# Questions tagged [truncation-error]

This tag is for questions relating to truncation error, which is the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation.

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### Order of accuracy of $0$

I had to calculate the truncation error, but the error gave me $0$. Then the question asked what is the order of accuracy, however since the error is $0$ should it be infinity? We use this difference ...
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### Find the strict bound for the error due to the truncation of an asymptotic series

As you may know, the asymptotic relations $y(t) \sim x(t)$ as $t\to 0$ tells us that $$\lim_{t\to 0 } \frac{y(t)}{x(t)} = 1.$$ If $x(t)=\sum_i^\infty \epsilon^i z(t)$ is an asymptotic series, then ...
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### Relative error of a number in machine epsilon units

I came across an estimation of the relative error between two representations of the same number, one implemented in C++ and another one via a computer algebra program, that was in units of machine ...
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### Truncation and rounding error bound derivation for finite difference approximation to the first derivative

Finite difference approximation to the first derivative: $$f'(x)= \frac{f(x+h) - f(x)}{h}$$ Heath's book on Scientific computing Section 1.2.4, Example 1.3 says truncation error for the finite ...
1 vote
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### Stirling's approximation fractional error

Given Stirling's approximation $\ln N!$ is approximated by $N\ln N-N+\frac{1}{2}\ln N$. I want to calculate the fractional error that comes from neglecting the third term $\frac{1}{2}\ln N$ for $N=10$ ...
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### Order of accuracy for non-smooth solutions and non-smooth local truncation errors

I'm working with numerical methods for solving PDEs (Linear Advection/Euler equations with temporal and spatial discretisation) using finite difference/finite volume methods. In these simulations I ...
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### Bounding the error for the remainder of $\log(x)$

We are asked to bound the error given by the remainder of the Taylor series of $\log(x)$ about some point $a>0$. Using the remainder as: $$R_n = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - a)^{n+1}$$ and ...
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### Global Error Bound for Implicit Euler

I'm trying to derive an error bound for the approximate solution obtained with Backward Euler to the IVP $$\frac{dy}{dt} = f(y), \hspace{.75cm} y(0) = y_0,$$ where $f \in C^2(\mathbb{R})$ with ...
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### Calculating the truncation error for exponential function in complex plane

Having seen this question on Taylor approximation of complex exponential function, I am looking for a solution this problem and would be great if I also knew the name of the paper. It is about ...
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### Nummerical truncation on MATLAB - Increase of precision

It might be not suitable for the current forum, but it is math-related. Than, I wonder if there is a MATLAB user among us. I currently implement multible degrees of Runge-Kutta methods on mentioned ...
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### How do I avoid significant rounding error in evaluating $(\ln(x) - \sin(\pi x))(1-x)^{-1}$?

How do I avoid significant rounding error in evaluating $$\frac{\ln(x) - \sin(\pi x) }{1-x}$$ This function causes error as $x\to 1$. How can this be avoided? I tried using taylor's expansion but I ...
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I wonder if the truncation error that I derived in the following approximation really has order 2? If so I also wonder what happens to the $u_{xxxx}$ term since it does not cancel out? $$u^n_{j+2}-4u^... 0 votes 1 answer 113 views ### How to prove finite difference approximation has error of order \mathcal{O}(\Delta x^2) I'm asked to prove that the finite difference approximation$$u_{xx}(x_i) = \frac{u_{i+1} - 2u_i + u_{i-1}}{\Delta x^2}$$gives a discretization error of order \mathcal{O}(\Delta x^2). My attempt: ... 1 vote 1 answer 243 views ### maximum error when rounding off multiple times So I am aware that when you round to n decimal places accuracy, the maximum error is ~0.5 × 10^n~ But if I use the rounded result, and use multiply it by another un-rounded number and round the ... 1 vote 0 answers 146 views ### Truncation error and non self-starting Heun I've seen two different truncation formulas for the midpoint rule. A common one is h^3 \frac{ f''}{24}. Another, referred to as open Newton Cotes, is h^3 \frac{f''}{3}. The Newton Cotes ... 1 vote 1 answer 73 views ### Truncation error when applying a finite difference scheme to solve u_t +Au_x = 0 The wave equation in one space dimension is given as$$ u_t + Au_x = 0 $$where$$ u := \begin{bmatrix} v(x,\, t) \\ w(x,\, t) \end{bmatrix}, \quad A = \begin{bmatrix} 0 & -1 \\ -1 & 0 \end{...
What techniques exist for bounding the error in a solution, to an IVP, produced by RK4? The below problem is intended to contextualize the question. Problem The $x$, $y$ and $z$ axes of a coordiante ...