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

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53 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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264 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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304 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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460 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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45 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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142 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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218 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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142 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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54 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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297 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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359 views

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

I have to minimize the function: $$F(x) = \sum_{i=1}^{M}\left\|x_{i+1} - x_i - K\left(\frac{x_{i+1} + x_i}{2}\right)\right\|^2 + \|x_1-c_1\|^2 + \|x_N-c_2\|^2,$$ where $x$ is a vector of $N$ scalars, ...
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92 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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155 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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151 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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154 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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106 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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435 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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7 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
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30 views

How to calculate $det(X^T X)$ efficiently, update one column of X each time

$X_{1} = (A, b)$, where $X_{1}$ is a $n\times p$ matrix, $A$ is a $n\times (p-1)$ and $b$ is $n\times1$. First calculate $\det(X_{1}^T X_{1})$, then update $b$ with $c$, st. $X_{2} = (A, c)$ and ...
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23 views

Efficient method to compute grand sum of a Vandermonde matrix?

Is there a computationally efficient method to calculate the sum of all elements (grand sum) of a Vandermonde matrix? Each row can be quickly calculated using the formula for a geometric progression. ...
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14 views

Evaluating a quadratic form with an inverse of a sparse PD matrix, comparison between using the inverse vs using a Cholseky decomposition

I have the following quadratic form I need to evaluate: $x^T A^{-1} y$, where $A$ is a sparse positive definite matrix, $x, y$ are sparse vectors. Now assume that I am given for free both $A^{-1}$ ...
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16 views

Approximation of Mahalanobis distance

If $A$ is a symmetric positive definite $n\times n$ matrix then the square Mahalanobis norm of a vector $v\in \mathbb{R}^n$ is given by $$\lVert v \rVert_A^2=v^t A^{-1} v.$$ Now I have a situation ...
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13 views

Proof that Householder Triangularization for QR is backward stable

How do you prove that QR factorization via Householder Triangularization is backward stable? Theorem 16.1 (From Trefethen and Bau): Let the $QR$ factorization of a matrix $A$ be computed by ...
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22 views

Linear algebra <perhaps an application of Gordan' Theorem>

Question. Let $a_1,...a_n\in\{0,1,-1\}^m$ and $\sum a_i=(1,...,1)$. Is there a permutation $\tau$ of $\{1,...,n\}$ Such that for each $k\in \{1,...,n\}$ the vector $\sum_{i=1}^k a_{\tau (i)}$ has ...
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18 views

Is my QZ-Step as postulated in the paper?

I am working with the paper An algorithm for generalized matrix eigenvalue problems from C.B. Moler and G. W. Stewart (paper available here: ...
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14 views

QZ-Algorithm: how to simultaneously generate the upper triangular form

I need to calculate the Eigenvalues of a generalized Eigenvalue Problem. To achieve this, i wanted to use the QZ-Algorithm, which avoids calculating the inverse of a Matrix. This quite useful, because ...
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35 views

Givens-Rotation from the right

i need to get a Givens-Rotation, which zeros a matrix entry when multiplied from the right side. I did already look at this topic givens rotation from right side but i could not really understand the ...
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8 views

Conversion of cylindrical harmonic field into space-harmonic field for plane waves

It is well known that a plane wave can be represented by an infinite sum of cylindrical wave function of the form $\varphi^i(\rho,\phi)=e^{\left(-j\beta \rho ...
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15 views

PCA when SVD is a skinny SVD

A = m * n matrix. When $m \ge n$, it is easy to see that the V matrix in the full SVD ($A = U*S*V^T$, where U and V are both orthonormal square matrix) and V in a skinny SVD are the same. When $m \lt ...
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81 views

Linear Algebra Book like Calculus Made Easy

Now, I know that there are a tons of reference requests for Linear Algebra books but mine is very specific: what is a nice, short, concise, simple, to the point book that gets at the heart of Linear ...
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41 views

Solving the linear system $XL + L^TX = M$ efficiently

I'm wondering of an efficient way to solve the following system for the symmetric matrix $X$, given a positive semi-definite matrix $S$ and any matrix $M$: $$ LL^T = S $$ $$ XL + L^TX = M $$ $$ (XL) + ...
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44 views

When to use iterative methods for solving systems of linear equation

Iterative methods such as Jacobi, Gauss-Seidel method and successive over relaxation have a very limited field of use - for diagonally dominant matrices. So how they could be used on practice? What is ...
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37 views

Implicit Solution of Linear Algebraic Equations with Discontinuities

I am trying to get a reliable algorithm for solving a set of linear algebraic equations involving implicit singularities/discontinuous function. The model equation is: $$ {\bf s}_{n+1} = {\bf s}_n ...
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26 views

Smallest square problem, $A^*A$ singular?

In our numerics class, we have to solve the smallest square problem $Ax = b$ with $$A = \left( \begin{matrix} 1 & 3 &-4\\ 3 & 9 & -2\\ 4 & 12 & -6\\ 2 & 6 & 2 ...
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14 views

Linear Inverse Problem with symmetry constraint

I'm not entirely sure if this is even a solvable problem: $\mathbf{A} = \mathbf{B} \mathbf{C}$ Knowns: $\mathbf{A} \in \Bbb{R}_{n\times m}^{+}$, $\mathbf{B} \in \Bbb{R}_{n\times m}^{+}$ An ...
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56 views

Determining Nullspace Basis so that only one column is deleted or added as a row is added or deleted, with remaining columns of basis staying the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...
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29 views

the SVD (singular value decomposition) of an augmented matrix

Suppose we have a $4\times 3$ dimensional matrix $A$. Denote the SVD of $A$ by $USV^T$, where $U\in R^{4\times 3}, S\in R^{3\times 3}, V\in R^{3\times 3}$. Then, we construct a new matrix $B=[A;0]\in ...
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24 views

Upperbound for a linear algebraic ratio?

Consider ($n\times 1$)-column vector $\mathbf{p} = (p_i)_{i=1}^n$ with $p_i > 0$ and a symmetric ($n\times n$)-matrix $\mathbf{A} = [a_{ij}]$ with $a_{ii} = 0$ and $a_{ij} \in [0,1]$ for $i \neq ...
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10 views

Backwards Stability of systems

Let $A$ be a nonsingular matrix, let $x_{k+1}$ be an approximation to the solution of $Ax=b$, and let $r^{k+1}=b-Ax^{k+1}$. Show that $x^{k+1}$ is $\epsilon$-backward stable approximate of ...
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28 views

Linearize discretized nonlinear system model

For the following nonlinear system I want to find the linearization after a discretization: $$ \begin{pmatrix} \dot{x_{1}} \\ \dot{x_{1}} \\ \dot{x_{1}} \end{pmatrix} = 1/A \begin{pmatrix} ...
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16 views

sparse, complex, unymetric test-matrix

Can anybody recommend me a sparse, complex, unsymmetric test-matrix (maybe from MartixMarket) which is solvable with a transpose-free QMR without preconditioning in under 1000 iterations?
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20 views

How many kind of basis function to approximate an arbitrary function

I am finding a list algorithm to approximate an arbitrary function. Such as Bernstein, he said that a linear combination of Bernstein basis polynomials $$B_n(x) = \sum_{\nu=0}^{n} \beta_{\nu} ...
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19 views

Implementing specific SVD algorithms

My goal is to learn to implement the two-sided Jacobi SVD, a method of SVD for bidiagonal matrices, and a method of SVD for tridiagonal matrices. Can anyone recommend a place to learn about these, or ...
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21 views

How to implement QR method for bidiagonal matrices?

My goal is to take the singular value decomposition of a (not necessarily square) matrix. I have a method to do bidiagonalization of a matrix, and I can chop the bottom rows of zeros. In order to find ...
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37 views

Help understanding Jacobi SVD

I found this link, and I want to complete this implementation of the Jacobi SVD method, but it isn't clear to me how to implement alpha, beta, and gamma. I think it's very clear that ...
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6 views

Find rank-vital rows (coloops) of a matrix

Let $A$ be a $m\times n, m\geq n$ matrix over a finite field. Coloop is any row of $A$, such that the rank of $A$ is decreased when that row is removed. What is an efficient algorithm to find all the ...
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51 views

QR method for Hessenberg matrices

In trying to implement the method, my approach is to use a reduction to Hessenberg form, and then to iterate using a QR method of Givens rotations. However, I am having trouble successfully ...