# Questions tagged [approximation]

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).

2,921 questions
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### Where can I find the proof of Laplace approximation?

I want to find the exact proof of Laplace approximation using big-O notation, approximating something like $e^{M(x)}dx$, but couldn't find the mathematical proof. I also have tried myself, but couldn'...
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### Log-Sum-Exp as an approximation of min function

I can prove that the function: $$f(\tau, x_1, x_2, ..., x_N) = -\tau \log \frac{1}{N} \sum_{i=1}^{N} \exp{\left(-\frac{x_i}{\tau}\right)}$$ converges to $\min(x_1, x_2, ..., x_N)$ for $x_i \geq 0$ as ...
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### $\{a_n\}$ be a sequence such that $a_{n+1}^2-2a_na_{n+1}-a_n=0$, then $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in…

Let $\{a_n\}$ be a sequence of positive real numbers such that $a_1 =1,\ \ a_{n+1}^2-2a_na_{n+1}-a_n=0, \ \ \forall n\geq 1$. Then the sum of the series $\sum_1^{\infty}\frac{a_n}{3^n}$ lies in......
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### What does “quasi-optimal” approximation mean in Finite Element Method (FEA)?

In FEA, we know that optimal approximation usually refers to that the approximate solution converges to the exact solution with optimal rates. However, what is "quasi-optimal" approximation mean ...
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### What is a good estimate for this log sum?

Given $n\gg0$ what is a good estimate for $$\sum_{i=1}^K\log\Big(\frac{n}{3^i}\Big)?$$ I am particularly interested in case of $K=O(1)$ and $K=O((\log n)^c)$ at a fixed $c>0$.
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### Approximating a discrete function with continuous function. [closed]

$$\sum^n_{i=1}\frac{x^i}{(n-i)!}$$ I tried approximating it but i am not able to find any series similar to this . I need to find a function which can be used in place of this series. Thanks!
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### Find a value of $\ln 1.2$ with accuracy of $10^{-4}$

I know the formula that helps find an approximate value. In this instance it would be like $\ln 1.2 = \ln (1 + 0.2) \approx 0 + 1 \cdot 0.2 = 0.2$. But I need to find the value more precisely. I ...
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### Infinitesimal approximations in triangle yielding differing results

I am trying to prove the equations of motion of a pendulum from its energy equation, and I am obtaining different results depending on which infinitesimal approximation I choose. The idea here is to ...
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### How does WolframAlpha or other software get such precise values for zeta(3), etc.?

I've been looking with a friend at the values of zeta at the odd integers. WolframAlpha can give us over 100 digits in a second or two, but it seems that if you take the sum out to n, say, then you ...
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### Uniform approximation of $L^2$ basis by smooth functions with bounded derivatives of all orders

Let $\mathcal{F}=\{f_i\}_{i\in\mathbb{N}}$ be an orthonormal Hilbert basis of $L^2[0,1]$. I am wondering whether it is possible to approximate the $f_i$ uniformly across $i$ in the $L^2$-norm by ...
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### Inconsistencies when approximating to the nearest whole number

Does 22.449 approximate to 22 or 23? If we see it one way $22.449≈22$ But on the other hand $22.449≈22.45≈22.5≈23$ Which one is correct?
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### Multiplying a normal distribution by a log-normal distribution

I need direction to approximate the resultant probability distribution of the product of two independent distributions: $N(\mu, \sigma^2)$ and $lognormal(\mu_{N}, \sigma_{N}^2)$, where $\mu_{N}$ is ...
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### Making sense of a probabilistic existence argument (set cover-like problem)

The following are excerpts from a paper by Karger and Motwani. Given this, they prove the following result: My question is: how do they derive the conclusion in the fourth sentence of their proof ...
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### Kalman filter parameters

I use a 1D Kalman filter for my task. Picture with filter. I get the source signal (orange) and filter it (black). Is it possible to set the filter parameters so that it display the this result - ...
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### Triangular waveform modelling of scatterplot

I have a scatter plot of Voltage vs. Time that looks like this: scatterplot of voltage vs time. The points were collected with analog to digital conversion with a data logger sampling at a constant ...
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### Approximating a Banach space “vector- valued function” by “simple functions”

I'm trying to prove the following claim: Let $T$ be a compact (metric) space, and let $\mathcal{X}$ be a Banach space over $\mathbb{K}$. Let $f : T \longrightarrow \mathcal{X}$ be a continuous ...
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### Approximating square root for long expressions

I'm currently working on a problem which asks me to calculate the potential energy of a three spring system arranged in an equilateral triangle constrained to move in the x-y plane. As a consequence ...