For questions related to approximations

learn more… | top users | synonyms

-3
votes
0answers
9 views

Proof of “Normal approximation to the log-normal distribution” [on hold]

I saw the post about the normal approximation to lognormal (Normal approximation to the log-normal distribution). The proof is shown as well. Yet as I'm looking for the proof in a journal article form ...
2
votes
1answer
48 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
0
votes
0answers
23 views

How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
0
votes
2answers
40 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
3
votes
0answers
65 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
0
votes
0answers
17 views

solve equation involving digamma function

I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast ...
1
vote
1answer
17 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
9
votes
1answer
89 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
0
votes
0answers
61 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
1
vote
2answers
37 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
0
votes
1answer
26 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
1
vote
1answer
39 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
0
votes
0answers
21 views

Spatial Basis Functions…any practical hints on choosing them?

I am not coming from a pure mathematical background, and now in my daily life as an engineering student, I am getting faced to spatial basis functions to approximates variables. The problem is: Are ...
3
votes
1answer
25 views

Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
7
votes
3answers
853 views

Is it possible to find square root using only rational numbers and elementary arithmetic operators

Suppose I have a number a How can I find it's square root using only +, -, /, ...
1
vote
1answer
17 views

Looking for a approximation/solution to my mortgage calculator function

I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
0
votes
1answer
47 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
1
vote
1answer
29 views

Estimating the absolute error of the function $f(x)=4x^2$

I have to estimate the value of $f(x)=4x^2$ for $x\in [1,2]$, and $x$ is unknown. the approximated value for $x$ is $\tilde x$, which is also in $[1,2]$. What is the maximum absolute error of $x$, ...
2
votes
0answers
31 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
0
votes
0answers
22 views

relative error of Poisson approximation to sum of Binomial

We have given $X_i\sim Bin(n_i,p_i)$ for $i \in \{1,...,m\}$ and are interested in $$P[X \geq x]$$ for $X=\sum_{i} X_i$. As we can approximate $X_i$ by $Y_i \sim Poisson(n_i p_i)$, I wonder, ...
2
votes
1answer
16 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
2
votes
1answer
48 views

Is this interpolation, does it have a name?

I was waching Signle Variable Calculus MIT lectures (I stop on 9 about linear approximation) I was also learn interpolation at my university and I thought that I'll create my own equation for ...
2
votes
1answer
47 views

How is equivalent to approximation property?

I'm trying to prove Lemma 4.8 of [1] online reading: Notation: $\tau_C(X)$ means the topology of uniform convergence on the compact subsets of $X$. Lemma 4.8. For a Banach space $X$ the following ...
9
votes
2answers
251 views

Difference between “≈”, “≃”, and “≅”

In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical Operators ...
0
votes
4answers
135 views
2
votes
0answers
18 views

Is $\frac{x^2(1-p)}{(n-x+1)p}\leq \frac{x^2}{(n-x)p}=O((np)^{-1})=o(1)$?

Is it true that for $n \rightarrow \infty$, $p \gg n^{-1}$, $0<p<1$ and $x=O(1)$, $$\frac{x^2(1-p)}{(n-x+1)p}\leq \frac{x^2}{(n-x)p}=O((np)^{-1})=o(1)?$$
1
vote
1answer
61 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
2
votes
0answers
27 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
1
vote
2answers
130 views

What is the sum that the square root button on calculator does so I can do it without the calculator button [duplicate]

I am not very good when it comes to Maths but the current work I am doing means I need to get better and quick. I have been teaching myself about areas, diagonals and square roots. However I am ...
1
vote
1answer
44 views

Understanding approximation $\sqrt{\frac{z}{z-2h}} \approx 1 + \frac{h}{z}$, $h>0, |z| \gg h.$

I am having troubles showing the above step regarding the approximation $$ \sqrt{\frac{z}{z-2h}} - \sqrt{\frac{z}{z+2h}} \approx \frac{2h}{z}, \quad h>0, |z| \gg h$$ given in an old exam. My ...
2
votes
2answers
42 views

Monotonically approximate $L^p$ function by step function

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is ...
1
vote
1answer
54 views

Meaning of $\sim$?

I often read $f(x) \sim g(x)$ and I wonder what the Standard Interpretation of this $\sim$ is. It seems to mean something like asymptotically equally distributed, something like $f(x)=g(x)(1+o(1))$. ...
1
vote
0answers
74 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...
0
votes
2answers
55 views

Newton Raphson Method for double roots

I am currently working on Newton Raphson Method. I am kind of facing a problem how Newton Raphson Method work on more than second order quadratic functions with double roots. I have googled and ...
1
vote
1answer
32 views

A better approximation of $({n \over e})^n$ than by $ \Gamma(1+n-1/2)$ ? (Focus is “reverse” to the Stirling approximation)

In a formula in my self-study of a summation method based on the matrix of Eulerian numbers (which I thus call "Eulerian summation") I am considering terms like $$ \Big({n \over e}\Big)^n \cdot {1 ...
1
vote
2answers
30 views

Approximate a surface by set of points

Given a set of points $(X, Y, Z)$ obtained from the experimental data that can be considered as a 3D surface. What is the common approach to get an approximating function Z=f(x,y) that describes the ...
-1
votes
1answer
35 views

Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...
0
votes
1answer
24 views

How to approximate the bounding region of a 2d differentiable mapping locally?

I have got a differentiable mapping $f:\Bbb R^2 \to \Bbb R^2$, Is the image of $f$ of a very small convex subset (e.g., a unit square) around any point, a bounded region? If it is bounded, can I ...
2
votes
0answers
34 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
0
votes
0answers
12 views

spatial-partitioning based physical simulation

I've learnt that spatial-partitioning based physical simulation is kind of "approximate" computation. Is it because: since the whole space is partitioned into cells, and only the interactions of ...
2
votes
1answer
24 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
1
vote
2answers
110 views

Using series find $\int_0^1 \sqrt{1+x^4}\hspace{1mm} dx$ up to $2$ decimal places

I cannot figure out an aesthetic way to do this. Can someone give a beautiful solution to this ugly question? This is what I have tried yet. I used the fact that $$x = ...
0
votes
0answers
13 views

Is it possible to approximate or bound this nonlinear mapping L h(x)?

I wanted to make sure whether the following approximation is possible. I have a vector ${\bf{x}} \in \mathbb{R}^N$ where the elements of the vector $\bf{x}$ are random variables. There is a matrix ...
0
votes
0answers
14 views

Using LLL to get approximate rational representations of numbers

Does anyone understand how is the LLL algorithm implemented to obtained the values of $(x,y)$ for approximating $\pi$ in this portion of text?
15
votes
1answer
199 views

Approximate value of a slowly-converging sum of $\sum|\sin n|^n/n$

In this question on Math.SE there appears this sum: $$ S = \sum_{n\geq1}s_n, \qquad s_n = \frac{|\sin n|^n}{n}, $$ which converges very slowly. What methods would you suggest for evaluating it ...
0
votes
0answers
13 views

Nonnegative Matrix Factorization in Machine Learning

I have a matrix $X^{m\times n}$, and I need to factorize it into $W^{m \times p}H^{p \times n}$, $p$ is the number of factors s.t. $p << min(m,n)$, and $m$ is the number of variables while $n$ ...
2
votes
0answers
39 views

Application of Weierstrass approximation theorem

How to approximate a continuous function to a desired accuracy using a polynomial? Theorem: For any $\varepsilon > 0$ and $f \in C([a,b])$, there exists a polynomial $p$ such that $\sup_{x \in ...
1
vote
0answers
31 views

Help understanding this approximation

In a paper that I'm reading, the authors write:- $$N_e \approx \frac{3}{4} (e^{-y}+y)-1.04. \tag{4.31}$$ Now, an analytic approximation can be obtained by using the expansion with respect ...
9
votes
6answers
179 views

When the approximation $\pi\simeq 3.14$ is NOT sufficent

It's common at schools to use $3.14$ as an appropriate approximation of $\pi$. However, here it's stated that for some purposes, $\pi$ should be approximated to $32$ decimal places. I need an example ...
1
vote
2answers
32 views

Poisson approximation to binomial distribution: $f(x)\geq g(x)$ or $f(x) \leq g(x)$

We have a random variable $X$ which has a Binomial Distribution Bin(n,p) and a random variable $Y$ which has a Poisson Distribution Poisson(np). We are interested in $$f(x):=Pr[X \geq x].$$ For ...