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Questions tagged [truncation-error]

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Truncation error when applying a finite difference scheme to solve $u_x +Au_t = 0$

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{...
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Bounding the error in a solution, to an IVP, produced by RK4.

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 ...
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23 views

What is the formula for the local truncation error in RK4?

The local truncation error of a one-step ODE solver is defined to be $$e_{i+1} = \lvert y(t_{i+1}) - \tilde{y}_{i+1}\rvert,$$ the absolute value of the difference between the correct solution of the "...
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1answer
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Questions about 0.999… equals 1 [closed]

Being 0.999... = 1, I expect that they have the same behaviour when applying the same algorithm/operation, but: If we define >, <, =, as checking digit by digit two number, we have that 0 < 1 ...
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Leading term of the truncation error and the order (with respect to the length Δy)

currently I am studying for my numerics exam. While studying I encountered a problem, which I couldn't solve. I hope you guys can help me as me test will be in two days. For the function φ=φ(x, y) ...
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23 views

Error expansion for Trapezoidal rule

From my lecture slides, the error expansion for the trapezoidal rule was stated as (even powers of h only) but from the Taylor series expansion of the exact and trapezoidal rule I can't ...
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1answer
35 views

How can I be sure that a partial sum is accurate to some number of decimal places?

In a homework assignment, I am asked how many terms of a series are needed to obtain four decimal places (chopped) of accuracy. There are certain tricks that I am supposed to use, where I can ...
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2answers
125 views

Deriving the central Euler method and intuition

My professor (Dutch) asked us to determine, among other things, the truncation error of the central Euler method. First of all, this is probably not the correct term, since there are very few results ...
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1answer
43 views

Finding error in a an approximation

We want to see the total error in approximating $$ f'(x) \approx \frac{ f(x+h)-f(x) }{h} $$ where $f: R \to R$ is differentiable. We can find $\theta \in [x,x+h]$ by Taylor's to that ...
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1answer
44 views

Local error per unit step

The solution of the ODE $$ y' = f(t,y)$$ is being seeked. Let $u_{m}$ be the numerical solution of a one step method and $y(t_m)$ its true solution. The local error $e_{loc} $ is then defined as $$...
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Show that Local Truncation Error is not $O(h^3)$ for any choice of constants

This is one of the exercise questions in the book Numerical Analysis by Richard L.Burden Show that the difference method $$y_0 = \alpha \\ y_{i+1} = y_i + a_1 f(t_i,y_i)+a_2 f(t_i+\alpha_2, y_i+\...
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1answer
95 views

Improved sieve for primes and prime twins?

Suppose we want to estimate the number of primes between $x$ and its square root, say for example between $10$ and $100$ with a sieve. There are $90 $ numbers so we estimate : $\pi(10,100) = 90(1-1/...
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1answer
86 views

Proof by using Taylor's Remainder Term: Truncation Error $=\frac{(b-a)h^2}{12} \max{|f''(z)|}$ for Trapezoidal rule of integration

I am trying to derive Truncation Error $=\frac{(b-a)h^2}{12} \max{|f''(z)|}$ for Trapezoidal rule of integration by using Taylor's Remainder term My Approach: In trapezoidal rule of integration, ...
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Testing convergence rates of numerical solution with no known solution

I am trying to test the accuracy of my code to a PDE with no numerical solution. I am using a Backwards time centred space finite difference method. I am calculating the error using; $\frac{u_{4h} - ...
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Truncation of infinite summation involving a probability mass function

Suppose we want to compute an infinite sum of the form $S := \sum_{x=0}^\infty f(x)p(x)$ within a tolerance error $\epsilon$, where $f : \mathcal{X} \to \mathbb{R}^+$ is some functional of interest ...
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Comparing Numerical Methods for Differential Equations

Why the 3-step Adams Moulton method is better than the 2-step Adams Moulton method? Noted that the local truncation error of 3-step Adams Moulton method is $O(h^4)$, while the local truncation ...
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Obtaining an upper bound on transition probabilities by truncation

Consider a two-dimensional Markov chain. Let's call the first dimension "Level" and the second dimension "Phase". The state space is $(\ell, p)$ such that $\ell \geq 0$ and $0 \leq p \leq h$. The ...
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1answer
63 views

Accuracy of LMM $y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}kf_{n+2}$

I am trying to determine how to find the accuracy of a LMM. Specifically the BDF2 method, $$y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}kf_{n+2}$$ for solving the problem $y'=f(y,x),y(0)=n$ ...
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About decimals and exponents

I was reading about the speed of convergence of iterative methods. Then I came across the following. How is it inferred here that "At least 126 terms are needed to ensure this accuracy for the ...
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One step method truncation error

I have this math problem. immage here https://imgur.com/a/FRj99re . The problem that I have is with the $f(t_{n-1},u_{n-1})$ at the end. What I tried to do is do a taylor expansion and then write all ...
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Local truncation error of Crank-Nicolson for PDE $u_t+au_x = 0$

Exercise 4: The Crank-Nicolson scheme for $u_t + a u_x = 0$ is given by $$ \frac{U_{j,n+1}-U_{j,n}}{\Delta t} + \frac{a}{2}\frac{D_xU_{j,n}}{2\Delta x} + \frac{a}{2}\frac{D_xU_{j,n+1}}{2\Delta x} =...
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Order of accuracy of least squares for computing gradients

This is more of a numerical methods question. Hopefully, this is the right stackexchange board to post this question. I am wondering if anyone is aware of the order of accuracy of using least ...
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208 views

Gaussian Quadrature Error Estimate

According to Chapter 5 of Numerical Methods and Software by Kahaner, et al. (1989), it can be shown that the error associated with Gaussian quadrature is $\displaystyle\int_a^b f(x)\,dx - \sum_{i=1}^...
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196 views

Local Truncation Error of the Midpoint Method.

I'm looking at a past paper and have been asked to show that the Midpoint method: $$w_{i+1}=w_{i-1}+2hf(t_{i},w_{i})$$ has a local second order truncation error. with the expansions: $$W_{i+1}= y + ...
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What are methods for correction of experimental measurements?

I have experimental data series collected from sensors, and I have some problems because the data is obviously digitally distorted. I don't know if it is a problem of rounding, or quantization, or ...
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Prove that $r_{0,n}$ and $r_{1,n}$ are convex on $(-\infty,0)$.

Prove that the following two functions are convex on $(-\infty,0)$: \begin{align} r_{0,n}(x)&=\sum_{i=0}^\infty \frac{x^i}{(n+i)!} & r_{1,n}(x)&=\sum_{i=1}^\infty \frac{x^i}{(n+i)!} &...
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1answer
243 views

Gaussian Quadrature and Error $O(h^4)$

I'm having a bit of confusion with a problem I'm trying to solve regarding the error. Given that the error of the second order composite Gaussian quadrature method should be $O(h^4)$, how am I able to ...
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307 views

Local truncation error for trapezoidal rule

Find the local truncation error for the Trapezoidal rule, and hence find the order of the method. What do you expect would happen to the local errors if we were to halve the step size h used ? Explain ...
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Prove that Runge Kutta Method (RK4) is of Order 4

Please somebody help me, recently we have been studying numerical methods for solving ODEs and we went over proofs for the Euler method being order 1 and Huen’s method being order 2. But our lecturer ...
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1answer
58 views

Value of $\log_e{1.2}$ correct to $7$ decimal places

My aim is to find the value of $\log_e 1.2$ correct to $7$ decimal places using the Taylor's series $$\log_e(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$$ For $7$ decimal places accuracy, the absolute ...
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1answer
290 views

Truncated binomial has limit?

When $n$ goes to infinity through the odd numbers, $n=1,3,5,7,\dots$ $$\dfrac{\sum\limits_{k=0}^{\frac{n-1}{2}} \binom{n}{k}(1+(\frac{a}{n})^{0.5})^k}{(2+(\frac{a}{n})^{0.5})^n}$$ seems to have a ...
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71 views

The order of accuracy when finding the LTE

When finding the Local Truncation Error (LTE) of a Linear multistep method (LMM), I'm aware how to taylor expand the expression, multiply by $1/H$, and then simplify as shown below: Now my question ...
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1answer
111 views

Error in numerical integration technique

I am reading about numerical integration techniques and the error in the approximations found. My notes give an example using the constant rule (I have read elsewhere it is called the rectangular ...
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336 views

Order of the local truncation error of Modified Euler (time integration)

I've got quite a long question. I would really appreciate it if you could help me out, because I'm stuck at the moment. I need to find the order of the local truncation error of Modified Euler, ...
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Error analysis of solving a linear system

Please enlighten me on the following issue. Let's say I'm supposed to numerically solve a linear system Ax = b. But since the space is too large I can only work on a truncated version of b, call it b'....
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Truncation error and the second centered difference approximation $\frac{d^2u}{dx^2}$ at $x = x_{j}$ ?.

The second centered difference approximation $\frac{d^2u}{dx^2}$ at $x = x_{j}$?. By expanding the terms $u(x_{j}+h)$ and $u(x_{j} - h)$ about the point $x_{j}$ with a Taylor series,we need to prove ...