# 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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### How many digits of of accuracy do I expect the have the solution $x$ of $||Ax-b||=0$

A least-square problem $||Ax-b||=0$ is solved using a backward stable algorithm (In my case, QR decomposition using householder projectors). The condition number is $\kappa(A)=10^5$. If the problem is ...
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### Condition number — what is it? [closed]

May someone tell me what the condition number of a matrix is and how it's related to SVD decomposition? I google a lot but got zero results on this.
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### Condition value of sparse matrix

I have a sparse matrix really ill-conditionned. I wondered if the places where the non zero values have an impact on the condition value. My matrix is PSD and what I'd like to know is if the condition ...
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### Condition inequality in perturbed LS

I have two matrices $A \in \mathcal{M}_{n,d}(\mathbb{R})$ and $B \in \mathcal{M}_{d,d}(\mathbb{R})$ with $B$ being symmetric definite-positive. I am trying to find a condition on $A$ for which I have ...
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### If $ax^2+bx+c$ has repeated root $x=r$, what is the condition number solving for $r$ if you only change one of $a,b,c$?

I will edit to be specific like Shaun said: If $ax^2+bx+c$ has repeated root $x=r$, what is the condition number solving for $r$ if you only change one of $a,b,c$? The book says that if $r$ is a ...
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### Calculate condition number

For $L1$ And $L2$: $L2=D-A(L1+I)^{-1}$ $L1=D-A$ Can we prove that: condition number of $L2$ < condition number of $L1$ ? if yes, how? Where $D$ is an in degree diagonal matrix and $A$ is an ...
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### Relative error bound for Ax=b

I have the matrix $A= \begin{pmatrix} 4-\alpha & 12+\alpha \\ 2-\alpha & 6+\alpha \end{pmatrix}$, a pair $\delta x,\delta b\in\mathbb{R^2}\backslash\{0\}$ satisfying $A\delta x=\delta b$, and ...
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### Condition number of principle submatrix

Suppose we have a $m\times n$ matrix. Consider its principle submatrices (removing the $i$th row and col). How do I find the principle submatrix with the smallest condition number? Currently I loop ...
### Prove this equality about condition numbers $\frac{1}{{\rm cond}(A)_2}=\frac{\lambda}{\|A\|_2}$
I am supposed to prove this equality. Let $A$ be an invertible square matrix over $\mathbb R$ $$\frac{1}{{\rm cond}(A)_2}=\frac{\lambda}{\|A\|_2}$$ where ${\rm cond}({}\cdot{})_2$ is the condition ...
Say I have three matrices $A_1$, $A_2$ and $A_3$ as given below- \$A _{1} = \begin{bmatrix} 1 & 5 & 8 \\ 7 & 3 & 4 \end{bmatrix}, \quad A ...