Questions tagged [accuracy]
For questions about characterizing or improving the accuracy of a computational method.
65
questions
5
votes
2
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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 ...
- 185
0
votes
0
answers
36
views
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, ...
- 1,016
0
votes
1
answer
107
views
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 ...
- 1
4
votes
0
answers
78
views
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 ...
- 2,055
6
votes
1
answer
187
views
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:
$$\log \left( \sum_{i=1}^{\infty}{I_{\nu+i}(2\...
- 163
0
votes
1
answer
583
views
Intuition for relative error for vectors
I'm trying to understand the notion of relative error for vectors in $\mathbb{R}^n$, but it's not "clicking" somehow.
$$\operatorname{\varepsilon-rel}(x_\text{approx}, x) = \frac{||x_\text{...
- 345
1
vote
2
answers
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views
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 ...
- 235
3
votes
1
answer
196
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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 ...
- 131
5
votes
3
answers
212
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 ...
- 10k
1
vote
0
answers
74
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 ...
2
votes
1
answer
126
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 ...
- 101
17
votes
1
answer
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, $\sqrt{1-x^{2}}$ ...
- 273
3
votes
1
answer
154
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 ...
user33042
2
votes
2
answers
178
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
$$
\frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
- 2,055
5
votes
1
answer
156
views
Accurate computation of Gauss-Kuzmin entropy
The Gauss-Kuzmin distribution gives the probability of an integer appearing as a partial denominator in the continued fraction of a real number $x$ as
$$
P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
- 2,055
2
votes
1
answer
268
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 ...
- 203
3
votes
0
answers
151
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 ...
- 2,055
0
votes
1
answer
61
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 ...
6
votes
1
answer
249
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 ...
- 2,055
0
votes
0
answers
19
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-...
- 2,055
1
vote
1
answer
132
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 ...
- 255
8
votes
1
answer
502
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 ...
- 1,320
2
votes
1
answer
98
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 ...
- 123
6
votes
1
answer
243
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 ...
- 63
6
votes
2
answers
2k
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
0
answers
56
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 = ...
- 321
2
votes
1
answer
201
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}\...
- 219
7
votes
1
answer
314
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 ...
- 345
6
votes
1
answer
194
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 ...
- 2,900
4
votes
2
answers
325
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 ...
- 956
6
votes
3
answers
165
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 ...
- 956
2
votes
0
answers
52
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
0
answers
115
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$ ...
18
votes
2
answers
3k
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 ...
- 439
2
votes
2
answers
541
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
0
answers
680
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 :)
...
- 128
4
votes
2
answers
554
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 ...
Shaz
0
votes
2
answers
79
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 $$
And a matrix-free operator:
$$\...
- 609
0
votes
1
answer
103
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:
$$ \nabla \bullet (k \nabla f) = \frac{\...
- 609
11
votes
2
answers
7k
views
Which Runge-Kutta method is more accurate: Dormand-Prince or Cash-Karp?
I simply want to know whether the Dormand-Prince Numerical Method
or the Cash-Karp Numerical Method
is more accurate.
- 213
3
votes
1
answer
193
views
Achieving high relative accuracy (vs. absolute accuracy) using spectral methods
Problem
I have a PDE that I'm trying to solve with spectral methods. The solution $y$ is always positive, and decays as $y \propto e^{-ax}$ for large $x$. The domain is $[0, \infty)$. (There are ...
- 161
0
votes
1
answer
752
views
Order of accuracy of DGFEM or FEM
I know that it is possible to determine the theoretical order of accuracy (B) of numerical solutions in FVM (for instance for a steady problem in which only the central differencing scheme (CDS) is ...
- 301
0
votes
0
answers
49
views
Order of accuracy of linearised vs non-linear system
Does the order of accuracy of a combination of schemes applied to solve a system of non-linear equations, match those of the same schemes applied to the linearised version of the system?
In other ...
- 301
4
votes
2
answers
603
views
Accuracy of numerical methods in finding eigenvalues
I have a problem with assesing the accuracy of my numerical calculation. I have a 2nd order ODE. It is an eigenvalue problem of the form:
$$ y'' + ay' + \lambda^2y = 0, $$
and the boundary condiations ...
- 59
5
votes
1
answer
236
views
Accuracy of finite differences
I'm currently reading this paper, and I'm a bit confused about how they compute the accuracy of their algorithm.
In the aforementioned paper, they investigate the regularized long-wave equation:
$$
...
- 265
1
vote
1
answer
180
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{\...
- 435
7
votes
1
answer
578
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 ...
- 73
2
votes
1
answer
6k
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^{...
- 435
3
votes
1
answer
6k
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 ...
- 435
3
votes
0
answers
230
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 ...
- 435