For questions on propagation of errors.

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4
votes
2answers
82 views

Avoiding loss of numerical accuracy

I need to to evaluate the function $f(x) = {1 - (1-A)^x \over A}$, where $0 < A \leq 1$ and $0 \leq x \leq 1$. A straightforward C implementation of $f(x)$ with floating-point arithmetic works ...
4
votes
1answer
387 views

Multiplication and division of values with geometric standard deviation

What is the geometric standard deviation of a value, which is the result of dividing two independent values, each of which has its own geometric standard deviation ? It is a frequent situation in ...
2
votes
1answer
76 views

Absolute and Relative Error of $x^y$

Suppose that we have two measured values $x$ and $y$ with maximum absolute errors of $e_x$ and $e_y$. Is there a formula to find a good upper bound for absolute and relative error of $x^y$?
2
votes
1answer
63 views

Margin of error for quotient of two measurements

I need to find the total margin of error for calculating velocity, while I have margins of error for time and distance. Actually the margins are the same (as both measurements were based on GPS - but ...
2
votes
1answer
64 views

Confused about the explicit formula for $\psi_0(x)$

In the explicit formula for $\psi_0(x)$ used in the PNT proof : $$\psi_0(x) = x - \sum_{\rho} \frac{x^{\rho}}{\rho} - \frac{\zeta'(0)}{\zeta(0)} - \frac{1}{2} \log (1-x^{-2}) $$ In particular the ...
2
votes
0answers
29 views

Monte Carlo with error on individual samples

I'm performing a Monte Carlo integration where the individual samples have an error, and I'm wondering how to estimate the final error. Some more detail: The integral E I'm after is estimated in the ...
2
votes
0answers
26 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
2
votes
1answer
34 views

Verlet method global error

I was trying to understand the global error calculation for the verlet method on Wikipedia but it's not so clear to me when it goes from: to Shouldn't be considered the error relative to x'' too? ...
2
votes
1answer
41 views

Expectation/probability question (a bit like Craps, I think)

I'll try to be as brief as possible: I have a number of events that can happen, say e1, e2...eN. N isn't particularly large. Each event has a probability of "failure". (Actually, there will likely ...
1
vote
2answers
3k views

Error propagation on weighted mean

I understand that, if errors are random and independent, the addition (or difference) of two measured quantities, say $x$ and $y$, is equal to the quadratic sum of the two errors. In other words, the ...
1
vote
2answers
48 views

How does big-O notation relate to the actual error involved in a numerical differentiation?

Suppose I have some position data ${x_1, x_2, ... x_n}$ that was sampled at an interval $h$. If I wanted the velocity data, I could apply a finite difference scheme: $ v_1 = \frac{x_2 - x_1}{h} + ...
1
vote
2answers
68 views

Propagation of Error

If I have a function $$f(x,y)=\sqrt{x^2+y^2}$$ with error in $x$ be $\Delta x$ and error in $y$ be $\Delta y$, then how do we calculate ${\Delta f}$? I know if we have $$f(x)=x^n$$, then I at least ...
1
vote
1answer
288 views

Error in a numerical derivative

I have a graph of data, say temperature ($T$) vs time($t$), I know the error bounds in each $\Delta T$. The range of t is from 0 $\to$ 1600 s, with small steps say 0.001 s. If I numerically take ...
1
vote
1answer
902 views

Error analysis - Bisection algorithm

I have a brief question related to an example in my textbook. In my book, the following theorem on Bisection Method is presented: If $[a_0,b_0], [a_1,b_1],. . .,[a_n,b_n]. . .$ denote the intervals ...
1
vote
2answers
38 views

Relation between condition number and perturbed matrix

Prove that if $A\vec{x} = \vec{b}$ and $(A+\delta{}A)(\vec{x}+\delta\vec{x}) = \vec{b}$, then $\dfrac{\|\delta\vec{x}\|/\|\vec{x}+\delta\vec{x}\|}{\|\delta{}A\|/\|A\|} \le \kappa{(A)}$, where ...
1
vote
1answer
56 views

Too small error on a physics lab

I have this function: $$\lambda=d \sin\left(\arctan\left(\frac{x}{z}\right)\right)$$ and I want to find its absolute error. d is a constant ($10^{-6}$), x is $(0.716 \pm 0.001) m $ , and z is $(1.000 ...
1
vote
1answer
51 views

How to estimate (if/any) displacement/rotations between 2d line segments taken from 2 data sets

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I derive these two line sets using image based (e.g. CD) and manual method ...
1
vote
1answer
68 views

Where does the error propagation formula comes from?

As an engineering student I have come several times across the formula $$\sigma_{f(\vec{x})}=\sqrt{\sum_{i} \big (\dfrac{\partial f}{\partial x_{i}}\sigma_{x_{i}}\big )^{2}}$$ for the propagation of ...
1
vote
1answer
137 views

Derivation of formula for estimating error in bulk-volume

In my textbook, a formula for estimating error in bulk-volume measurements is derived, but I don't quite follow one step in the derivation. The book writes the following: The bulk volume of a ...
1
vote
1answer
34 views

bound on matrix inverse with different elements

I'm hoping that someone can point me to some literature on the following. Is there a way to bound the inverse of a matrix if I change the value of 1 element in that matrix. Let's say I have a matrix ...
1
vote
1answer
64 views

Calculating Covariance. (Multiplication of Two Covariance Matrices)

I have an equation T3=T1*T2 where T is 3*3 Transformation matrix representing position of an object in 3D. Now each of these position has some error in the form of 3*3 covariance matrix i-e ∑. My ...
1
vote
1answer
54 views

How is statistical uncertainty calculated for the modulus function?

I know it's an unusual function to calculate an uncertainty for, but I haven't been able to figure out a reasonable means for calculating derivatives for it to do so myself. I know modular arithmetic ...
1
vote
0answers
40 views

Finite difference method - implementation and stability

I have a PDE for a function $P(r,t)$ in spherical co-ordinates of the form $P_{t} = D(P_{rr} + (2/r)P_{r}) - a\Omega\left(\frac{P}{P + k} \right) $ where subscripts denote partial derivative with ...
1
vote
0answers
57 views

When examining global error bounds for Euler method, can I rescale the domain limits?

I'm looking at provable global error bounds of the Euler method for the first time and I was surprised to find that the bound grows exponentially in the amount of time (the domain size) propagated ...
1
vote
0answers
160 views

Problem Condition and Algorithm Stability

Consider 2 mathematical problems: $$ f_1(x) = a - x \\ f_2(x) = e^x -1 $$ The condition number for a function is defined as follows: $$ k(f) = \left| x \cdot \frac{f'}{f} \right| $$ Lets analyze ...
1
vote
0answers
53 views

How to estimate error pattern of a set of line segments with respect to given reference segments (2D case)

I am having set of pair of line segments (2D). Though each pair should be coincided on top of each other they are not so. I have been the reference data and then I extracted other line segments ...
0
votes
1answer
20 views

Mean of data with uncertainties and confidence interval

My question sounds rather simple, but sadly I have not been able to find an answer online (which is rather strange as this seems very basic, so I apologize if I simply haven't looked well enough) The ...
0
votes
1answer
193 views

What is the error in Newton's Method for Matrix Inversion?

I need it to invert a matrix. Wikipedia explains that there is a generalization of the Newton Method for matrices. However, there is nothing mentioned about the error bounds. Suppose we have, as ...
0
votes
1answer
119 views

errors, random and approximations

Hello good evening all! If a reading is reported as R = 200.045 + 0.001 or 200.045 - 0.001 Ohm. Does +0.001 or -0.001 Ohm represents a systematic or random error? Thanking you.
0
votes
1answer
20 views

Error of function with errors on arguments and size of arguments

EDIT: my question is not how to derive the formula below (I think the derivation is more or less what I guessed, like the answer below supports), but whether it can be made valid for the case where ...
0
votes
1answer
71 views

How to calculate error size from division of two ratios

How one would calculate an error from division of two rations? I am given 1/y and x/y as their decimal representation (numbers ...
0
votes
1answer
94 views

Error Propagation in Successive Least Square Adjustment

I have a certain problem in surveying in which I'm trying to do some error analysis. I'll layout the problem first. My current goal is to evaluate the variance-covariance matrix for the position of ...
0
votes
1answer
39 views

What is the proper way to determine error in my measurements?

How do I know what the error is in measurements I took using an oscilloscope? In the image below you will see on channel #1 of the oscilloscope (yellow) there is a pattern. I manually measured the ...
0
votes
1answer
630 views

Proving the relative error of division.

The problem says to show that the relative error for division on a computer is $$Rel(\frac{x_{A}}{y_{A}})=\frac{Rel(x_{A})-Rel(y_{A})}{1-Rel(y_{A})}$$ $$\approx Rel(x_{A})-Rel(y_{A})$$ provided ...
0
votes
1answer
135 views

Angle between two vectors with noisy data?

Given two vectors V1=(x1,y1,z1) and V2=(x2,y2,z2) I can calculate the angle between the two vectors using the dot product of the two vectors and their magnitudes. This approach, however, is only ...
0
votes
1answer
47 views

I don't understand where some numbers come from when calculating the stop-and-wait frame that reaches 50% efficiency.

I am working on homework assignment and I found a similar question and the answer that I can refer to. On this website I am being presented the question and the solution. Question 1A: ...
0
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0answers
16 views

what are some good ways to define the errors between two functions?

I have two functions. One is the original function (that contains 4 variables), and the second one is the approximation to the first one (also contains 4 variables). The question is, if I want to ...
0
votes
1answer
15 views

What is the error on a counting variable?

I don't know if a variable that acts as a counter has an associated error. I don't see it as a measured quantity so I think that it has no error. The problem is this: you have an image of a star and ...
0
votes
0answers
6 views

To find error in time histogram

I have a data which is recorded from a detector. Whenever the detector produce signal it records the time. I have recorded the data for several cycles, one cycle is 0 to 1 second. Finally I made the ...
0
votes
0answers
17 views

Monte carlo error: Combining “experimental” and statistical errors

I'm doing a slightly involved Monte Carlo approximation of a quantity $E$ where I end up with the following formula: $E=\frac{\sum_{i=1}^np_ie_iG_i}{\sum_{i=1}^np_iG_i}\ .\ \ \ \ \ \ \ \ \ $ (1) ...
0
votes
1answer
13 views

Error propagation for ratio data

I'm working with some published ratio data for isotopes of uranium (238U) and lead (208Pb and 206Pb). The data are published in ratios of 238U/208Pb and 206Pb/208Pb, but I need the data to be in the ...
0
votes
0answers
11 views

looking for asymmetric probability distribution

In physics, quantities are sometimes measured with both an upper and lower error. For example, I might say that an object's mass is $m = 10^{+0.1}_{-0.2} \text{ kg}$. In my case, these arise from ...
0
votes
0answers
32 views

Standard deviation on absolute square of complex average

I am calculating a quantity $q=|c|^2$ where I obtain $c\in\boldsymbol C$ as an average of a collection of estimates with errors: $\langle c\rangle=\sum_{j=1}^nc_j$, and the question is what error to ...
0
votes
1answer
33 views

Help in Measuring Error on Estimates of Differential Equations

I am working on a project for class where I have to estimate the solutions to a damped harmonic oscillator ($x''+2 \gamma x'+ \omega^2 x=0$) and compare three methods for doing so (Third Order ...
0
votes
1answer
44 views

Propagation of Error question?

If I have a function $$f(r,V,B)=\frac {2V}{r^2B^2} = 2Vr^{-2}B^{-2}$$ what is the propagation of error? If I use the power rules and multiplication rules described here ...
0
votes
1answer
39 views

Error bound of the Euler method

I am self studying working through the book "A First Course in the Numerical Analysis of Differential Equations" and have come to a deadend on q 1.2. The linear system $y' = Ay, y(0) = y_0$, where ...
0
votes
0answers
34 views

Error Propagation - functions of the mean.

Given a number of measurements $\{x_i\}$ with values distributed according to a (known) probability distribution $\rho(x)$ with a theoretical mean $\langle x\rangle = \int dx x\rho(x) = f(y)$ and a ...
0
votes
0answers
26 views

Error of the norm of solution in linear least-squares

How can we estimate the solution norm ($\Vert x \Vert$) error, separate from the solution ($x$) error in solving $Ax=y$ (linear least-squares problem)? Is the error of $\Vert x \Vert$ higher or lower ...
0
votes
0answers
65 views

Numerical Error in Computation - What Are the Students' Expected to Know?

I'm going to conduct an educational research about math undergraduates' conceptions about "numerical error." So I'm making a list of items I think someone with a BSc in mathematics is expected to know ...
0
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0answers
23 views

Am I correctly analyzing this data?

I've found myself having a difficult time analyzing some recent data due to some confusion in my statistics knowledge. I'm taking dose rate measurements with a detector for 10 trials. I expose the ...