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.

Filter by
Sorted by
Tagged with
0
votes
1answer
16 views

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?
0
votes
1answer
31 views

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....
0
votes
0answers
19 views

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 ...
0
votes
1answer
39 views

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 ...
0
votes
1answer
64 views

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 ...
1
vote
1answer
44 views

Is the condition number of a 2x2 block symmetric matrix greater than the condition number of its upper left hand block?

Is there any known relation between cond(M) and cond(Q) when $$M=\begin{bmatrix}Q&A^T\\A&0\end{bmatrix}$$ and Q is symmetric positive definite and A is rectangular full row rank? From the ...
0
votes
0answers
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)...
2
votes
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 ...
2
votes
0answers
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 &= ...
2
votes
0answers
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 ...
1
vote
0answers
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 $||.||$...
0
votes
1answer
46 views

Condition number of dot product of vectors

I was wondering if anybody knows what is the relative condition number of dot product of vectors and how to compute it. I'm just reading about this stuff, but don't really understand how to compute it....
0
votes
1answer
70 views

Proving an inequality based on condition number.

I was trying to prove this inequality, by taking $K(A) = ||A|| ||A^{-1}||$ and also the error $A(x -\hat{x}) =e$, I am thinking how to get those terms, estimates? any help in ideas to proceed.
0
votes
1answer
44 views

For an orthogonal matrix $Q$, prove $\operatorname{cond}(Q)=1$

Given an orthogonal matrix $Q$, prove $$\|Q\|_2\cdot \|Q^{-1}\|_2=1$$ I succeed to solve it with eigenvalues but I'm looking for an easier way.
2
votes
1answer
59 views

Solution to a modified linear system using two methods

I am trying to obtain the solution to a modified linear system. I am comparing two methods to solve this modified linear system, and I'm noticing some issues with one of the methods. A linear system ...
2
votes
0answers
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. ...
2
votes
2answers
166 views

Different condition numbers of $\begin{pmatrix} a & b \\ b & c \end{pmatrix}$

Let $a,b,c \in \mathbb{R}$ and $A := \begin{pmatrix} a & b \\ b & c \end{pmatrix}$ and $\det(A) \neq 0$. Find the condition number with respect to the 1-, 2- and $\infty$-norm and discuss ...
0
votes
0answers
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 ...
2
votes
1answer
135 views

Infinite norm of a vector

While reading the book Numerical Linear Algebra by Trefethen and Bau, I came across the following example. The authors indicate that $\|J\|_{\infty} = 2$, however if I recall the definition of $\|\...
4
votes
1answer
94 views

Is there any inequality between 2-norm condition number and Frobenius norm condition number for rectangular matrix?

What I have found in [1] Condition number inequality between Frobenius norm and 2-norm for square matrix, Consider a full rank matrix $X \in \mathbb{C}^{n \times m}$, $m=n$, then we can have, $$n - ...
0
votes
0answers
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 $$\kappa = \frac{\|x'\|}{\|x\|}\cdot \|p\|=\frac{\|-e^p\|}{\|1-e^p\|}\cdot \|p\|$$ ...
0
votes
1answer
47 views

Prove that $cond(A)\ge \frac{||A||}{||A-B||}$ for any induced matrix norm

Prove that for any induced matrix norm: $cond(A)\ge \frac{\left\lVert A \right\rVert}{\left\lVert A-B \right\rVert}$ Where $A$ is an invertible matrix, and $B$ is a singular matrix. The condition ...
2
votes
3answers
51 views

What's the different between necessary, sufficient, necessary and sufficient condition?

1)The range of values of "$a$", such that $|x-2|< a$ is a necessary condition for $x^2-3x-10<0$ 2)The range of values of "$a$", such that $|x-2| < a$ is a sufficient condition for $...
1
vote
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?
1
vote
0answers
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 ...
0
votes
2answers
20 views

Upper bound of the norm of a matrix difference using an absolutely converging geometric series and Neumann's theorem

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{...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 \\...
1
vote
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 ...
1
vote
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 ...
0
votes
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?
0
votes
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} &...
0
votes
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 ...
2
votes
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 ...
1
vote
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 ...
0
votes
0answers
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 ...
1
vote
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 $...
0
votes
0answers
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 $\mathcal{K}(A):=\|A\|\cdot\|A^{-1}\|$ ...
1
vote
0answers
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 ...
1
vote
1answer
339 views

Finding the relative condition number given a function.

I'm teaching myself how to find the relative condition numbers and I am struggling with connecting it to something basic like scalar multiplication. For example, the first problem of my text book ...
0
votes
1answer
46 views

Condition number in terms of inner products for $A = I - uv^T$

I need help with the following problem: $A = I - uv^T$, where u and v are vectors in $R^n$. Find and expression for $\kappa_2(A)$ in terms if the inner products $r = (u, u)$, $s = (v, v)$ and c = $(...
0
votes
0answers
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: ...
1
vote
1answer
81 views

Create a matrix with a given sparsity pattern and whose condition number is low

In Matlab, I need to create a symmetric matrix with a given sparsity pattern and whose condition number is low ($\leq 10$). The matrix is sparse (more than half of its entries are zero). From what I ...
3
votes
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 $...
2
votes
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 ...
0
votes
0answers
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 ...
1
vote
1answer
86 views

Compute the condition number of a $3 \times 3$ matrix.

Compute the condition number of the matrix $B$ : $$ B=\begin{bmatrix}3&7&1\\5&8&0\\6&3&2\end{bmatrix} $$ in terms of $\|\cdot\|_{1}$ and $\|\cdot\|_{\infty}$. Important: Our ...
0
votes
0answers
132 views

Finding the condition number of a matrix

I'm having trouble calculating the condition number of a matrix. I'm trying to get ahead of my class material. Any help would be appreciated. Question: Calculate the condition number $$k(A) = \|A\|...