For questions related to approximations

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1answer
84 views

Mysterious subleading corrections to sums with internal dependence on limit

Is there a standard method for finding expansions in $N$ of sums like $$S(N)=\sum_{n=0}^N \sqrt{N^2-n^2}$$ beyond the first term? It is easy to compute here that $$S(N)=N^2 \int_0 ^1 \sqrt{1-x^2} ...
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0answers
39 views

Numerically integrating in to Chebyshev polynomial

I'm trying to find the Chebyshev interpolate for an ODE in a given interval. That is, given an ODE that looks something like: $$y'' = g(y) \ y'$$ I want to numerically integrate it inside the ...
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0answers
31 views

Zeta function universality: How to compute the shift parameter for simple functions?

I've come across Zeta function universality. For a nice function $f$ in a nice subset $U$ of the complex strip between real $0$ and $1$, one can find a real $t$, such the zeta function $\zeta$ shifted ...
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2answers
29 views

When to consider an approximation as a Good approximation?

What is the criteria for the Good approximation ? e.g. we can approxiamte $\sin(x)$ to $x$ for $x<0.16 rad $ why 0.16 ? why not 0.23 enother e.g. $\tanh(x/2s)=x$ for $x<s$ and so on .. how ...
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1answer
103 views

The approximation of first-ordered modified Bessel function of the second kind

After analysing the outage probability of a single relay selection system, I got to the following form: $P = 1 + \sum\limits_{k = 1}^K {\left( \begin{array}{l} K\\ k \end{array} \right){{\left( { - 1} ...
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1answer
110 views

Stable algorithms from a backwards recurrence?

This is homework, so please only hints. We see that the recurrence $$I_n=\frac{1}{n}-\alpha I_{n-1}$$ can be used to compute the value of an integral, but the algorithm is unstable when $|\alpha ...
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1answer
43 views

Summing up Standard Deviations - best approximation

For a long data series the overall Standard Deviation shall be collected. However, due to memory constraints the data has to be aggregated per day - in a way that only one number is stored per day ...
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0answers
34 views

Approximating sums

I got a general question, that is motivated by a recent problem. So let me first describe the problem and then add the general part: I got a rather simple (using only basic elements) equation, which ...
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0answers
29 views

How to approximate derivative inside derivative

I am using a box-scheme for solving partial differential equation. The function is approximated with: $$ f_p=\psi\cdot\left(\theta\cdot f_{j}^{k+1}+(1-\theta)\cdot f_{j}^{k} \right) + (1-\psi)\cdot ...
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3answers
47 views

Name of the generalization of quadtree and octree?

What is the name of the equivalent of quadtrees and octrees in n-dimension ?
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4answers
268 views

Why is this number so close to $1$?

The only positive solution of the equation $\sin (\tan x) = x$ is at a number $a = 0.999906...$. Is it a coincidence that the number $a$ is so close to $1$, or is there a conceptual explanation? It ...
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1answer
213 views

Midpoint approximation over/under estimation

So left handed approximation underestimates the area under a increasing curve and over estimates for decreasing curves. And right handed approximation overestimates for increasing curves and ...
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1answer
39 views

Taylor polynomial approximation (Little $o$)

Suppose $(a_1,...,a_n) \in {\mathbb{R}_{*}^{+}}^n$, how can I prove that using the little $o$ (Taylor polynomial approximation): $\displaystyle\lim _{ x\to +\infty }{ \left( \cfrac { \sum _{ i=1 }^{ ...
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1answer
40 views

Chebyshev coefficients from a polynomial

Is there an efficient algorithm for finding the coefficients in a Chebyshev basis of a polynomial? That is, given the set of $a_k$ such that: $p_n(x) = \sum_{k=0}^N a_k x^k $ Find the set of $c_k$ ...
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3answers
68 views

Factoring/approximating an apparently simple formula

Does anyone know if the following formula can be factorized or approximated: $a^3 + b^3 + c^3 + a^2b + ab^2 + a^2c + ac^2 + b^2c + bc^2 + abc$ It looks a lot like $(a + b + c)^3$, except for the ...
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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}) \, ...
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2answers
39 views

Confusion regarding an alleged hyperbola

While studying a chapter called price elasticity of demand in my economics course, I have been presented with something called a unit elasticity curve (some sort of a hyperbola), which has supposed ...
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4answers
108 views

linear or quadratic approximation for exp(-x) for large x

Is there any linear or quadratic approximation of $exp(-x)$ where $0<x<L$ ? $L$ is large, may be 40 (say) i.e. $x$ is not close to zero.
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1answer
87 views

Showing (1 - polynomial fraction) raised to a polynomial power is a negligible function

Let $P(k)$ and $Q(k)$ be two polynomials ($k>0$). Let $\mathrm{neg}(k)$ be a negligible function for sufficiently large $k$ (see Appendix on question for definition). Does someone know how to show ...
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0answers
19 views

Iterative approximation of non-constant values in linear equation

The issue regards an algorithm for iterative approximation of unknown transaction values. For each iteration (each day), we are give the total revenue of all transactions for that day, and we have the ...
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1answer
55 views

Approximating a Gaussian Integral: Can you do better?

I have attempted to approximate this Gaussian: $$ I =\int_{0}^{\lambda}dx\left(r+x\right)\exp\left(-\rho\left(ax^{1/2}+bx^{3/2}\right)\right)\ $$ using $$ I ...
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1answer
191 views

How to calculate APR using Newton Raphson

I'm have a computer program to calculate apr using Newton Rhapson. I imagine most mathletes can code so i dont imagine the coding being an issue. The solution is based on this initial formula ...
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4answers
488 views

Evaluating the precision in the calculation of $\mathrm{e}$

I'm calculating $\mathrm{e}$ using a computer like this: $$ \mathrm{e} \approx \sum\limits_{i=0}^n {1\over i!} $$ I'm storing it as a rational number. I was wondering, if I write down my rational ...
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1answer
156 views

Find derivation (dB/decade) for given amplitude characteristic of low pass filter [Hz, -]

I am trying to find derivation (differential attenuation) for frequency's 600 and 2000 Hz for given amplitude characteristic of low pass filter, which look like this: I assume, that I should ...
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1answer
49 views

Determining the surface with given polynomial borders

Let's say we'd like to guess the shape (I'm not sure the word 'approximate' is appropriate here) of some surface when we are given its borders via third order polynomials (i.e. we are given their ...
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3answers
57 views

I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator.

Let $f(x)=\ln x$. Then the Taylor polynomial of degree 2 at $x=e$ is $P(x)=1+\frac1e(x-e)-\frac1{2e^2}(x-e)^2$ I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator. ...
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2answers
39 views

Help evaluating or approximating this integral

For a thermodynamics project I'm working on, I need to evaluate this integral: $\int \frac{(a-bx)(x-c)^d}{x^3}dx$, where $a,b,c,$ and $d$ are all positive constants. I tried evaluating it on ...
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1answer
42 views

Newton Raphson Method: approximating root

How do we start from approximating a root using this technique? I know of two, viz - a table of x vs f(x), and see where f(x) changes sign - plot a graph, and see where the graph cuts the x axis But ...
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1answer
48 views

Normal approximation for binomial distribution isn't giving correct result, z score comes out 0

I'm trying to use the normal distribution to calculate approximate values for the (cumulative) binomial distribution with large values (since it's impractical to evaluate the factorials). I'm very ...
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2answers
34 views

Finding a line of approximation using the normal equations for $A\vec{x}=\vec{b}$

To find the line $y=ax+b$ that best approximates the data points $\{(-2,3),(0,5),(1,7)\}$ I need to use the equation $$A\vec{x}=\vec{b}\ \ \ (\mbox{where}\ \vec{x}=\left({a\atop b}\right))$$ Then ...
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1answer
30 views

Check my short proof - asymptotic approximation, which function is bigger

The goal of this exercise is to show that $\ln(n+1)-\ln(n) = O(\frac{1}{n})$ what I did is: I used the fact that if $f=O(g)$ then $\frac{f}{g}=O(1)$. $\ln(n+1)-\ln(n)=\ln(\frac{n+1}{n}) = \ln(O(1))$ ...
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4answers
164 views

Using $(1+x)^k \approx 1+kx$ to approximate?

Use the approximation $(1+x)^k \approx 1+kx$ to estimate $(1.0003)^{26}$ and $\sqrt[4]{1.006}$. I know how to solve this step-by-step, but I don't understand what I'm doing exactly: why does ...
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0answers
66 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: ...
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0answers
69 views

Density of $C_0$ space in $L^p$ space for Hilbert-space-valued functions

The theory says that the space of continuous functions with compact support is dense in Lp space for functions taking value in finite dimensional space (or maybe just in real or complex space?) But, ...
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3answers
33 views

How to include the fee of a sell order in itself.

Sorry for the poor title, I am sure there is a name for this problem (and an easy solution) I have an account balance of \$1000. When I want to buy some EUR, I need to sell USD with a fee of 1%. That ...
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0answers
35 views

Bounding a convolution with a maximal function

Consider a family of kernels $\{K_{\epsilon}\}_{\epsilon>0}$ such that: $\int_{\mathbb{R}^d}K_{\epsilon}\ dx=1$ $|K_{\epsilon}(x)|\leq A\delta^{-d}$ for all $\delta>0$ $|K_{\epsilon}(x)|\leq ...
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0answers
72 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
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1answer
68 views

Find an upper bound on the absolute error of 3.141 as an approximation to π

Q: Find an upper bound on the absolute error of 3.141 as an approximation to π I have no idea what to do... :( What I know: absolute error = real value - approximate value Help :)
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3answers
310 views

Approximating $1/z$ by polynomials

Let $C=\{\mathrm e^{\mathrm it}, 0\le t\le 3\pi/2\}$ and $f(z)=1/z$. By Runge's theorem, there is a sequence of polynomials $p_n(z)$ such that $$\lim_n p_n(z)=f(z)$$ uniformly on $C$. Does anyone ...
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2answers
77 views

How to approximate $n \int_{0}^{1} [1 - x^m ]^n x^m dx $ near infinity?

I have a hypothesis that if: $$ I_{n,m} := n \int_{0}^{1} [1 - x^m ]^n x^m dx $$ where $m,n \in \mathbb{N}$ then $$ \lim_{n \rightarrow \infty} \frac{I_{n,m}}{n^{-\frac{1}{m} } } = c_m $$ But I ...
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1answer
45 views

Approximating $\tanh(B\sqrt{A} )$ for small $A$ and arbitrary $B$ by correcting for asymptotic behaviour of $\tanh$

I would like to approximate a function containing terms of the form $\tanh( B\sqrt{A})$ for small $A$. I have tried doing a Taylor series, but I consistently find that it is not only $A$ that has to ...
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1answer
63 views

How do I show that these two definite integrals are approximately equal

Consider the following two integrals: $I_1=\int\limits_{0}^{1/x}G(s)\ ds$ $I_2=\Big|\int\limits_0^\infty G(s)\exp[-ixs]\ ds\Big|$ where $G(s)$ is a monotonically decreasing positive function, ...
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1answer
25 views

How to determine coefficients of $p(x) = x^6$ with the Chebyshev processing

I want to calculate the coefficients of $p(x) = x^6$ with the Chebyshev processing. How to do that? Following question would be, how to estimate the error in $[-1,1]$, if i only use terms until ...
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1answer
37 views

Asymptotic behaviour of the area of a 2-dimensional flat subset of $\mathbb{R}^3$

I am interested by the area of the $2$-dimensional flat subset of $\mathbb{R}^3$ defined by the following equations with one parameter $t>1$: $x,y,z>0$ (positive octant) $x+y+z=t$ (hyperplane ...
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0answers
69 views

Real approximation to the maximum using Laplace's method integral

The Laplace's Method states that under some conditions, it holds that: $ \sqrt{\frac{2\pi}{M(-g''(x_0))}} h(x_0) e^{M g(x_0)} \approx \int_a^b\! h(x) e^{M g(x)}\, dx \text { as } M\to\infty$ Where ...
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4answers
62 views

Approximation of $\frac{1+a}{1+b}$

I've found the following assertion on an economics book: For $r$ and $g$ small enough, $\frac{1+r}{1+g}\approx 1+r-g$ (where $r$ is the interest rate and $g$ is the growth rate of the economy) ...
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1answer
48 views

How to find a sequence by its limit?

Is there any way to construct non-trivial sequence by its limit? Something like $\begin{cases} a_1=2 & \\ a_{n+1}=\dfrac1{2}\left(a_n+\dfrac2{a_n}\right) \end{cases}$for $\sqrt2$. I'm especially ...
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1answer
86 views

A curious partitions coincidence $\sum_{n=0}^\infty P(n) q^{n+1}$?

Given the partition function $P(n)$ and let $q_k=e^{-k\pi/5}$. What is the reason why, $$\sum_{n=0}^\infty P(n) q_2^{n+1}\approx\frac{1}{\sqrt{5}}\tag{1}$$ $$\sum_{n=0}^\infty P(n) ...
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1answer
17 views

How do I know that this is the density of the Chebyshev Points?

By knowing that a discrete distribution of points go asymptotically to the density: $\displaystyle p(x)= \frac{1}{\pi \sqrt{1-x²}}$ in $[-1, 1]$ I am able to conclude that interpolating at those ...
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1answer
73 views

Minimizing the $L^2$ error when approximating with trigonometric polynomials

I want to find approximations ${\rm g}_{n}\left(x\right) \in T_{n}$ of $\,\,{\rm f}\left(x\right)$, so that the error $$ \left\vert\left\vert\,{\rm f} - {\rm g}_{n}\,\right\vert\right\vert^{2} = ...