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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135 views

Condition number of $A^{-1} B$ where $A$ and $B$ are banded Toeplitz matrices.

I'm looking at a filtering problem with feedback, which can be represented by the equation $A\underline{y} = B\underline{x}$, where $A$ and $B$ are lower triangular banded Toeplitz matrices and $\...
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1answer
131 views

Condition number of $AA^T$ when $A$ is polynomial Vandermonde

Suppose I'm doing polynomial regression of degree $m$ $$p(x, \mathbf{w}) = w_0 + w_1x + \dotsb + w_mx^m$$ given training data $(x_1, t_1), \dotsc, (x_N, t_N)$. Suppose I'm using the loss function $...
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187 views

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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25 views

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 ...
2
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1answer
37 views

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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59 views

Solving numerically a strongly stiff nonlinear ODE system with ill-conditioned Jacobian

Using Matlab, I am trying to solve numerically the following nonlinear system of ODEs: $$\begin{aligned} \dot B &= -\alpha B -\nu BV \\ \dot X &= A-\mu_1 X -c E(B)VX \\ \dot Y &= ...
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33 views

Show that $ \kappa_2(A) \leq [\kappa_1(A) \kappa_{\infty} (A)]^{1/2}$

Suppose for a matrix $ A \in \mathbb{R}^n$, we have $ \ ||A||_2 \leq ||A^TA||^{1/2}$, where $||.||$ is a norm on $\mathbb{R}^n$ associated to matrix norm on $\mathbb{R}^{n \times n}$ and $||.||_2$ is ...
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99 views

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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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 ...
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1answer
45 views

Measure for how singular a square matrix is in the range [0,1]

I am interested in estimating how close a square matrix is to being singular such that I can compute a value $s \in [0,1]$ where $s=1$ would mean the matrix is singular, and $s=0$ means it is as far ...
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2answers
78 views

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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228 views

Relative condition number, Ill conditioned, Well conditioned

I'm currently learning about relative condition number (K), and how they are considered as well conditioned or ill conditioned. From my understanding, a large K value represents ill-conditioned, ...
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1answer
251 views

Find the condition number of $A$

Find the condition number of $$A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2}& 0& 0& 0 \\ \end{...
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31 views

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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1answer
124 views

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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24 views

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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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 ...
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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 ...
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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 ...
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249 views

Condition number, well conditioned or ill condition

Determine the condition number for taking the square root of a number $x$. Is this problem well-conditioned or ill-conditioned I know that conditioning measures how sensitive a problem is to a small ...
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50 views

Condition number for a nonlinear matrix

Is there such a thing as a condition number for a nonlinear (polynomial) matrix in multiple variables, and if so, is it something simple (I doubt it) or maybe there are multiple methods? Equations ...
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100 views

At which point is a condition # considered ill-conditioned?

At which point is a condition # considered ill-conditioned? In my opinion, a condition # of $10^{10}$ corresponds to ill-conditioning obviously. But what about a condition # of say, $2000$? $200$? $...
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33 views

If $\lambda*I - A$ ill-conditioned, then is $\lambda$ an eigenvalue of A?

I am trying to show the following logic: if $\lambda \in \mathbb{R}$ and $||(\lambda I-A)^{-1}||_2$ is $||(\lambda I-A)^{-1}||_2 > \frac{1}{\eta}$ where $\eta$ is machine precision, then $\...
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364 views

Proving the condition Number of a Matrix

The theorem I am trying to prove is that the condition number of matrix A is: $$\operatorname{cond}(A) = \|A\|\cdot\|A^{-1}\|$$ where $\|A\|$ signifies the operator norm. From the properties of the ...
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41 views

Is stiffness of an equation related to variable coefficients

I have a system of four coupled nonlinear ODEs. Three of them are of first order, while the fourth equation is of second order. The latter also incorporates a variable coefficient, $a(x)$, which can ...
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330 views

Show when the inequality for matrix-vector multiplication for the 2 norm is an equality?

I am asked to show for which vectors the inequality $||Ax|| \le ||x||||A|| $ is an equality. My intuition tells me that this happens when $x$ is in the direction of the right singular vector ...
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131 views

Give a vector pair $(b,δb)$ for which $\dfrac{\|δx\|_2}{\|x\|_2}=k_2(A)\dfrac{\|δb\|_2}{\|b\|_2}$ holds

The condition number of a matrix gives a sharp estimate of the sensitivity of $x$ with respect to perturbations of $b$ when solving $Ax = b$, this means there exists a right hand side $b$ and a ...
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124 views

What linear transformations are singular values and/or condition numbers invariant under?

This is a purely academic question. Given the singular value decomposition of a matrix, $A=USV^T$, what is the largest set of linear transformations which leave the singular values invariant? I can ...
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1answer
539 views

How to estimate condition number based on SVD of submatrix?

Given an $m\times n$ ($m\geq n$) real valued matrix, $A$, its SVD, and an $n$-dimensional real valued vector, $x$, is there a computationally efficient way to accurately estimate the condition number ...
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341 views

Condition number example

I have been learning about the condition number, i.e $$cond_p(A) = ||A||_p||A^{-1}||_p.$$ I've been considering the $2$ norm, and have been thinking about when the condition number is equal to $1$. ...
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1answer
82 views

Various Interpretations of Condition Number

Condition number of a matrix $A$ signifies how quickly solution $x$ changes in $Ax = b$ as we make small changes in $b$. This is given by $||A||.||A^{-1}||$. When we consider spectral norm, this turns ...
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40 views

Why should I overdetermine my inverse problem?

It is often said that inverse problems, say the generation of an image from some remote sensing method, benefit from overdetermination by multiple measurements. It matches intuition, but other than a ...
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1answer
73 views

How to tell when residual error, and norm of eror is to small in Linear algebra

I am taking a course in linear algebra and I am having some trouble understanding the norm of error and the residual error. Our teacher define the norm of error as $$e=||\tilde X -x|| $$ And he ...
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98 views

Conditioning in positive-definite Toeplitz systems

For the testing of a program of mine, I was trying to generate poorly-conditioned matrices. The context requires symmetric, real, Toeplitz, and positive definite matrices; moreover, these matrices ...
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1answer
363 views

Show property of condition number

The condition number $\kappa(A)$ of a matrix $A$ is defined as $\kappa(A) = \| A \| \cdot \| A^{-1} \|$, where $\left\|A\right\|=\max_{x\neq 0}\frac{\left\|Ax\right\|}{\left\|x\right\|}=\max_{\left\|...
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1answer
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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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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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20 views

Equivalent to a condition number inequality but for singular matrices

When we have a linear system ($AX=b$) ($A$ an invertible matrix of size $n\times n$ and $X,b$ vectors of size $n$) and there's a disturbance in $b$ (say $b+\delta b$) we get $A(X+\Delta X)=(b+\delta b)...
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42 views

Do BFGS and L-BFGS methods converge when the matrix $H=B^{-1}$ is ill-conditioned?

I am using BFGS and L-BFGS to solve an unconstrained optimization problem. The objective function is the Mean Euclidean Error. The output is given by an Artificial Neural Network. The line search ...
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38 views

Is the problem well conditioned or not $x=1-e^p$.

For large numbers $|p|\gg1$ we want to compute $x=1-e^p$. Is the problem for $|p|\gg 1$ well conditioned. We have $$\kappa = \frac{\|x'\|}{\|x\|}\cdot \|p\|=\frac{\|-e^p\|}{\|1-e^p\|}\cdot \|p\|$$ ...
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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 ...
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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 ...
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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 \\...
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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} &...
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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 ...
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100 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 ...
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108 views

A lower bound for the condition number matrix

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 $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...
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59 views

Matlab - eigenvalue doesn't zero the characteristic polynomial

I created a symmetric $21 \times 21$ matrix $A$ with condition number $7.5044$. I wrote the following code: ...
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33 views

How to test a function for condition and stabilty?

How can I test the condition and stability of the following function, for values $x=0$, $x=0.25$ and $x=10^{-5}$? $$ f(x) = \frac{1-cos(2x)}x $$ What should one do here? For condition, I think I ...