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

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2
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0answers
55 views

Solution to pertubed linear system

Suppose one has the following system of linear equations $$(A + \Delta A) x = b$$ where $A$ and $\Delta A$ are large sparse matrices and $\Delta A$ is "small" compared to $A$, furthermore vector $x$ ...
2
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0answers
156 views

Showing that the least-square method minimizes error

Assume that the relation between temperature and time is defined as follows: $$T = A^kC$$ We can find parameters $A$ and $C$ using the least-square method. The given relation is not linear, but we can ...
2
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0answers
56 views

commuting a LU factorisation

Consider the permutation matrix $P= \begin{pmatrix} 0 & \cdots & 0 & 1 \\ 0 & \cdots & 1 &0 \\ \vdots & \ddots & \vdots & \vdots \\ 1 & \cdots ...
2
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0answers
169 views

Check for Ill Conditioned matrix

How can I efficiently check if a tridiagonal system with 1's in diagonal is ill-conditioned or not ? The common way is to get the ratio of largest and smallest singular values and see if its greater ...
2
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0answers
26 views

Re-calculate solution after altering some elements in a linear system

Problem I have a linear system: $$ Mx = b $$ $M$ is like a Band Matrix. And assume I have a solution $x_{init}$ at beginning. There will be some operations which are going to alter some elements in ...
2
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78 views

Spectral/ Eigen-Value solution with a linear constraint?

Is there a spectral or eigen-value solution to finding $X$ such that $Tr(CX^TMX)$ is minimum for a symmetric matrix $C$ and a p.s.d matrix $M$. Also there is a linear constraint on the minimization ...
2
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137 views

The norm of the matrix

This problem is in Trefethen'book Numerical Linear Algebra Suppose the $m\times n$ matrix $A$ has the form $A=\begin{pmatrix}A_1\\A_2 \end{pmatrix}$ where $A_1$ is a nonsingular matrix of dimension ...
2
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88 views

Need little hint to prove a theorem .

I have an iterative method \begin{eqnarray} X_{k+1}=(1+\beta)X_k-\beta X_k A X_k~~~~~~~~~~~~~~~~~ k = 0,1,\ldots \end{eqnarray} with initial approximation $X_0 = \beta A^*$ ($\beta$ is scalar ...
2
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2k views

What is the algorithm for LU factorization in MATLAB?

What is the algorithm for LU factorization in MATLAB, i.e. [L,U] = lu(a)? After searching for many examples and trying to compare the result with MATLAB, they are ...
2
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168 views

(Experimental) Can it be shown that this extension of the secant-interpolation has quadratic convergence?

Background: I needed some efficient but simple interpolation-methods aside of Newton's iteration because I want to have it in contexts, where the derivative of a function is not always known. So an ...
2
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0answers
945 views

QR with column pivoting

Golub and van Loan's algorithm 5.4.1 for QR factorization is suitable as a rank revealing algorithm. The results are R, Q with the subdiagonal elements stored in "factored form" and the column ...
1
vote
0answers
27 views

Generate a random neutrally stable matrix

I need to generate random real matrices such that all eigenvalues have real part equal to 0 -- i.e. random neutrally stable matrices. What's the simplest way to do this? Note that I don't care about ...
1
vote
0answers
13 views

SVD of a block partitioned matrix

Given a block partitioned matrix $\boldsymbol{A}$ $$ \boldsymbol{A} = \begin{bmatrix} \boldsymbol{A}_{1,1} & \boldsymbol{A}_{1,2} & \cdots \\ \boldsymbol{A}_{2,1} & ...
1
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0answers
45 views

Really confused about LU decomposition and Doolittle algorithm

I'm really confused about the Doolittle algorithm, so I need some help. At the end of the description at wikipedia, it says It is clear that in order for this algorithm to work, one needs to ...
1
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0answers
33 views

Question on the uniqueness of LU decomposition

Let $A \in \mathbb{K}^{n \times n}$ a matrix, and suppose that we can run the Gaussian elimination on $A$ without row or column interchange, so there exist the $LU$ decomposition of $A$. We define ...
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41 views

Using Cholesky factorization to solve the system AXA=B

I have been given a problem of solving X, which is an unblurred image, in the system: $$B = A X A \iff X = A^{-1} B A^{-1}$$ Where the matrix A describes the blurring of an image and the matrix B is ...
1
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0answers
42 views

Least-squares solution to a transformation between coordinate frames

Suppose I have four coordinate frames in 3D space: A, B, X and ...
1
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0answers
27 views

LU-factorisation of a square matrix

I need to show that the following matrix cannot be factor into the product LU. \begin{equation} A=\begin{bmatrix}1&2&-1\\2&4&0\\ 0&1&-1\end{bmatrix} \end{equation} I did the ...
1
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0answers
39 views

What's the point of 1-norm matrix estimation? Why not brute force?

Calculating (brute-force) 1-norm of a square matrix should take $O(n^2)$ operations, with a small factor involved. Apparently, there is an algorithm (link) for estimating 1-norm that takes $O(n^2 t)$ ...
1
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0answers
33 views

Is it better to compute $A^tA$ once and then $Ax$ several times or compute $y=Ax$ and then $A^ty$ every time?

So I have this algorithm which given a matrix $A$ it assigns $A=A^tA$ outside the loop and then on the algorithm loop it solves multiple instances of $Ax$ for different $x$s, (meaning that it's ...
1
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0answers
42 views

matrix with positive diagonal elements

I was wondering if a symmetric matrix with positive elements only in the diagonal (negative elsewhere) is any special beside the symmetry. Thanks in advance
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62 views

Rayleigh quotient ($|r(q)-\lambda|=O(||q-x||_2^2)$ ?)

how to show $|r(q)-\lambda|=O(||q-x||_2^2)$ $r(q)=q^*Aq/(q^*q)$ and $\lambda$ is an eigenvalue, A is a Hermitian matrix. x is the unit eigenvector corresponding to $\lambda$. and q is a unit vector. ...
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0answers
86 views

Meaning of singular Jacobian and workarounds to Newton's method

I'm currently working with Galerkin's method to solve differential equations and I have to retrieve unknown coefficients for the truncate expansion. This is just to set the background for why I need ...
1
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0answers
21 views

LU Decomposition for the solution of two linear systems

Let's say I have the following linear system: \begin{equation} \left[ \begin{array}{cccc} S&&L^{T}&&A^{T}&&0\\ L&&0&&0&&0\\ ...
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0answers
42 views

the Gauss-Jordan algorithm requires how many multiplications/divisions and add/subtractions

I am trying to show this following result. The Gauss-Jordan algorithm requires $\frac{n^3}{2}+n^2-\frac{n}{2}$ multiplications/divisions and requires $\frac{n^3}{2}-\frac{n}{2}$ ...
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0answers
85 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 ...
1
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0answers
58 views

Does this function have a unique fixed point?

Consider the following equations in fixed-point form: $$\mathbf{x}=F(\mathbf{x})=[f_1(\mathbf{x}),f_2(\mathbf{x}),\cdots,f_n(\mathbf{x})]^T$$ where the function $f_i$ is defined as ...
1
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0answers
19 views

Speed of pseudo-inverse (with possibly ill-conditioned matrices)

I am computing the pseudo-inverse of several matrices of identical size $m \times n$ . However, computation (e.g. with the LAPACK pinv) seems to be much slower in some cases (5 to 10 times slower). ...
1
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0answers
57 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} ...
1
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0answers
39 views

Write down a linear programming problem

I want to replicate a linear programming problem.I have the following information, for the background." A fuzzy regression analysis with only one independent variable X results in the following ...
1
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0answers
47 views

Solve linear system with matlab

In my problem, $A$ is a $m \times n$ matrix with $m \geq n$ and $\mathrm{rank}(A)= n$. Let $\Gamma$ be the $(m+n) \times n$ matrix defined by : $$ \Gamma = \begin{bmatrix} A \\ \mathrm{I_{n}} ...
1
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0answers
102 views

stability of FTCS scheme for parabolic equation

Can you suggest any method for stability analysis of FTCS scheme for the the following parabolic equation ? D.E: $u_{t}=a(x,t)u_{xx}+f(x,t,u)$, $0<x<1$, $0<t<T$, $T>0$ BCs: ...
1
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0answers
53 views

Setting up Kernel to Numerically Solve Fredholm Equation of Second Kind

I am looking to confirm if what I am doing is the proper procedure. I writing a program to discretely solve a Homogeneous Fredholm Equation of Second Kind that is set up as follows: $ \int ...
1
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0answers
25 views

Numerical methods for computing exponential, if I have computed an exponential of a perturbated matrix

I need to compute the product $e^{H_1}\,e^{H_2}\,\ldots\,e^{H_n}$ for antihermitian matrices $H_j$ that do not commute and $H_i-H_{i+1}$ is small. Is there a numerically convenient way to compute ...
1
vote
0answers
23 views

Transformation between Ideal and Warped Surface

I work on manufacturing metal panels with holes drilled in them. Suppose I have an ideal 3D surface from CAD. I want to compare it to the actual part using reference points to compare between the two. ...
1
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0answers
47 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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0answers
43 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 ?
1
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0answers
44 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 ...
1
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0answers
40 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 & ...
1
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0answers
54 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 ...
1
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0answers
94 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 ...
1
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0answers
64 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, ...
1
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0answers
42 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 ...
1
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0answers
34 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 ...
1
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0answers
95 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 ...
1
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0answers
91 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 ...
1
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0answers
79 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 ...
1
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0answers
64 views

Numerical algorithm to solve quadratic eigenvalue problem.

Given the equation $$-4 \left(a^2+a (n-1) (2 t^2-1)\right) \left(\sum _{i=0}^n \alpha_i t^{2 i}\right)^2 \\ +\frac12 \left(\sum _{i=0}^n \alpha_i t^{2 i}\right) \left(t \left(8 a (t^2-1)+1\right) \sum ...
1
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0answers
69 views

Kalman Filter Predict Update of LDL Decomposition of a Covariance Matrix

From the state predict equation: http://en.wikipedia.org/wiki/Kalman_filter# $$P_{n+1}=AP_nA^T + Q$$ Suppose the $LDL^{T}$ ( http://en.wikipedia.org/wiki/Cholesky_decomposition#LDL_decomposition_2 ) ...
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0answers
25 views

Algorithm for finding only the $k$-th singular vectors

I know that we have truncated SVD that can compute the first say $k$ largest singular values (and corresponding singular vectors). However, I'd like to know if there is an algorithm that can find only ...