# What is Condition: Definition and 636 Discussions

In the field of numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the input, and how much error in the output results from an error in the input. Very frequently, one is solving the inverse problem: given

f
(
x
)
=
y
,

{\displaystyle f(x)=y,}
one is solving for x, and thus the condition number of the (local) inverse must be used. In linear regression the condition number of the moment matrix can be used as a diagnostic for multicollinearity.The condition number is an application of the derivative, and is formally defined as the value of the asymptotic worst-case relative change in output for a relative change in input. The "function" is the solution of a problem and the "arguments" are the data in the problem. The condition number is frequently applied to questions in linear algebra, in which case the derivative is straightforward but the error could be in many different directions, and is thus computed from the geometry of the matrix. More generally, condition numbers can be defined for non-linear functions in several variables.
A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned. In non-mathematical terms, an ill-conditioned problem is one where, for a small change in the inputs (the independent variables) there is a large change in the answer or dependent variable. This means that the correct solution/answer to the equation becomes hard to find. The condition number is a property of the problem. Paired with the problem are any number of algorithms that can be used to solve the problem, that is, to calculate the solution. Some algorithms have a property called backward stability. In general, a backward stable algorithm can be expected to accurately solve well-conditioned problems. Numerical analysis textbooks give formulas for the condition numbers of problems and identify known backward stable algorithms.
As a rule of thumb, if the condition number

κ
(
A
)
=

10

k

{\displaystyle \kappa (A)=10^{k}}
, then you may lose up to

k

{\displaystyle k}
digits of accuracy on top of what would be lost to the numerical method due to loss of precision from arithmetic methods. However, the condition number does not give the exact value of the maximum inaccuracy that may occur in the algorithm. It generally just bounds it with an estimate (whose computed value depends on the choice of the norm to measure the inaccuracy).

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Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.
3. ### I Understanding Diffraction Condition in Kittle's Intro to Solid State Physics

I am going over the diffraction condition section in Kittle's Introduction to Solid State Physics physics and I am having a hard time understanding why the phase difference angle for the incident wave is positive while the phase angle difference for the diffracted wave is negative. Thank you...
4. ### I Understanding Euler Method: Finding Initial Condition of y(0)=1

The Euler method is straightforward to me; i.e ##y_{n+1}=y_n+ hf(t_0, y_0)## where the smaller the steps i.e ##h## size the better the approximation. My question is 'how does one go about in determining the initial condition ##y(0)=1## in this problem? am assuming that this has to be a point...
5. J

### Does induced drag theory include the Lift = Weight condition?

This is usual induced drag diagram. I have 2 questions: From Kutta–Joukowski theorem Fr is always perpendicular to effective airflow. 1. Does it mean for case without effective airflow(zero induced downward velocity), Fr is perpendicular to freestream airflow,so drag is zero? When effective...
6. ### Mathematica Select[ list, condition] with a parameter in the condition

This works: a=0.4 Select[ list, #[[2]] > a-0.025 && #[[2]] < a+0.025 & ] {{401803.,0.42485,3.33299,0.776904,0.277985},{402066.,0.40333,9.23462,0.381478,0.397121},{402872.,0.41899,3.47237,0.742789,0.27385}} But why doesn't this work? :- Select[ list, #[[2]] > b - 0.025 && #[[2]] < b +...
7. ### I When and how can I apply Born rigidity condition?

Hello, I try to better understand how and when I can apply the Born rigidity condition. So, for the following example: We've two space probes (Pa and Pb), that travel at an exact equal and same proper acceleration. At a given time tb0 in Pb, and as measured by Pb, the distance is Lba0 (it's...
8. ### Solution to Differential Equation with Limit Boundary Condition

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9. ### Probability that A will win given a condition

I tried to solve this problem using the chart given below. But I get a different answer of ##\frac {2}{3}## rather than ##\frac {3}{4}##. Maybe the answer given is incorrect? I determine from the chart the number of ways in which A could win given that A has already won 2 of first 3 points...
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11. ### Vector space of functions defined by a condition

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12. ### Dynamical Systems - Chaos: Stability condition for a 2-cycle system

Hi, (This question is part of the same example as a previous post of mine, but I have a question about a different part of it) I was looking at a question from an exam for a course I am self-teaching. There is a sub-question which asks us to find the values of a parameter for which the 2-cycle...
13. ### I Correct Formula for "No Fringe Condition" (Michelson Interferometer)

In two different textbooks, there are two different formulas with different derivation styles for the "No Fringe Formation" Condition. In approach (a), they use an amalgamation of bright and dark for 2 wavelengths having very minute difference in the following manner: 2dcostheta=n*λ(1)...
14. ### Finding A Relative Condition Number

Hm I'm new to these concepts, and I want to make sure I am on the right track, would the relative condition number be: k=(x/2)((1/sqrt(x+1))-(1/sqrt(x))(1/(sqrt(x+1)-sqrt(x))). Or would I have to solve the limit as x approaches 0? Thank you.
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16. ### Motion of Rolling Cylinder in Fixed Cylinder: Confusing Constraint Condition

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18. ### MHB 1.4.1 complex number by condition

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19. ### Boundary condition: null traction on the boundary of an elastic block

Hi everyone, I'm trying to understand the rationale behind the boundary condition for the problem "Finite bending of an incompressible elastic block". (See here from page 180).Here we have as Cauchy Stress tensor (see eq. (5.82)): ##T = - \pi I + \mu (\frac{l_0^2}{4 \bar{\theta}^2 r^2} e_r...
20. ### Condition to form a primordial black hole that I don't understand

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21. ### Laplace's Equation and Boundary Condition Problem

I really have no idea as to how to attack the problem in the first place. I am here to ask for some generous help on how to start. The figure is shown below for reference.
22. ### I The diffusion equation with time-dependent boundary condition

Hi everyone, I am trying to solve the 1 dimensional diffusion equation over an interval of 0 < x < L subject to the boundary conditions that C = kt at x = 0 and C = 0 at x = L. k is a constant. The diffusion equation is \frac{dC}{dt}=D\frac{d^2C}{dx^2} I am using the Laplace transform method...
23. ### Condition to three vectors being collinear

Now i am rather confused, the answer apparently is that ##(w-u) = \lambda(u-v)## But, i could find a way that disprove the answer, that is: Be u v and w vectors belong to R2, a subspace of R3: What do you think? This is rather strange.
24. ### I Quasilinear Equation but with non-zero initial condition?

The way I was taught to solve many quasi-linear PDEs was by harnessing the initial condition in the characteristic method at ##u(x,0) = f(x)##. What if however I need use alternative initial conditions such as ##u(x,y=c) = f(x)## for some constant ##c##? Can the solution be propagated the same way?
25. ### Showing that a state is unentangled under a certain condition

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26. ### How to 'shift' Fourier series to match the initial condition of this PDE?

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27. ### A Energy condition respecting warp drives in Einstein Cartan theory

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28. ### Initial condition of Wave functions with Yukawa Potential

Hello, I was going to solve with a calculator the eigenvalues problem of the Schrödinger equation with Yukawa potential and I was thinking that the boundary conditions on the eigenfunctions could be the same as in the case of Coulomb potential because for r -> 0 the exponential term goes to 1...
29. ### Weird condition describing symmetry transformation

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30. ### Will the pebble meet with the block according to the given condition?

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31. ### Topological insulators and their optical properties

I have tried to write down the boundary conditions in this case and looked into them. As conditions i) and ii) were trivial, i looked into iii) and iv) for information that I could use. But all I got was that for the transmitted wave to have an angle, the reflective wave should also have an...
32. ### A Quantum Mechanic/wave function/junction condition

May i know how do i eliminate C and D and how do i obtain the last two equations? Are there skipping of steps in between 4th to 5th equation? What are the intermediate steps that i should take to transit from 4th equation to the 5th equation?
33. ### In short circuit condition, what is wrong with following derivation

V=ir considering the whole circuit Now we knowV=E-ir Then ir=E-ir Therefore i=E/2r Therefore,V=E/2
34. ### MHB Condition for A Quartic Equation to have a Real Root

Show $20a^2+20b^2+5c^2\ge 64$ if $y=x^4+ax^3+bx^2+cx+4$ has a real root.
35. ### MHB Is the Condition Number of A'A Related to its Matrix Norm?

Hey! :o We have a matrix $A\in \mathbb{R}^{m\times n}$ which has the rank $n$. The condition number is defined as $\displaystyle{k(A)=\frac{\max_{\|x\|=1}\|Ax\|}{\min_{\|x\|=1}\|Ax\|}}$. I want to show that $k_2(A^TA)=\left (k_2(A)\right )^2$. We have that...
36. ### Enthelpy change in isentropic condition (air-con)

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37. ### Rearranging the equation for the cutoff condition in optical fibers

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38. ### What is the continuity condition for the heat flux through a boundary?

Assume there is a boundary separates two medium with different heat conductivity [κ][/1] and [κ][/2]. In one medium, the temperature distribution is [T][/1](r,θ,φ) and on the other medium is [T][/2](r,θ,φ). What is the relationship between [T][/1] and [T][/2] ? Is it - [κ][/1]grad [T][/1]=-...
39. ### I Sufficient condition for a vector field to be conservative

Homework Statement:: F is not conservative because D is not simply connected Relevant Equations:: Theory Having a set which is not simply connected is a sufficient conditiond for a vector field to be not conservative?

42. ### I Condition for a pair of straight lines

While determining the condition for the pair of straight line equation ##ax^2+2hxy+by^2+2gx+2fy+c=0## or ##ax2+2(hy+g)x+(by^2+2fy+c)=0 ## (quadratic in x) ##x = \frac{-2(hy+g)}{2a} ± \frac{√((hy+g)^2-a(by^2+2fy+c))}{2a}## The terms inside square root need to be a perfect square and it is...
43. ### Solving the Mystery: Exploring E << m Condition for Carbon 12 Ions

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44. ### B The Strong Energy Condition in General Relativity

Hello. I've recently been reading this paper... https://arxiv.org/pdf/gr-qc/0001099.pdf ...in the hope that I can begin to understand some the role of the energy conditions in General Relativity. But I'm not making much progress and so I've turned to this paper...
45. ### I Condition for delta operator and total time differential to commute

While deriving continuity equation in Fluid mechanics, our professor switched the order of taking total time derivative and then applying delta operator to the function without stating any condition to do so(Of course I know it is Physics which alows you to do so) . So,I began to think...
46. ### Discrete-time Signal & Periodicity condition

Namaste I seek a clarification on the periodicity condition of discrete-time (DT) signals. As stated in Oppenheim’s Signals & Systems, for a DT signal, for example the complex exponential, to be periodic, i.e. ej*w(n+N) = ej*w*n, w/2*pi = m/N, where m/N must be a rational number. Above is...
47. ### Condition for f(x,y,z) = f(x,y,z(x,y)) being extremized

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48. ### A What type of function satisfy a type of growth condition?

Let ##f:\mathbb{R}^n\rightarrow\mathbb{R}^n##. Is there any class of function and some type of "growth conditions" such that bounds like below can be established: $$||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),$$ with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
49. ### Condition for AB+A+B=0 where A and B are matrices

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50. ### How does the retarded scalar potential satisfy the Lorentz gauge condition?

As homework, I shall show that the retarded scalar potential satisfîes the Lorentz gauge condition as well as the inhomogenous wave equation. We saw in class how to do it. But I was thinking about this, and it seems to me that it's redundant to prove both of those things. For, if the scalar...