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).
Homework Statement
Find all the holomorphic functions ##f: \mathbb C \to \mathbb C## such that ##f'(0)=1## and for all ##x,y \in \mathbb R##,
##f(x+iy)=e^xf(iy)##
I am completely stuck with this exercise, for the second condition, I know that...
Homework Statement
Show that the stability condition for a circular orbit of radius a, i.e.
f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0
is equivalent to the condition
\frac{d^2V(r)}{dr^2} > 0
for r=a where V(r) is the effective potential given by
V(r) = U(r) +...
Homework Statement
How to derive equation (22) on page 31 of Kittel's Intro to Solid State Physics 8th edition.
The equation is: 2\vec{k}\cdot\vec{G}+G^2=0
Homework Equations
The diffraction condition is given by \Delta\vec{k}=\vec{G} which from what I can surmise is the starting...
Homework Statement
a small object is mounted to the perimeter of a hoop of radius r. The mass of the object and the hoop is the same. The hoop is placed into a fixed semi-cylinder shaped rough trough of radius R, such that the small mass is at the top. Find the least R/r ratio such that the...
Hi all,
Generally boundary condition (Dirichlet and Neumann) are applied on the Load Vector, in FEM formulation.
The equation i solved, is Generalized eigenvalue equation for Scalar Helmholtz equation in homogeneous wave guide with perfectly conducting wall ( Kψ =λMψ ), and found, doesn't...
Homework Statement
U= { (x1, x2, x3, x4) | x1 x3 ≥ -5 }
The Attempt at a Solution
Let x = (1,2,3,4) and y = (1,2,3,4)
x+ y = (2,4,6,8)
x1x3 = 2x6 = 12
12 >-5 so closure by addition is fulfilled.
I've been hearing contradicting information-some state that any 1 test of...
If a vector field ##\vec{v}## is non-divergent, so the identity is satisfied: ##\vec{\nabla}\cdot\vec{v}=0##;
if is non-rotational: ##\vec{\nabla}\times\vec{v}=\vec{0}##;
but if is "non-linear"
Which differential equation the vector ##\vec{v}## satisfies?
EDIT: this isn't an arbritrary...
If a holomorphic function is a function that \frac{\partial f}{\partial \bar{z}} =0
Thus, an antiholomorphic function is a function that \frac{\partial f}{\partial z} =0 ?
Homework Statement
Stuck on two similar problems:
"State the normal stress boundary condition at an interface
x_3-h(x_1,x_2,t)=0between an invisicid incompressible fluid and a vacuum. You may assume that the interface has a constant tension."
The second question in the same but the fluid is...
Homework Statement
Consider a ##j=1, SU(2)## representation (or fundamental ##S0(3)## representation). Suppose that ##a_i, b_i## and ##c_i## (i=1,2,3) are vectors transforming under this representation i.e ##a_i' = [\rho_1 (x)]_{ij} a_j = \rho_{ij} a_j## and similarly for b and c. Consider...
I see on StackOverflow people doing stuff like
for (int k = 0; k < something.length(); k++)
where the body of the for loop never has any chance of modify something's length. I thought that was considered bad coding practice. The implementation of something's class might not return a private...
Hello guys,
I'm trying rather hard to find a real square matrix that will satisfy this ##AA^T=-I##. The first thing that comes to my mind is of course orthogonal matrix. But clearly it isn't. In fact, ##A^T=-A^{-1}## in order for it to work. The condition is pretty strict if you also consider...
Hi,
It is a well known fact that in an inverse linear problem low condition numbers have low noise amplification and therefore decrease the error.
So I wanted to test this: I draw random (skinny) matrices A, calculate y=A*c where c is a known coefficient vector, add some noise and...
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i'm using Matlab/simulink to create a microgrid model which simulate dynamic control of power based on the loads. the loads are changing based on the realtime. I use several voltage sources to power the loads. I would like to be able to turn on/off the generator based on the loads. for...
Homework Statement
Find the deflection at x=L/4 and x=L/2 for the beam
Homework Equations
See attached pic.
The Attempt at a Solution
So I have the solution derived in class. Only 0<x<L/2 is derived because since the load on the beam is at L/2, the equation is valid for the...
Greetings! :biggrin:
Homework Statement
Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials:
\int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'}
Hint: Use integration by parts
Homework Equations
P_l=...
Homework Statement
A particle is in a 1D harmonic oscillator potential. Under what conditions will the
expectation value of an operator Q (no explicit time dependence) depend on time if
(i) the particle is initially in a momentum eigenstate?
(ii) the particle is initially in an energy...
Homework Statement
dy/dx =4yx^3-y y(1)=-3
dy/y = (4x^3-1)dx
ln(y) = x^4-x+C
y = e^(x^4-x+C)
But an answer source says that after the integration I get
ln(y) = x^4 - x + ln(C)
so then..
ln(y/c) = x^4 - x
y = Ce^(x^4 - x)
which makes it much easier to solve for the constant given...
Hi,
Given an over-determined system of linear equations y=A c, the condition number of matrix A essentially says how good vector c can be restored from measurements y.
Changing the order of rows clearly does not change the condition number.
But is there information/literature on how to...
Homework Statement
0<\left|x+3\right|<1/4
Homework Equations
The Attempt at a Solution
(-13/4)<x<(-11/4) and x\neq-3
Thanks in advance. This is my first post and I am unfamiliar with formatting this kind of stuff so I will work on getting better at that aspect.
Homework Statement
now I have a PDE
$$u_{xx}+u_{yy}=0,for 0<x,y<1$$
$$u(x,0)=x,u(0,y)=y^2,u(x,1)=0,u(1,y)=y$$
Then I want to know whether there are some method to make the PDE become homogeneous boundary condition.
$$i.e. u|_{\partialΩ}=0$$
(If the equation below do not appear correctly, you can read all of the question in the attached file.)
Solving the time dependent 1D Schrödinger equation, one can show that in all points (x,t),
i\bar{h}\frac{\partial}{\partial t}\Psi(x,t)=-\frac{\bar{h}^2}{2m}\frac{\partial^2}{\partial...
Homework Statement
Given the tridiagonal matrix T_{n}
2+x 1 0 0
1 2+x 1 0
0 1 2+x 1 etc.
Derive an upper bound on the infinity norm ||T_{n}^{-1}|| and also derive an upper bound on the condition number of the matrix
The Attempt at a Solution
This is not...
What does it mean for a condition to be "open"? E.g. it is said that det(A)≠0 is an open condition for a matrix group.
Furthermore, this implies that GL(n) has the same dimensions as the group of all nxn matrices as, and I quote, "the subgroup of matrices with det(A)=0 is a subset of measure...
Homework Statement
Solve the following DE
y'' + 8y' − 9y = 0, y(1) = 1, y'(1) = 0
Homework Equations
Homogenous DE with constant coefficients
The Attempt at a Solution
Well, i solved it normally using a CE and having
yH= c1 e^t + c2 e^(-9t) ..
y' = c1 e^t -9 c2 e^(-9t)...
Hello guys!
Homework Statement
The question is like this:
If ##z=\frac{a}{b}## and ##\frac{1}{a+b}=\frac{1}{a}+\frac{1}{b}##, find ##z##.
The Attempt at a Solution
This question is challenging for me because I don't know exactly where to start. The latter condition stated, the sum of...
I need to know how I can prove the existence and uniqueness of a solution (using Lipschitz condition and well-posedness, stability analysis, etc.) for a system of 12 ordinary differential equations. I have the theorem that I need to use, but the number of calculations and work that I would have...
As far as I know there is no explicit formulas but is this true? I've tested it in Matlab with random matrices and It seems true!
cond(A+B) =< cond(A) + cond(B)
Where can I find a proof for this hypothesis?
I would like to know if the second part of this question is asking something different.
**Problem:** Consider the linear system $19x_1+20x_2=b_1, 20x_1+21x_2=b_2$. Compute the condition number of the coefficient matrix. Is the system well-conditioned with respect to perturbations of the...
Homework Statement
They give me y ' = (xysinx)/ (y+1) , y(0) = 1
Homework Equations
So I just separated and integrated
The Attempt at a Solution
I'm 99 percent sure I'm OK up till here. I just wanted to get an explanation for something.
I was wondering my answer is y + ln(y)...
Hello,
According to Stephen Hawking no boundary condition universe does not have any boundary in space time.If it is so then it is like earth.You can not go north to north pole.Earth does not have any edge or boundary.So universe is like closed structure like earth.Means after some times it...
For commutator, HQ-QH = 0 .
But for this case as shown below, complex ψQHψ - HcomplexψQψ= 0?
If the operator Q is in term of (∂/∂t) and (∂/∂x) ,then the HQ-QH may not be zero.
Is there any restriction for Q operator?
It says that |xy| < p. But I don't understand why even after reading the proof. If I have a four state DFA, whose last state is the one that is going to repeat for a given input string of length p, |x| is already going to be four, since it represents the states necessary to reach the repetition...
Starting from the Cauchy definition of convergence of a series :
\forall N,\epsilon>0,\exists N_0 | k>N_0\Rightarrow |\underbrace{\sum_{n=1}^{N+k}u_n-\sum_{n=1}^k u_n}_A |<\epsilon
rewriting A in terms and considering a positive decreasing sequence :
A\Rightarrow \epsilon>u_{N+k}+\ldots...
when we electric field between two conductors in certain direction the current density should pass in its direction why current density direction change at boundary although the direction of electric field is the same for both conductors
Although I understand the derivation of boundary condition in case of steady electric current but I did not understand, that the electric field which is in direction of J current density which is generated from permanent potential to have a current in a conductors that is applied between two...
Given
\begin{align}
L\phi &= \lambda\phi\\
\phi_t &= M\phi
\end{align}
where \(L\) and \(M\) are operators and \(\lambda\) a constant.
I want to show the compatibily condition is \(L_t + [L,M] = 0\) where \([,]\) is the commutator.
\[
(L\phi)_t = L_t\phi + L\phi_t = \lambda\phi_t = \lambda...
Hi All
I am among the people who bought the nexus 7 without the data capability and pretty much the first generation of its kind when it first appeared at Google's store. After almost a year and a half I have noticed problems with the battery. It's not much of a problem as it is my annoyance...
Hi all,
I'm asking a question about the number of the boundary conditions in high-order PDE. Say, we are solving the nonlinear Burger's equation
\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=\nu \frac{\partial^2 u}{\partial x^2} subject to the initial condition u(x,0)=g(x)...
Hi all,
It may be a trivial question. But, if I have a PDE of variable u(x,t)
--------------------------------
\dot{u} = f(u,\partial_x{u},..)
with boundary condition :
u(0,t) = u(L,t) =0.
--------------------------------
Now I need to calculate
\partial_x{u}
for that can I define the...
Measure Theory, Caratheodory condition
The set E \subset ℝ^{p} satisfy Caratheodory's condition if:
\forall A \subset ℝ^{p}
m_e (A) = m_e(A \cap E) + m_e(A \cap E^c)
Prove that if E is measurable then E satisfy the Caratheodory's condition.
I know
m_e (A) \leq m_e(A \cap E) + m_e(A \cap E^c)...
q)give the converse ,the contrapositive and inverse of these conditional statements
a)if it rains today,then i will drive to work
b)if |x|=x then x>=0
c)if n is greater than 3,then n^2 is greater then 9
Homework Statement
A small rock is thrown vertically upward with a speed of 17.0m/s from the edge of the roof of a 30.0m tall building. The rock doesn't hit the building on its way back down and lands in the street below. Air resistance can be neglected.
Homework Equations
Acceleration...
I read from the PDE book about Laplace equation in static condition ie ##\frac {\partial U}{\partial t}=0##.
But is it true that even if U is time varying ie ##U=U(x,y,z,t)##, you can still have Laplace and Poisson's equation at t=k where k is some fixed value. ie...
As per orthogonality condition this equation is valid:
\int_0^b xJ_0(\lambda_nx)J_0(\lambda_mx)dx = 0 for m\not=n
I want to know the outcome of the following:
\int_0^b xJ_0(\lambda_nx)Y_0(\lambda_mx)dx = 0
for two cases:
m\not=n
m=n
Homework Statement
I have applied separation of variables to a transient radial heat equation problem.
T is a function of r and t.
I have reached the following step:
Homework Equations
T_2(t,r) = \sum_{m=1}^ \infty c_m...
Assume that a point x is an interior point of domain of some function f:[a,b]\to\mathbb{R}, and assume that the limit
\lim_{(\delta_1,\delta_2)\to (0,0)} \frac{f(x+\delta_2)-f(x+\delta_1)}{\delta_2-\delta_1}
exists. What does this imply?
Well I know it implies that f'(x) exists, but...
I want to verify this, according to Griffiths p 557:
\vec{F}(\vec r')=-\nabla U+\nabla\times\vec A
ONLY if both ##\nabla U\; and \; \nabla\times\vec A\rightarrow\;0## faster than ##\frac 1 {r^2}## as ##r\rightarrow\;\infty##.
But the requirement of ##\vec{F}(\vec r')## is only...