3
votes
1answer
56 views

9 point stencil for Laplacian operator

Given the following 9 point Laplacian \begin{align} -\nabla^2u_{i,j} = \frac{2}{3h^2}\left[5u_{i,j} - u_{i-1,j} - u_{i+1,j} - u_{i,j-1} - u_{i,j+1} - u_{i-1,j-1} - u_{i-1,j+1} - u_{i+1,j-1} - ...
2
votes
1answer
114 views

A Beautiful Determinant!

Find the determinant of the following matrix in the terms of $a_1,a_2,\cdots,a_n$ explicitly, $$ \begin{bmatrix} a_1 & a_2 & a_3 & \cdots & a_n\\ a_2 & a_3 & a_4 & \cdots ...
0
votes
1answer
311 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
1answer
54 views

Homework for Gauss Seidel method

Let A be a strictly diagonally dominant matrix. Suppose we use Gauss Seidel method to solve $Ax=b$, a sequence of vectors {$x_{0},x_{1},...,x_{k},...$} is obtained (where $x_{0}$ is the initial guess) ...
0
votes
2answers
55 views

Legendre Polynomial Orthogonality and Size

Show $(P_i,P_j)=\begin{cases} 0& i \neq j \\ \frac{2}{2j+1} & i = j\end{cases}$ for $0 \leq i, j\leq2$ I'm just not sure exactly what I'm supposed to do. Do I plug in values of i and j and ...
0
votes
1answer
35 views

matrix exponential and Spectral abscissa

Prove that $\lim_{t \rightarrow \infty} \|e^{tA}\| = 0$ if and only if $\alpha(A) < 0 $, where $\alpha$ is the Spectral abscissa, defined as $\max{Re(\lambda_i)}$. I tried to approach this ...
0
votes
0answers
11 views

need to determine weights so that quadrature formula holds

Let $l$ be an interval on the real axis, $t_1,...,t_n$ be distinct $n$ points, then there exists n numbers $m_1,...,m_n$ such that the quadrature formula, $\int_l p(t)dt = m_1p(t_1) + ... + ...
0
votes
1answer
23 views

Residual norm for iterative scheme

Consider a linear system $A\vec{x} = \vec{b}$, where $A \in \mathbb{R}^{m\times{}m}$ is non-singular and positive definite. Given the following iteration scheme $\vec{x}^{(k+1)} = \vec{x}^{(k)} + ...
1
vote
2answers
38 views

Relation between condition number and perturbed matrix

Prove that if $A\vec{x} = \vec{b}$ and $(A+\delta{}A)(\vec{x}+\delta\vec{x}) = \vec{b}$, then $\dfrac{\|\delta\vec{x}\|/\|\vec{x}+\delta\vec{x}\|}{\|\delta{}A\|/\|A\|} \le \kappa{(A)}$, where ...
0
votes
1answer
64 views

Dense symmetric positive definite matrix

How could one define a dense symmetric positive definite matrix (dimension $1000 \times 1000$) with uniformly distributed eigenvalues (with the smallest eigenvalue $1$ and the condition number $100$) ...
2
votes
1answer
117 views

Help in this exercise about Richardson extrapolation.

We know $F(h)=a_0 +a_1h + a_2 h^3$ $F(1)=4$; $F(1/2)=21/8$; $F(1/4)=145/64$ Find a approximation of $F(0)=a_0$ with Richardson extrapolation method with an absolute error less than $10^{-2}.$ ...
3
votes
1answer
84 views

Computational cost, power method and page rank

When solving the PageRank problem for $n$ web pages, it is necessary to find a solution of the eigenvector equation $$(fM)*p = p,$$ where $$fM = dM + (1 - d)Z$$ $$Z =\frac{1}{n}*ee^T$$ $$e =[1, 1, ...
1
vote
2answers
43 views

stability function

I have an exercise which asks me to find polynomials $P$ and $Q$ with a degree $2$ that satisfy $$\exp(z)= \dfrac{P(z)}{Q(z)} + O(z^5)\ \text{for} \ z\to 0$$ My question is: Are they actually unique ...
1
vote
1answer
99 views

Richardson Iteration

Given the Richardson Iteration, $x_{n+1} = x_n + \alpha(b-Ax_n)$ (with $\alpha$ a scalar constant). To which polynomial $p(A)$ at step $n$ does this iteration correspond to? My first idea ...
0
votes
1answer
45 views

Generalized minimal residuals: eigenvalues and sets of functions

Can someone help me on this exercise (2 parts)? Thanks! Suppose that $S \subseteq \mathbb{C}$ is a set whose convex hull contains $0$ in it's interior (so $S$ is contained in no half-plane ...
0
votes
1answer
90 views

On eigenvalues, hermitian matrices and SVD

Are my ideas on the following "true or false"-statements correct? If $A$ is hermitian and $\lambda$ is an eigenvalue of $A$, then $|\lambda|$ is a singular value of $A$. My answer would be ...
1
vote
0answers
72 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 ...
0
votes
1answer
69 views

Backward stable algorithm

Assume we have fixed unitary matrices $Q_1, \dots, Q_k \in \mathbb{C}^{m,m}$ and a matrix $A \in \mathbb{C}^{m,n}$ which can be perturbed. How can we proof that the algorithm on computing the product ...
0
votes
1answer
16 views

Stability and complexity of some functions

Can someone check if my solutions/arguments on this exercise are correct? Thanks! Are the following statements true or false? $\sin (x)=\mathcal{O}(1)$ as $x \rightarrow \infty$ $\sin ...
-3
votes
1answer
61 views

Conditional number: exercise

Let's say we've got a $202 \times 202$ matrix $A$ for which $||A||_2=100$ and $||A||_F=101$ (the Frobenius norm). How can we find the sharpest bound (lower) on the 2-norm condition number of $A$? ...
0
votes
1answer
80 views

Singular value decomposition: rotation

Suppose that $A \in \mathbb{C}^{m \times n}$ and $B$ ($ \in \mathbb{C}^{n \times m}$) is the matrix obtained by rotating $A$ ninety degrees clockwise. Do $A$ and $B$ have the same singular values? ...
3
votes
1answer
268 views

Polynomial Condition Number

I have a question, from "Applied Numerical Linear Algebra"(James W. Demmel), that I don't know how to do. Consider $\mathbb{R^{d+1}}$ as the set of polynomials of degree $\leq d$ and $S_a$ the set of ...
1
vote
1answer
59 views

Hessenberg Matrices

$$A=\begin{bmatrix} 2&3&4\\ 3&-5&5\\ 4&5&0\end{bmatrix}$$ Find a unitary matrix $Q$ such that $A = QHQ^{H}$, where $H$ is Hessenberg. I am having a little trouble ...
3
votes
1answer
32 views

Arrange the computation of $f(x)$ so that the loss of significant figures can be reduced using $4$-digit decimal arithmetic

For the function $f(x)=\sqrt{x^2+1}-x$, have computed $f(100)=0$ using 4-digit decimal arithmetic (rounding after every intermediate calculation). The value $f(100) = 0.0049998750$ (to 8sf) is given ...
2
votes
0answers
41 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
votes
0answers
87 views

QR decomposition help

What do Q and R stand for? Why must the diagonal entries of R be positive instead of just nonzero?
1
vote
1answer
152 views

Proof of GMRES convergence

I am working on a homework, where we have to proof that GMRES finds an exact solution of $Ax = b$ in at most m steps (with A being an $m \times m$-matrix). The proof is split up into several steps, ...
2
votes
1answer
530 views

Matlab Matrix Multiplication Calculate Significant Figures

First off, long time reader, first time poster. Thanks in advanced for all the help this site has offered! So the question! I have two matrices in the form of the variables ...
0
votes
1answer
287 views

SVD and linear least squares problem

Edit: I've actually found an error: Instead of full SVD I had to use, "economy size" SVD, where $U$ has only first $n$ columns, and $\Sigma$ becomes a square matrix. I also forgot to take the ...
1
vote
1answer
58 views

Eigenvalues, Eigenvectors and Eigendecomposition

If there is a symmetric matrix, say $$B = \left[\begin{array}{cc} 0 & A\\ A^T & 0 \end{array}\right]$$ where $A$ is a $m\times n$ submatrix with $m \geq n$. Is it possible to express the ...
0
votes
2answers
71 views

solving a matrix equation $X-I=a \cdot (X\cdot U^T + U \cdot X)$

I am trying to solve the $n \times n$ diagonal matrix $X$ in the following equation: $$X-I=a \cdot (X\cdot U^T + U \cdot X)$$ where $0<a<1$ is a given scalar, $U$ is a $n \times n$ given ...
4
votes
1answer
127 views

Show the space spanned is an invariant subspace

Let $A$ be real and let $\lambda = \alpha + i \beta$ be a complex eigenvalue of $A$ with eigenvector $x + iy$, show that the space spanned by $x$ and $y$ is an invariant subspace of $A$. What I ...
1
vote
1answer
150 views

Proving the SVD Theorem by induction on the rank of $A$

This is an exercise and it is divided into steps. The first step says: Suppose $A\in\mathbb{R}^{m\times n}$ has rank 1. Let $u_1\in\mathbb{R}^n$ be a vector in $R(A)$ such that $\left \| u_1 \right ...
0
votes
1answer
92 views

Generating Gauss-Seidel hard system

I am writing various Gauss-Seidel algorithm parallel implementations using different programming techniques for my assignment. I have created a MATLAB script for generating strictly diagonally ...
1
vote
2answers
278 views

Given orthogonal basis what is the orthogonal complement?

The question states: Let $q_1,q_2,\dots,q_n$ be an orthogonal basis of $\Bbb R^n$ and let $S = \operatorname{span}\{q_1,q_2,\dots,q_k\}$, where $1 \le k \le n-1$. Show that $S^\perp = ...
3
votes
2answers
142 views

How can I prove $\mathrm{maxmag}(A)=\frac{1}{\mathrm{minmag}(A^{-1})}$, and $\mathrm{maxmag}(A^{-1})=\frac{1}{\mathrm{minmag}(A)}$?

Using $\mathrm{maxmag}(A)=\max_{x\neq 0}\frac{\|Ax\|}{\|x\|}$, and $\mathrm{minmag}(A)=\min_{x\neq 0}\frac{\|Ax\|}{\|x\|}$ I found this quite simple to prove using a proposition stating that ...
1
vote
2answers
83 views

Calculate sum of ONB

Here's a homework question: Let ${u_1, \ldots, u_n}$ be an ONB in $C^n$. Assuming that $n$ is even, compute $$||u_1 - u_2 + u_3 -\cdots - u_n||$$ I have no idea how to solve this. Can anyone help? ...
0
votes
1answer
242 views

Norm $\|A\|$ is not induced by any vector norm [duplicate]

Possible Duplicate: Subordinate matrix norm I have a question in my homework for Numerical Linear Algebra, which is as follows: Show that the norm $\|A\| = \max \limits_{i, j} |a_{i,j}|$ ...
3
votes
2answers
148 views

Fast inversion of a triangular matrix

I need to inverse a matrix $A$ given its $QR$ decomposition. It's a numerical task. It is told that the inversion should be "possibly cheap". But it does not look like I can do something more ...
3
votes
1answer
213 views

How to Store a Banded Matrix by Diagonal

I'm taking a graduate level independent study course this semester in Matrix Computations. I'm not getting much support from the professor, so am turning to the excellent StackExchange community for ...