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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5
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0answers
54 views

Pointwise approximation of a closed operator

If $T:\mathcal D(T) \rightarrow \mathcal Y$ is a closed operator from a Banach space $\mathcal X$ to a Banach space $\mathcal Y$, is it possible to find bounded operators $T_n\in \mathscr B(\mathcal ...
5
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5answers
2k views

Numerically Efficient Approximation of cos(s)

I have an application where I need to run $\cos(s)$ (and $\operatorname{sinc}(s) = \sin(s)/s$) a large number of times and is measured to be a bottleneck in my application. I don't need every last ...
4
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2answers
111 views

Approximation of DE

It depends on my previous question. Closed form solution of DE I don't want to deal with Airy functions. How can I approximate this DE in continous domain $[0,1]$? $$y''(x)+(x+1)y(x)=0\quad\text{ ...
4
votes
1answer
494 views

Hermite Interpolation of $e^x$. Strange behaviour when increasing the number of derivatives at interpolating points.

I am trying to understand Hermite Interpolation. Here is my pedagogical example. I want to approximate $f(x)=e^x$ on the domain $[-1,1]$ using Hermite interpolation. I choose the Chebyshev zeros ...
3
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2answers
143 views

Bounds on $ \sum\limits_{n=0}^{\infty }{\frac{a..\left( a+n-1 \right)}{\left( a+b \right)…\left( a+b+n-1 \right)}\frac{{{z}^{n}}}{n!}}$

I have a confluent hypergeometric function as $ _{1}{{F}_{1}}\left( a,a+b,z \right)$ where $z<0$ and $a,b>0$ and integer. I am interested to find the bounds on the value it can take or an ...
3
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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 ...
3
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1answer
954 views

With Euler's method for differential equations, is it possible to take the limit as $h \to 0$ and get an exact approximation?

I was recently watching a tutorial on Euler's method for approximating differential equations, and the whole time I was thinking "why can't you just take the limit of the step size $h$ as it goes to ...
3
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2answers
384 views

Asymptotic number of unlabeled graphs

A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be $$c(n) = 2^{n^2}/n!$$ because there are $2^{n^2}$ labeled graphs, almost all of them ...
2
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0answers
74 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
2
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4answers
139 views

How to extract fraction from a floating point number

I'm making some tests with float type (floating point number) with programming and in some of my tests I need to extract the fraction that originates the float value. Let $ x $ be a floating point ...
2
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1answer
43 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
2
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5answers
81 views

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$ I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$ Also a tighter upper bound is appreciated.
2
votes
1answer
266 views

Error analysis of exponential function

By definition: $$ e^x = \lim_{n \rightarrow \infty} ( 1 + \frac{x}{n} ) ^ n$$ I am interesting in calculating the error $$\left | e^x - \left( 1 + \frac{x}{n} \right) ^ n \right|$$ for some fixed $n ...
2
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1answer
81 views

Does projection onto a finite dimensional subspace commute with intersection of a decreasing sequence of subspaces: $\cap_i P_W(V_i)=P_W(\cap_i V_i)$?

Let $V$ to be an infinite dimensional linear space over some field $k$. (you can take $k=\mathbb{C}$, or further assume $V$ is a complex Hilbert space). And assume $W$ is a finite dimensional ...
2
votes
1answer
80 views

Interpretation of the variational principle for the Ritz approximation, solid Mechanics

Below $U$ and $V$ are recpectively the internal and external energy components of a given structural element: $$U+V=W$$ Expressing $U$ in terms of the strains $\varepsilon$ and the material ...
2
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1answer
51 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
2
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3answers
799 views

How to calculate the area of bizarre shapes

I'm looking for an algorithm to calculate the area of various shapes (created out of basic shapes such as circles, rectangles, etc...). There are various possibilities such as the area of 2 circles, 1 ...
2
votes
2answers
372 views

Triple Recursion Relation Coefficients

I am reading Atkinson's "An Introduction to Numerical Analysis" and I am trying to understand how a certain equation was reached. It is called the "Triple Recursion Relation" for an orthogonal family ...
2
votes
1answer
268 views

Solving an integral with Laplace method

I'm trying to approximate the sum $$\sum_{\alpha=1}^{\mu} \Big(1-\frac{(\alpha(2 \mu-\alpha))^2 \gamma_1 \gamma_2}{2n^2 \mu^4}\Big)^{\frac{\lambda}{2}}$$ with an integral ...
1
vote
1answer
66 views

Least absolute deviation for item prices

How would I calculate the values of $A,B,C$ using least absolute deviation? $R = 1$ $2A + B$ = $C + R$. $B + C$ = $5A$. $A + C + 2R$ = $B + 4R$. $A + B + C$ = $6.33R$. Using least squares ...
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2answers
40 views

Least Squares approximation for item prices

Let's say that $A$, $B$, $C$ are different items with different values. $R$ is a unit of currency, for simplicity I'll let it be $1$. Traders frequently trade these items on an open market. Price is ...
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2answers
45 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 ...
1
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1answer
114 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 $f(x)=g(x)(1+o(1))$. ...
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2answers
77 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 ...
1
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1answer
46 views

Why does the asymptotic equation of the modified Bessel of the second kind (Iv) have an imaginary part?

This is a follow up to this question. How does one arrive at the asymptotic expressions for the bessel functions? After looking at: G. N. Watson, "A Treatise on the Theory of Bessel Functions", 2nd ...
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2answers
106 views

How does one arrive at the asymptotic expressions for the bessel functions?

It is known that Bessel functions for large arguments will behave as exp or cos/sin however I was wondering how does one arrive at those results. The motivation being that I would like to use these ...
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0answers
405 views

Simpson's Rule derived from Trapezoidal Rule

I was just wondering if I could have some assistance in regards to the Trapezoidal Rule and Simpson's Rule. I have a question where it asks to generalize the Trapezoidal Rule to the case of ...
1
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1answer
367 views

Ei[x] Approximation

I'm working with the function $F(x)=e^{-k(x+1)}\int_1^x\frac{N^2}{t(N-t)}e^{kt}dt$. Breaking it down into into single fractions helps a little, yielding: $F(x)=Ne^{-k(x+1)} \int_1^x [\frac{1}{t} ...
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2answers
247 views

Continued Fractions Approximation

I have come across continued fractions approximation but I am unsure what the steps are. For example How would you express the following rational function in continued-fraction form: $${x^2+3x+2 ...
1
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1answer
309 views

acoustics under water

I've got the following problem that is taken from the numerical analysis book by Kahaner-Moler-Nash (P8-15): The speed of sound in ocean water depends on pressure, temperature and salinity, all ...
1
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1answer
200 views

Quadratic approximation of a cost function with a Taylor expansion

See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92. Given an instantaneous cost ...
1
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2answers
163 views

approximation formula for the integral

Get an approximation formula for the following integral: $$ \sum_{k=1}^n \left( \frac{1}{35} \right)^{k-1}\int_0^{\frac{\pi}{2}}\cos^{2(n-k)+1}(y) \cdot \sin^{2(k-1)}(y) \, dy $$
1
vote
1answer
877 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...
1
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3answers
1k views

efficient and accurate approximation of error function

I am looking for the numerical approximation of error function, which must be efficient and accurate. Thanks in advance $$\mathrm{erf}(z)=\frac2{\sqrt\pi}\int_0^z e^{-t^2} \,\mathrm dt$$
0
votes
2answers
77 views

Is there a good approximation for this?

What is a good approximation for $\dfrac{k!}{\binom{k^2}{k}}$ as a function of $k$? Is there a $k_0\in\Bbb N$ such that for all $k\gt k_0$, ...
0
votes
1answer
19 views

Ratio of step sizes in Richardson extrapolation for numerical integration

When using Richardson extrapolation for numerical integration, are there any criteria whether ratio between the steps should be or does it not matter what step size I use? For an integral I can write ...
0
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0answers
14 views

What is the distribution of 'max of some normaldistributions'?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
0
votes
1answer
29 views

How to construct a continuous function that is (mean) convergent to a given square integrable function

(In the Riemann Sense, this is a lemma before the Fouriers-Mean-Convergence Theorem) Suppose we have a square integrable function f:$[0,2\pi]\rightarrow \mathbf{C}$. We know that $\int_{a}^{b} f^2 ...
0
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1answer
35 views

Can a comparison network obtain all the n! permutations of a vector?

I want to permute a vector using comparison networks. This is the only method I have at my disposal. My original idea is to use a sorting network like Batcher or Bitonic. Basically I place my vector ...
0
votes
1answer
61 views

Uniform convergence and relative error?

I never took a class where uniform convergence was introduced, so I only know the Definition and not much more about it. I have an Approximation of some sequence of functions $f_n$ by some function ...
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0answers
41 views

Applications of low-rank matrix approximation

There was a similar question here Use of low rank approximation of a matrix that has unfortunately remained unanswered. Although being along the same lines, my question will be formulated in a little ...
0
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1answer
82 views

Maximum likelihood estimators

I have $X_1,X_2,\dots,X_n$ as random samples from a binomial distribution, with probability function: $$p_X(x) = Pr(X=x) = {m \choose{n}}\alpha^x(1-\alpha)^{m-x},x=0,1,2,\dots,m$$ where $m$ is given ...
0
votes
1answer
58 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
0
votes
2answers
60 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 ...
0
votes
3answers
127 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
0
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1answer
178 views

Proximal functions

I am a little bit new to proximal functions and I am currently stuck with the following problems How would I derive the prox function for the regularizer (h(x) function) : $\alpha\sum_{k+} $ and for ...
0
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1answer
840 views

Normal approximation to the log-normal distribution

Intuitively, it seems that a lognormal distribution with a tiny $\sigma/\mu$ ratio might look quite a bit like a normal distribution. Can this be formalized in any way (e.g., by stating upper bounds ...
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1answer
44 views

a question on decreasing sequence of subspaces II

This is related to this question see here Let $V= \mathbb{Q}^{\mathbb{Z}}$, considered as an infinite dimensional linear space over $\mathbb{Q}$. And assume $W=\mathbb{Q}^F$ is a finite dimensional ...
0
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1answer
150 views

Approximation for expected number of distinct values

When I draw n evenly distributed integer random numbers from an range of [0,m], what is the expected number of distinct values? I am aware of this answer, but is look expensive to compute. Is there ...
0
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3answers
688 views

Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$

How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?