For questions on propagation of errors.

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What are disadvantages of solving difference equations backwards?

Say you have the initial value of 100th and 99th term and you want to see how much error is in 1st term instead of picking values from the table for 1st and 2nd term and propagating to 99th and 100th ...
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Solution to recursive equation

what will be the form of solution for this kind of recurrence equation? $$P_{n+1} + \dfrac{2n P_n}{x} - P_{n-1} = 0$$ $x$ is a constant. Will a guess solution of form $\lambda^n$ work? I need to ...
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16 views

error calculation when the error is not constant

I have to calculate the error on the following quantity: $$f(\epsilon^M,\epsilon^S)= \sqrt{ \frac{1}{N}\sum_{i=1}^N (\log{\epsilon_i^M} - \log{\epsilon_i^S})^2 } $$ Usually I would use this standard ...
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2answers
22 views

Proof of a result in numerical analysis, error bound.

I would like to proove the Lemma 3.1. in this book. My attempt... I want to split the lemma into several parts. Part 1: $$\prod_{j=1}^{n} (1 + \epsilon_j) = 1 + \sum_{j=1}^n \epsilon_j + O(|u|) = 1 ...
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1answer
32 views

scientific computing problem, error analysis and writing algorithm

For $f(x)=(1-\cos(x))/x^2$, (a) Analytically evaluate $\lim_{x→0} f(x) = L$. (b) As $x→0$, at what rate does $f(x)→L$? (c) Suppose that we are able to represent floating point numbers with $N$ ...
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26 views

Error propagation when function contains (or is) a derivative

I am familiar with the usual error propagation formula, but am unsure how to proceed when the function itself contains a derivative. I will present the simplest case here: My function f(x) is simply ...
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41 views

Error in distance between points in spherical coordinates

I have two points with spherical coordinates: $a=(r_1,\theta_1,\phi_1)$ and $b=(r_2,\theta_2,\phi_2)$. The cartesian coordinates of the points are: $$ (r_i \cos\theta_i \cos\phi_i, r_i \cos\theta_i ...
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1answer
56 views

Find the absolute and relative error for a calculator with incorrect rounding

A calculator is out of order. The calculator will round up every single number to the nearest integer if the value at the first decimal digit is 6 and above, or else it rounds down the number to be ...
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1answer
36 views

Propagation of Uncertainties [closed]

I have five values for the volume of sodium hydroxide needed to neutralise a fixed quantity of hydrochloric acid, each trial with an uncertainty of 0.05 mL. If I take the average of these five values, ...
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Do theorems involving chaotic mappings hold in a finite precision context?

Chaotic mappings are known as highly sensitive to their initial state. It is well known that the first type of Chebyshev polynomials is chaotic, e.g. this. This mapping is defined recursively as ...
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15 views

Do Monte Carlo perturbations capture all the uncertainty in prediction?

I have a model $M$ that I use to predict a value $y = M(\vec x)$. I have known one-$\sigma$ error bars on each input $x_i \in \vec x$. I want to know the one-$\sigma$ error bar on my prediction $y$. ...
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10 views

Statistical error on the inverse of a variable

Doing a report for my studies I had to calculate error of $\frac{1}{T}$, where T is the temperature I measured with a systematic error of 0.1. I came up with a formula $\frac{1}{T-0,1} - \frac{1}{T}$ ...
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1answer
23 views

Error Propagation

I hope I am right in this section. I am unsure with error propagation. When calculation the error in a titration, many errors has to be taken into account: Error in Glassware/ Error in Balance/ ...
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27 views

How to propagate uncertainty into the prediction of a neural network?

I have inputs $x_1\ldots x_n$ that have known $1\sigma$ uncertainties $\epsilon_1 \ldots \epsilon_n$. I am using them to predict outputs $y_1 \ldots y_m$ on a trained neural network. How can I obtain ...
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1answer
33 views

Show that the ratio between two numbers is always proportional to the the maximum percentage error of their average?

The question is this: if we have a set of any random consecutive numbers, for example {1, 1.2, 4.2, 4.8, 5.6, 7.4, 9.8} then how can we prove that calculating the ratio between each of the numbers and ...
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32 views

Calculate Standard Deviation in Multiplication

I have a question that asks to find the result (in sig figs) as well as the standard deviation for the following $$(23\pm 8)\times(99\pm 11)\times(11\pm 3)\times(8\pm 4)$$ I began this question by ...
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1answer
70 views

Simpson's Rule for IVP. Truncation Error proof

Edit: replaced all c's with y's as the c just denotes replacing a series of coupled linear equations Ay with uncoupled equations $\Lambda c$ no biggie. Im working through the lecture notes for a ...
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1answer
47 views

Determining percentage errors for inverse trig functions in conjunction with other errors

Let's say I have the following equation in which the unknown is $θ$: $$tan(θ)=\frac{a}{b}$$ $$tan(θ)=\frac{5}{3}$$ $$θ=arctan(1.667)$$ $$θ=59.036°$$ Let's say the absolute errors ($∆S$) and ...
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51 views

Calculating percentage error for sums when there aren't absolute values?

So, let's say that I measured the lengths of two objects which measured 20 mm and 30 mm. I used the same ruler for both measurements and it has an absolute error of ±1 mm. If I wanted to calculate ...
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1answer
25 views

Does the number of significant digits of a measurement affect the relative error of the measurement?

Reading this document, when measuring a thing we have $$\text{relative error} = \frac{\text{absolute error}}{\text{value of thing measured}}.$$ Let's say we have a measurement of the height of an ...
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1answer
15 views

Propagating error for a circle

The radius of a circle is measured to be 12.1 cm plus or minus .030 cm What is the uncertainty in the area of this circle? Thank you for your help!
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1answer
22 views

Estimating errors from optimization? (Genetic algorithm or otherwise)

I have a vector of observations $\vec x_{\text{obs}}$ that have been measured with known uncertainties $\vec \sigma_{x}$. I have a model $f$ that takes parameters $\vec \theta$ and produces values ...
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1answer
28 views

Error propagation of complex quantity in polar coordinates?

Suppose I have a complex number $C = \left | C \right | e^{i\phi}$ where | | denotes modulus and $\phi$ is the phase angle. Now, I know the error in the modulus, $\delta |C|$ and error in the phase, ...
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22 views

Finding the weighted mean with varying errors

Simply enough, I've a set of numbers with varying errors: 2.02 +- 0.10 , 2.26 +-0.30 , 3.24 +- 0.30 , 3.33 +- 0.30 , 3.92 +- 0.30 , 4.02 +- 0.10 I'm certain there's a simple formula to compute the ...
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2answers
31 views

What does it mean to calculate a number to $n$ decimals of exactness.

I was asked to numerically calculate Bessel functions for certain points and report their values to "6 decimal places of exactness". I did this in matlab and there's no truncate function, so I was ...
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61 views

Uncertainty of Multiplication using Absolute Uncertainties

I am about to calculate the uncertainty of a measurement device. The formula of the result can be written as $$P=\frac{V}{R \times G}$$ All three values are uncertain; for $R$ and $G$, both the ...
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12 views

Error propagation with repeated values in the numerator and detonator

How would you do error propagation on an equation like this: $C=\left(\frac{A-B}{A B}\right)$ where A has error σa and B has error σb I would guess you use standard text book relations but since A ...
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1answer
14 views

How to estimate the error on the position of the point x where y is maximal in quadratic relationships?

I would like to estimate the elevation at which species richness is expected to be maximal. The relationships between species richness ($y$) and elevation ($x$) follows a second order polynomial ...
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19 views

Error propagation for multiparameter-non-linear fit with covariance matrix.

I have created a multiparameter-fitted-function for some data: $𝒇(𝒙)=𝑨𝒙^{(𝑪+𝟏)}𝒆^{−(\dfrac{(𝑪+𝟏)𝒙}{𝑩})}+𝑬𝒙^{(𝑮+𝟏)} 𝒆^{−(\dfrac{(𝑮+𝟏)𝒙}{𝑭})}$ I know the values and absolute ...
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57 views

If $\theta = (59.3\pm 1.2)^{\circ} $, find $\tan(\theta)$ and it´s uncertainty.

How can I find the uncertainty? I need to use this formula: The general formula for uncertanties is: $\sigma_q = \sigma_q (a,b,c,...)$ and $\mathrm{Cov}(a,b,c,...)=0$ $\,\,$then $$\sigma_q ^2 = ...
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1answer
97 views

How to propagate uncertainties in the dependent variable when doing linear regression?

Let's say I have an independent variable $\vec x$ and a dependent variable $\vec y$ and measurement errors on my dependent variable that I know to be $\delta y$. For the sake of simplicity, let's say ...
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21 views

What is the source of the error in the Sherman-Morrison formula application?

The Sherman-Morrison formula results in small errors in relation to the standard matrix inverse operation after each application, as shown here. From what I understand, the Sherman-Morrison identity ...
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1answer
22 views

About absolute error calculation

Given a prism immersed in the air with an opening angle equal to $\alpha = \bar{\alpha} \pm \Delta\alpha = 60° \pm 1°$ and $\delta = \bar{\delta} \pm \Delta\delta = 45° \pm 2°$ is the angle of minimum ...
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18 views

To what set do measured values belong?

This question is more conceptual than practical. It seems that when we apply mathematics to measured values, we treat them like real numbers. When measured values take error into account using ± ...
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15 views

Measure of how well a 3D function represents experimental data

I have experimental data of a one dimensional heat equation and corresponding values for a predicted temperatures. Is there any method in which I could statistically analyse the data to determine if ...
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35 views

Error propagation with dependent errors

I have a function $f(x_1,\ldots,x_n)$ where the variables $x_k$ have errors $\delta_k$. If these errors are independent I can add them root mean square: $\delta ...
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1answer
34 views

round-off error of sample

Basic question: If I have a large set of numbers where the third decimal has been rounded off. For example $a_1 = 5055.29$ $a_2 = 1755.13$ $a_3 = 1083.03$ $a_4 = 335.99$ $(...)$ If I sum all ...
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14 views

low number poissonian errors: is it possible for a measurement of 4 counts to have a significance higher than 2 sigma?

What is the best way to measure statistical significance of an overdensity in counting experiment where you have small numbers? Rather than bore you with my actual problem, please consider the ...
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18 views

Error on determinant from statistical errors on complex matrix elements

Say I have a complex matrix $A$ whose elements $A_{ij}$ have statistical error $\delta_{ij}$. I need to figure out from these errors what will be the error on the determinant $|A|$. If the matrix was ...
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31 views

Calculating absolute error. Teacher distributes the abs value signs.

Ill illustrate my confusion with an example: It can be shown, assuming $E_xE_y=0$ that the error in an arithmetical multiplication will be: $E_{xy}=xE_y+yE_x+\mu$ Where $\mu$ is the so called ...
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22 views

Propagating uncertainties in Gaussian fit

I'm doing an analysis where I have a set of random variables with some known uncertainties (the uncertainties are different for each random variable). The random variable is roughly Gaussian ...
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38 views

Geometric Mean of Random Variables

I measure a series of $n$ objects [O_1, O_2, O_3, ..., O_n]. Because those measurements are quite hard to perform, I have quite a lot of measurement error and ...
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29 views

Uncertainty in distance from uncertainty in coordinates

I know this is basic, but I've managed to get myself confused. So, I have an object at location $(x,y,z)$ with uncertainty in the location of $(\delta x, \delta y, \delta z)$. What is the ...
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1answer
29 views

Standard Error for Weighted Values

I want to calculate the standard error for an experimental measurement. The data is stored as a 2D image which is circularly symmetric about a center point. To reduce the data we radially integrate ...
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21 views

Statistical uncertainty on a percentages inferred from an A/B test

Question: What is the statistical uncertainty on percentages inferred from an A/B test? Example: $N = 100$ people are presented with an A/B choice (e.g., jumping into shark-infested water, choosing ...
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2answers
39 views

How can I calculate the uncertainty of two combined uncertainties?

If uncertainty can be calculated as half the range, and percentage uncertainty is the uncertainty over a mean all multiplied by 100, how can I find what the uncertainty is of A^2 * B^3 when A = 21.3 ...
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2answers
47 views

Prove that the rounding error can contaminate half the digits of computed root

I am trying to resolve the following problem: If $b^2 \approx 4ac $ the rounding error can contaminate half the digits of the root computed with the formula: $\dfrac {-b \pm \sqrt {b^2 - 4ac}} ...
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2answers
49 views

Prove graphically that the Lambert equation has exactly zero, one or two roots

I need some help on the below problem. Consider the Lambert equation: $xe^x = a$ for real values of x and a (a) Show graphically that the equation has exactly one root $ \xi(a) \ge 0 $ if $ ...
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1answer
50 views

How does one bound computational error for a finite difference approximation of the second derivative?

I'm trying to wrap my head around ways to minimize total computational error (defined as a sum of the bounds on the truncation and rounding errors) by taking a differentiable function $f : \mathbb{R} ...
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1answer
38 views

Statistical methods in experiments

I'm currently facing a huge problem in manipulating raw data from an experiment. Suppose I have the set of data: $$T_1=6.12 \pm 0.001s,\\ T_2=4.90875 \pm 0.001s, \\ T_3=3.9 \pm 0.001s, \\ T_4=3.69 ...