Questions tagged [accuracy]
The accuracy tag has no usage guidance.
57
questions
5
votes
3answers
139 views
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 ...
1
vote
0answers
45 views
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 ...
1
vote
1answer
106 views
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 ...
2
votes
2answers
138 views
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
$$
\frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
7
votes
1answer
275 views
Accurate and efficient computation of the inverse Langevin function
The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ occurs in computations related to elastomers and paramagnetic materials. It is easily computed accurately and with high ...
16
votes
1answer
3k views
How can I avoid catastrophic cancellation?
I have the following formula that I need to rewrite in order to avoid catastrophic cancellation.
$$y =\sqrt{\frac{1}{2}\left(1-\sqrt{1-x^{2}}\right)}$$
As $x$ becomes smaller, $\sqrt{1-x^{2}}$ ...
3
votes
1answer
141 views
How to get a more accurate cancelation
I shall try to get to the point, so let me know if there is something left and you need more details.
I am solving a couple of equations that are not coupled explicitly, but their corresponding ...
5
votes
1answer
151 views
Accurate computation of Gauss-Kuzmin entropy
The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as
$$
P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
2
votes
1answer
211 views
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 ...
3
votes
0answers
140 views
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 ...
0
votes
1answer
49 views
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 ...
5
votes
1answer
185 views
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 ...
0
votes
0answers
18 views
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-...
1
vote
1answer
89 views
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 ...
7
votes
1answer
174 views
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 ...
9
votes
2answers
6k views
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.
2
votes
1answer
90 views
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 ...
6
votes
1answer
213 views
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 ...
6
votes
2answers
1k views
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:
...
1
vote
0answers
52 views
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 = ...
2
votes
1answer
102 views
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:
$$\frac{d\rho_j}{dx}\approx\frac{2-\eta}{3}\frac{\rho_{j+1/2}-\rho_{j-1/2}}{\Delta x}+\frac{1+\eta}{3}\...
4
votes
2answers
238 views
Intervals where the sign of a polynomial can be computed reliably
This is a follow-up of a previous question.
Let $p$ be a polynomial with floating-point coefficients.
Is there a method for finding intervals where evaluating $p$ in floating-point arithmetic always ...
6
votes
1answer
151 views
Does the IEEE-754 standard mandate that exp2 is rounded correctly?
The IEEE Standard for Floating-Point
Arithmetic section "9.2 Recommended correctly rounded functions" lists functions that are recommended (but not required) by a language standard to provide, among ...
6
votes
3answers
149 views
Accurate evaluation of the sign of a polynomial
Let $p$ be a polynomial with floating-point coefficients and let $a$ be a floating-point value.
Is there a method for accurately evaluating the sign of $p(a)$ in floating-point arithmetic?
I don't ...
1
vote
1answer
173 views
Debugging an implemented numerical method: which term gives the drop in accuracy?
I have a numerical scheme to solve an hyperbolic system of equations of this type (this a more simple version just for clarity):
$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\...
2
votes
0answers
49 views
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 ...
3
votes
0answers
107 views
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 $n$ ...
17
votes
2answers
2k views
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 ...
2
votes
2answers
462 views
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/...
5
votes
0answers
536 views
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 :)
...
4
votes
2answers
476 views
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 ...
0
votes
2answers
73 views
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:
$$\...
0
votes
1answer
90 views
2nd order accurate finite difference method variable material properties near boundary
I'm aware that a 2nd order accurate finite-difference method using variable properties for central differencing can be written in a finite-volume type way:
$$ \nabla \bullet (k \nabla f) = \frac{\...
10
votes
4answers
3k views
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:
$$
F_m(t) = \int_0^1 u^{2m} e^{-tu^2} d u
= {\gamma(m+{1\over 2}, t)\...
3
votes
1answer
179 views
Achieving high relative accuracy (vs. absolute accuracy) using spectral methods
Problem
I have a PDE that I'm trying to solve with spectral methods. The solution $y$ is always positive, and decays as $y \propto e^{-ax}$ for large $x$. The domain is $[0, \infty)$. (There are ...
9
votes
0answers
437 views
Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy
I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states:
$$ (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1} $$
I wish to perform small ...
4
votes
2answers
484 views
Accuracy of numerical methods in finding eigenvalues
I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form:
$$ y'' + ay' + \lambda^2y = 0, $$
and the boundary condiations ...
0
votes
1answer
530 views
Order of accuracy of DGFEM or FEM
I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
0
votes
0answers
47 views
Order of accuracy of linearised vs non-linear system
Does the order of accuracy of a combination of schemes applied to solve a system of non-linear equations, match those of the same schemes applied to the linearised version of the system?
In other ...
5
votes
1answer
200 views
Accuracy of finite differences
I'm currently reading this paper, and I'm a bit confused about how they compute the accuracy of their algorithm.
In the aforementioned paper, they investigate the regularized long-wave equation:
$$
...
7
votes
1answer
518 views
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 ...
3
votes
1answer
5k views
Why a finite difference scheme would give second order of accuracy in norm L2 but 1.5 with L1 (while 1 with Linf)?
My finite difference scheme for the 2D Euler equations is second order accurate in theory, since all the terms are second order accurate, with the advective terms being third order. So I expect a rate ...
2
votes
1answer
4k views
Correct way of computing norm $L_2$ for a finite difference scheme
I am computing the rate of convergence of my finite difference scheme in norm $L_2$. Which is the correct way to compute it? This:
\begin{align}
L_2 &= \frac{1}{N}\sqrt{\sum_{j=1}^N(u^{...
3
votes
0answers
212 views
Computing accuracy of my finite difference scheme for uniform grid on a non-uniform grid
I have a ADI finite difference scheme for the 2D Navier-Stokes equations that uses a second order accurate (central) approximation for the advective terms. I am ignoring the diffusive terms for now. I ...
2
votes
3answers
217 views
Compute accuracy order as mesh gets refined?
I have implemented a FVM code and now I need to plot the accuracy of the method as the mesh gets refined. Having a very fine mesh, my idea is to compare what is the error between the coarser and fine ...
4
votes
1answer
370 views
Finite differences vs. elements: Accuracy and implementation
I am trying to solve the 2D Poisson equation numerically:
$ \frac{\partial ^2 \phi}{\partial x^2} + \frac{\partial ^2 \phi}{\partial y^2} = 1 $
with the Dirichlet boundary condition $\phi = 0$.
I ...
2
votes
0answers
183 views
boundary conditions with non-constant coefficients in cell centred finite volume method
Suppose am solving the heat conduction equation in 1d with
Dirichlet boundary conditions. The thermal conductivity $k$ is a non
constant function.
So $-(k(x)u'(x))' = f(x)$
The value of $k$ enters the ...
5
votes
0answers
79 views
Order of convergence of Scrodinger eq. with CN scheme
I'm trying to solve numerically the 1-dim time dependent Schrodinger equation using the Crank Nicolson scheme and the Thomas algorithm to solve the tridiagonal matrix.
The physical system consists of ...
3
votes
1answer
37 views
Known Algorithm to compute errors of given nature
I have three sets of data, measured by three different devices: A,B and C of air balloon whose fall is influenced by wind and ...
11
votes
2answers
1k views
How do you improve the accuracy of a finite difference method for finding the eigensystem of a singular linear ODE
I am attempting to solve an equation of the type:
$ \left( -\tfrac{\partial^2}{\partial x^2} - f\left(x\right) \right) \psi(x) = \lambda \psi(x) $
Where $f(x)$ has a simple pole at $0$, for the ...
