# Questions tagged [accuracy]

For questions about characterizing or improving the accuracy of a computational method.

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### Estimating the rate of convergence of Projected Gradient Descent on constrained polynomial objectives

I am estimating the order of convergence of Projected Gradient Descent (GD) on quadratic polynomials with random coefficients independently drawn from Uniform(-1,1) and bounded by a unit hypercube. I'...
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### numerical schemes for 1D PDE: for smaller grid size there is an increased roundoff error, larger size more truncation, so sweet spot in between?

I had a discussion with a colleague today. He claimed that usually for a general numerical scheme for solving a general 1D PDE, for smaller grid size there is an increased roundoff error because of ...
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### Unexpected result when summing sorted (and unsorted) positive floating point numbers

I am exploring Higham's excellent Accuracy and Stability of Numerical Algorithms and chapter 4 is dedicated to summation. So I decided to test the most basic thing. Summing positive random numbers ...
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### Is computing the 2-norm of a vector numerically stable?

Is computing the 2-norm of a vector $x$, computed by setting $\alpha = \max_i x_i$ and then computing $|\alpha| \cdot ||x / \alpha ||_2$ (e.g., as in this link) numerically stable? There are two parts ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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