# Tagged Questions

For questions that involve concrete approximations, such as finding an approximate value of a number with some precision. For questions that belong to the mathematical area of Approximation Theory, use (approximation-theory).

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### Optimal approximation of spline curve using linear interpolation

I have a parametric cubic spline which I need to draw in graphics. I am restricted to using a set of lines to draw this, and for performance reasons I need as few lines as possible. So I need an ...
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I found the following approximation for the function $$f=\ln(x+1)$$ $$f\simeq\Psi\left(x+\dfrac{3}{2}\right)-2+\gamma+\ln(2)$$ where $\Psi(x)$ is the 'Digamma' function: $\Psi(x)=\dfrac{\dfrac{d}{dx}\... 1answer 63 views ### A curious approximation to$\cos (\alpha/3)$The following curious approximation$\cos\left ( \frac{\alpha}{3} \right ) \approx \frac{1}{2}\sqrt{\frac{2\cos\alpha}{\sqrt{\cos\alpha+3}}+3}$is accurate for an angle$\alpha$between$0^\circ$... 0answers 22 views ### Reduced Chebyshev approximation? Few days ago my teacher mentioned a method of approximation for a function. I think it was called "reduced chebyshev approximation", where you find the taylor series of degree$N$then subtract from ... 1answer 37 views ### What is an accurate approximation and asymptotic for this function? $$f(n)=\Bigg(1-\Big(1-\frac{1}{2^{n/2}}\Big)^n\Bigg)^{n^7}$$ I am interested in large$n$. The number$7$can be replaced by any fixed integer. I have $$f(n)\rightarrow\Bigg(1-e^{-n2^{-n/2}}\Bigg)^{... 1answer 44 views ### Upper bound for ratio of modified Bessel functions of second kind I was wondering if someone has an idea if for 0 < x < y, and 0< \nu \leq \frac{1}{2}, one can obtain an upper bound for the ratio$$ \frac{K_{\nu}(x)}{K_{\nu}(y)} $$Thanks. 1answer 88 views ### Alternate proof of the integral: \int_0^1 x^x(1-x)^{2x}\,dx\neq3/8 I am looking into the integral:$$I=\int_0^1 x^x(1-x)^{2x}\,dx\neq\frac{3}{8}$$How might you prove this to be true? What's tough is that the integral$$3/8\lt I<0.37503$$numerically. I managed ... 1answer 54 views ### Proof that Newton's Method gives better and better approximations with each iteration? I've seen this question and answer: Why does Newton's method work? It gives some geometric intuition as to what is going on when applying Newton's method, but what I really need to know is why it ... 1answer 56 views ### How to prove that your approximation using Newton's Method is correct to x decimal places? I just watched a video about a problem stated as: "Find where f(x)= x^7-1000 intersects the x axis. Find solution correct to 8 decimal places." The author uses Newton's Method, repeating the ... 0answers 56 views ### Ideas on how to simplify or approximate this nasty sum I have a sum (let's call it p):$$p:= \frac{1}{n!}\sum_{i=2}^l (i-1)\frac{(n-k)!}{(n-k-i+2)!}(n-i)!$$where l, n, k are fixed positive integers, and k \leq n. I'd like to either simplify or ... 1answer 32 views ### Optimization algorithms for Distribution and Logistics scenario I am looking for a way to express the following logistics/distribution problem as an equation that can be run thru a solver to find an approximate solution. The problems is described as follows: ... 2answers 45 views ### Binomial theorem estimate for very large samples I have around 2^{105} balls, of which 1 in 20 is white. I expect that when I draw a random sampling of them, roughly 5% of all balls drawn would be white. What is the probability that, if I draw ... 2answers 65 views ### When is \frac{dx}{dt}=\frac{\Delta x}{\Delta t} a valid approximation? It is often said that when the change in e.g. \Delta x is small than we can make the approximation:$$\frac{dx}{dt}=\frac{\Delta x}{\Delta t}$$But it is not enough to say \Delta x is small ... 1answer 12 views ### Approximately equal too symbol operations permitted I have to doubt to clear if A is approximately to 2/3 B can I write it as A approximately equal to 4/6 B? Since it is approximate relation is it permitted to multiply both numerator and denominator as ... 1answer 37 views ### Why is \prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}? I came across this approximation in the book Principles of Population Genetics by Hartl and Clark (page 130).$$\prod_{i=1}^{k-1}\left(1-\frac{i}{2N}\right) \approx 1 - \frac{{k \choose 2}}{2N}$$... 0answers 19 views ### What is the name of this approximation? I remember studying a while back about an approximation method where the error is calculated using$$ E_{n}=M_{n+1}-a_{n+1} \widetilde{T}_{n+1} $$Where \widetilde{T}_{n}=\frac{{T}_{n}}{2^{n-1}}, ... 1answer 29 views ### Solve using succesive approximation methods Consider the equation f(x) = xe^x-1=0. We wanna solve it using successive approx method, solving the fix point problem x=e^{-x} I don't understand Banach's theorem, how can I solve this? Any idea,... 2answers 44 views ### Separation of integral by approximation I'm working with the following integral \displaystyle\int_0^y \frac{dx}{x \sqrt{1-ax-bx^2}} and would like to split it in something like$$\int_0^y\frac{dx}{x \sqrt{1-ax}}+\int_0^y\frac{dx}{x \sqrt{... 0answers 17 views ### methods function approximation My goal is to modelize the temperature of a motor: I want to be able to calculate the temperature of a motor at a given moment, with a given torque and a given speed, in a constant environment. For ... 3answers 50 views ### Why transform degrees into radians when computing linear approximation to find$\tan{44^\circ}$? I am asked to find the linear approximation of$\tan{44^\circ}$. Why should I transform degrees into radians to do that? I understand that using degrees would give me a wrong solution (which would be ... 1answer 33 views ### Linear approximation to find$\frac{1}{4.002}$I am asked to find$\frac{1}{4.002}$using linear approximation. The way I proceeded was: $$f(x) = \frac{1}{x} \quad a = 4 \quad f(a) = f(4) = 0.25\\ f'(x) = - \frac{1}{x^2} \quad f'(a) = f'(4) = - ... 1answer 47 views ### Is there a simple way of computing when a^n=b^m I don't want exact equality just close enough to be useful in approximation. i.e. 2^{10 }= 10^3 is very useful and used daily for an approximation. Is there a do this efficiently? Is there a way ... 1answer 36 views ### Proportion in sets We have 3 sets of positive integers.$$A = \{x_1,y_2,z_3\},\quad{} B = \{x_2,y_2,z_2\}, \quad{} C= \{x,y,z\}$$Which proportion do we use for adding A and B (x_1+x_2 and so on), so the ... 1answer 29 views ### Local quadratic approximation I wanted to implement some penalized regression parameter estimation algorithm by Fan&Li (http://sites.stat.psu.edu/~rli/research/penlike.pdf, section 3.3, [1]), but cannot catch the idea of some ... 4answers 143 views ### Approximation of \sqrt{2} I got the following problem in a chapter of approximations: If \frac{m}{n} is an approximation to \sqrt{2} then prove that \frac{m}{2n}+\frac{n}{m} is a better approximation to \sqrt{2}.(... 1answer 19 views ### Inequality insde algorithm design (kleinberg), Disjoint Path I am reading the "Algorithm Design" (Kleinberg et al). Inside this book, at the chapter 11.6 there is an inequality I have failed to resolve. Considering \beta=m^{(1/3)} and |I|>=1 (it is a ... 0answers 13 views ### How do I show I've rounded to a certain decimal? I noticed that we could say \sqrt{0.9}\approx0.9 if we round to one decimal, or \approx0.95 if we round to two decimals. But what is the correct way to show how many decimals we rounded to? ... 9answers 3k views ### How is the derivative truly, literally the “best linear approximation” near a point? I've read many times that the derivative of a function f(x) for a certain x is the best linear approximation of the function for values near x. I always thought it was meant in a hand-waving ... 1answer 72 views ### Approximation of a polynomial with fractional power I have a polynomial I need to find the roots of, the major difficulty is that this polynomial has fractional exponents. I have made an approximation and I would like to have some idea of the error I ... 1answer 31 views ### How do I calculate this limit when two terms tend to infinity at similar rates In a particular problem that I am currently trying to solve, I have the following expression (this is not the entire expression, I have included only the terms involving a_1 and b_1), \lim_{(... 0answers 16 views ### Computing the partition-function of an exponential family member I am working on an Monte Carlo Expectation Propagation problem. In that context I got the following property: I = \sum\limits_i w^{(i)} \log p_\eta(x^{(i)}) where \{w^{(i)}\}_i are weights, ... 0answers 60 views ### What's about \sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}, where H_n is the nth harmonic number? I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for ... 1answer 49 views ### B-Spline approximation deviates a lot while increasing the number of control points??? I'm dealing with a problem to approximate some data points with B-Spline. I follow the method and implemented the algorithm from this site: Curve Global Approximation. 1) The first step is to ... 3answers 53 views ### Question for the estimation of \sum_{i=1}^x \frac{1}{w+i} as x \to \infty. I have a question of the estimation of this summation:$$ \frac{1}{w+1}+\frac{1}{w+2}+\cdots+\frac{1}{w+x}$$Which is:$$\sum_{i=1}^x \frac{1}{w+i}$$What I have tried: applying limit to the ... 1answer 56 views ### Approximation of product of Bernoulli with different proportions I want to update a variable Y with Beta (uniform for simplicity, Y \sim U(0, 1)) distribution, with Bernoulli information each period... But each period the proportion parameter of the Bernoulli ... 3answers 115 views ### Approximating the normal CDF for 0 \leq x \leq 7 In answer http://stackoverflow.com/a/23119456/2421256, an approximation of the complementary normal CDF (ie \frac{1}{\sqrt{2 \pi}} \int_x^{+\infty} e^{-\frac{t^2}{2}} dt) was given for 0 \leq x \... 0answers 6 views ### Other types of mean error Let \tilde f be an approximation of the function f(x) = \arccos(x). I'm using MATLAB to figure out how good this approximation is by calculating a mean error. My first idea was to use this formula ... 0answers 5 views ### Equi-oscillation condition EOC(n) for best polynomial approximations. I'm given the funciton f(x) = x^3 on [-1, 1] and am told to find the best order one polynomial approximation to this. The solution says that p(x) = \frac 34 x is this polynomial (and I verified ... 2answers 32 views ### Is it true that \sum_{i=1}^m \frac{1}{\sqrt{i}} = O \left( \sqrt{ m-1 } \right) ? Is it true that?:$$ \sum_{i=n}^m \frac{1}{\sqrt{i}} = O \left( \sqrt{ \frac{m-n}{n}} \right) $$In special case if we have n = 1, is it true that?:$$ \sum_{i=1}^m \frac{1}{\sqrt{i}} = O \left( \... 3answers 65 views ### Probability of alternating sequence from uniform distribution Say I sample a discrete uniform distribution$U$(say$U$has a support of$N$elements, and there is a total order on the elements) a number$K$times, resulting in a random sequence$A_{i}$. What ... 0answers 38 views ### Upper-bounding$\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}}$? Suppose$a_1, ..., a_n \in \mathbb{N}$are arbitrary integers. Is it possible to bound $$A =\sum_{i=1}^n \sum_{j = i}^{i+a_i} \frac{1}{\sqrt{j}}$$ with either of the following: $$B = c\sqrt{\sum_{... 0answers 23 views ### Bernstein polynomial in Banach For a continuous function f : [0,1] \to R, there exists a sequence of polynomial functions: P_n(x)=\sum_{k=0}^n C^k_n x^k(1-x)^{n-k} f(\frac{k}{n}) (Bernstein's polynomes) which converges ... 0answers 52 views ### Understanding a medieval approximation A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is 5000 square cubits. It reads in relevant part: … that the ... 2answers 76 views ### Arriving at the asymptotic \int \limits_\lambda^\infty e^{-t^2/2}dt \sim \frac{e^{-\lambda^2/2}}{\lambda} In the book "The Probabilistic Method", the integral \int_\lambda^\infty e^{-t^2/2}dt is said to be "approximately equal" to \frac{e^{-\lambda^2/2}}{\lambda} for large \lambda. I assume what is ... 1answer 35 views ### How can I express this in terms of Gauss-Hermite Quadrature? I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it?$$f_Z(z)=\int_0^{\infty}\... 0answers 31 views ### Approximate$e^{0.01} \sin(0.02)$by using its linearization$f(x,y) = e^x \sin(y)f_x = e^x \sin(y)f_y = e^x \cos(y)L(x,y) = f(0,0) + f_x(0,0)x +f_y(0,0)y$Solving:$f(0,0) = e^0 \sin(0) = 0f_x(0,0)x = e^0 \sin(0) \cdot x = 0f_y(0,0)y = e^...
Find uniform approximation up to order $O(\epsilon)$: $$\begin{cases} \epsilon y''+\epsilon y' - y^2=-1-x^2 \\ y(0)=2 \\ y(1)=2 \end{cases}$$ At $\epsilon=0$ solutions $\pm \sqrt{1+x^2}$ don't ...