# 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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### Why does finer mesh mean worse condition number?

Suppose I am working on the finite element approximation of a problem. My understanding is that the condition number of the resulting algebraic system becomes worse when the mesh becomes finer. What ...
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 $... 0answers 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 ... 0answers 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 ... 1answer 6k views ### Condition number of a rectangular matrix From what I understand, the condition number of a rectangular matrix$Ais its largest singular value divided by its smallest nonzero singular value $$\kappa(A) := \frac{\sigma_1 (A)}{\sigma_n (A)}... 3answers 534 views ### How to measure how far a matrix is from being singular? What would be the best mathematical tool/concept to measure how far a matrix is from being singular? Could it be the condition number? 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 ... 1answer 159 views ### Condition number of matrix is 1 iff A^TA=\alpha I I am trying to prove the following: The condition number \kappa_p(A)=1 for A an matrix iff A^TA=\alpha I for some scalar number \alpha\neq 0. I read somewhere on the internet that I ... 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 ... 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 \|\... 2answers 346 views ### How is f(x)=x+1 not backwards stable if I consider the error propagated in the addition? Many sources claim that f(x)=x+1 is not backwards stable. That is, it does not give an exact solution to a slightly perturbed (or "nearby") problem. e.g. https://www.cs.usask.ca/~spiteri/CMPT898/... 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 ... 2answers 136 views ### Condition numbers: Find vector pair for which the equality holds I have that Ax = b. I would like to find a vector pair (b, \delta b) for which the following equality holds:$$\frac{||\delta x||_2}{||x||_2} = \kappa_2(A) \frac{||\delta b||_2}{||b||_2},$$... 1answer 37 views ### Discretising the Fourier Integral gives a high condition number I have the following integral equation$$ \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$$where f is to be found, and g is known. I believe this problem is ... 0answers 59 views ### Solving numerically a strongly stiff nonlinear ODE system with ill-conditioned Jacobian Using Matlab, I am trying to solve numerically the following nonlinear system of ODEs:$$\begin{aligned} \dot B &= -\alpha B -\nu BV \\ \dot X &= A-\mu_1 X -c E(B)VX \\ \dot Y &= ... 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$is ... 0answers 99 views ### Find the relative condition number of$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}$Find the relative condition number of with respect to the 1-norm. ... 1answer 147 views ### Suppose ||.|| is an induced matrix norm, A is non-singular, and B is singular. Prove$\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$is non-singular ... 1answer 45 views ### Measure for how singular a square matrix is in the range [0,1] I am interested in estimating how close a square matrix is to being singular such that I can compute a value$s \in [0,1]$where$s=1$would mean the matrix is singular, and$s=0$means it is as far ... 2answers 78 views ### How to choose two diagonal matrices minimizing the condition number I have a matrix$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)$should be minimal. How to provide such diagonal matrices? 0answers 228 views ### Relative condition number, Ill conditioned, Well conditioned I'm currently learning about relative condition number (K), and how they are considered as well conditioned or ill conditioned. From my understanding, a large K value represents ill-conditioned, ... 1answer 251 views ### Find the condition number of$A$Find the condition number of $$A = \begin{bmatrix} 0 & 0 & -10^4 & 0 \\ 0 & 0 & 0 & -10 \\ 0 & 10^{-3} & 0 & 0 \\ 10^{-2}& 0& 0& 0 \\ \end{... 1answer 494 views ### Condition number of the product of two s.p.d. matrices The condition number for an invertible matrix A is defined as follows$$\mathcal{k}(A) := \|A^{-1}\| \|A\|$$where \|\cdot\| is the Euclidean norm. If A is symmetric, then$$\mathcal{k}(A)= \... 3answers 121 views ### multiplying a linear system by an invertible diagonal matrix let$Ax = b$be a linear n by n system if we multiply this system by a non-singular diagonal$D$I can say that the new system still has the same solution as the previous one , right ? I ran some ... 1answer 83 views ### Does it hold that$\kappa(A^2) = \kappa(A)^2$? Let$A \in \mathbb{R}^{n \times n}$be invertible and$b\in \mathbb{R}^n$. Let$x \in \mathbb{R}^n$be the solution of$Ax=b$. Let$\kappa(A)$denote the condition number of matrix$A$. Does the ... 1answer 64 views ###$\text{cond}(A)\gt \text{cond}(A+B)$for$AA^T=I$Let$A$a matrix such that$AA^T=I$. Is there a matrix$B$such that $$\text{cond}(A)\gt \text{cond}(A+B)$$ If so give numerical exmples for this, otherwise prove that there isn't. The condition ... 1answer 70 views ### Is$A \mapsto A^+$well-conditioned? Let$A$have reduced SVD$A = \tilde{U} \tilde{\Sigma} \tilde {V}^*$, and define the Moore-Penrose pseudoinverse of$A$as$A^+ = \tilde{V} \tilde{\Sigma} \tilde{U}^*$. In the$l^2$norm$\| \cdot \|$,... 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 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 ... 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 ... 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 ... 3answers 1k views ### Sensitivity of the least squares method and matrix condition number It is my understanding that if we have a simple linear system such as $$Ax = y$$ The condition number of$A$provides an indication of how sensitive the solution$x$will be in relation to changes ... 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 ... 1answer 101 views ### Condition number of a polynomial root problem I dont't understand how the condition number is defined for a problem such as:$x^2-2xp+1=0,\ p\geq1$Here there are two roots$x_-=p-\sqrt{p^2-1}$and$x_+=p+\sqrt{p^2-1}$I understand that the ... 1answer 1k views ### Condition number of a diagonal matrix Let$\|\cdot\|$be any norm on$\mathbb{C}^n$. Let$A\in \mathbb{C}^{n\times n}$We define the matrix norm by$||A||=\max_{||x||=1}||Ax||$. If$A=diag(\lambda_1,...,\lambda_n)$and it is invertible, ... 1answer 397 views ### Condition number of matrix multiplied with permutation matrix Given a$n \times n$matrix$A$with condition number$1$, prove or disprove$\mbox{cond} (A) = \mbox{cond} (PA)$, where$P$is is any permutation matrix. My attempt:$cond(A) = ||A||*||A^{-1}||$... 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 ... 0answers 32 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 \ $. Here$||.||$... 1answer 124 views ### Is division ill-conditioned when divisor is close to zero? My intuition is that division of real numbers is ill-conditioned when divisor is close to zero. Is this intuition correct? 0answers 24 views ### what is the significance of$\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 ...