# Questions tagged [condition-number]

The condition number of a matrix is the ratio of the largest to the smallest singular value in the singular value decomposition of a matrix.

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### Minimizing the condition number of a block matrix

Let $$A = \left[ \begin{array}{ccc} & B_{(n(m+1)-m) \times (n(m+1))} & \\ \hline Z_{m \times (nm)} & | & S_{m \times n} \end{array} \right]$$ be a block ...
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### Matrix Solvers for High Condition Numbers

Suppose I wish to solve the following square system of equations: $$A x = b$$ Suppose that $A$ is modestly large and sparse (problem size $\sim 10^{3-4}$). Suppose that $A$ is banded primarily ...
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### preconditioners for matrices arising out of PDEs

Suppose I have the heat the following one dimensional PDE for the heat equation: $$\frac{\partial u}{\partial t } = \alpha \frac{\partial^2 u}{\partial t^2 }$$ which I discretized in the spatial ...
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### Discretising the Fourier Integral gives a high condition number

I have the following integral equation $$\int_0^1 e^{-2\pi i s t} f(t)\, \text{d} t = g(s), \hspace{3em} -1/2 \leq s \leq 1/2$$ where $f$ is to be found, and $g$ is known. I believe this problem is ...
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### Prove the inequality for Condition number of matrix

Let $A \in \mathbb{R}^{n \times n}$ be a non-singular matrix. Let $\hat A=A+\delta A, \ \hat x=x+\delta x, \ \text{and} \ \hat b=b+\delta b$ with $Ax=b$ and $\hat A \hat x=\hat b \$. Here $||.||$...
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### Is division ill-conditioned when divisor is close to zero?

My intuition is that division of real numbers is ill-conditioned when divisor is close to zero. Is this intuition correct?
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### what is the significance of $\kappa$ in the proof of inconditional stability of the Crank-Nicolson scheme

When using a centered difference approximation $$\frac{\partial}{\partial t}u(t,x) = \frac{u(t + \Delta t/2,x) - u(t - \Delta t/2, x)}{\Delta t} + O((\Delta t)^2)$$ It is an approximation of the ...
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### Wavelets for preconditioning in MATLAB

I have been trying to understand the Wavelet transform in order to use it as a precondioner for ill-conditioned linear problems of the form $A\vec{x}=\vec{b}$. I have come across this paper that is ...
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### Condition number on the DFT-like complex vandermonde matrix

Given $M \in \mathbb{N}$ and $0 < L \le M$, $L \in \mathbb{N}$ consider a set of $L-1$ integers, such that $0 \le i_1 < i_2 \ldots < i_{L-1} \le M$ Note that this index set has symmetry ...