0
votes
1answer
44 views

Compute $\lim\limits_{x \to 1}[f(x)]$ and $\lim\limits_{x \to -1}[f(x)]$ for $f(x)=\frac{1}{1-x}-\frac{1}{1+x}$

Is it possible to rewrite expression $\frac{1}{1-x}-\frac{1}{1+x}$ in order to be able to find its values near $x=1$ and $x=-1$ more precisely? This is a question in a numerical methods course. Is the ...
0
votes
1answer
93 views

Computing infinite product over primes

How can I compute $$ \prod_p \left(1+\frac{k}{p}\right)\exp(-k/p) $$ where $0<k<e$ and the product is over all primes $p$? Background L. G. Sathe proved [1] that there are $$ ...
2
votes
0answers
10 views

Remainder of the minimax approximation polynomial - number of extrema

Recall some definitions. Let $f \in C [a,b]$. The minimax polynomial $p_n$ is the polynomial $p_n (x) $ of degree $\leq n$ that minimizes $||f-p_n||_\infty $. It can be proved that this polynomial ...
0
votes
1answer
28 views

Is the assumption $f \in C^4$ necessary for the composite Simpson's rule to be of order $p=4$?

In my introductory numerics class, we wanted to integrate a function $f \in C[a,b]$ numerically. After developing the Simpson's rule, we proved that if $f \in C^4$ then the composite Simpson's rule ...
0
votes
1answer
26 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
0
votes
0answers
34 views

Is the following statement on the stability of the forward Euler method true or false?

My text asks whether the following statement is true or false: The forward Euler method for approximating the solution of $x'=\lambda x$ is stable for all $\lambda \in \mathbb R$ and all step ...
1
vote
0answers
31 views

Sequence and intermediate value theorem

For part (a), By intermediate value theorem, there exist c between 0 and 1 such that $f(c) = 0$ Now, I supposed that there also exist d between 0 and 1 such that $ f(d) = 0 $ and $ c \neq d $ I am ...
0
votes
0answers
29 views

Condition under which newton raphson converges

I see in a book that under the following condition newton-raphson method (for finding the zero of a function) converges: 1) The function is continuously differentiable 2) The function is positive in ...
3
votes
1answer
215 views

Numerical verification of solution.

I have the non-linear equation \begin{align} &\left( {x}^{2}-1 \right) \left( -\frac{1}{4}\left({\frac { \left( 4\,{x}^{3}+2\, ex \right) ^{2}}{ \left( {x}^{4}+e{x}^{2}+f \right) ...
1
vote
0answers
44 views

Continuation fixed points of parameter dependent Newton

Suppose I have the iteration operator of the Newton method for some $\beta$-parameter dependent function $g_{\beta}: \mathbb{R} \rightarrow \mathbb{R}$. Let us assume that $g_\beta$ is in ...
-1
votes
2answers
47 views

How to prove that there is no interval that maps to itself under a function

I have the function $ g(x) = x^3 + 3x^2 - 3 $ and I need to show that there is no interval $ [a,b] $ such that $ g:[a,b] \mapsto [a,b] $. How do I go about this? Thanks a lot
1
vote
1answer
45 views

Numerical Solution of an Equation with Multiple Roots

Let me consider an equation $f(x)=0$ which I know to have a solution $x=x_0$, but I need to find its another solution. So I might consider finding root for the equation $$\frac{f(x)}{x-x_0}=0$$ but I ...
5
votes
0answers
95 views

What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
0
votes
0answers
19 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
2
votes
2answers
102 views

Integration problem: $\int _ {-\infty} ^ {\infty} \frac {e^{-x^2}}{\sqrt{\pi}} e^x\ dx$

I have to integrate $$\int _ {-\infty} ^ {\infty} \frac {e^{\large-x^2}}{\sqrt{\pi}} e^{\large x}\ dx.$$ I've already done by numerical approximations, like Simpson's rule and Gauss-Hermite, but I ...
1
vote
0answers
21 views

Computing area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions

I need to compute the area/ space of intersection between a pair of Beta distribution/ Dirichlet distributions. As I am a non mathematics guy, it will be great if someone helps me out with the ...
0
votes
1answer
42 views

Numerical approximate a convergent series

Consider i have a series $\sum_{i=1}^{\infty} X_i $ which i know converges in $\mathbb{R}$,but don't know exactly where. I am trying numerically approximate to the convergence point but not sure when ...
2
votes
3answers
92 views

Numerical Solution of $\frac{x}{1-e^{-x}} -5 = 0$

I am working on a problem at the moment which cuts down to the following question: How do I get a numerical solution for: $$\frac{x}{1-e^{-x}} -5 = 0?$$ I've been thinking about using Newton's ...
1
vote
1answer
65 views

Root of a function? (Proof with Banach theorem)

Given is a function $f$: $\left [ -1,1 \right ]\rightarrow \mathbb{R}$, which is continuous and differentiable. The function $g$: $\left [ -1,1 \right ]\rightarrow \left [ -2,2 \right ]$ is a ...
-1
votes
1answer
41 views

A problem on numerical analysis [closed]

Let $f(x)=e^x$ be approximated by Taylor's polynomial of degree n at the point $x=\frac{1}{2}$ and on the interval $[0,1]$. If the absolute error in this approximation does not exceed $10^{-2}$ , then ...
2
votes
0answers
51 views

Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
1
vote
1answer
47 views

Rate of convergence to find root of $x^2-5$

I want to find out the rate of convergence to find the root of $f(x)=x^2-5$ using fixed point iteration with \begin{align} x_k = \Phi(x_{k-1}) &&&&&&\Phi(x)=1+x-x^2/5 ...
0
votes
2answers
57 views

How to understand or show this?

We have $$F=ABh^{p_1}+\theta (h^{p_2})$$ $$G=Ah^{p_1}+\theta (h^{p_2})$$ We $A$,$B$ are real numbers, $h$ positive, $|h|\leq 1$ , $p_1<p_2$ natural numbers and $\theta(h)$ means that it is of ...
1
vote
0answers
24 views

nonlinear sequence to sequence transformations

i know matrix methods such as Cesaro,Holder,Riesz are regular linear sequence transformations. i wonder if there is any regular nonlinear sequence transformation?
1
vote
0answers
74 views

Newton-Rhapson for reciprocal square root

I have a question about using Newton-Rhapson to refine a guess of the reciprocal square root function. The reciprocal square root of $a$ is the number $x$ which satisfies the following equation: ...
0
votes
0answers
78 views

Any idea for this integral?

I'm trying to solve this integral with respect to a and b two positive reals. I think it's hopeless to find a closed form solution nevertheless if anyone has an idea to approximate analytically its ...
1
vote
1answer
94 views

Analysis for Engineers: Where Do You Start?

Having taken none of the prerequisite rigorous treatments of mathematics during my undergrad years, I feel at a disadvantage to the people in my major what do have that analysis/abstract math ...
0
votes
1answer
42 views

Prove that $a(u-u_{h},u-u_{h})\ge 0$

Assume that $a$ is bilinear, symmetric and positive definite form, $u\in X$ and $u_{h}\in X_{h}\subset X$. I know the following fact: $$a(u-u_{h},u_{h})=0$$ Frm positive definiteness ...
0
votes
1answer
83 views

Bisection method absolute error

I know that $\varepsilon \le 2^{-n-1}(b_0 - a_0)$, how to conclude from this that I need $n = \lfloor log_2{\frac{b_0 - a_0}{2\varepsilon}}\rfloor+ 1$? Using logs I get $ n \le ...
0
votes
2answers
53 views

Newton method to find $\frac{1}{\sqrt{a}}$ [duplicate]

What function should I use to find $\frac{1}{\sqrt{a}}$ without using division?
1
vote
2answers
29 views

Reverse of number (numerical)

You can find reverse number $R$ using $x_{n+1} = x_n(2 - x_n \cdot R) \ \ $ where $ n = 0,1,..$ Prove it using Newton method for finding $0's$ of some function $f$ Anyone have idea what that ...
3
votes
1answer
91 views

Why below sequence is diverge?

This problem maybe simple for you,but i dont know that why below sequence is diverge?please help me about this: why $‎\lbrace\mid x_{k}\mid‎\rbrace$ with below definition is diverge? $x_{k+1}:= ...
1
vote
0answers
36 views

Determine which one more accurate approx $f''(x)$

Derivatives can be written 1.) $$f'(x) = lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ 2.)$$f'(x) = lim_{h\rightarrow 0} \frac{f(x)-f(x-h)}{h}$$ Also $$f''(x) = lim_{h \rightarrow 0} \frac{f'(x+h) - ...
1
vote
1answer
55 views

When $ e^x$ ~ $ e^{-2x}$ ? - Numerical analysis

For what $x$, $ e^x $ ~ $e^{-2x}$ ? And how one can change this expression to avoid significant digits loss? I am able to think only about $x =0$, but then both are equal and you lose nothing.
0
votes
1answer
58 views

$a + b = a$ in machine precision [closed]

I have the following statement: "If $a + b = a$, then $b = 0$" may not true with the floating point operations. Actually, if $|y| ‎< (\varepsilon / B) |x|$, then $fl(x+y) = x$, where ...
1
vote
1answer
97 views

contraction point?

This is an interesting question I saw in a book online: Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence ...
1
vote
2answers
117 views

Sequences and Contraction of a fixed point

Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence $x_0, x_1, x_2,\ldots$ given by $x_n = g(x_{n-1})$. ...
0
votes
1answer
33 views

Proving something is a spline of degree $n$

I'm trying to understand this theorem: In the proof they only explain why $S_{(n)}(x)$ is identically $0$ outside the interval $(0,(n+1)h)$. But it is not clear to me that this is a spline of ...
0
votes
2answers
102 views

Simplifying equation into Newton Raphson form

Given the equation $\displaystyle{\int_{-x}^x\exp({-t^2})dt}=-\ln(x)$: a. Simplify the integral using Gauss method with 3 points. b. Solve given equation by Newton Raphson iterative ...
4
votes
1answer
131 views

g continuous on [a,b] using intermediate value theorem

Suppose that g is continuous on an interval [a,b] and that $g(x) ∈ [a,b]$ for all $x ∈ [a,b]$. (a) Use the intermediate value theorem to prove that is at least one number $c ∈ [a,b]$ with $g(c) = ...
1
vote
2answers
59 views

Finding A,B,C s.t $f'(a)+O(h^2)=\frac{Af(a)+Bf(a+2h)+Cf(a+3h)}{h}$

Find constants A,B,C s.t for differtiable three times function f, $f'(a)+O(h^2)=\frac{Af(a)+Bf(a+2h)+Cf(a+3h)}{h}$ I know that $f'(a)+O(h^2)=\frac{f(a+h)-f(a-h)}{2h}$ so I need to solve ...
1
vote
0answers
340 views

Finding error and proving Romberg Integration Method

Let $f$ be a function, which its integral has to be approximated by using romberg method.The $n\cdot m$th cell in romberg matrix (a lower triangular matrix) is given by ...
2
votes
2answers
88 views

Rate and order of convergence of $\sum_{k=0}^n \frac{x^k}{k!}$ to $\operatorname{exp}(x)$ as $n \rightarrow \infty$

Since $\lim_{n\rightarrow\infty} \sum_{k=0}^n \frac{x^k}{k!} = \operatorname{exp}(x)$, I was wondering how fast this series converges to $\operatorname{exp}(x)$. I'm suggesting that this series ...
0
votes
0answers
110 views

If $f$ is a differentiable function and $f'$ is bounded show that there exists a unique fixed point

let $f$ be a continuously differentiable function from $\mathbb{R}$ to $\mathbb{R}$ and such that $|f'(x)| \leq 4/5$ for all $x \in \mathbb{R}$. Show that there exists a unique $x$ in $\mathbb{R}$ ...
2
votes
2answers
180 views

Is there a mean value theorem for higher order differences?

The standard mean value theorem tells us $\frac{f(x+h)-f(x)}{h} = f'(c)$ for some $c$ between $x$ and $x+h$. Rewriting this, we may see it as $\frac 1h\Delta_h f(x) = f'(c)$. This makes me wonder if ...
11
votes
5answers
412 views

How to calculate $2^{\sqrt{2}}$ by hand efficiently?

I've been trying to calculate $2^{\sqrt{2}}$ by hand efficiently, but whatever I've tried to do so far fails at some point because I need to use many decimals of $\sqrt{2}$ or $\log(2)$ to get a ...
0
votes
0answers
72 views

Small symbols behind parantheses

I am currently reading the following paper where the author uses constantly small symbols after parantheses, but I do not know what this means. I am particularly interested in equation (23), so you ...
4
votes
4answers
202 views

How calculate $\pi$ to an accuracy of 10 decimal places?

Let $a=3.00000000001234...$ (irrational number) If $\overline{a}=3.00000000001$ (approximation $11$ places) then $|a-\overline{a}|<10^{-11}$ Note that the reciprocal is not satisfied: If ...
0
votes
2answers
62 views

Proving the continuity of a function at a given point - help needed

I have come across this question and am not sure as to how to go about finishing it. I have started off with working at out the limit at $x=2$ and this is $-4$. How then (or what do I use) to equate ...
1
vote
1answer
430 views

Numerical integration over a surface of a sphere

I am integrating a double integral in spherical coordinates over the surface of a sphere in MATLAB numerically. Although I have changed the relative and absolute tolerance I get the feeling that ...