# 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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### Is there a threshold above which a matrix is ill-conditioned?

My matrix has a condition number of $45.678$. Is it an ill-conditioned one? Is there a threshold above which a matrix is ill-conditioned?
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### How to choose two diagonal matrices minimizing the condition number

I have a matrix $A \in R^{n×n}$. I would like to choose two diagonal matrices $D_1,D_2 \in R^{n×n}$ such that $\text{cond}(D_1AD_2)$ should be minimal. How to provide such diagonal matrices?
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### Prove that $\kappa_F(A) \ge \sqrt{n}$

I reached at a point after solving it $$\kappa_F(A) = \sqrt{\mbox{tr}(A^H A) \cdot \mbox{tr}(A^{-1}(A^{-1})^H}$$ Now I am stuck. How to proceed? Or, alternatively, a new approach is also appreciated....
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### 1/Infinite-norm condition number and the QR decomposition

I have seen that often the condition number of a matrix, $A$, may be estimated by taking the QR decomposition of $A$, $A=QR$, and using numerical methods to estimate the condition number of the upper ...
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### Use SVD to reduce condition number of matrix

I need to compute the inverse of an ill-conditioned matrix. Since condition number is ratio of high/low singular values. I am approximating the matrix by removing small singular values. But the ...
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### Condition number of a matrix-vector product

If $A$ is an $m \times n$ matrix and $x$ is an $n \times 1$ vector then the linear transformation $y=Ax$ maps $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$, so the linear transformation should have a ...
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### Find the relative condition number of $f(x,y) := y e^{4x^2}$ with respect to the 1-norm.

Let $f: \mathbb{R} \to \mathbb{R}$ (I guess it's supposed to be $\mathbb{R}^2 \to \mathbb{R}$) be defined by $f(x,y) := y e^{4x^2}$ Find the relative condition number of with respect to the 1-norm. ...
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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 ...
I am having trouble proving the following statement: Let $\mathbf{A}\in\mathbb{C}^{n\times n}$ be a square matrix such that $\|\mathbf{A}\|<1$, for some induced norm $\|.\|$. Then, $\|(\mathbf{... 0answers 35 views ### Approximate symmetric matrix by minimizing condition number We want to approximate A symmetric semi definite positive by another X that's symmetric and whose condition number$\frac{\lambda_{\max}(X)}{\lambda_{\min}(X)}$. The optimization problem can be ... 0answers 25 views ### Wavelets for preconditioning in MATLAB I have been trying to understand the Wavelet transform in order to use it as a precondioner for ill-conditioned linear problems of the form$A\vec{x}=\vec{b}$. I have come across this paper that is ... 0answers 31 views ### Is the computed condition number reliable when the output of Matlab cond > 1e16 (inverse of machine precision)? I read from the an excellent recent paper on solving ill-conditioned Vandermonde matrix, they are outperforming the state-of-the-art, i.e., Bjorck and V. Pereyra method An interesting quote from the ... 1answer 63 views ### Solving an apparently “simple” system of linear equations I have to solve this apparently simple system of linear equations: $$a t^5+b t^4+c t^3+d t^2+e t=0 \\ a (t/2)^5+b (t/2)^4+c (t/2)^3+d (t/2)^2+e (t/2)=0 \\ a/6 t^6+b/5 t^5+c/4 t^4+d/3 t^3+e t^2/2=0 \\... 0answers 46 views ### Condition number on the DFT-like complex vandermonde matrix Given M \in \mathbb{N} and 0 < L \le M, L \in \mathbb{N} consider a set of L-1 integers, such that 0 \le i_1 < i_2 \ldots < i_{L-1} \le M Note that this index set has symmetry ... 0answers 24 views ### Relative Condition Number of f(x_1,x_2)=x_1/x_2 Find the relative condition number of f.$$f(x_1, x_2)= \frac{x_1}{x_2}$$So, when I use the definition of the relative condition number \kappa, I get:$$\kappa(f,x)= \lim_{\epsilon \rightarrow ... 3answers 4k views ### Condition number of a product of two matrices Given two square matrices$A$and$B$, is the following inequality $$\operatorname{cond}(AB) \leq \operatorname{cond}(A)\operatorname{cond}(B),$$ where$\operatorname {cond}$is the condition number, ... 2answers 69 views ### What is the probability of getting at least 50 questions of 100 right? If there are 100 MCQs with 4 options each. The probability that a person gets an question right is 0.25. What is the probability of getting at least 50 questions of 100 right? 0answers 86 views ### Condition number of augmented matrix I am trying to solve following problem. Let$A$be a$m$by$n$($m\geq n$) full rank matrix. What is then condition number of its augmentation$M$: $$M = \begin{bmatrix} I & A \\ A^{\top} &... 2answers 36 views ### Condition of \log(x) around x =1. As far as my understanding of condition numbers go, they represent how much an error in input can change output. What I don't understand is why for values of \log(x) around x=1 is the condition ... 1answer 860 views ### Matrix condition number and loss of accuracy There are quite a few sources online that say something along the lines of : "As a rule of thumb, if the condition number \kappa(A)=10^k then you may lose up to k digits of accuracy on top of ... 1answer 147 views ### Suppose ||.|| is an induced matrix norm, A is non-singular, and B is singular. Prove \frac{1}{\kappa(A)}\leq\frac{||A-B||}{||A||}. \|\cdot\| is the induced norm for n\times n matrices in \mathbb{C}, with respect to some vector norm (\mathbb{C}^n\to\mathbb{R}). A and B are n\times n matrices where A is non-singular ... 1answer 45 views ### Computing the condition number of a matrix Given$$A=\begin{bmatrix}23.89&-36.48&1.432&21.65\\-36.48&54.58&-5.193&-34.45\\1.432&-5.193&-1.0717&1.937\\21.65&-34.45&1.937&20.50\end{bmatrix}.$$I am ... 0answers 101 views ### how to understand and use condition number in under-determined and over-determined linear system? I know that the condition number of a numerical problem measures the sensitivity of the answer to small changes in the input. I also have a sense that for under-determined system we would have large ... 1answer 71 views ### Prove that$k(A) \geq \rho(A)/\min |\lambda|$and that$k(A) \geq \rho(A) \rho(A^{-1})$I'm trying to solve this question here. Thank you in advance for your help. Prove that$k(A) \geq \rho(A)/\min |\lambda|$and that$k(A) \geq \rho(A) \rho(A^{-1}) $We assume the matrices$A$and$...
I have the following proposition: Theorem: For every invertible matrix $A\in\mathbb{R}^{n\times n}$ and every matrix norm $\|\cdot\|$, then the condition number $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...