0
votes
1answer
23 views

Tridiagonalize matrices with Householder transformation

I know that it is possible to tridiagonalize symmetric matrices by using a Householder trafo. I also found that we can get any matrix to Hessenberg form by using Householder trafos, but I still don't ...
0
votes
0answers
13 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 ...
0
votes
0answers
51 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 ...
1
vote
0answers
25 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: ...
0
votes
0answers
51 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 ...
1
vote
1answer
14 views

Condition number of a matrix bounded from below and above?

Is condition number of an invertible matrix bounded from below? And is condition number always bounded from above for an invertible matrix?
0
votes
2answers
95 views

Different method for QR decomposition - is it possible

This method could also possibly be applicable to matrices of higher dimension, but for the simplicity of my question i will only ask it for $2$x$2$matrices. Suppose $A=\begin{pmatrix} a_{11} & ...
1
vote
0answers
21 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 ...
0
votes
0answers
12 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 ...
2
votes
0answers
64 views

Numerically approximate the maximum of an element of a vector after a series of matrix multiplications.

Where S is a sigmoidal function, A_i is a matrix, and x is an input vector, and ...
0
votes
0answers
40 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 ...
0
votes
0answers
34 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. ...
0
votes
1answer
48 views

Solving 2x2 diagonally dominant matrix systems (non-symmetric)

I have a linear system of the form $Ax=b$ where $A\in \mathbb{R}^{2\times2}, b\in \mathbb{R}^{2\times1}$. A is diagonally dominant and non-symmetric. This is a "kernel" that I am using to solve a ...
0
votes
1answer
41 views

Application of Conjugate Gradient Method to non-symmetric matrices

I am currently working on a problem in which I am using the Conjugate Gradient method to solve for the steady state solution of a continuous time Markov chain. I am applying the algorithm found in ...
4
votes
2answers
73 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
0
votes
0answers
22 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 ...
2
votes
1answer
33 views

A generalization of GMRES

In oder to solve $Ax=b$, GMRES method finds $x_n$ in the $k$-th Krylov subspace i.e.: $$K_n=span\{b,Ab,...,A^{n-1}\}$$ and we have the condition: minimize $\|r_n\|_2$, which $r_n=b-Ax_n$ Now we ...
0
votes
0answers
17 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 ...
0
votes
2answers
43 views

Least Squares Solution Confusion

Say if I have an overdetermined system $A\vec x=\vec b$, I can use the normal equations $\implies$ $A^TA\vec x=A^T\vec b$. If I solve for $\vec x$ I will get a "solution" with an error. It says in ...
2
votes
1answer
41 views

QR transformation with Householder transformation

It's a task i do to understand minimizing the error including the QR transformation with the help of Householder transformation. I think i really do something wrong but i dont get it running i hope ...
3
votes
1answer
81 views

I want to study Numerical linear algebra [closed]

Would you like to recommend a book to me? the proof is explicit and easy to understand is preferred.
2
votes
2answers
60 views

Determinant of an ill conditioned matrix

I have the following ill conditioned matrix. I want to find its determinant. How is it possible to calculate it without much error \begin{equation} \left[\begin{array}{cccccc} ...
0
votes
0answers
35 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 ...
0
votes
0answers
37 views

condition number of orthogonal matrix

Let $A\in M_n(\mathbb R)$ be an orthogonal matrix. Then: $cond (A) =1$. I tryed to use facts about the eigenvalues but is did not help. In 2-norm it is easy to prove it since $||A||_2 = \sqrt{\rho ...
3
votes
2answers
59 views

Power iteration

If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration Especially, I often see that we ...
2
votes
1answer
36 views

Can an iterative method converge for some initial approximation?

Studying iterative methods for solving(or approximating) linear equation systems, I came accross the following theorem$^1$: Let the following be an iterative method: $$x^{(0)},\qquad known\\ ...
2
votes
0answers
44 views

Effective computation of matrix commutator

Is there a faster way to compute the commutator of large (at least one of them sparse) matrices $[A,B]$ then to compute $AB$ ,$BA$ and subtract them?
1
vote
0answers
28 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
vote
0answers
63 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
vote
0answers
24 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
vote
0answers
25 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 ...
0
votes
0answers
20 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 ...
3
votes
1answer
99 views

Prove $|\det A| \leq \prod_{j=1}^n ||a_j||$

Let's say A is a square n by n matrix. ||$x$||=$x^T x$ and x is a real n-column norm. How would you show this? I tried to use the QR factorization here in showing that ||$a_j$||=||$r_j$||, but ...
3
votes
1answer
217 views

Numerical verification of solution.

I have the non-linear equation \begin{align} &\left( {x}^{2}-1 \right) \left( -\frac{1}{4}\left({\frac { \left( 4\,{x}^{3}+2\, ex \right) ^{2}}{ \left( {x}^{4}+e{x}^{2}+f \right) ...
1
vote
0answers
48 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 ...
7
votes
1answer
145 views

Fast algorithm for approximating Eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16}$x$2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather ...
0
votes
1answer
21 views

How to evaluate the accuracy for sparse linear system solver

I'm currently trying to do some experiments on linear solver. However, it's a little hard to get the sense of the numbers. For example, I know large condition number is bad, but how large is bad? ...
2
votes
1answer
219 views

equations solved with Newton's method by finding the zeros of functions?

I found this statement in one paper I read recently: This problem can be solved by finding the zero of functions: ...
1
vote
2answers
90 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
2
votes
1answer
116 views

Finding vector $x$ so that $Ax=b$ using Householder reflections.

Assume $n\times m$ matrix $A$ and vector $b$ are given. I am looking for $x$ that satisfies $Ax=b$ in terms of linear least squares problem. Let $A=\begin{bmatrix} 1 & 1 & 1 \\0 & 1 & ...
0
votes
0answers
44 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 ...
1
vote
1answer
25 views

QR decomposition

Assuming that we have a QR decomposition of a matrix $A \in \mathbb{R}^{m \times n}$, $Q \in O(m)$ and $R \in \mathbb{R}^{m \times n}$ such that $A=QR$, where R is an upper triangular matrix. Now ...
0
votes
1answer
113 views

Trapezoid Rule to Simpson's Rule Extrapolation

I need to show that one extrapolation of the trapezoid rule leads to Simpson's rule. I've looked through the other posts on ME, specifically the post with the same title, and this for help, but I ...
2
votes
1answer
104 views

(A + D)x = b … efficiently!

let say we have $A$ to be symmetric positive-definite (SPD), moreover block tridiagonal Toeplitz matrix and $D$ is block diagonal SPD (both with full rank). Let say we know everything about $A$ and ...
0
votes
0answers
43 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 ...
3
votes
1answer
54 views

Necessary and Sufficient conditions for convergence of matrix iterations

I need some help figuring out how to go about the iteration part of the problem...I don't really know where to start. If someone can please help take me through it that would be greatly appreciated. ...
0
votes
1answer
31 views

Estimating rates of convergence

If I have a set of data points obtained from a numerical approximation say 15.3828 15.2458 15.2095 15.2003 how can I estimate the rate of convergence?
0
votes
1answer
53 views

How does LU decomposition work?

I'm interested in the algorithm of LU decomposition in order to solve a LSE like $Ax=b$, where $A$ is a square matrix. My question is: When I compute $PA=LU$ do I also need to interchange rows in $L$ ...
0
votes
1answer
800 views

Gauss Seidel iteration in matlab

I've posted this question before for crout factorization. Now, I need help with Gauss-Seidel iteration. Write a program that takes a value for n and solves for x using the following method: ...
1
vote
0answers
25 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...