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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2answers
60 views

find which two points an arbitrary point is nearest to

I have a line segment of connected points (a path in 2D), and a point $P$ that is not calculated based on this segment, although I can guarantee that the point will be placed along the path. Based on ...
3
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1answer
46 views

Alpha max plus beta min algorithm for three numbers

There exists fast algorithm to approximate length of 2D vector - Alpha max plus beta min algorithm. It says that $\alpha\cdot\max(x,y)+\beta\cdot\min(x,y)\approx\sqrt{x^2+y^2}$ for some constants ...
2
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2answers
38 views

$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$

I seek to prove the identity $$\int_2^x\frac{dt}{\log^kt}=O\left(\frac{x}{\log^kx}\right)$$ I was given the following hint: Split the integral into $\int_2^{f(x)}+\int_{f(x)}^x$ for a ...
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0answers
8 views

Approximating a grid-valued signed distance function with a continuous function

I want to solve a continous optimization problem using IPOPT. My optimization involves a signed distance function whose values are defined on a 2D grid. Since IPOPT can't handle piecewise functions, I ...
1
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1answer
32 views

Using SVD to approximate matrix-vector multiplication?

Given some matrix A, is it possible to use Singular Value Decomposition to approximate Ax for some vector x within some error bound? According to Efficient low rank matrix-vector multiplication, it ...
1
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1answer
40 views

Expectation of trigonometric functions involving random variables.

This is more a formulation question. I need help making a sales pitch (lol). I am working on an practical engineering problem where I encounter functions of the form: $\cos(\phi + d_\phi)$, $ ...
1
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0answers
21 views

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c)

how to approximate compute p(a|b, c) , using p(b|c), p(a|*, c), p(a), p(b), p(c) p(a|*, c), * means anythins. p(a|*, c) = $\sum_i p(a|i, c) $ error = | p(a|b, c) - f(p(b|c), p(a|*, c), p(a), p(b), ...
3
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1answer
29 views

Trapezoidal Rule in Stirling's Approximation

https://en.wikipedia.org/wiki/Stirling's_approximation Here $$\sum \limits_{i=1}^n log(i) $$ is approximated as $$\int^n_1log(x) dx\ +\ 1/2\ log(n)$$ but I would approximate it as ...
4
votes
3answers
108 views

Calculate fractional part of square root without taking square root

Let's say I have a number $x>0$ and I need to calculate the fractional part of its square root: $$f(x) = \sqrt x-\lfloor\sqrt x\rfloor$$ If I have $\lfloor\sqrt x\rfloor$ available, is there a ...
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0answers
7 views

Mapping Function from Generated Data

Assuming solution space data is 1:1 mapped to domain space, is there a good/well known approach to flushing out a mapping function/approximation having access to lots of mapped data? I imagine I am ...
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1answer
31 views

Finding the integer parts of irrationals

When working with continued fraction expansions, I sometimes have to calculate the integer part of irrationals quickly without a calculator, what would be an effective way to do this? For example, ...
1
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1answer
22 views

Approximation Error of Stirlings Formula

Stirlings Approximation : $n! \approx \sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$. So $100!$ has an approximate percentage error of about $\frac{100}{12n} = \frac{1}{12}$. Using this information, how does ...
0
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2answers
57 views

what is the best approximation for sine?

can you tell me which is the best approximation for cosine/sine functions. It should also reduce the computational complexity. I've already tried the Bhaskara-1 approximation. Can you suggest me ...
2
votes
1answer
36 views

Approximating non-rational roots by a rational roots for a quadratic equation

Let $a,b,c$ be integers and suppose the equation $f(x)=ax^2+bx+c=0$ has an irrational root $r$. Let $u=\frac p q$ be any rational number such that $|u-r|<1$. Prove that $\frac 1 {q^2} \leq |f(u)| ...
1
vote
1answer
14 views

Linearization of an implicitly defined function

$f(x,y,z)=e^{xz}y^2+\sin(y)\cos(z)+x^2 z$ Find equation of tangent plane at $(0,\pi,0)$ and use it to approximate $f(0.1,\pi,0.1)$. Find equation of normal to tangent plane. My attempt: I found that ...
7
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1answer
157 views

Accelerating approximations for arccos

I have recently built a method to accelerate drastically the accuracy of the following approximation of $\arccos(x)$ : $f_n(x)=2^n\sqrt{2-2g^{n-1}(x)}$ where $g(x)=\frac{1}2\sqrt{2+2x}$ and ...
1
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3answers
44 views

Curve fitting as a linear least squares approximation problem?

So I have a problem from a textbook that will count for a bonus homework assignment, but I am having some trouble knowing where to start. Some more difficult curve fitting problems can be ...
0
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0answers
14 views

Is there a way to calculate expected entropy and confidence bounds in this unknown multinomial parameter problem?

Given low-count and fairly low dimensional count data (non-negative integers) $n_1,\ldots,n_d$ where say $d \leq 5$ and $n_i \leq 10$, the goal is to compute the posterior mean and posterior ...
2
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1answer
23 views

Assymptotically approximating a sum similar to binomial.

I am, through some combinatorial problems which I'm working on, trying to figure out what the following sum becomes as $n\rightarrow \infty$: \begin{equation*} \sum_{i=1}^{n-1} ...
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0answers
38 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...
0
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1answer
61 views

How to approximate Heaviside function by polynomial

I have a Heaviside smooth function that defined as $$H_{\epsilon}=\frac {1}{2} [1+\frac {2}{\pi} \arctan(\frac {x}{\epsilon})]$$ I want to use polynominal to approximate the Heaviside function. ...
4
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2answers
2k views

Is this a valid attempt at the Riemann Hypothesis? [closed]

From Marcus Du Sautoy's book “The music of the primes”, there is a method of finding a very long list of N consecutive numbers which are not primes. e.g $101!+2, 101!+3,...,101!+101$ all of which are ...
1
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1answer
25 views

Origin of divergence in a divergent field (2D)

I have a field of measured vectors, see example of four vectors in image below. If there was no noise they would all point outward exactly from one "central point". i.e. there would be a circle whose ...
1
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0answers
24 views

Speed of the usual approximation of the exponential

Let's consider the usual approximation of the exponential function $f_n(x)=(1+\frac{x}n)^n$. What do we know about its speed of convergence to the exponential? That is to say, how can we characterize ...
2
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4answers
260 views

Explain this inequality, related to logarithms

I am trying to understand a proof of Stirling's formula. One part of the proof states that, 'Since the log function is increasing on the interval $(0,\infty)$, we get $$\int_{n-1}^{n} \log(x) dx ...
0
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1answer
32 views

approximation of $x^2$ in hilbert spaces

use the least squares to find the best linear approximation to $f(x)=x^2$ on [-1,1]. that is find the line $y=a_0+a_1x$ that minimizes $\int_{-1}^1|f(x)-y(x)|^2$ solution I used the theory of ...
1
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0answers
37 views

How to find the upper bound of a binomial coefficient by using binomial theorem?

I have a task to find a upper bound of the binomial coefficient for all $r \leq \frac{n}{2}$. I've already obtained by using such relation: $$\frac{\sum_{k=0}^{r}\binom{n}{k}}{\binom{n}{r}}$$ Which ...
3
votes
3answers
70 views

How can a binomial coefficient can be approximated by using Stirling's formula?

I've met some difficulties with such question: How can we approximate a binomial coefficient by using a Stirling's factorial approximation. I've evaluate a little bit and got this How can I ...
1
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2answers
45 views

Can every continuous function on complex domain be approximated by polynomials pointwise?

Do you know any theorem that will help me with this question: Let $f$ be any continuous function on complex plane. Show that there is a sequence $(P_n)$ of polynomials such that $P_n$ converges ...
1
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1answer
56 views

Maclaurin Series Approximation of $\sin{x}$

Use first ten terms of the Maclaurin series for $\sin{x}$ to find an approximation to the values of both $\sin{\left(\frac{6\pi}{7}\right)}$ and $\sin{\left(\frac{20\pi}{7}\right)}$? One can say that ...
0
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1answer
21 views

Local Approximation of Real Valued Functions

I'm unsure where to begin. Any guidance would be greatly appretiated. Suppose that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ has continuous second order partial derivatives, and at the origin ...
0
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1answer
45 views

Show that S is a cubic spline (natural or clamped)

Please see question. I believe the answer should be: $S_0(2)=\frac12(x^3-3x+2)=2$ $S'_0(2)=\frac12(3x^2-3)=\frac{9}{2}$ $S''_0(2)=\frac12(6x)=6$ $S_1(2)=\frac12(x^3-12x^2+45x-46)=2$ ...
0
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0answers
24 views

expectation approximation

Note: You don't have to understand Approximation Algorithms to answer this Hello. I need to prove an algorithm approximation by using expectation. The algorithm takes $x_i \in {0,1,2}$ such that ...
0
votes
1answer
13 views

error in approximation a monotonic function in L^1

I am trying to solve this problem, but I am getting an incorrect solution. Here is the problem and my approach: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize ...
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
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1answer
29 views

How to approximate a normal distribution?

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
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1answer
27 views

Finite differences matrix and integrals

Let $f:[a,b]\to \mathbb{R}$ a smooth function. Consider a partition $a=x_1<x_2<\ldots<x_n=b$. If we put $X=(f(x_1), f(x_2), \ldots, f(x_n))$, where $x_{i+1}-x_i=\Delta x$ then: $ (f'(x_1), ...
2
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0answers
35 views

Can trigonometric functions for double precision be implemented in terms of those for single precision?

In some program environments like GLSL there is support for single and double precision numbers for arithmetic and square roots computation, but only single precision trigonometric functions are ...
2
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0answers
22 views

Extension of laplace method's

It is well known that the integral $\int_a^b e^{-n \cdot f(x)}dx$ can be approximate by $\sqrt{\frac{2\pi}{n|f''(x_0)|}}e^{-n\cdot f(x_0)}$ at $x_0$ the maximum of $f(x)$ in $(a,b)$ (for large $n$..). ...
1
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1answer
39 views

Angle between slopes of a curve

I am trying to understand what the change in angle of the slope of a curve means. It is hard to explain with words so here's an image that should help. The red curve has had its derivative ...
2
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3answers
121 views

Approximation of a sum $\sum ^{\sqrt{n} }_{k=5}\frac{\log\log(k)}{k\log(k)} $

What method could I use to obtain an approximation of this sum $$\sum^{\sqrt{n}}_{k=5}\frac{\log\log(k)}{k\log(k)}$$ Should I proceed by an integral? How can I calculate its lower and upper bound?
0
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1answer
16 views

approximation using floating point arithmetic

Let $x=2.14366$ and $y=2.14363$ and $d=x-y.$ If $d*$ is the value of d computed using $5-$digit decimal floating point arithmetic, find the relative error. For this question I know how to calculate ...
0
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3answers
42 views

Approximation of an expression (no calculator please!)

Today I had my college admission exam, It was good, but there was a question which I found a bit interesting (but unable to solve at the moment). It says, Question: Find the positive integer which ...
0
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2answers
37 views

Can step functions approximate trigonometric functions?

I have read a notion from a number of different sources simply stating that a step function can approximate any trigonometric function. I am not convinced by simply reading this notion, for example ...
0
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0answers
55 views

Equivalence of two linear least squares problem

Here I want to build a subspace representation $Uq$ to approximate $x$, where $q$ is the reduced coordinates. We know that the best approximation to $x$ is the linear least squares solution $q_1 = ...
2
votes
1answer
32 views

Numerical approximation of differentiation

I have the following task to solve: Let $b>x$ be defined, determine $w_0,w_1$ and $w_2$ in dependency of $b$ such that the approximation $f''(x) \approx w_0 f(x-h) + w_1 f(x) ...
3
votes
3answers
260 views

Integral involving Bessel functions of the first kind

I am stuck with the following integral. Does it converge? $$ \int_{0}^{\infty}\left(J_1(x)^2+J_1(x)J_1(x)^{''}\right)\text{d}x $$ According to tables I find that the first term is divergent, so I ...
1
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1answer
67 views

Need help with a Crank Nicholson Method example problem.

I have an exam coming up and the professor released the sample test containing a Crank Nicolson question. I was out of town for those two lectures, so I missed the information. Even though I have ...
0
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1answer
25 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
2
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1answer
36 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...