# Tagged Questions

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### 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} - ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### QR decomposition help

What do Q and R stand for? Why must the diagonal entries of R be positive instead of just nonzero?
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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? ...
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### 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 $\|A\| = \max \limits_{i, j} |a_{i,j}|$ ...
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 ...