Questions tagged [accuracy]

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25 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
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1answer
46 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
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1answer
108 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 ...
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0answers
13 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
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1answer
64 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 ...
2
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1answer
91 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 ...
2
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1answer
84 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
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1answer
200 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
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2answers
461 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: ...
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0answers
51 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
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1answer
85 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}\...
6
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1answer
131 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 ...
6
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1answer
118 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 ...
4
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2answers
118 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
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3answers
128 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 ...
2
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0answers
45 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 ...
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0answers
96 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 $n$ ...
15
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2answers
731 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
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2answers
349 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/...
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0answers
301 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
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2answers
353 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 ...
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2answers
67 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: $$\...
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1answer
73 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{\...
7
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2answers
3k 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.
3
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1answer
167 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 ...
0
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1answer
310 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 ...
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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 ...
4
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2answers
279 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 ...
5
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1answer
167 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: $$ ...
1
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1answer
162 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{\...
7
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1answer
471 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 ...
2
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1answer
2k 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
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1answer
4k 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 ...
3
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0answers
175 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
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3answers
195 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
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1answer
342 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
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0answers
167 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 ...
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0answers
335 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
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0answers
75 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
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1answer
36 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 ...
3
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0answers
166 views

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 ...
11
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1answer
244 views

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....
10
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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)\...
8
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1answer
439 views

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 ...
11
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2answers
828 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 ...
10
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2answers
1k views

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 ...
25
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5answers
639 views

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 ...
9
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2answers
944 views

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 $t = \frac1{x}$ ...
23
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4answers
1k views

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 ...