Questions tagged [accuracy]
For questions about characterizing or improving the accuracy of a computational method.
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Is there software that can autogenerate numerically-accurate floating point C routines from symbolic formulae?
Given a real function of real variables, is there software available that can automatically generate numerically-accurate code to calculate the function over all inputs on a machine equipped with IEEE ...
- 353
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4
answers
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When is a high order method useful for computational fluid dynamics simulations?
Many numerical approaches to CFD can be extended to arbitrarily high order (for instance, discontinuous Galerkin methods, WENO methods, spectral differencing, etc.). How should I choose an ...
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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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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
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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.
- 233
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Are there Improved ways of computing $p \log(p)$?
Most math libraries have a number of versions of logarithm functions.
Most of the time we assume them to be perfect, but actually quite a lot of them just offer a certain number of digits of precision....
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2
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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 ...
- 1,155
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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:
$$
F_m(t) = \int_0^1 u^{2m} e^{-tu^2} d u
= {\gamma(m+{1\over 2}, t)\...
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Are 8 Gauss points required for second order hexahedral finite elements?
Is it possible to get second order accuracy for hexahedral finite elements with fewer than 8 Gauss points without introducing unphysical modes? A single central Gauss point introduces an unphysical ...
- 3,909
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numerical integration with possible division by 'zero'
I am trying to integrate
$$\int^1_0 t^{2n+2}\exp\left({\frac{\alpha r_0}{t}}\right)dt$$
which is a simple transformation of
$$\int^{\infty}_1 x^{2n}\exp(-\alpha r_0 x)dx$$
using $t = \frac1{x}$ ...
- 2,119
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Updating matrix diagonal with Woodbury matrix identity and maintaining numerical accuracy
I have a dense matrix A and its corresponding inverse $A^{-1}$. The Woodbury matrix identity states:
$$ (A + UCV)^{-1} = A^{-1} - A^{-1}U(C^{-1} + VA^{-1}U)^{-1}VA^{-1} $$
I wish to perform small ...
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votes
1
answer
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Accurate and efficient computation of the inverse Langevin function
The Langevin function $\mathcal{L}(x) = \mathrm{coth}(x) - \frac{1}{x}$ occurs in computations related to elastomers and paramagnetic materials. It is easily computed accurately and with high ...
- 1,825
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Demonstrating that the time step size is small enough in a code with automatic step size selection
I recently inherited a large body of legacy code that solves a very stiff, transient problem. I would like to demonstrate that the spatial and temporal step sizes are small enough that the ...
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What is the numerical difference between abs(z)^2 and z x z*, where z is a complex number
I am doing research in electromagnetics (a branch of physics) and I deal with complex numbers in my computing tasks (I use MATLAB). Let's say $z$ is a complex number. To calculate the square of the ...
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What is a good definition of "accuracy to N digits"?
Let's say I have two numbers $x$ and $y$ which I'd like to compare to see whether they are equal up to their first $N$ digits.
For instance $ 1.002 \approx 1.001$ and $1002 \approx 1001$ up to 3 ...
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Matrix multiplication accuracy Matlab vs Python
I am translating some Matlab code into Python and I having some problems regarding matrix multiplication accuracy. Assuming we have following data:
...
6
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1
answer
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Whittaker-Shannon interpolation: Accuracy dies with speedup; can it be fixed?
With a truncated Whitaker-Shannon series (cardinal series)
$$
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)}
$$
we can naively ...
- 2,125
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votes
1
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Is it possible to proof a-b+b = a for all double floating-point numbers?
I want to know whether the equation : a-b+b = a is always true for a, b belongs to double precision floating-point number and |a|>=|b|.
If the equation is true, how can I proof it?
If not, what ...
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General approach to infinite sums
My question is specific to algorithms and models of computation.
I would like to write code to evaluate the following expression quickly and accurately:
$$\log \left( \sum_{i=1}^{\infty}{I_{\nu+i}(2\...
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votes
3
answers
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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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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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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
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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
$$
P(a_k = k) = -\log_2\left(1 - \frac{1}{(k+1)^2}...
- 2,125
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Accuracy of finite differences
I'm currently reading this paper, and I'm a bit confused about how they compute the accuracy of their algorithm.
In the aforementioned paper, they investigate the regularized long-wave equation:
$$
...
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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 ...
- 11.1k
5
votes
0
answers
709
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 :)
...
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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 ...
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2
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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 ...
Shaz
4
votes
2
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 ...
- 966
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votes
1
answer
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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 ...
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2
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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 ...
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0
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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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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
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1
answer
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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
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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 ...
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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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votes
0
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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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0
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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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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 ...
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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 ...
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How to accurately decompose positive semidefinite matrix and use the lower triangular part in linear equations
I have $n$ arbitrary $p\times 1$ vectors $x_i$, and $p\times k$ matrices $A_i$, and $n$ $p \times p$ positive semidefinite matrices $S_i$, where some (often most) of the $S_i$'s are same (for example ...
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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^{...
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1
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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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1
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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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3
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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 ...
- 1,019
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2
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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
$$
\frac{1}{1 + \frac{1^2}{1 + \frac{2^2}{1 + \frac{3^2}{1+\...
- 2,125
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votes
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Precision loss in Matrix-Vector product when applying Finite-Difference scheme
I am applying a 6th order Finite-Difference differentiation scheme as seen in http://www.scholarpedia.org/article/Method_of_lines/example_implementation/dss006
There seems to be severe numerical/...
2
votes
1
answer
294
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Order of Accuracy Measurements on 1D Advection Methods
I am trying to learn about basics of computational fluid dynamics, at the moment on the simple example of linear advection in 1D.
I am am currently testing the theoretical predictions of the order of ...
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1
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Computational Fluid Dynamics: Question on a third-order accurate finite difference approximation
According to this paper the following finite difference approximation is third-order accurate:
$$\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}\...
- 219
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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 ...
