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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1answer
27 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 ...
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1answer
29 views

An obstacle encountered in a proof of the existence of a best approximating polynomial of degree $\leq n$

Let $n \in \{0, 1, 2, \dots\}$, let $a, b \in \mathbb{R}$ be such that $a < b$ and let $f \in \mathcal{C}[a, b]$ be a real function that is continuous on the non-degenerate, compact interval $[a, ...
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2answers
86 views

Terms needed to approximate with given error?

How many terms of this series would one need to add to get an approximation of $\pi$ with error less than $10^{-4}$? $$ 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots $$ So far, I wrote the ...
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1answer
14 views

Approximation of monthly payment using Taylor expansion

I am trying to understand what does APR(annual percentage rate) and how it is calculated. Thanks to Wikipedia, I got the formula of monthly payment for a fixed rate multi-year mortgage in the ...
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0answers
20 views

Numerical integration to find mean arterial pressure, as used in cardiovascular physiology

In physiology the mean arterial pressure (MAP) is calculated as $MAP = DP + \frac1{3}PP = DP + \frac1{3}(SP-DP)$ (where PP is pulse pressure, DP is diastolic pressure and SP is systolic pressure), and ...
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3answers
2k views

How to convert $\pi$ to base 16?

According to this Wikipedia article $\pi$ is approximately 3.243F in base 16 (i.e. hexadecimal). Can someone explain this? (Note: I understand how to convert an integer to base 16) Thanks
3
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1answer
49 views

Bounds on Maclaurin series of $e^{-x^2}$

This is a problem from a textbook: By taking the 4th degree Maclaurin polynomial for $e^{-x^2}$ find an approximation to $\int^1_0 e^{-x^2} \text{dx}$. Place bounds on the error in this ...
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0answers
17 views

Maclaurin polynomial error term

this is a problem from a textbook, What degree Maclaurin polynomial of $e^x$ must be taken to guarantee an estimate of $e$ to within $1 \times 10^{-6}$ The answer from textbook is $n=17$, but I ...
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0answers
27 views

Computing integrals in order to find an approximation function

For a project in scientific computing I am trying to find an approximation of an unknown function $f(x)$. Given: data points $(x, f(x))$ A basis with which we can approximate $f(x)$ consists ...
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0answers
33 views

Uniqueness of best approximation. (the sketch of the proof)

Let $X$ be a compact Hausdorff space. Let $A = C(X)$, the space of real-valued continuous functions with supremum norm. Prove : if $X$ has at least 2 points, then there is a one-dimensional subspace ...
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2answers
55 views

Integral approximation - [closed]

Whole day I can not figure out how can be proved the equality: $$\int_0^1 x^2 dx = \frac{1}{n} \sum_{i=1}^n \left(\frac{2i-1}{2n}\right)^2 + \frac{1}{12n^2}$$ Can someone help me, what should I use ...
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0answers
22 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
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0answers
22 views

the set of Best Approximation is a norm-closed convex set.

Let $V$ be a normed vector space. Let $W$ be a proper closed subspace of $V$. Let $M$ be the set of best approximations in $W$. Prove that $M$ is a norm-closed convex set. I've shown that the ...
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0answers
15 views

how to understand 'expected max approximation error'

The background is that: E() denotes the expectation and $y$ satisfies a certain probability distribution $g(y)$, then we independently sample $y_1,y_2$ from $g(y)$. It is assumed that $E(y_1-E(y))=0, ...
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0answers
18 views

Approximation of a trisecting an angle

I learned a proof that it is impossible to trisect an angle. Is there some research that if we have been given an angle, a ruler and a compass and we are allowed to draw $m$ circles and $n$ lines/line ...
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2answers
48 views

Why does this approximation of square roots using derivatives work this way?

I came up with this way to estimate square roots by hand, but part of it doesn't seem to make sense. Consider how $f(n) = \sqrt{n^2+\varepsilon} \approx n$ when $\varepsilon$ is small. Therefore, ...
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0answers
83 views

set cover problem: d-cover

I've the following problem: Let the universe U and a list of subsets (S_1,..., S_k), be the usual input for the (unweighted) SET COVER PROBLEM, Consider the following doubling scheme for producing a ...
0
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1answer
54 views

Error of lagrange interpolation

If the original function I want to approximate using Lagrange interpolation is a polynomial the error function $(x-x_{0})...(x-x_{n})\frac{f^{(n+1)}(\xi)}{(n+1)!}$ is not working because the $n+1$ ...
1
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1answer
20 views

Can $e^\frac{1+z}{z-1}$ be uniformly approximated in the disc by harmonic continuous functions

Ie: For $f(z)=e^\frac{1+z}{z-1}$ is it true: $\inf\{||f-u||: \mbox{u is harmonic in the unit disc and continuous on the closed unit disc}\}=0$. Note the infinum is taken over the interior (ie: ...
2
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1answer
23 views

Boundary of the limit when $a<<1$

Let's say I have the expression $-b+\frac{3}{2}b\cdot a^2$. Can I say that by taking the limit when $a<<1$, that expression is $\approx -b$ ? can a constant number, like $\frac{3}{2}$ can "ruin" ...
1
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1answer
38 views

Numerical Approximation of Differential Equations with Midpoint Method

I want to proof that the local truncation error of the Midpoint Method is $d_{k+1}=O\left(h^{3}\right)$ Approach The local truncation error is defined as: ...
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3answers
32 views

Use a linear approximation to estimate the given number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
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2answers
51 views

Use a linear approximation to estimate the number $64.07^{2/3}$

That's all they give you. I tried putting it into the linear approximation equation of: $$ f(a)+f'(a)(x-a) $$ but I get almost the same value as $64.07^{2/3}$, which is around $16.0116$. Just not ...
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0answers
15 views

How to analytically find these rounding issues

Let's say we have a fixed yearly amount that we have to divide equally among an amount of days. For instance for $1,600 we may have: ...
1
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1answer
24 views

Approximate using differentials when partial derivatives are given?

I have ran into this problem on my online math assignment, this week we are covering partial derivatives and higher order partial derivatives, but I don't think I have learnt anything that can help me ...
2
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1answer
74 views

least squres vs. lagrange interpolation

can some one tell me the differences between these two approximation techniques, what are the strengths and weaknesses of each, and which better one to use. Thanks
0
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1answer
27 views

How does $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximate $\mathbf{H}$?

Page 3 of a guide to Levenberg-Marquardt optimization says that $\left[\nabla f\left(\mathbf{x}\right)\right]^\mathrm{T} \nabla f\left(\mathbf{x}\right)$ approximates the Hessian matrix of $f$. I do ...
3
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0answers
21 views

Estimating special values of the Riemann zeta function on the critical line

If $p,q$ are primes, is it necessarily true that $$\left|\zeta\left(\frac{1}{2} + i\frac{p}{q}\right)\right| > (p+q)^{–(p+q)} ?$$ (Here $\zeta$ is the Riemann zeta function.)
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0answers
37 views

Approximation to the negative-binomial and negative-hypergeometric distributions

It is known that a binomial distribution can be approximated by a normal variable with the same mean and variance, for sufficiently large $n$. What approximations are known for the negative binomial ...
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1answer
48 views

Approximate an integral using Monte Carlo method

I have a question on an assignment Calculate the value of the integral I = $\int_0^\pi sin^2(x)dx$ using the Monte Carlo Method (by generating $ 10^4 $ uniform random numbers within domain [0, ...
2
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3answers
32 views

Errors and Taylor Polynomials

For $g(x)=x^{1/3}$, $a=1$, degree $3$ I found the Taylor polynomial: $$p_3(x) = 1 + (x-1)/3 - ((x-1)^2)/9 + (5(x-1)^3)/81$$ How do I use the error formula for the Taylor polynomial of degree 3 to ...
0
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1answer
43 views

if $x\ll 1$ is it safe to assume that $x\ll \frac{1}{2}$

I know that: if $x\ll 1$ then we can write $\frac{x}{x+1}\rightarrow x$ but is it safe to write $\frac{2x+1}{x+1}\rightarrow 1$?
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0answers
53 views

Historical calculations of $tan^{-1}x $ and $e^x$

SineBhaskara_I One reads that $tan^{-1}(x) $ series expansion existed in early (Indian) history. But like the Sine trigonometric function, did any similar approximation exist as well? The query ...
2
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4answers
137 views

Parabolic sine approximation

Problem Find a parabola ($f(x)=ax^2+bx+c$) that approximate the function sine the best on interval [0,$\pi$]. The distance between two solutions is calculated this way (in relation to scalar ...
0
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1answer
35 views

Uniform Approximation by Finite Wavelet Sum

Suppose $\psi:\mathbb{R}\rightarrow\mathbb{C}$ is a rapidly decreasing, bounded function with zero integral. For $j,k\in\mathbb{Z}$ define a function $\psi_{jk}(x):=\psi(2^{j}x-k)$. Suppose is the ...
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1answer
23 views

Second Order Approximation for a Polynomial

if I have an expression: $L=\frac{12a^3d^3-4wa^3d^2+16a^2d^2-4wa^2d+6ad+1}{12a^3d^3-4wa^3d^2-4a^2wd+16a^2d^2+7ad-aw+1}$ what is the second order approximation in $\frac{d}{w}$? I know that ...
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0answers
16 views

Approximating an integral with a change of integral

(I have previously found out $x_1 = -\frac{1}{\sqrt{3}}$ and $x_2 = \frac{1}{\sqrt{3}}$ ) Approximate an integral using the 2-point rule, with an appropriate change of integral, to approximate ...
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0answers
28 views

Trapezoidal Rule Error Bounding with the Absolute value of x

So, I am attempting to find a large enough n to allow for the Error from evauluating the Trapezoidal Rule to be less than 1/100. I know the equation is K(a-b)^3/12n^2 > 1/100 however I am running into ...
2
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1answer
43 views

Estimating accuracy of Taylor series approximations with 2 bounds

I have a question from a previous exam as such: Use Taylor's Inequality to estimate the accuracy of the approximation $f(x) \approx T_{3}(x)$ when $0.8 \leq x \leq 1.2$. I computed from an ...
3
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5answers
133 views

Approximation of $\sqrt{ x + y } - \sqrt{ x - y }$

I've been struggling to try and find a way to approximate the function: $\sqrt{ x + y } - \sqrt{ x - y }$ I should mention that $y$ is positive and a small number, so that $0<y<<1$. What ...
3
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2answers
51 views

function approximation using series

I was trying to approximate a function by another one using some kind of a series. Let $$ f(x) = m\cdot \left(1-\sqrt{\frac x2\biggm/\left(1+\frac x2\right)}\right) $$ I'm trying to approximate ...
0
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1answer
22 views

For small x, Taylor Series to determine constants n and C in the approximation

cos(x) - e^(-(x^2)/2) I have tried writing the taylor expansion for cosine and e but do not know how to combine them in one. In addition to this, i do not know how to approach the problem whilst ...
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4answers
182 views

How to compute a lot of digits of $\sqrt{2}$ manually and quickly?

After having read the answers to calculating $\pi$ manually, I realised that the two fast methods (Ramanujan and Gauss–Legendre) used $\sqrt{2}$. So, I wondered how to calculate $\sqrt{2}$ manually in ...
0
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1answer
39 views

Messing with the linear approximation… I don't get Taylor's formula??

Alright, so I was messing around with the linear approximation for a function; I wanted to see if I could get the formula for the 2nd degree approximation from it. I went about it like this: $$ f(x + ...
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0answers
25 views

Superstrong approximation

I am reading up on research in approximate groups and have noticed that one of the reasons for doing this research is because it has applications in superstrong approximation theory. I'm more or less ...
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0answers
26 views

Approximation using Stirling

In the article "Series evaluation of Tweedie exponential dispersion model densities" by Peter Dunn and Gordon Smyth it is stated that $$-\log\Gamma(1+j)-\log\Gamma(-\alpha j)\approx ...
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1answer
63 views

Approximation for $ e^{ - x^2 } $ , x>0.

what is the good approximate so that it works for a large range of values. My purpose is to calculate logarithm of likelihood ratios. $ \log \left( {\frac{{e^{ - x_1 ^2 } + e^{ - x_3 ^2 } }} {{e^{ - ...
0
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1answer
62 views

how to approximate this expression $\frac{1}{8}x^2(1-\frac{1}{12}x^2)/(1-\frac{1}{4}x^2)$

when x is small, for example <1, then the expression can be approximate by (from a book) $$ g(x)= \frac{-x^2}{8}{\frac { \left( 1-1/12\,{x}^{2} \right) }{1-1/4\,{x}^{2}} }= \frac{-x^2}{8} ...
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0answers
55 views

How to derive this integral?

I have relatively simple integral but I could not figure out how to solve. It is $$ \int_{0}^{1}dz \frac{(1-z)(1-z^2)A^2}{B^2z+(1-z)^2A^2} $$ $$B>>A$$ EDIT: You should use an approximation to ...
2
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2answers
155 views

Approximating $\ln(1+\exp(x)+\exp(y))$

Ok, so I know that $\ln(1+e^x)\approx x$ when $x$ is large. But what about $\ln(1+e^x+e^y)$ when both $x$ and $y$ are large? I can figure out cases when $x\gg y$ or $y\gg x$ since that simplifies ...