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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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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943 views

If $\kappa (A) > \kappa (B)$, show $\kappa (B^{-1}A) < \kappa (A)$

Let $A$ and $B$ be a toeplitz and symmetric positive definite $NxN$ matrices. If $\kappa (A) > \kappa (B)$, how to show that: $$\kappa (B^{-1}A) < \kappa (A)$$ where $\kappa $(X) is ...
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Condition number problem

I am given the function $$f(x) = \frac{1}{1+2x} - \frac{1-x}{1+x}$$ and I am asked the following: Explain why for $x \approx 0$ there is a numerical problem. Is the problem in the neighbourhood $x \...
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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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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 ...
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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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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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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 ...
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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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1answer
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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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Relating condition number of hessian to the rate of convergence

While minimizing a lipschitz continuous strongly convex functions, the rate of convergence of gradient descent method depends on the condition number of the hessian of the function where high ...
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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.
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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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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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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 ...
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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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47 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....
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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.
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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 ...
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Improving the Condition while keeping the column space

For a vector $b \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n \times n}$ the Krylov subspace is the subspace $\mathcal{K}_k(A,b) = \operatorname{span}\{b,Ab,\dots ,A^kb\}$. I am currently working with ...
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2k views

Condition number of matrix inversion with respect to spectral norm

I would like to show that "the condition number for inversion of $A$, with respect to the spectral norm is $k(A)=\rho(A)\rho(A^{-1})$" for $A\in M_n$ as nonsingular and normal matrix . Can anyone ...
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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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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 $\|\...
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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 ...
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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 - ...
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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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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 $...
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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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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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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{...
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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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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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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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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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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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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, ...
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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?
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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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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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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 ...
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1answer
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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

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