2
votes
1answer
19 views

Find the largest value for $x_1$ in (0,1) such that $f(0.5)-P_2(0.5) = -0.25$ (interpolation)

I'm not really sure where to go with this problem and I'm hoping you can help. The problem states: Let $f(x) = \sqrt{x - x^2}$ and $P_2(x)$ be the interpolation polynomial on $x_0 = 0, x_1$, and ...
0
votes
1answer
23 views

Lagrange interpolation: Getting a bound and finding the error

I am struggling to understand this: The problem asks me to find the lagrange error of the polynomial approximation given the nodes $x_0 = 1, x_1 = 1.25, x_2 = 1.6$ with $x = 1.4$ The function I am ...
0
votes
2answers
42 views

Numerical integration tolerance pitfalls

Consider that we want to estimate $$\int_{\pi/2}^{\pi/2+8\pi}sin(x)dx$$ (the value of this integrate is obviously zero) with the Midpoint rule. We start with the endpoints $a=\pi/2$ and $b=\pi/2+8\pi$ ...
0
votes
0answers
35 views

estimate an upper bound for the error of an interpolation polynom

The task is to estimate the error of an interpolation polynom $p(x)$ to an function $f(x)$. The sampling points are $x_0 = -1,\ x_1= 0,\ x_2=1,\ x_3=3$ So i already calculated the polynom which does ...
3
votes
1answer
366 views

What initial guess is used for finding n-th root using Newton-Raphson method?

I would like to know what is an optimal initial guess for use with Newton-Raphson method when finding n-th root. I develop some program which uses GMP C++ library. GMP manual says: The initial ...
2
votes
1answer
485 views

error of the composite trapezoidal rule

Let $f\in C^2[a,b]$. The composite trapezoidal rule is given by $$T_n[f]:=h\left(\frac{f(a)+f(b)}{2}+\sum_{k=1}^{n-1}f(x_k)\right)\;\;\;\;\;\left(h:=\frac{b-a}{n},\;x_k:=a+kh\right)$$ First, I've ...
0
votes
2answers
135 views

How do we prove the error estimation of the rectangle method

Let $f\in C^2[a,b]$. An approximation of the integral over $[a,b]$ is given by $$I[f]:=\int_a^bf(x)\text{ dx}\approx \frac{b-a}{n}\sum_{i=1}^nf\left(a+\frac{2i-1}{n}(b-a)\right)=:M_n[f]$$ I've spent ...
3
votes
1answer
59 views

Error of apprximation

Have anyone read book "Paul Glasserman Monte Carlo MIFE", it's good, but i'm stuck in chapter 6 page 341 let $$ dX_t=a(X_t)dt+b(X_t)\,dW_t $$ they said that $$ \int_{t}^{t+\Delta t}a(X_{u}) \, ...
2
votes
1answer
22 views

How to show this estimation?

i have this polynom $$p(x) = \sum_{i=0}^{m}a_ix^i$$ I want to show, that if $\tilde{z}$ is the approximation to the simple zero digit $z \neq 0$ in first approximation, the following estimation ...
0
votes
1answer
24 views

How to show, that a relative mistake of a special function can be estimated in a given way

how to show, that if you have a function like this $$ y = f(x_1,...,x_m) := c \frac{x_1 *...*x_r}{x_{r+1},...,x_m}, \quad 1 < r \leq m,$$ the relative mistake in first order can be estimated ...
1
vote
1answer
28 views

Estimating error in calculation

I'd like somebody to verify my solution of the following problems: Let's assume, that float arithmetics $fl()$ has precision $\nu$ for standard operations $(+\ -\ \cdot \ \div)$. a.) Estimate ...
0
votes
1answer
117 views

How can I make estimates on large powers and logarithms such as $e^{10}$?

Just wondering, are there any useful tricks to make estimates of large powers or logarithms just by hand such as for $e^{10}$? Any such ways to get an error less than 1?
3
votes
2answers
2k views

Modern formula for calculating Riemann Zeta Function [duplicate]

Possible Duplicate: How to evaluate Riemann Zeta function I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would ...
1
vote
2answers
678 views

Help with Chebyshev Economization of $\exp(x)$?

This may be a stupid question, so I apologize in advance if it is. This is a very common example of Chebyshev Economization, but I still do not understand how the coefficients are found. I want to ...
3
votes
0answers
81 views

Numerically estimate $a^b$ [duplicate]

Possible Duplicate: How can I calculate non-integer exponents? What is the most efficient way to estimate $a^b$ ($a > 0$) numerically? My goal is not to use built-in math functions (like ...
4
votes
1answer
885 views

Method for estimating the $n^{th}$ derivative?

When using numerical analysis, I often find that I am required to estimate a derivative (e.g. when using Newton Iteration for finding roots). To estimate the first derivative of a function $f(x)$ at ...
40
votes
10answers
2k views

What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...