For questions on propagation of errors.

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1answer
21 views

What should be the expected error of a 4th order Runge-Kutta integration in a multivariable (non-linear) context?

I am writing a program where the user input $n$ variables, their initial values and differential equations. They may be non-linear. To find the value of the variables at a time $(t_0 + T)$, I use 4th ...
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1answer
29 views

Error propagation - implicit functions

I have a little problem that I should solve quickly and I'm a little bit on pressure, so that any help/tip would be of great help. I have two nonlinear equations with two unknown variables x and y ...
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0answers
24 views

Significance and Poisson processes

Must I include the Poisson error on a new observation when considering whether it is consistent with a distribution? For example, the number of rain drops that fall into a cup per minute are recorded....
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1answer
19 views

Minimum error in floating point approximation of an elementary function.

I need a confirmation of a thing that probably is silly. Let $x$ a floating point number representable using $e$ bits for exponent and $m$ bits for mantissa, let $f$ a be an elementary function, you ...
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8 views

How to make a covariance matrix from multiple observations of different objects?

I have $N$ objects. From each object, I sample $M$ values $(x,y)$ like so: ...
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0answers
34 views

Calculating a three sigma limit on data

So I have a list of data points each with their own error (eg. 3 +/- 0.2 units) and I want to determine which of the data points are within 3 sigma. To do this do I just calculate the standard ...
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0answers
10 views

Input Error estimation

I was wondering what are the methods used to detect the input's error when having the output's error of a model. I thoroughly searched on google, but I failed to find a well explined method, or ...
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0answers
18 views

Taylor series approximation of inverse trigonometric function

Suppose we have a function of three variables $a,b,c$ defined as, $f(x,y,z)=\arctan\left(\frac{\sqrt{x^2y^2-z^2}}{y^2-z}\right)$. Suppose $x=a, y=b, z=c$ satisfy the following property: (1) $a,b,c>...
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1answer
27 views

Rounding error of trapezoidal method

I'm working with the Modified Euler method sometimes called Heun's method or explicit trapezoidal method. I have a book on ordinary differential equations numerical analysis that claims: The ...
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0answers
57 views

What does this notation mean? $∂df^2 / ∂x$

I am given variance $\sigma_x$ and function $y=f(x)$ According to my book, the following equation gives the new variance of $y=f(x)$. But I'm not sure what this notation means, as in what this ...
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2answers
33 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
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1answer
24 views

Is $\frac{\rm d}{\rm d\omega(t)}\int_{t_0}^t\omega(t')\rm dt'=\int_{t_0}^t\frac{\rm d \omega(t')}{\rm d\omega(t')}dt'=t_0-t?$

I'm trying to calculate an error propagation, but the expression in the most LHS of the equation in the title crops up. Are you allowed to simply exchange the order so that the operations cancel? $\...
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26 views

Nonlinear equations algorithm - Newton method

Some time ago I posted a question regarding the simple case of finding the intersection point when I have only two functions, and with your help I found an answer. It was this case: $f(x) = a + \...
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0answers
16 views

Error propagation in complex formula

I'm currently trying to use error propagation formulae to calculate an estimate for the error in the following molecular dynamics formula: $C_v^* = \frac{3}{2}\bigg[1-\frac{2}{3NT^{*2}}\big\langle (\...
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0answers
21 views

Uncertainty propagation and division

I am getting confused about a super basic issue. I have two quantities: $6\pm4$ and $4\pm3$ . So let's say $x = 6$, $\Delta x = 4$, $y = 4$, $\Delta y = 3$ Now I want to calculate the uncertainty ...
1
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1answer
37 views

Error in average of $x^2$ from error in average of $x$?

Is there an easy way to obtain the error in $\langle{x^2}\rangle$ from $\langle{x}\rangle$ or are they independent? The values of x are from a molecular simulation application, I obtained a set of ...
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0answers
25 views

How should I compensate for numerical inaccuracy in fixed-point multiplication?

I am experimenting with fixed point arithmetic. I found a library for doing fixed-point math using vectors of 32-bit integers . At the end of the mulfpu (unsigned fixed point multiplication) ...
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3answers
36 views

Uncertainties in angle measurement

I wonder why uncertainties in angle measurement MUST be in radians. For example, I want to calculate the uncertainty in measuring the function $y= \sin (\theta)$ when the angle is measured $\theta = ...
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0answers
13 views

Main differences between Monte carlo and Law of propagation of variance

I need to know the main differences/limitations between/of Monte-Carlo simulation technique and law of propagation of variance. Can someone briefly describe it or is there any good reference or link ...
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0answers
22 views

Propagation of Errors, Variance of a $N$ dimensional vector

I am working on a calculation and I'm trying to see if the last step is true. I tried to simplify all the details of the problem. Let $\mathbf{W}$ be a $N\times N$ matrix, $a=\{a_{1,}a_{2},a_{3}\}$ ...
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0answers
14 views

Two questions about the propagation of uncertainty: e^x, and a complex iterative equation

I'm trying to do some calculations involving the propagation of uncertainties in complex equations as part of my PhD research, and I'm running into brick walls. One advisor isn't around and the other ...
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1answer
84 views

Truncating a taylor expansion for a recurrence relation?

Let's say I have a function $N$ whose future value at a time $t + t_{d}$ obeys the relation $N(t + t_{d}) = A(t)N(t)$ where $A(t)$ is also a function of $t$ whose value can be calculated. One can ...
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1answer
22 views

propagation of error from product of Taylor Series

Say I have two functions $f(x)$ and $g(x)$, both of which I will be approximating with Taylor series $T_f(x)$ and $T_g(x)$ respectively. Lets say $f(x)$ is order $O(x^{n_1})$ and $T_f(x)$ has error of ...
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1answer
23 views

Numerical analysis: what is the error term for the rule…?

The question goes: derive the error term for the rule $phi$ to approximate the third derivative of f(a). I have attached a screenshot I understand how to take the Taylor series in the hint, but the ...
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0answers
53 views

Looking analytically if one formula is better than another one

I'm studying errors in functions on numerical methods. On my notes, I've written the Heron's Formula: Let $a\geq b\geq c$:$$A=\sqrt{p(p-a)(p-b)(p-c)}\ \ ,$$ where $$p=\frac{a+b+c}{2}.$$ This formula ...
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1answer
30 views

Magnitude of floating point rounding errors in operations

I'm doing the following exercise: Consider $$ f(x)= \begin{cases} \displaystyle\frac{1-(\cos(x))^3}{x^2}, \quad if\ x\neq 0\\ \displaystyle\frac{3}{2}, \quad if\ x= 0\\ \end{cases} $$ Calculate $f(...
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1answer
28 views

The area of a triangle ABC is given by S = 0.5abSin(C) calculate the max percentage error of S

$S=0.5abSin(c)$ calculate the maximum percentage error of S when a,b and c have a percentage error within 1% and c = 45 degrees. I do not understand where I went wrong the answer is supposed to 2.8%. ...
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0answers
18 views

Error propagation: add errors in quadrature, or use a weighted standard deviation?

I have a measurement $x$ with a known uncertainty $\sigma_m$. I have a black box that can take an error-free measurement $x$ and produce a value $y$ with a known uncertainty $\sigma_{b}$ (which is ...
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0answers
39 views

Find the error in the Maclaurin series for $\ln\left(\frac{1+x}{1-x}\right)$.

I have already that the series is, $$\ln\left(\frac{1+x}{1-x}\right)\approx 2\left(x+\frac{x^3}{3}+\frac{x^5}{5}+...+\frac{x^{2n+1}}{2n+1}\right).$$ The remainder is equal to, $$f^{n+1}(c)\frac{(x)^{...
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15 views

Inverse Propagation of Uncertainty

Can any one help me find a reference under the title of " Inverse Propagation of Uncertainty". I am starting a research on this topic after I had studied the forward propagation of uncertainty. ...
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2answers
33 views

Operations with truncation error

Assume I'm given an equation $a\frac{\partial p}{\partial x}+b\frac{\partial q}{\partial x}$. I approximate $\frac{\partial p}{\partial x}$ using a truncated taylor series to first order, such that ...
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1answer
114 views

Accuracy of GPS bearing, given two locations and their accuracies

For an Android app I need to determine the accuracy of the bearing returned by the GPS. Android supplies the following data, among others: The GPS position The accuracy of the GPS position, ...
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0answers
24 views

Composite Spherical Harmonics expansion and error propagation

Let's assume that $f$, $g$ and $t$ are three functions defined over the surface of a sphere. In particular $f$ is defined using $g$, $t$ and the integral operator as follows: $$f(\omega)=\int_{\...
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0answers
17 views

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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0answers
21 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
25 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
36 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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0answers
31 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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0answers
66 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
73 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
50 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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0answers
13 views

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 $T_n(...
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19 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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0answers
14 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
31 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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0answers
41 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 ...
0
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1answer
45 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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0answers
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
137 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 ...
0
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1answer
103 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 ...