrained optimization problem, using Newton's method, I am faced with a Hessian matrix that is very large (10^8 by 10^8) but very, very sparse - Non zero elements along the ...
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43 views

Induced matrix p-norm

Let $\|\cdot\|_p$ denote the $p$ norm $(p≥1)$ defined for every vector $x=(x_1,x_2,\ldots,x_n)^t\in\mathbb C^n$ by $\|x\|_p=(\sum|x_j|^p)^{1/p}$ and let $|||\cdot|||_p$ denote the matrix norm defined ...
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33 views

update cholesky factorization

I need to compute cholesky(H'*H) where H is a big sparse rectangular matrix. After that H is modified by adding several lines. That is Hn = [H ; line_1 ; ... ; line_n] in Matlab. How can I recompute ...
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18 views

order of convergence for approximations

Let $u \in L^{2}(0,1)$ and $0 < x_{1}< x_{2}<... < x_{n} = 1$, where x$_{k}$ = k$\cdot$h, n$\cdot$h = 1, a partition of the interval [0,1]. Define I$_{k}$(x) = 1 if x $\in$ [x$_{k}$, ...
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18 views

Location and perturbation of eigenvalues

This is a problem from Horn and Johnson's Matrix Analysis. I'm having trouble showing the bolded parts in the following paragraphs. In fact, I don't really understand what the sentences mean. I would ...
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26 views

A problem on Gersgorin cirle passing through the eigenvalue of an absolute matrix

I'm having trouble solving the following problem. I think I need to show that the matrix $D^{-1}|A|D$ has property SC, but I can't come up with a way to show it. I would really appreciate any ...
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30 views

Is the following matrix Upper Hessenberg?

Does $$ A = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$$ properly satisfy the definition of upper Hessenberg?
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55 views

Applying Central Difference (Finite Difference Method) in MATLAB

I was given a rather complicated few problems to solve in MATLAB using the central difference method, and I'd like some help figuring out how to translate this into code. The goal is to discretize ...
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32 views

Show that Newton’s Method is well-defined for all k and converges to 0 for $x_0>0$

Let $f : R → R$ with $f$ twice continuously differentiable, $\gamma > f''(x)>\delta, f(0)=0,f'(x)>\rho $ for $x ≥ 0$. Show that for any $x_0 > 0$ that Newton’s Method is well-defined for ...