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ill-conditioned
ill-conditioned, a. (ˈɪlkənˈdɪʃənd) [f. ill condition + -ed2.] Having bad ‘conditions’ or qualities; of an evil disposition; in a bad condition or state. In Geometry, applied to a triangle which has very unequal angles, such as that by which a star's parallax is determined.1614 Raleigh Hist. World i... Oxford English Dictionary
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Well-posed problem
Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the An ill-conditioned problem is indicated by a large condition number. wikipedia.org
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Ill-Conditioned & Condition Number - Statistics How To
The condition number is the ratio of the change in output for a change in input in the 'worst-case'—that is to say, at the point when the change in output is largest per given change in input. If a function is differentiable and in just one variable, the condition number can be calculated from the derivative and is given by (xf′)/f.
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Ill-conditioned matrices and their singular values For ill-conditioned matrices, must it be that the smallest singular value is arbitrarily close to $0$? I know that $K_2(A) = \frac{\sigma_{max}}{\sigma_{min}}$ where ...
Of course. Think of the matrix $A$ defined as $$ A = \mathrm{diag}(1, 2, \ldots, N), $$ where $\mathrm{diag}$ is the diagonal matrix. For any value $N$, the condition number of the matrix will be $K_2(A) = N$ and the smallest singular value $\sigma_{\min} = 1$. The matrix can also be small, for exam...
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Ill-conditioned matrix > **Possible Duplicate:** > Inverse matrices are close iff matrices are close Consider this problem: **$Ax = b$** I want to solve it/find x and the matrix A is ill-conditioned. Why is the ...
Because if one knows the coefficients of $A$ only up to a given precision, a small variation in them will cause a huge variation in the coefficients of $A^{-1}$, hence, presumably, in the solution $x=A^{-1}b$. Alternatively, even if $A^{-1}$ is known with an absolute precision, if one knows the coef...
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Geometric interpretation of an ill-conditioned matrix Given a non-singular matrix $A\in$ $\Bbb R^{n\times n}$ (invertible) with SVD decomposition $(U, \Sigma, V)$, how would you interpret geometrically $A$ being ill c...
The worse-conditioned the matrix, the "skinnier" the ellipsoid.
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Condition number of preconditioned system Suppose we are solving an ill-conditioned system $Ax = b$, and we are trying to solve it using preconditioned technique. Given $\kappa (T)\approx \kappa(A)$, where $\kappa(A)$...
You need more information to eliminate pathological cases such as $T = \text{diag}(1,1,1,....,1,\epsilon)$ and $A = \text{diag}(1,1,1,...,\epsilon^{-1})$. Both matrices have condition number $\epsilon^{-1}$, but \begin{equation} T^{-1}A = \text{diag}(1,1,1,...,1,\epsilon^{-2}) \end{equation} has con...
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Why subtracting 1 is considered ill-conditioned? I was reading the following to better understand stability: > Consider evaluating $f(x) = \sqrt{1+x}-1$ for $x$ near $0$. $C_f(x) = \frac{\sqrt{1+x}+1}{2\sqrt{1+x}}$ s...
When you subtract two nearly equal floating point numbers you lose precision. In your example suppose we are working with seven place base $10$ numbers and let $x=10^{-4}$. Then $t_1=1+x=1.000100$ is exact. $t_2=\sqrt {t_1}=1.000050$ is within $\frac 12LSB$. But when we subtract $t_2-1$ we get $5.0 ...
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Large and ill-conditioned quadratic convex problem I need to solve a convex quadratic problem numerically: $\min f(x) = \frac{1}{2} x^\top A x - b^\top x$, where $A$ is a very large and ill-conditioned semi positi...
I assume you are looking to minimize $f(x)$, and that $A$ is symmetric and positive semi-definite. * If $b \notin Null(A)^{\perp}$ then there is a vector $v \in Null(A)$ such that $b^Tv \neq 0$. Without loss of generality assume $b^Tv >0$ (else multiply $v$ by $-1$). Then for all real numbers $\thet...
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谱聚类
基本算法 计算拉普拉斯矩阵 （或归一化的拉普拉斯矩阵） 计算前 个特征向量（这些特征向量对应 的 个最小的特征值） 考虑由这 个特征向量组成的矩阵，矩阵的第 行定义了图节点 的特征 根据这些特征对图节点进行聚类（例如使用k-均值聚类） 大型图的（归一化）拉普拉斯矩阵通常是病态的（ill-conditioned wikipedia.org
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Is there a relation between Ill-posed problems and Eigenvectors. One can easily explain an ill-posed problem with an equation AX=b. The following link is an good example: < 1) Can there be a class of ill-posed probl...
It could be ill-conditioned if the condition of $A$, i.e., the quotient of its largest and smallest eigenvalue, is large. 2) No. Ill-posedness is connected to the absolute values of eigenvalues. If $A$ is only semi-definite (not positive definite), then $Ax=b$ is ill-posed.
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Generate arbitrary numerically invertable matrix I'm designing a unit-test for a matrix inversion function. Currently I make a random matrix as a test case by generating its elements with random numbers uniformly dist...
It sounds like you want the Gershgorin circle theorem. Since all the off-diagonal matrix entries are at most $1$, you can guarantee that no eigenvalue of your matrix is too small by ensuring that all the diagonal entries are substantially larger than $N-1$. That is, taking $c=N$ should suffice.
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On the conditioning of a Symmetric Toeplitz Matrix I have the following problem, which I hope is enough interesting for you to help me. I have a matrix $A$ which is **Toeplitz, Symmetric and Positive definite**. Such...
Your matrix $A$ is numerically singular, and the problem $Ax = b$ cannot be solved, by any algorithm. Any preprocessing, postprocessing, iterative refinement, etc. will be useless. Possible solutions: 1. use high precision arithmetics, 2. reformulate the problem. As the second method you may conside...
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Why is Lagrange interpolation numerically unstable? Here is my understanding of the polynomial interpolation problem: Interpolating by inverting the Vandermonde matrix is unstable because the Vandermonde matrix is il...
Instability does not always refer to numerical issues in an algorithm. It can refer to the effect of perturbations in data inputs to the output solution. For example, in the picture below we fit a 4th degree polynomial to two sets of 5 data points. The data sets match except that the data point at t...
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3D surface fitting I am attempting to find the mathematical representation of a surface given a set of (x,y,z) data points. I recently tried using the method of least-squares which worked well for most of my situation...
Try using Moore-Penrose pseudoinverse $A^+\\!$, it coincides with $(A^TA)^{-1}A^T$ when the inverse exists but is defined for any $A$. When $A^TA$ is singular there are multiple least square fits, and pseudoinverse will give you the one with the smallest Euclidean norm. Stable numerical methods for ...
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