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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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Computing the relative error of two Runge Kutta Methods for Convergence Analysis

I am currently endeavoring to assess the relative error between the classical Runge-Kutta (RK4) method and another RK variant. I've opted to employ the Ordinary Differential Equation (ODE) governing a ...
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Why does the graph of the absolute error vs x for the three-point central difference formula have this shape?

Using the three-point central difference formula, $$f'(x) \approx \frac{f(x+h)-f(x-h)}{2h}$$ to approximate the $f'(x)$ of $f(x)= e^{-x} sin(x)$ for the interval $[0,15]$ using different step sizes ($...
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Why is this the expression for the rounding error in the three-point midpoint formula for the approximation of the first derivative?

so I understand that in using the three-point midpoint formula, $$f^{(1)}(x)= \frac{y_{+1}-y_{-1}}{2h} + \frac{e_{+1}-e_{-1}}{2h} - \frac{h^2}{6} f^{(3)}(\xi^*), \ \ \ \ \ \xi^* \in [a,b]$$ round-off ...
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Find the local truncation error (LTE) of $y'=e^{t-y}$ using Euler's Method

Find the local truncation error (LTE) of $y'=e^{t-y}$ using Euler's Method, knowing that $y(0)=1$, $t=[0,1]$, $h=0.25$ I know the local truncation error (LTE) introduced by the Euler method is given ...
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Estimation of the number of terms required to evaluate an infinite series function up to a given precision

Consider the following series function defined in terms of an infinite series of the form $$ f(x,y) = \sum_{m=0}^\infty f_m(x,y) , $$ where $$ f_m(x,y) = (2m+2)(2m+3)(2m+5) \, \Gamma \left( m + 7/2 \...
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Obtaining the compound Simpson's rule from simple SR.

in this book or here on the page $58$ I was trying to obtain the compound Simpson's rule $(2.2.4)$ from the Simple Simpson's approximation as in $(2.2.2)$. I understand that the compound rule on $-1,.....
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Total error for the finite difference approximation

I was reading Scientific Computing, An Introductory Survey, by Michael Heath. In the Example 1.11, he made a Finite Difference Aproximation, with the usual approximation : $f’(x)\neq \frac{f(x+h)-f(x)}...
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Algorithmic error computing $\frac{e^x -1}{x}$

It is well know that in order to calculate the algorithmic error of a function one can use backward analysis with using the visual representation of a graph with nodes the $i-$th step of the algorithm ...
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Strange results for the error of numerical integration: The Newton-Cotes rules

I was using the Maple software for the computation of the numerical integration error. I'm obtaining two strange properties which I'm unable to explain. First for items $(72)$ and $(73)$ below while ...
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Why can we assume that squaring makes things smaller in numerical methods and Taylor expansion?

If we square a number <1 it becomes smaller, but a number >1 becomes larger. 0.01 second is 100 milliseconds. 0.01^2=0.0001 (second^2) and 100^2=10000 (millisecond^2). So, the same quantity can ...
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Von Neumann stability for inhomogeneous PDE

I've got an inhomogeneous PDE of the following form: $$\alpha\partial^2_xu+\partial_tu=f$$ with $\alpha<0$ and a source term $f$. I descretise $u$ according to $u_{m,n}=u(m\Delta t,n\Delta x)$ ($f$ ...
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How is error in numerical methods defined (Big O notation, O(h))?

I am trying to understand how the numerical method error is measured in O(h). In my understanding, the big O notation is depend on the dominant term of equation, for example f(x)=x^2+x, the big O ...
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Local truncation error vs Global truncation error

I know that global truncation error is proportional to $h^p$ while local truncation error is proportional to $h^{(p+1)}$, where $h$ is the step size. But where does this relationship come from and how ...
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relative error of division and subtraction

My task is to calculate the relative error of (1) $\frac{1}{n} - \frac{1}{n+1}$ (2) $\frac{1}{n(n+1)}$ with the definition $|\frac{z-rd(z)}{z}|$ for the relative error and $rd(x+y)=(x+y)(1+e)$ with $|...
Alina Grünaug's user avatar
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Computing the error of an approximation

I had an idea to approximate a function $f(t)=\frac{v(t)}{k}$ where I now want to find out how big the error is. My idea was to approximate the function by a polygonal chain: The time $t$ gets parted ...
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How to find local truncation error in Forward Euler Method without knowing the actual value

In my high school project, I am solving differential equations using the forward Euler numerical method and this is because the equations were too hard to solve analytically. To find the local error ...
Gaussian 123's user avatar
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1 answer
108 views

Where can I find "detailed" error analysis of modified Euler's methods?

I'm studying the local truncation error of each Heun's, Midpoint, and Ralston's methods. For Heun's method, I found a material in here. However, I don't get how the following is derived. $$ f(t+h, y(t+...
Minsik Seo's user avatar
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Are there any ways to increase the precision in MATLAB without built in functions?

I am a beginner learning about MATLAB scientific computation, floating point numbers, and numerical error. When I am using a very small $x$ value for some equations, such as $y(x) = (\exp(x)-1-x)/x^2$,...
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High order Runge Kutta methods and global error

I understand that if we use a high-order Runge Kutta method like RK4 the rate of convergence of the error as the stepsize $h$ tends to $0$ should be of order $h^4$. Does that necessarily imply that ...
Rudinberry's user avatar
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Bound for the truncation error

I am trying to find the minimal bound for the truncation error of the following problem https://i.sstatic.net/eNMmc.jpg
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Truncation error - subtraction

i try to eliminate truncation error with subtraction in calculating root of function where$$x_1=\frac{-b-\sqrt{b^2-4c}}{2},\quad b<0,\quad 0<c\ll 1.$$Does someone have an idea of how to change ...
carl_799's user avatar
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Is relative error a useful measure for quantification of error? Or is it just an approximation for perhaps a better error quantification method?

Consider the following situation. There is a partial differential equation (PDE) that has a known analytic solution, and one is solving that equation numerically on a computer. Assuming an ...
Joseph Robert Jepson's user avatar
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Show that a linear multistep method is of rank $p$ $\Leftrightarrow$ its local trunctation error is equal to 0 for any $x \in \pi_p$

We have the following ODE: $$x'(t) = f(t,x) \quad \wedge \quad x(t_0) = x_0$$ Let's say we have a linear multistep ($q$ steps) method: $$\sum_{j=0}^q \alpha_j x_{k+j} = h \sum_{j=0}^q \beta_j f_{k+j}$...
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Finding the Local Truncation Error for the Explicit Euler Scheme

I need to demonstrate finding the LTE for the Explicit Euler scheme when i) $\mu=\frac{1}{6}$ and ii) $\mu\neq\frac{1}{6}$. I have been looking for a reference text/video and I looked at the ...
DoonieCaan's user avatar
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191 views

Truncation error in solving Differential Equation

Tomorrow I've got a test for my Numerical Analysis course and there's this one type of question I cant seem to understand. An example: We have the Differential Equation $u'(t) = f(u(t))$ and we're ...
Jord van Eldik's user avatar
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Can the left side of the equation be truncated to the first term?

So, I am deriving a high-order compact finite difference scheme, and got into the equation below: \begin{multline} \delta t^2 \{1 + \dfrac{h^2}{12} (\delta x^2 + \delta y^2 + \delta z^2) + ...
Vitorotri's user avatar
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1 answer
128 views

Computation of global error bound for Euler's formula [closed]

I am trying to calculate the global error bound for Euler's method, but I am having trouble. I am given the formula $|y(t_{i}) - u_{i}| \leq \frac{1}{L}(\frac{hM}{2} + \frac{\delta}{h})(e^{L(t_{i}-a)}-...
ImNotGoodAtDis's user avatar
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Consistency finite differences vs finite elements

Whether a discretisation is consistent (and the order of consistency) in the FDM setting is defined by the the truncation term, e.g.: $$\partial_{xx} = \frac{1}{h^2}\begin{bmatrix}1 & -2 & 1\...
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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 ...
Jorge Correa Merlino's user avatar
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1 answer
933 views

Is the Global Error of the Taylor Series Second Order Method $O(h^2)$?

To solve an initial value problem $\frac{dx}{dt}=f(t,x(t))$ with $t\in[t_0,t_N]$, $x(t_0)=x_0$ we can use a Taylor series method of second order, with step size $h$: $$x_{i+1}=x_i+hf(t_i,x_i)+\frac{h^...
maths54321's user avatar
1 vote
2 answers
189 views

Are expressions including radicals particularly difficult to integrate numerically?

I noticed some unusual result errors when numerically integrating using a TI-84 Plus CE. This answer explains that the TI-84 uses a 15-point Gauss-Kronrod method with a maximum error of $10^{-5}$ to ...
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Deriving bounds for the remainder of asymptotic series

Given a sequence $(g_m)_{m\geq 1}$ with the asymptotic series $$g_m=\frac{a_1}{m} + \frac{a_2}{m^2} + \frac{R_m}{m^3}$$ where the numbers $a_1$, $a_2$ are known and we know $0<R_m<0.35$. Given $...
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How does truncating a series affect "upstream" values in the series?

It is known that truncating the Gregory’s series to 5,000,000 terms leads to an "almost but not quite" value for π: $$ \pi=4 \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2 k-1}=4(1-1 / 3+1 / 5-1 / ...
Cybernetic's user avatar
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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 ...
hal's user avatar
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1 answer
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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 ...
Argha Chakraborty's user avatar
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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How to calculate the centered approximation of ${\frac{{df}}{{dt}} = - \frac{f}{\tau }}$?

In the textbook of An Introduction to Computational Methods in Hydrodynamics by Willy Benz. 2.2 Differential Equations. I know how to derive the general formula above $$ f\left( {{x_0} + \Delta x} \...
Nor.Z's user avatar
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1 answer
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Local Truncation Error on Improved Euler's Method

I need to reach the conclusion that $$\tau(x,h) = \frac{\phi(x + h) - \phi(x)}{h} - \frac{1}{2}[f(x,\phi(x)) + f(x+h,\phi(x) + h f(x,\phi(x)))] = O(h^2)$$ So I tried using Taylor expansions and ...
Felipe Machado's user avatar
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Numerical Methods - Relative Error

In our lectures, we've been looking at relative error and in our recent problem sheet we were asked to prove the following: Let $\hat{x} \neq 0$ be an approximation of a non-zero quantity $x$. ...
user839136's user avatar
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What is the error of FEM-like spline discretizations?

Let's have $f(x)$ a nice*, scalar valued function. It is approximated by $$ f(x)\approx g(x)=\sum_i f(x_i) N_i(x), $$ where $x_i = x_0+i \Delta x$ are uniformly spaced points, $N_i(x) = N(x-x_i)$, ...
shinjin's user avatar
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Find the local truncation error of the exponential-Euler approximation.

The Problem I have a differential equation of the following form $$\dot{y} = A(y)y+B(y)$$ Provided that $A$ does not depend on $y$ (or depends very weakly on $y$), I can approximate by using the ...
kostas1335's user avatar
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Truncate a sum in order to speed up an algorithm (Big O)

Let $k\leq n$, I am looking to truncate this sum (with a Big O or Little o) in order to compute just few terms $$ \sum^k_{i=1} (k-(i-1))^{p}\sqrt{y_i} $$ where $y=(y_i)_{i\leq n}$ is a bounded ...
Al Bundy's user avatar
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1 answer
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Numerical analysis - Error analysis for the second order Taylor method

I'm trying to prove the following result for the second order Taylor method: $f$ is continuous and satisfies a Lipschitz condition with constant L on $D=\{(t,y)\mid a\leq t \leq b, -\infty<y<\...
Kim's user avatar
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Simple loss of significance question.

I know that (because of the finiteness of computer arithmetic) for large values of $x$, the function $$f(x)=\log(x+1)-\log(x)$$ will be subtracting two very close values. Can I get around this by re-...
Alex D's user avatar
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3 answers
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Forward and centered finite difference give same error plot: why?

Let us consider the following standard approximations of the first derivative of a function $$FD = \frac{f(x+h)-f(x)}{h}$$ $$CFD = \frac{f(x+h)-f(x-h)}{2h}$$ The first is first order accurate, while ...
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What factorial number to get a good approximation of the matrix exponential?

$$e^X = \sum_{k=0}^N{1 \over k!}X^k$$ Assume that $X \in \Re^{nxn}$ is random matrix. What number $N$ should I use to get a good accuracy compared if $N = \infty$? Is this possible to measure?
euraad's user avatar
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Relative error from truncation error in Taylor series expansion

Given the function $f(x)=\sin x$, please expand it about $x=0$ using Taylor series and truncate the series to $n=6$ terms then find the relative error at $x=\pi/4$ due to truncation found?
Robert Mdee's user avatar
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2 answers
44 views

Closely minimize error bound $\frac{4 \pi \exp(\cosh(a))}{\exp(a N) -1}$

I'm trying to minimize an error bound $$\frac{4 \pi \exp(\cosh(a))}{\exp(a N) -1},$$ where $N$ is the step size for the trapezoidal rule and $-a < Im < a, a > 0$ is a strip bound which may ...
Peter Manfred's user avatar
2 votes
1 answer
199 views

forward - backward differencing = central differencing

From Taylor series, we can derive: Forward Differencing Formula: $$ f'(x_{i}) = \frac{f(x_{i+1}) - f(x_{i})}{h} - \frac{f''(x_{i})h}{2!} $$ $$\tag 1 f'(x) \approx \dfrac{f(x+h)-f(x)}{h}$$ (1) ...
x89's user avatar
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2 answers
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Approximate the integral $\int_0^{0.5}{x^2e^{x^2}}dx$ correct to four decimal places using a Maclaurin series.

I got $$\int_0^{0.5}{\sum_0^\infty}\frac{x^{2n+2}}{n!}dx$$ for the taylor series representation, but I'm not sure what to do next. Do I use 0 and 0.5 as bounds for z for the Lagrange Error Bound? And ...
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