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).
Consider a bearing joint together with a long pipe (with radius a) by using shrink-fitting. The grip between the pipe and the inner ring of the bearing give rise to the surface pressure p at the interface. If a moment M now is applied to the pipe, what will the slip condition between the two...
I have a general question about how to construct nonlinear ODE systems with given condition such as # of critical points with certain characteristics of the phase portrait of each critical point.
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A function f is both integrable and infinitely differentiable, i.e. f\in L_1(\mathbb{R}) \cap C^{\infty}(\mathbb{R}). Is it correct to say that this implies that the derivatives of f are also in L_1(\mathbb{R})? My reasoning: we have I<\infty, where
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The laplace equation(or possion) expressed as (in circular annulus),
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Assume that there are two domains where the boundary of...
Homework Statement
A spherical billiard ball radius a and mass M is sitting on a flat surface. The coefficient of friction between the ball and table is u. The moment of inertia for the billiard ball is
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The equation is:
u'''(t) + u/2 u''(t) = 0
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I need to get both analytic solution and numerical solution. For the...
I would like to check my reasoning for this problem to make sure I understand what an orthogonal matrix is.
Homework Statement
Determine if the matrix is orthogonal. If orthogonal, find the inverse.
\begin{pmatrix}
-1 & 2 & 2\\
2 & -1 & 2\\
2 & 2 & -1
\end{pmatrix}
Homework...
Anyone know how to solve this one?
Homework Statement
Determine weights w1 and w2 that cause the tension T in the horizontal cable to be 100 N.
Here's the diagram:
http://tinypic.com/m/f56cyb/2Homework Equations
All I know is the component method but. . .I'm not sure how to apply it here...
I am solving the heat equation in a non comercial C++ finite elements code with explicit euler stepping, and adaptive meshes (coarse in the boundaries and finer in the center). I am aware the CFL condition for the heat equation depends on dt/h**2 for the 1D, 2D, 3D case. When I solve the...
Homework Statement
m=6kg
r=0.18m
Fapp=33N
I'm struggling to understand how to answer this question and correlate the linear force applied to rotation without being given the coefficient of friction causing the rotational motion.
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Hello all:
I would very much appreciate advice on setting up a problem. Apologies in advance... This is probably a silly question--I'm more of a chemist than an engineer/math person!
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Homework Statement
Consider the PDE xu_x + y u_y = 4 u, -\infty < x < \infty, -\infty < y < \infty. Find an explicit solution that satisfies u = 1 on the ellipse 4x^2 + y^2 = 1.
Homework Equations
The Attempt at a Solution
The characteristic curves are
x(t,s) = f_1(s) e^t...
Homework Statement
Consider a cyclist turning around a circular track with radius r and speed v as shown
in attached fig.
Let W be the weight of the system (cyclist + bicycle) (W = mg)
N = normal force by ground on the system
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θ = angle of...
I'm asked to determine if for the solution
y=c_{1}e^{x}cos(x)+c_{2}e^{x}sin(x)
for:
y"-2y'+2y=0
whether a member of the family can be found that satisfies the boundary conditions:
y(0)=1, y'(\pi)=0
Not quite sure what to do here. The examples in my book give boundary conditions for the same...
Is there a term to describe something like a boundary condition but which can be applied within the domain, not just on the boundaries?
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Usually positive Ljapunov exponent is said to be a condition for chaos,
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PLEASE...
theorem states that for y'' + p(x)y' + q(x)y = r(x);
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in y'' - y'/x = 0;
both y = x^2 and y = 0 and y = k x^2 satisfy above with ..
y(0) = 0; and...
The question is x^2dy/dx + y^2=0 , y(1)=3
I re-arrange the equation to get -1/y^2dy=1/x^2dx
Seperated them, then I integrate both sides to get 1/y=-1/x + c
Now I don't get how they got the answer y=3x/(4x-3), as when I try use the condition I get a different answer, could anyone help? I...
Hello,
I want to calculate the conditioning of a matrix, therefore I use the cond() commando in Matlab. 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.
If I calculate the condition number of...
Lets us say I have a cube and I apply to a face of the cube a heat flux of 100 watt/m^2.
Lets us say i divide the face of the cube into say 10 elements (area of each face of the element is 1 m^2).
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Sorry for a...
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Advanced thanks for any enlightenment!
see this image
https://docs.google.com/leaf?id=0B5ZURGN4IHlTZWIxN2Q5NzEtNzQxZC00NTFkLWFkNDAtMTZmMDU5ZGIwOGQx&hl=en_GB&authkey=CL3RqN4O"
something about question:=> a block M1 placed over M2 Nd M2 placed on the earth.A constant force F is applied in upward direction. You have to tell about...
Hi All ,
what I am trying to do is, after getting so many solution I just want to print only the intger solution:
For[n = 0, n ≤ 10, n++
Do[Print[{n, (n - Sqrt[Sqrt[
2]Sqrt[3n^2 - 9n +
8] + 3n - 4])/2,
OTHER SOLUTION, (n - Sqrt[-Sqrt[2]...
http://www.math.northwestern.edu/~clark/285/2006-07/handouts/lin-constraint.pdf
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1. From about...
Hi,
Can anybody tell me the difference between a Cauchy Boundary condition and a combined Dirichlet/Neumann Boundary Condition for PDEs?
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What is the condition for the continuous time signal x(t) to be periodic if it is the linear combination of n periodic signals.
where
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where
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Faraday's law has an integral and a differential version:
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When I use the differential version I always have a constant of...
Hi
I am studying magnetic vector potential from Griffiths book. The eq 5.76 in his book gives
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\frac{\partial \vec{A_2} }{\partial n}- \frac{\partial \vec{A_1} }{\partial n}=-\mu_o \vec{K}
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Can someone state clearly a sufficient condition for a 4-tuple of quantities to be a four vector? For instance I saw the naive definition of speed four vector is not but I couldn't really understand when a 4-tuple is or is NOT a four vector.
Homework Statement
The von Laue condition for x ray diffraction can be written dΔκ =2πn where d is a displacement vector between indentical atoms and n an integer What is the smalest angle θ through which x -rays of wavelength 0.12 nm may be scattered by a sample of polonium ,a simple cubic...
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Dear all,
I have a question regarding the usual Goldstone theorem, which states that, for a system with continuous symmetry breaking, massless bosons must appear. However, if you look at the derivations of this theorem [1], the crucial assumption seems that, the conserved quantity associated...
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Homework Statement
State the condition for the equation px^2 + qx + r = 0 to have:
a)two different real roots
b)two equal real roots
c)no real rootsHomework Equations
b^2-4ac maybe?The Attempt at a Solution
I missed a class due to ailments and now have to catch up on missed work. I can't find...
Hi Guys,
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Given a bounded domain with the homogeneous Neumann boundary condition, show that the Laplacian has an eigenvalue equal to zero (show that there is a nonzero function u such that ∆u = 0, with the homogeneous Neumann B.C.).
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∫...
i can't see how they got X and Y expressions
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'A' and phi are needed to be found
but when i don't know how to innpur the initial condition to find A_x and A_y
because when we integrate x' expression bot sides i get
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PDE--initial condition
Homework Statement
Solve the equation u_x+2xy^2u_y=0 with u(x,0)=\phi (x). Strauss PDE 2nd ed., chapter 1.5, exercise 6.
Homework Equations
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The...
Euler's method?
Describe how to modify Euler's method to deal with the initial condition of y(5)=3 instead of y(0)=3.
Use this on the equation y'=1+t using a time step of 0.1 and initial condition of y(5)=3 and find y(5.4) .
??
Now find the exact solution of this equation using calculus...
Hi,
I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the Neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...
Hi,
I am trying the solve the Poisson equation in a domain with curved boundaries using the Finite Difference Method (second order accurate). I need to apply the neumann condition on the curved boundary. I have used bilinear interpolation to do this but this causes the resultant scheme to be...
Hey,
For a problem I am working on I need to know if the energy-momentum tensor of the Reissner- Nordstrom metric obeys the weak energy condition. Since I need the result for the following calculations, I just want to be sure that it is correct. Does anyone of you know the correct answer and...