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).
My attempt:
$$
\begin{vmatrix}
1-\lambda & b\\
b & a-\lambda
\end{vmatrix}
=0$$
$$(1-\lambda)(a-\lambda)-b^2=0$$
$$a-\lambda-a\lambda+\lambda^2-b^2=0$$
$$\lambda^2+(-1-a)\lambda +a-b^2=0$$
The value of ##\lambda## will be positive if D < 0, so
$$(-1-a)^2-4(a-b^2)<0$$
$$1+2a+a^2-4a+4b^2<0$$...
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##.
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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...
Hello,
I try to better understand how and when I can apply the Born rigidity condition.
So, for the following example:
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##f : [0,2] \to R##. ##f## is continuous and is defined as follows:
$$
f = ax^2 + bx ~~~~\text{ if x belongs to [0,1]}$$
$$
f(x)= Ax^3 + Bx^2 + Cx +D ~~~~\text{if x belongs to [1,2]}$$
##V = \text{space of all such f}##
What would the basis for V? Well, for ##x \in [0,1]## the basis for ##V##...
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)
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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)...
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.
A hypersurface being spacelike (a local condition - every tangent to the surface being spacelike) does not preclude that points on it cannot be causally connected (one is in the future or past light cone of the other). A classic example is a spacelike spiral surface. Typically, for foliating a...
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1.4.1 Miliani HS
Find all complex numbers x which satisfy the given condition
$\begin{array}{rl}
1+x&=\sqrt{10+2x} \\
(1+x)^2&=10+2x\\
1+2x+x^2&=10+2x\\
x^2-9&=0\\
(x-3)(x+3)&=0
\end{array}$
ok looks these are not complex numbers unless we go back the the...
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...
Hi guys, I'm new here.
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[Talking about a spherically density perturbation]
The rarefaction wave starts at the surface...
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.
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...
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.
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?
This is an iff statement, so we proceed as follows
##\Rightarrow## We assume that ##|\phi \rangle## is uncorrelated. Thus the state operator must be of the form ##\hat \rho = \rho^{(1)} \otimes \rho^{(2)}## (equation ##8.16## in Ballentine's book).
The spectral decomposition of the state...
Hi,
Question: If we have an initial condition, valid for -L \leq x \leq L :
f(x) = \frac{40x}{L} how can I utilise a know Fourier series to get to the solution without doing the integration (I know the integral isn't tricky, but still this method might help out in other situations)?
We are...
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Question 1:
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Question 2:
From question 1 I have got the value of V which is 9 m/s. By using v= ω√(A^2-x^2), I have got the value of x. Now, do I need to add it with 2.5(distance...
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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?
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...
h and s can be obtained from "Saturated refrigerant-134a—Pressure table"
however, how to get h2? it is not on the curve, and neither p or dV is given in the question. Thank you
Hello!
In Optical fibers, let ##k_1## and ##k_2## be respectively the propagation constants in core and cladding, ##\beta## the propagation costant of a mode along the direction ##z##, ##a## the radius of the fiber. Using the normalized quantities ##u=a \sqrt{k_1^2 − \beta^2}## and ##w=a...
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]=-...
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?
Hello!
I am new here, and I need (urgent) help regarding the following question:
Let $\boldsymbol{A}_{(n\times n)}=[a_{ij}]$ be a square matrix such that the sum of each row is 1 and $a_{ij}\ge0$$(i=1,2,\dots,n~\text{and}~j=1,2,\dots,n)$ are unknown. Suppose that...
253 Which of the following is the solution to the differential equation condition
$$\dfrac{dy}{dx}=2\sin x$$
with the initial condition
$$y(\pi)=1$$
a. $y=2\cos{x}+3$
b. $y=2\cos{x}-1$
c. $y=-2\cos{x}+3$
d. $y=-2\cos{x}+1$
e. $y=-2\cos{x}-1$
integrate
$y=\displaystyle\int 2\sin...
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...
Clarification:
The statement in the title is actually from the solution to the homework question, as given by the textbook (you can see the whole thing below under "Textbook solution"). The solution doesn't explain everything, which is where my confusion comes from. Usually in my classes we...
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...
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...
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...
As far as I know when a function is extremized its partial derivatives are all equal to 0 (provided we aren't dealing with a constraint)
##\left(\frac{\partial f}{\partial x} \right)_{yz} = \left(\frac{\partial f}{\partial y}\right)_{xz} = \left(\frac{\partial f}{\partial z}\right)_{xy} =0##...
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:
\begin{equation}
||f(x)||\geq g\left( \text{dist}(x,\mathcal{X})\right),
\end{equation}
with ##\mathcal{X}:= \{x:f(x)=0\}## (zero...
I first tried by assuming the matrices but it was becoming complicated so i tried taking transpose on both sides,it also did not help.So now i could not think of what to do further.Help please.
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...