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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Bounds on error on solving $Ax=b$ for perturbed system
I am interested in solving $x$ from $Ax=b$, where $A \in \mathbb{R}^{m \times n}$ and $m > n$, it is rectangular and even rank-deficient. This means that we can only solve it in the least-square ...
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Does an equivalent of the condition number for the error in system coefficients of a linear equation system exist
Let's assume that a system
$$
[A]\overline{x} = \overline{b}
$$
is given.
The normal condition number assigns an upper bound to the relative error in $\overline{x}$ based on the error in $\overline{b}$...
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Upper bound on the condition number of the similarity transformation matrix
In the context of square matrices, we have been given
$$A = T\Lambda T^{-1}$$ where $\Lambda$ is a known diagonal matrix. It is also known that the condition number of A is bounded above. Say,
$$\...
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Suitability of QR factorization for solving a ill-conditioned linear system.
I am trying to solve a linear system $Ax = b$ where
$$
A =\begin{bmatrix}
2& 9& 2& 1& 4& 1& 0& 0& 0 \\
9& 65& 9& 1& 4& 1& 0& 0&...
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How does Self-Scaling Fast Givens QR work for (Regularized) Linear Least Squares
Basic Problem
I need help to understand the Self-Scaling Fast Givens QR decomposition proposed in
Anda A. A. & Park H., Self-Scaling Fast Rotations for Stiff and Equality Constrained Linear Least ...
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Find the condition number of a normal matrices.
Find the condition number of a normal matrices.
My attempt:-
I know condition number of $X\in \mathbb C^{n,n}$ is defined by $\kappa(X)=||X|| \cdot ||X^{-1}||.$ Definition of Normal matrix is given by ...
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Relation between condition number and relative error
For the equation $Ax=b, A \in \mathbb{R}^{m*n}, x \in \mathbb{R}^n, b \in \mathbb{R}^m$, the condition number of $A$ gives an upper bound of the relative error in $x$ given a relative error in $b$:
$||...
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How to find the condition number of \begin{bmatrix} I & B\\ B^{T} & I \end{bmatrix}
I want to find the spectral condition number of this matrix, notice that this matrix is symmetric, hence the spectral condition number can be wriiten as $$\frac{\max_{1\leq i \leq n }|\lambda_{i}| }{\...
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Minimum singular value vs condition number to determine closeness to singularity
Which criteria should I use to determine if a matrix is close to singularity or not?
My application is to try to find an optimal matrix that maximizes the minimum singular value/minimizes the ...
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The definition of the conditional number for a matrix
According to https://en.wikipedia.org/wiki/Condition_number, the conditional number for the matrix $M$ is $||A|| \times ||A^{-1}|| = ||A A^{-1}|| = 1$. In another word, is $||A||$ the absolute value ...
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Derivative of 2-norm of a matrix.
I have a function:
$$f(A)=cond(AB)=\left \|{AB}\right \|_2 \left \| {(AB)^{-1}} \right \|_2 $$
We know that:
$$\left \| X \right \|_2= \sqrt{\lambda_{max}(X^TX)}$$
So:
$$f(A)=\sqrt{\dfrac {\lambda_{...
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Can the computation of condition numbers be written in functional form？
Given an $n \times n$ matrix M, I want to optimize the condition number $\text{cond}(AM)$, where $A$ is a $m\times n$ ($m>n$) matrix.
I should write the computation of condition number $\text{cond}(...
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Are there any numerically stable methods for computing the condition number of a near-singular matrix that are fast?
I know of a number of techniques for computing the condition number that are either slow (doing a whole singular value decomposition) or fast (ratio of norm of matrix and its inverse obtained via LU ...
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How to approximate the condition number of a matrix using its component
Given a psd matrix $H=[A, B; B^\top, C]$. I can find bounds for singular values and the norm for $A, B, C$. How can I construct a bound for the condition number of $H$? I have tried using interlacing ...
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Uncertainty Principle in Kernel-based Interpolation
If one wants to interpolate or reconstruct a function $f:\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$ on a finite Set $X_n:=\{x_1,\ldots,x_n\}\subset\Omega$ using translates of a ...
