1
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1answer
66 views

How do I construct such a numerical method for solving ODE?

I am asked to expand $x(t+h)$ and $x(t+2h)$ around $t$ up to the rest term of the third order, find $A, B, C \in \mathbb R$ such that $$x'(t)=\frac{Ax(t)+Bx(t+h)+Cx(t+2h)}{h} + O(h^2)$$ and based on ...
1
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1answer
17 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
1
vote
1answer
27 views

Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
3
votes
1answer
75 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
0
votes
2answers
39 views

Taylor polynom, residual for symmetric values

When creating the taylor polynom for a $C^3$-function around a certain point i get the formula $f(z+h)=f(h)+hf'(z)+\frac{h^2}{2}f''(z) + \frac{h^3}{6}f'''(z) + R$ Now lets say I create the polynom ...
0
votes
0answers
58 views

How many terms to use in a Taylor series for local truncation error

So from my understanding for a finite difference approximation, you're supposed to expand the series "about" the point $x$, e.g., $$u(x+h) = u(x) + h \ u'(x) + ...
3
votes
0answers
94 views

Loss of Significance problems - Taylor Expansion

(2) This question addresses the notion of loss of significance. You are encouraged to revisit the Taylor series expansion that you have learned in calculus, as you will need to apply it here. Explain ...
3
votes
3answers
181 views

Find an accurate value of $f(x)=\sqrt{4x^2+x}-2x$ for large values of x. Calculate $\lim_{x\to\infty}f(x)$

My works: $x^2$ can be very large if x is large, thus the function has lose-of-significance error and we need to reformulate it. $$ ...
1
vote
1answer
67 views

Use Taylor polynomials with remainder term to evaluate the following limits $\large\frac{e^x-x-1}{x^2}$

My work: Since $\large e^x=\sum\limits_{j=0}^\infty \frac{x^j}{j!}$, then $\large\frac{e^x-x-1}{x^2}=\sum\limits_{j=2}^\infty \frac{x^{j-2}}{j!}=\sum\limits_{d=0}^\infty \frac{x^{d}}{(d+2)!}$. (Let ...
2
votes
0answers
193 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
0
votes
3answers
240 views

solve the initial value problem ,by Taylor's method of order $N=3$

solve the initial value problem ,by Taylor's method of order $N=3$ $y'(t)=ty(t)+(1-t)e^t,0\le t\le 2,y(0)=1$ with an accuracy of $5 \times10^{-3}$ first we consider the taylor expansion of $e^x$ $ ...
0
votes
0answers
82 views

Order of accuracy of the following approximation?

This is a computational analysis class, the answer shouldn't be complicated. Given the ODE: $$\begin{cases}y'=f(t,y)\\y(t^0)=y^0\end{cases}$$ What would be the order of accuracy (using Taylor ...
1
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1answer
334 views

Numerical analysis Taylor's method question: Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$.

Let $f(x)=\tan^{-1}(x)$ Let $P_n(x)$ be the $n$th Taylor polynomial for $f(x)$ about $x_0=0$ Find a value of $n$ necessary for $P_n(x)$ to approximate $f(x)$ within $10^{-6}$ on $[-0.5,0.5]$. Is ...
0
votes
1answer
57 views

Applying the Taylor series

For the initial value problem $\dot{y} = f(y), y(t_0) = y_0$, where $f(y)$ is smooth, we look at the discrete evolution $\Psi^t := y_1 = y_0 + h f((1-\Theta)y_0 + \Theta y_1)$, where $\Theta \in ...
2
votes
1answer
145 views

Numerical Analysis best estimate on polynomial order

I need to determine the best integer value of $k$ for the equation: \begin{equation} \arctan(x) = x + O(x^k) \text{ as $x\to 0$} \end{equation} Taylor's Theorem with Lagrange Remainder would ...
1
vote
3answers
69 views

Numerical Analysis and Big O

How can I show that $e^x -1$ is not $O(x^2)$ as $x\to0$ I'm not sure where to start. We can use Taylor's Theorem with remainder: \begin{equation} e^x = \sum\limits_{k=0}^n\dfrac{x^n}{n!} ...
0
votes
0answers
45 views

How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
0
votes
1answer
731 views

estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
2
votes
1answer
99 views

Don't understand how to start with this assignment question

I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term ...
2
votes
0answers
296 views

Consistency order of backward Euler method

How can I proof that backward Euler method has consistency order 1? Implicit function theorem states that for a sufficiently small $h$, $$ \vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1) $$ has a unique ...
0
votes
1answer
302 views

When finding upper bound for error, can $\xi$ be different from $x$?

The question is to find $P_3(x)$ for $ f(x) = (x-1) \; \ln x $ about $x_0 = 1$ and find the upper bound on the error for $P_3(0.5)$ used to estimate $f(0.5)$. I got $$ f^{(4)}(x)= \frac{2}{x^3} + ...
1
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1answer
67 views

Combining error terms from two Taylor expansions

When deriving the five-point differentiation formula as shown in this book, the IVT was used to combine $ f^{(5)} (\xi_1) $ and $ f^{(5)} (\xi_2) $ into one error term, $ f^{(5)}(\tilde{\xi}) $ As ...
3
votes
1answer
1k views

Truncation error using Taylor series

How can we use Taylor series to derive the truncation error of the approximation $$f^\prime(x)\approx\frac{f(x+h)-f(x-h)}{2h}$$