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).

learn more… | top users | synonyms

3
votes
0answers
61 views

When can the commutator of two matrices be neglected in series expansions?

Under what conditions can the higher order commutators in the Baker–Campbell–Hausdorff formula be neglected when the commutators does not vanish exactly and there is no small parameter in the ...
3
votes
1answer
142 views

Iterative model fitting

I have a sequence of points $\{(x_k,y_k,z_k)\}$ and I need to fit some $2D$ model $P(x,y)$ that approximates $z$ in some sense. The $z_k$$'s$ are noisy samples of some $2D$ function $z_k = f(x,y) + ...
4
votes
5answers
384 views

Improving Newton's iteration where the derivative is near zero?

I'm implementing a root-solver for finding x coordinates of a function f(x), after I have an y-coordinate. The function is periodic, roughly sinusoidal with constant amplitude but non-linearly ...
8
votes
1answer
178 views

Is this sequence convergent?

I heard this question from a professor a couple years ago. I still think about it... Does the sequence $(a_n)_{n\in \mathbb N}$ with $$a_n=\sqrt[n]{|\sin(n)|}$$ converges ( to $1$ ) ? I believe ...
2
votes
1answer
94 views

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 ...
15
votes
1answer
384 views

A cute approximation for $\cot(2\pi x)$(!?)

Numerical calculations and some theory leads to the suggestion that $$\cot(2\pi x) \rightarrow\frac{1}{2\pi}\sum_r \frac{1}{x-r}$$ where $r$ ranges over all the roots of $B_{2n+1}$ (Bernoulli ...
0
votes
1answer
34 views

approximation on a graph

I expect my question to sound very naive, please excuse for this. I have a set of $(x, y)$ points on a graph, which are the measurements of a real-life process. Now I want to draw an "approximation ...
2
votes
1answer
984 views

Method of dominant balance

Find the leading asymptotic behaviour as $x \rightarrow \infty$ of $$x^2y'' + (1 + 3x)y' + y = 0 $$ Can someone kindly explain me how to solve this problem? Im learning asymptotic analysis, and I ...
1
vote
1answer
128 views

approximating function $a+b\cdot 2^x$

Hi I need to implement a function $a+b\cdot 2^x$ in a highly resource constrained device. a and b are constants and x is a variable. How do I go about finding a simplified version of this expression ...
3
votes
1answer
87 views

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$.

If $f \in\operatorname{Lip}_K[a, b]$, show that $f$ can be uniformly approximated by polynomials in $\operatorname{Lip}_K[ a, b]$. Context: $f \in \operatorname{Lip}_K[a,b]$ then it is ...
2
votes
1answer
487 views

Rounding .5 - why isn't rounding away from zero the 'right' answer?

I am familiar with the issue of 'how should one roung .5?', and I am familiar with the conventional solutions, but I don't understand why there isn't a correct answer. When you're formulating a ...
2
votes
1answer
152 views

approximate error between integral an sum

I am new here. My problem: There is an integral $I:=\int_0^1 f(x)\,dx$ for $f\colon [0,1]\to\mathbb{R}$ and I want to compute it by ...
0
votes
1answer
92 views

Approximating a simple sum

Can somone help me find an assymptotic formula for n, for fixed x , for this sum , perhaps an inequality would be even better, or some bound on the error. $$\sum_{k=1}^n \frac{1}{\log(kx)}$$ I need ...
1
vote
1answer
99 views

understanding the least squares criterion

I was given 20 data points and asked to choose the most suitable lowest degree polynomial to fit them using the least-squares criterion. I looked it up, but what i found seems far too complex or just ...
12
votes
1answer
255 views

Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.

This integral is very closely related to the sophmores dream that states $$ \int_0^1 x^{-x}\mathrm{d}x = \sum_{n=1}^\infty n^{-n} = 1.27\ldots $$ For example here ...
2
votes
0answers
570 views

Approximation of integral using series expansion of the integrand.

I have a smooth function $x \rightarrow f_\epsilon (x)$ on $x\in[-1\ldots 1]$ (dependent on the continuous parameter $\epsilon$) and I want to approximate the integral $$ I=\int_{-1}^1 f_\epsilon ...
2
votes
1answer
45 views

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, ...
1
vote
0answers
29 views

clairification on standard deviation

I have a homework question that gives me a set of $x$ values and their respective $f(x)$ values and asks me to find the line which best fits the data. I have to do this by finding the estimated ...
2
votes
2answers
301 views

Using differentials to approximate a function

So I have a homework problem that I cannot figure out. I am supposed to approximate the value of $\sqrt{(4.98)^2-(3.03)^2}$ using differentials. What I have so far is $$f(x,y)=\sqrt{x^2-y^2}$$ ...
4
votes
2answers
127 views

Linear regression where undershooting isn't as bad as overshooting

Given a set of points $(x_i, y_i)$, least-squares linear regression finds the linear function $L$ such that $$\sum \varepsilon(y_i, L(x_i))$$ is minimized, where $\varepsilon(y, y') = (y-y')^2$ is the ...
2
votes
5answers
993 views

Rough y(x) approximation for simplified Cubic Bezier curve

I need to get a very rough (and fast) $y(x)$ approximation of a simplified Cubic Bezier curve to use in my animation code, where there's only one control variable: $$ P_0 = (0, 0)\\ P_1 = (0, 0)\\ ...
8
votes
3answers
340 views

difference of square roots approximation

In two of my physics courses in the past week, I've come across an approximation for the difference of two square roots for large radicands: $\sqrt{x+a}-\sqrt{x+b}\approx\frac{a-b}{2\sqrt x}$ for ...
2
votes
1answer
426 views

Approximating Stirling's number of the second kind to allow for large inputs

I'm looking for an approximation for Stirling's number of the second kind, $S_2(n,k)$, which counts the ways to partition a set of $n$ objects into $k$ non-empty subsets: ( ...
1
vote
1answer
107 views

Approximate function from sample data

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function. I don't have function definition. It's described as a fuzzy inference system. I have the inference system and can manipulate sample data for each ...
0
votes
1answer
496 views

Try to find an approximation by logarithm function.

Recently I am thinking about this question: Assume $x$ is real, $x\geq0$, $c$ is a positive constant number and $z$ is also a real constant between $3.5$ and $4$. Now there is a function: $$ ...
1
vote
0answers
201 views

Chebyshev Rational Approximation

Find the Chebyshev rational approximation of degree 4 with m=n=2 for $f(x)=\sin(x)$ . Now I do have a program that can evaluate this But its asking me to find the coefficients of the Chebyshev ...
1
vote
2answers
248 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 ...
3
votes
0answers
281 views

Minimizing maximum absolute column sum norm of the residual between a matrix and its $k$-rank approximation

Let $X \in \mathbb{R}^{m\times n}$ be a matrix with rank $r$. How can we find the optimal $\tilde{X} \in \mathbb{R}^{m\times n}$ whose rank is $k$ where $k\leq r$ and the reconstruction error in ...
2
votes
2answers
128 views

Approximating $\pi$ in Binary

I am interested in creating a Java program that generates digits of $\pi$ (in Binary though). To be clear, the number I'm looking for begins: $11.00100100 \dots$ I am unsure of the most efficient way ...
0
votes
1answer
296 views

Techniques for bounding a sum

I have a very messy function. It consists sums four levels deep, and the inner-most term is itself quite messy. $$ \sum \sum \sum \sum (\mbox{stuff})$$ I can't find a closed form for this function. ...
1
vote
1answer
312 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 ...
0
votes
0answers
179 views

When does distribution bootstrap mean converge to distribution sample mean?

Let $\bar{X}_n$ denote the sample mean of n iid random variables. Let $\bar{X}^*_n$ be the bootstrap sample mean. Does $\left|\mathbb{P}\left(n^{1/2}\bar{X}_n\leq ...
1
vote
0answers
63 views

How to approximate $\sqrt[3]{x}$ when $x$ is rational number

One wants to approximate the real value of $\sqrt[3]{x}$ when $x$ is rational number. One want to approximate to two decimal digits. Is there any way to do this quickly?
2
votes
3answers
236 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
2
votes
3answers
1k views

Approximate a convolution as a sum of separable convolutions

I want to compute the discrete convolution of two 3D arrays: $A(i, j, k) \ast B(i, j, k)$ Is there a general way to decompose the array $A$ into a sum of a small number of separable arrays? That is: ...
4
votes
2answers
532 views

How to obtain an approximate expression for $\sqrt{\varepsilon}$ where $\varepsilon \ll 1$?

Is there a way to obtain an approximate expression for the square root $\sqrt{\varepsilon}$ of a small number $\varepsilon \ll 1$? To be more precise, I would like to have an expression which (1) I ...
1
vote
1answer
123 views

How to show that linear span in $C[0,1]$ need not be closed [duplicate]

Possible Duplicate: Non-closed subspace of a Banach space Let $X$ be an infinite dimensional normed space over $\mathbb{R}$. I want to find a set of vectors $(x_k)$ such that the linear ...
3
votes
2answers
6k views

What's the name of the approximation $(1+x)^n \approx 1 + xn$?

A good approximation of $(1+x)^n$ is $1+xn$ when $|x|n << 1$. Does this approximation have a name? Any leads on estimating the error of the approximation?
1
vote
0answers
243 views

Approximate function using non-orthogonal basis

I'm currently trying to wrap my head around somebody's (very concise) description of Finite Element Analysis (FEA): ...
0
votes
1answer
60 views

How to approximate this infinite integral over $1 \leq x \leq y < \infty$

Does anyone know how to approximate this infinite integral $$ H(\beta, i, l) = \iint_{1 \leq x \leq y < \infty} (1-x^{-\beta})^{i-1} (x^{-\beta} - y^{-\beta})^{2(l-i)} x^{-\beta} y^{-\beta i} dxdy, ...
0
votes
3answers
2k views

How to find the first-order approximation around a given point?

Lets pretend i have some function $f(x) = 2*x_1 + 3*x_2$, and says find first order approximation around some point [a b]. I know the formula $f_a (x) = f(x') + f(x)'*(x-x')$, but do not know how to ...
1
vote
1answer
2k views

numeric approximation to find maximum of a function

is there a fast numeric algorithm to approximate the maximum of an given function in an interval [x,y] without calculate the derivation of the function? I only know about solver to calculate the ...
6
votes
1answer
1k views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
3
votes
1answer
260 views

What is the proof of the rules of significant figures?

I wanted to know how do we know that the rules that we follow when doing arithmetic with significant figures are correct? Like why when adding or subtracting we keep the same number of decimal places ...
3
votes
3answers
63 views

Approximation for $2^r\ln \frac{2^r}{2^r-r}$

I know the function $$2^r\ln \frac{2^r}{2^r-r}$$ is about linear in $r$, but I need an argument that an undergraduate could follow. Is there a simple way to explain this? I'd be happy with a simple ...
2
votes
3answers
372 views

Simple test if point is above or below sine curve

Is there any simple formula or algorithm for determining if a point lies above or below the sine curve? For instance, if I have a point $(x, y)$, how can I test whether or not $y > \sin(x)$? ...
1
vote
0answers
89 views

Rapidly convergent series for $\sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)}$ (rigid rotor)

I need to evaluate this series for arbitrary $\beta > 0$: $ Q = \sum_{J=0}^{\infty} (2 J + 1) e^{-\beta J(J+1)} $ Is it related to a known transcendental function? From the research I did, it ...
2
votes
2answers
738 views

How many bits are in factorial?

I am interested in good integer approximation from below and from above for binary Log(N!). The question and the question provides only a general idea but not exact values. In other words I need ...
1
vote
0answers
198 views

Approximation algorithm for sine function

What is the most common approximation algorithm for sine function which can be implemented with self-organizing maps algorithm? I was thinking of least-squares approximation but I can't find a ...
2
votes
1answer
138 views

Upper bound for $\Gamma(x+y)$

Let $x, y \geq 1$ be two real numbers. I am wondering if one can get an upper bound for $\Gamma(x+y)$ in terms of $\Gamma(x)\Gamma(y)$? Any references or ideas are very appreciated. Thank you.