Mathematics Stack Exchange
Mathematics Stack Exchange

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).

Approximations are representations of numbers that aren't exact, for example $\sqrt{2}\approx 1.41$. Such representations may be obtained using differentials (more generally, Taylor's formula), linear interpolation, etc.

4607 questions
2
votes
3 answers

Approximation of $\frac{1-x}{1+x}$ if $x \ll 1$?

In an exercise in general relativity, I am trying to show that in the limit as $\hat{r}\rightarrow\infty$, \begin{equation*} \frac{1-GM/2\hat{r}}{1+GM/2\hat{r}} \approx 1 - \frac {2GM}{\hat{r}},\ \ \text{or}\ \ \frac{1-\phi}{1+\phi} \approx 1-4\phi\…
Frank Harris
  • 131
2
votes
1 answer

Basel Problem approximation error bounded by $\mathcal O(1/x)$?

In this answer it is stated that $$ \sum_{n\geq1}\frac{1}{n^2}=\sum_{n\leq x}\frac1{n^2}+\mathcal O(1/x). $$ Is this statement true as $x\to\infty$? What I've done: If $x$ is fixed, then I think the answer is almost trivial, because we may set…
nfhplrpd
  • 55
2
votes
0 answers

numerical approximation to logarithm

we know that $$ \ln(x)=\frac{1}{x} \int_{0}^{1}dt \frac{1}{1+xt} $$ then given a cuadrature formula inside $(0,1)$ is that true $$ \ln(x)= \frac{1}{x}\sum_{i}\frac{w_{i}}{1+xt_{i}} $$ wht other rational approximations of logarithm are useful based…
Jose Garcia
  • 8,506
2
votes
1 answer

Computing the specific relative error

Suppose that $x$ is the exact value of some quantity and $\hat{x}$ is the approximate value of this quantity. The numbers $$\Delta(\hat x):=|x-\hat x|,$$ $$\delta(\hat x):=\frac{\Delta(\hat x)}{|\hat x|}$$ are called absolute and relative errors,…
RFZ
  • 16,814
2
votes
1 answer

Understanding weighted linear least square problem

I am having difficulty in understanding about weighted linear least squares. Could anybody explain me instead of minimizing the residual sum of squares why we need to minimize the weighted sum of squares? Further, I want to know about the term…
Srijan
  • 12,518
  • 10
  • 73
  • 115
2
votes
1 answer

Approximating a log-power function

I can't figure out how the following approximation has been done, I would appreciate any…
astroboy
  • 209
2
votes
1 answer

Approximating the solution to $2^x=x^{200}$

I'm wondering if there exists some numerical method (maybe using a series) for approximating the solution to an equation like $2^x=x^{200}$. By rewriting both sides of this equation we have $e^{x\ln 2}=e^{200\ln x} $, so the curves intersect when…
math_guy
  • 83
2
votes
1 answer

Approximation of a function

Let $f:[0, 1] \rightarrow [0, \infty]$ be a function of $x$, with a parameter $\theta > 0$, such that $f$ is continuous $f$ is strictly decreasing $f(0) = \infty$ $f(1) = 0$ For example, $f(x) = (- \log(x))^{\theta - 1}$. For a given $\theta$, is…
user7064
  • 313
2
votes
2 answers

best real approximation to complex numbers

I have a system of equations and its answers are complex, but I want real numbers. Is there any way to find the best real approximation to a complex number?
sima bh
  • 21
2
votes
3 answers

Why is $\sqrt{1 + x^2}$ approximately equal to $1 + \frac{x^2}{2}$?

I saw this in Shankar’s Physics book and couldn’t make out the reasoning behind it. I would assume the dx and derivative have nothing to do with it. https://i.stack.imgur.com/D21AW.jpg
Jon Person
  • 97
2
votes
0 answers

Is there a name for this iterative formula for finding near-miss approximations for the square root of X?

I noticed a pattern in near-miss approximations for $\sqrt 2$ and eventually figured out a general iterative formula for the square roots of whole numbers: $\frac {Nn + XDd} {Dn + Nd}$ Where $X$ = the value we are approximating the square root…
Tory
  • 93
2
votes
0 answers

EXP(x) approximation in old 1980's computer ROM

The presentation : Old 1980’s ROM (Apple 2e, Commodore 64, ...) uses a Taylor’s series-like to evaluate the exponential function EXP(x) : EXP(x) = 1 + x + 1/2! x^2 + ... + 1/7! x^7 x is in [-1, 1], thanks to a preliminary work. Another preliminary…
jpcohet
  • 29
2
votes
2 answers

Smooth approximation of three-phased linear models

I am looking for a smooth (continuous differentiable) approximation of the following two three-phased functions with breakpoints at $B_1$ and $B_2$:$$ y_1(x, B_1, B_2, a, b) = \begin{cases} a; & x < B_1\\ a + b(x - B_1); & B_1 \leqslant x \leqslant…
Tom Wenseleers
  • 103
  • 8
2
votes
1 answer

approximate a square function with a linear one

I have to code a function in matlab (F1) whose values range from 0.740261423849103 to some where around 0.95. Then there is another function (F2) which is usually the square of F1. Is there any way I can replace F2 [ which is F1^2] by a linear form…
user13267
  • 365
2
votes
1 answer

Finding a value of $a$ to satisfy an expression of the form $a*(1-\frac{1}{b})^{(a-1)} = r$

Consider the following expression: $a*(1-\frac{1}{b})^{(a-1)} = r$ Provided some real number value for $b$, I need to find a positive real number $0 < a \leq b$ to satisfy the above equation, where $0 < r < 1$. Must we appeal to an approximation for…
PartiallyCovered
  • 23
1 2 3
…
16 17