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

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2
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0answers
56 views

Customising force-directed graph layout

I would like to implement a variant of the force-directed graph layout where some nodes are constrained to moving only along a predefined curve (e. g. circle). I looked at some implementations using ...
2
votes
0answers
162 views

Orthonormal Matrix weighted regression

$Q$ is a rectangular matrix with orthonormal columns. A linear system composed of $$Qx= b$$ is really easy to solve as: $$Q'Q=I$$ hence: $$x=Q'b$$ Given that $Q$ is orthonormal can this be used to ...
2
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0answers
186 views

$l^1$ norm estimate for inverse of Vandermonde matrix

As title, I would like to know the known upper bound for the $l^1$ norm for inverse of Vandermonde matrix. A quick search gives this paper by Gautschi 40 years ago, but it deals with the infinity norm ...
2
votes
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
158 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
57 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
179 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
votes
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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0answers
80 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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0answers
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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0answers
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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0answers
3k 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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0answers
171 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
votes
0answers
1k 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
35 views

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $?

How can I write $\prod\limits_{i = 0}^n {\left( {x - {x_i}} \right)} $ in terms of a polynomial as $\sum\limits_{i = 1}^n {{a_i}{x^n}} $? In the other words, is there a way to write $a_i$ in terms of ...
1
vote
0answers
15 views

Conditional expectation of a set of Gaussian variables

I was wondering if there is an efficient way to compute the conditional expectation of every element in a Gaussian random vector ? Specifically: For a pair of Gaussian random variables $[x,y]$, the ...
1
vote
0answers
26 views

How to apply Runge-Kutta to an implicit scheme?

I see there are some differences in the solution as I increase the resolution of my grid. I'm using Operator Splitting to solve Diffusion Reaction equation \begin{equation} \frac{\partial ...
1
vote
0answers
11 views

Relationship between QR and LU factorization

Both algorithm return very similar results in terms of having a upper/right triangular matrix as one of the factors. What is the relationship between Q and L, and between R and U? What is the ...
1
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0answers
23 views

How can I solve this specific set of equations?

Here are the equations: $$\sum_{k = 1}^n i_k + Y_n u_n = J \quad \quad (1)$$ $$i_1 + Y(u_1 - u_2) = J \quad \quad (2)$$ $$i_k - Y(u_{k - 1} -2u_{k} + u_{k + 1}) = 0, \quad \quad k = 2, ..., n - 2 ...
1
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0answers
42 views

Numerically stable SVD

In this question regarding SVD, it is explained why eigen decomposition of $ A^tA $ is not numerically stable compared to "direct SVD algorithms". Since the former is the algorithm I'm most familiar ...
1
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0answers
28 views

Numerically finding eigenvalues of a Volterra operator of first kind

I'm looking for a solution to the following problem - $\int_{-\infty}^{\infty} K(x-y) f(y) = \lambda f(x)$ Consider $K(x-y) = \left\{ \begin{array}{lr} e^{-(x-y)} & : x > y \\ 0 & : x ...
1
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0answers
26 views

Ideas for expressing the inverse of matrix quadratic form $CAC^T$

I want to find an expression for the inverse of the matrix system $Z=CAC^T$, where $A \in \mathbb{C}^{n \times n}$ is block diagonal with dense blocks, and $C \in \{-1,0,1\}$ with dimension $m \times ...
1
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0answers
38 views

Eigenvalues after Givens-Rotation

Im just validating my own Code of a Givens-Rotation in Matlab. Therefore i let matlab compute the Eigenvalues after each Givens-Rotation. I am wondering why the Eigenvalues computed by matlab are ...
1
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0answers
42 views

Simplifying the Generalized Eigenvalue Problem

Let $\Sigma_1$, $\Sigma_2$ be symmetric positive-definite real $n\times n$ matrices. We want to solve the generalized eigenvalue problem $$ \Sigma_1V=\Lambda\Sigma_2V, $$ where $\Lambda$ is the ...
1
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0answers
54 views

complex rank-one update

I'm trying to find the eigendecomposition of a rank-one update to a complex matrix $D + uv^T$. The matrix $D$ is diagonal, but not the identity. It has unique imaginary entries along the diagonal. ...
1
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0answers
39 views

Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay 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, ...
1
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0answers
22 views

Overdetermined system with discrete data.

The setup I have a set of experimental data (subscript 1) which calculates two variables $u_1(x,y,z)$ $v_1(x,y,z)$ I can calculate the three spatial gradients for my two variables ($u_1$ and ...
1
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0answers
26 views

QR Algorithm fails under certain conditions

First of all, i have to admit that i am really knew to this numeric stuff. I have to detect two complex Eigenvalues of a Matrix and therefor i implemented some easy QR-Algorithm with MatLab. I am ...
1
vote
0answers
31 views

$Az + B\overline{z}$ as a linear operator

Given two matrices $A,B \in \mathbb{C}^{n\times n}$ with fixed $n\in\mathbb{N}^+$, let us consider the operator $$ L:\mathbb{C}^n \to \mathbb{C}^n,\\ L(z) = Az + B\overline{z}. $$ This operator is not ...
1
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0answers
22 views

Row degeneracy in systems of linear equations

I am trying to understand the concept of row degeneracy in a system of linear equations, but having trouble understanding this problem. \begin{align} x+2y+z &= 2 \tag{1} \\ 2x+y+3z &=5 ...
1
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0answers
13 views

Enforcing additional constraints in linear equation

In a finite element context, I come up with a sparse "stiffness matrix" $A$ and a corresponding RHS $b$. The goal is now to solve $$Au = b$$ Where $u$ is a coefficient vector of the solution. Now I ...
1
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0answers
35 views

Solving System of Linear Equations

These are the two known equations: $$\frac{(I_2+I_3)-(I_1+I_4)}{I_1+I_2+I_3+I_4} = \frac{2x}{L}$$ $$\frac{(I_2+I_4)-(I_1+I_3)}{I_1+I_2+I_3+I_4} = \frac{2y}{L}$$ where I know the values of $(x,y,L)$. ...
1
vote
0answers
25 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
88 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
vote
0answers
36 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 ...
1
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0answers
52 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
45 views

Least-squares solution to a transformation between coordinate frames

Suppose I have four coordinate frames in 3D space: A, B, X and ...
1
vote
0answers
29 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
40 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
48 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
1
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0answers
65 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. ...
1
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0answers
120 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
22 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\\ ...
1
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0answers
47 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 $\frac{n^3}{2}-\frac{n}{2}$ ...
1
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0answers
60 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
21 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
74 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
43 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
55 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}} ...