Questions tagged [accuracy]
For questions about characterizing or improving the accuracy of a computational method.
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Accurate and efficient computation of the inverse Langevin function
The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ occurs in computations related to elastomers and paramagnetic materials. It is easily computed accurately and with high ...
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votes
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Oscillation in non-linear porous flow solved by finite difference
I am trying to solve numerically the flow of a gas through a porous, spherically symmetric body. The non-dimensional equations read:
$$
\frac{\partial\rho}{\partial t}+\tau\frac{1}{r^2}\frac{\partial}{...
5
votes
2
answers
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What is the best method of computing $a^{(k)}/k!$?
I have the following expression
$$
\frac{a^{(k)}}{k!}
$$
where $a^{(k)}$ is the rising factorial. Is it better to evaluate it using floating-point arithmetic separately, that is, call a function that ...
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B-spline abscissa values for numerical stability
I am working with spline functions using the B-spline basis,
$$
f(x) = \sum_{i=1}^n c_i B_i(x;\mathbf{t})
$$
where the $B_i$ are cubic B-splines with a knot vector $\mathbf{t}$. For my application, ...
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What is the rationale of second-order finite volume discretization?
When it comes to a second-order accurate finite volume discretization of Navier-Stokes equations, which one of the two following rationales is adopted?
1- Second-order accuracy is a direct consequence ...
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1
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How to correctly plot Order of accuracy for different finite difference schemes
I have implemented Upwind, Lax, Lax-Wendroff, Leapfrog and macCormak method for the linear advection equation with Dirichlet boundary conditions. I am trying to create the order of accuracy plots for ...
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Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?
I simply want to know whether the Dormand-Prince Numerical Method
or the Cash-Karp Numerical Method
is more accurate.
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answers
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Numerical algorithms made stable by unums which are unstable on IEEE floats
For unums, there is good evidence (see figure 5) that accuracy is better than IEEE floats. (Note: I use the term "unum" broadly to refer to any of the various iterations and revisions to the ...
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votes
1
answer
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General approach to infinite sums
My question is specific to algorithms and models of computation.
I would like to write code to evaluate the following expression quickly and accurately:
$$\log \left( \sum_{i=1}^{\infty}{I_{\nu+i}(2\...
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votes
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answer
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Intuition for relative error for vectors
I'm trying to understand the notion of relative error for vectors in $\mathbb{R}^n$, but it's not "clicking" somehow.
$$\operatorname{\varepsilon-rel}(x_\text{approx}, x) = \frac{||x_\text{...
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How to find the optimum finite difference method for derivatives?
Related to: What are the negatives of using higher order finite diference schemes?
Problem: I have some discrete data of a trajectory $x_t$ with errors $\delta x_t$ of a physical system sampled at ...
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votes
3
answers
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Automatic finite differences
Given numbers $x, y \in \mathbb{R}$ where
$$\frac{|y-x|}{|x|}$$
is small, and code that implements the function $f$ with a sequence of arithmetic operations, I would like to compute to high accuracy ...
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vote
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Comparison between stability and accuracy of various Finite Difference schemes
I`m Analyzing the 1D convection equation (PDE) for stability, consistency, and accuracy.
I know Both upwind schemes (explicit and implicit) show better stability with a higher number of waves for ...
2
votes
1
answer
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Accuracy gap for apparently stable solution
I was reasoning about the behaviour of the methods I'm using for my simulation and I noticed that, considering $h_s$ as the timestep over which I have unstable solutions and $h_a$ as the timestep ...
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votes
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Accelerating convergence of a generalized continued fraction
I wish to compute
$$
\frac{1}{1 + \frac{1^3}{1 + \frac{2^3}{1 + \frac{3^3}{1+\cdots} } } }
$$
to high accuracy. To start, I tried computing
$$
\frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
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answer
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How can I avoid catastrophic cancellation?
I have the following formula that I need to rewrite in order to avoid catastrophic cancellation.
$$y =\sqrt{\frac{1}{2}\left(1-\sqrt{1-x^{2}}\right)}$$
As $x$ becomes smaller, $\sqrt{1-x^{2}}$ ...
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How to get a more accurate cancelation
I shall try to get to the point, so let me know if there is something left and you need more details.
I am solving a couple of equations that are not coupled explicitly, but their corresponding ...
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votes
1
answer
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Accurate computation of Gauss-Kuzmin entropy
The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as
$$
P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
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answer
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Order of Accuracy Measurements on 1D Advection Methods
I am trying to learn about basics of computational fluid dynamics, at the moment on the simple example of linear advection in 1D.
I am am currently testing the theoretical predictions of the order of ...
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Derivative of Whittaker-Shannon interpolant
Last time we looked at how to improve the accuracy of Whittaker-Shannon interpolation, where user njuffa demonstrated that judicious use of sin_pi could greatly ...
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Training accuracy improves but test set accuracy remains the same
I have built an ANN model with 5 hidden layers and 100 nodes in each layer to solve a multilabel classification problem. After the first run, I get a training accuracy of ~66% and a test set accuracy ...
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votes
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Whittaker-Shannon interpolation: Accuracy dies with speedup; can it be fixed?
With a truncated Whitaker-Shannon series (cardinal series)
$$
f(t) = \sum_{j = 0}^{n-1} y_{j} \frac{\sin\left(\pi( \frac{t-t_0}{h} -j)\right)}{\pi\left(\frac{t-t_0}{h}-j\right)}
$$
we can naively ...
- 1,825
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votes
0
answers
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Cost functions to judge time/memory/accuracy tradeoffs
I am working on an interesting algorithm: Its absolute error is exponential in a parameter $j \in \mathbb{N}$, and for a given $j$, I have complete freedom to choose between an $\mathcal{O}(1)$ time-...
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vote
1
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Adam Bashforth 4 method: how to determine starting values and stil keep the the order of accuracy
I am using an Adam Bashforth 4 method to solve an IVP problem so I need other numerical method to estimate the first 3 values. I am very much interested in finding a way to estimate the first 3 values ...
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votes
1
answer
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What is a good definition of "accuracy to N digits"?
Let's say I have two numbers $x$ and $y$ which I'd like to compare to see whether they are equal up to their first $N$ digits.
For instance $ 1.002 \approx 1.001$ and $1002 \approx 1001$ up to 3 ...
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votes
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answer
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When writing a research article, how to deal with deviations from theoretical expectations due to numerical errors?
We research the aerodynamics of the vehicle. Usually, there exists turbulence, which will cause the time-varying measurement of the force. The unsteady flow field determines we could not get a ...
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Is it possible to proof a-b+b = a for all double floating-point numbers?
I want to know whether the equation : a-b+b = a is always true for a, b belongs to double precision floating-point number and |a|>=|b|.
If the equation is true, how can I proof it?
If not, what ...
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Matrix multiplication accuracy Matlab vs Python
I am translating some Matlab code into Python and I having some problems regarding matrix multiplication accuracy. Assuming we have following data:
...
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Decrease in slope during convergence analysis
I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides.
I used 5 refinements:
$dx = dy = dz = ...
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votes
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answer
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Computational Fluid Dynamics: Question on a third-order accurate finite difference approximation
According to this paper the following finite difference approximation is third-order accurate:
$$\frac{d\rho_j}{dx}\approx\frac{2-\eta}{3}\frac{\rho_{j+1/2}-\rho_{j-1/2}}{\Delta x}+\frac{1+\eta}{3}\...
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votes
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answers
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Intervals where the sign of a polynomial can be computed reliably
This is a follow-up of a previous question.
Let $p$ be a polynomial with floating-point coefficients.
Is there a method for finding intervals where evaluating $p$ in floating-point arithmetic always ...
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votes
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answer
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Does the IEEE-754 standard mandate that exp2 is rounded correctly?
The IEEE Standard for Floating-Point
Arithmetic section "9.2 Recommended correctly rounded functions" lists functions that are recommended (but not required) by a language standard to provide, among ...
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Accurate evaluation of the sign of a polynomial
Let $p$ be a polynomial with floating-point coefficients and let $a$ be a floating-point value.
Is there a method for accurately evaluating the sign of $p(a)$ in floating-point arithmetic?
I don't ...
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vote
1
answer
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Debugging an implemented numerical method: which term gives the drop in accuracy?
I have a numerical scheme to solve an hyperbolic system of equations of this type (this a more simple version just for clarity):
$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\...
CommunityBot
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votes
0
answers
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Quick evaluation of floating point Absolute Error
I need to to find a quick and dirty way to estimate the absolute error introduced by a series of agebraic operations of IEEE single precision floating point numbers, a pessimistic result is ok.
The ...
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answers
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How can I evaluate the accuracy of my n-body simulation?
I am making an n-body simulation in python. There are many different methods to numerically solve the system of differential equations governing the gravitational interactions between the $n$ ...
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answers
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Practical example of why it is not good to invert a matrix
I am aware about that inverting a matrix to solve a linear system is not a good idea, since it is not as accurate and as efficient as directly solving the system or using LU, Cholesky or QR ...
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votes
2
answers
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Precision loss in Matrix-Vector product when applying Finite-Difference scheme
I am applying a 6th order Finite-Difference differentiation scheme as seen in http://www.scholarpedia.org/article/Method_of_lines/example_implementation/dss006
There seems to be severe numerical/...
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Order of accuracy of FVM discretization
I've recently got interested in CFD and started a small project by solving the radial Reynoldsequation. Why the Reynoldsequation? I recently encountered it through my studies and somehow got stuck :)
...
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Numerical differentiation of non-linear functions
In my efforts to write a code for a calculation I have encountered a problem of numerically differentiating a non-linear function at different points on a grid. I used the simple forward finite ...
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Accuracy between ill-conditioned matrix-free vs. matrix-based operators
As far as I know, precision errors become larger as the condition number of a matrix increases.
Consider a matrix-based operator:
$$A = \nabla \bullet k \nabla $$
And a matrix-free operator:
$$\...
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2nd order accurate finite difference method variable material properties near boundary
I'm aware that a 2nd order accurate finite-difference method using variable properties for central differencing can be written in a finite-volume type way:
$$ \nabla \bullet (k \nabla f) = \frac{\...
- 1,226
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votes
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answers
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Fast and accurate double precision implementation of incomplete gamma function
What is the state of the art way of implementing double precision special functions? I need the following integral:
$$
F_m(t) = \int_0^1 u^{2m} e^{-tu^2} d u
= {\gamma(m+{1\over 2}, t)\...
CommunityBot
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3
votes
1
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Achieving high relative accuracy (vs. absolute accuracy) using spectral methods
Problem
I have a PDE that I'm trying to solve with spectral methods. The solution $y$ is always positive, and decays as $y \propto e^{-ax}$ for large $x$. The domain is $[0, \infty)$. (There are ...
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Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy
I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states:
$$ (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1} $$
I wish to perform small ...
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Accuracy of numerical methods in finding eigenvalues
I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form:
$$ y'' + ay' + \lambda^2y = 0, $$
and the boundary condiations ...
CommunityBot
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Order of accuracy of DGFEM or FEM
I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
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votes
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Order of accuracy of linearised vs non-linear system
Does the order of accuracy of a combination of schemes applied to solve a system of non-linear equations, match those of the same schemes applied to the linearised version of the system?
In other ...
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votes
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answer
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Accuracy of finite differences
I'm currently reading this paper, and I'm a bit confused about how they compute the accuracy of their algorithm.
In the aforementioned paper, they investigate the regularized long-wave equation:
$$
...
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What is the numerical difference between abs(z)^2 and z x z*, where z is a complex number
I am doing research in electromagnetics (a branch of physics) and I deal with complex numbers in my computing tasks (I use MATLAB). Let's say $z$ is a complex number. To calculate the square of the ...
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