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).
QUESTION:
Solve the separable differential equation
dy/dx = sqrt(4y+64), Initial Condition: y(4)=9,
and find the particular solution satisfying the initial condition.
MY ATTEMPT:
(dy/dx)^2 = 4y+64
((dy/dx)^2)-4y = 64
,/' (((dy/dx)^2)-4y) dx = ,/' 64 dx
Is this the right method...
I have a question regarding the slide:
http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture3.pdf
On page 18-21 it gives the proof of the theorem that | \psi_i^{~} \rangle and |\phi_{i}^{~}\rangle generate the same density matrix iff |\psi_{i}^{~}\rangle = \sum_{j} u_{ij}...
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Homework Statement
"Solve for t > 0 the one-dimensional wave equation
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The boundary condition in x = 0 is u(0,t) = 0.
Assume that the initial values u(x,0) and...
Please teach me this:
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Thank you very much in advance.
Hi there,
I am working on a problem in Fluid dynamics. I've written a code in MATLAB which finds the shape of the free surface of the liquid based on Newton Relaxation. The code is fairly robust and produces good results which agree with current papers on the same problem.
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http://amath.colorado.edu/~baldwind/sous-vide.html
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1) Consider that one way light speed is anisotropic.
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Homework Statement
Given the Lagrangian density
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Homework Statement
http://img189.imageshack.us/img189/9859/mathlm.th.jpg
(Click the thumbnail)Homework Equations
When taking the laplace transform of derivatives L{f^n(t)=(s^n)*F(s) - s^(n-1)*f(0)...}; the step function u(t-a)
The Attempt at a Solution
we have never done a step function...
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Hi All,
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Homework Statement
This is what my professor wrote on the board.
Suppose the condition is as follows
y(0) = y(L) = 0
link to img http://www.izhuk.com/painter/image2.php?id=1286387629-69-86-215-102&md5=c054d375567134a9faa133b47fe690b2
This is the fundamental harmonic - with nodes appear at x...
Homework Statement
I need to prove the equation attached. I also have to describe why the integrals vanish.
Homework Equations
The Attempt at a Solution
I am not sure how to begin. Our teacher told us this equation is known as the orthogonality condition for sines. I also know...
Homework Statement
\psi(x,t) = \psi_1 e^{-i E_1 t/2} + \psi_2 e^{-iE_2 t/2}
under what conditions is the probability density time independent?
Homework Equations
|\Psi(x,t)|^2 = \psi(x,t)* \psi(x,t)
The Attempt at a Solution
i found a statement in pg 71 of prof Richard...
Let's consider a second order differential equation
f(x,\dot x,\ddot x,t)=0
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Are you satisfied with your current weight and level of fitness? If so, what do you do to maintain it? If not, why do you think you have fallen from it (assuming you were once there)?
Homework Statement
For a plane stress condition (stress z-axis = 0 ), prove the following relations if strain x-axis and strain y-axis are determined by experiments.
σ_x=(∈_x+v∈_y/1-v^2)E
&
σ_y=(∈_y+v∈_x/1-v^2)E
where:
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σ_y = stress in y-axis
∈_x = strain...
Homework Statement
Ok so this may get a little drawn out here because my book only gives me one example and I guess I can't decipher its meaning. So here is the example they give:
For \;\; \lim_{x \to 2} x^{2} = 4
Find a \;\; \delta > 0 \;\; such that whenever
0 < |x-2|< \delta, \;\;\...
Dear all,
why is it needed in the diff manifold definition that the base set M is topologically Hausdorf ?
Since M is locally homeomorphic with Rn as metric space is Hausdorf, shouldn't this condition be automatically satisfied?
Thanks.
Goldbeetle
hello,
I would like to know if following reactions are possible/correct and under what condition:
2NaOH+K->KOH+NaOH
and
HCN+K->KCN+H
Theoretically it should be possible but I'm not sure,Please help me understand
Thanks
It's obvious that if Maxwell equations are fulfilled by some E(x,y,z,t)
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Homework Statement
stand with one arm and the side of one foot pressed against a wall. can u raise the other leg side ways?if not then why not?
Homework Equations
The Attempt at a Solution
i tried myself but couldnot raise.i m not getting the reason.
Homework Statement
consider two conditions x2-3x-10 < 0 and |x-2| < A on a real number x, where A is positive real number
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Hi...:)
I need help as to how to solve the Reynolds equation:
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Thanks for replay.
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Apparently Fluent has this thing called an "overflow" boundary condition. I tried searching the help files and found absolutely nothing about it. Does anyone know what this is and why someone would want to use it?
Homework Statement
A string is attached to a ring of mass m which is free to move up and down a frictionless pole. The string is subject to tension T and its mass per unit length is \rho. The displacement of the string from its equilibrium position along the x -axis is y(x,t).
The boundary...
Homework Statement
A Sample of gas 250 mL at 37 C and 730 torr . What volume would gas occupy at standard conditions ?
Homework Equations
pv = nRT
V2 = T2P1V1
-------
T1P2
The Attempt at a Solution
Sample Gas
V = 250 mL ( 0.250 L )
P = 730 torr ( .960...
Homework Statement
what is the condition for function fg^(-1) to exist ?
Homework Equations
The Attempt at a Solution
i understand that g^(-1) only happens when its one-one , how about fg^(-1) ?
Hi everyone,
This is related to my previous https://www.physicsforums.com/showthread.php?t=392069"
Let A=(a_{ij}) be a symmetric (i.e., over reals) PSD matrix with the following conditions on Leading Principle Minors (determinant of the submatrix consisting of first i rows and i...
Hi everyone,
Let A=(a_{ij}) be a symmetric (i.e., over reals) PSD matrix. Then is the following correct?
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I can not prove or disprove it..any help?
By the way how...
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Consider a box with length, width and height given by L. The box encompasses the region...
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State whether there is any difference between the general conditions of equilibrium of a system of coplanar forces acting at a point and acting on a rigid body .If so,where is the difference?
Please provide me the answer of the above question.
In Peskin's textbook, he uses the spin sum as \sum u\bar{u} = \gamma^{\mu}p_{\mu} +m ;
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section formula ?
Hey all,
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