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
262 views

How do I determine appropriate rational approximations to a sum of square roots in order to bound the error accumulation?

I have two numbers, $A$ and $B$, that are sums of integer multiples of a set of square roots of small primes (and 1) and their products: $A = a_0 + a_1\sqrt 2 + a_2\sqrt 3 + a_3\sqrt 5 + a_4\sqrt 6 + ...
1
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0answers
107 views

Intuition for approximating Ei(x)

I'm working with a function that is defined to be the sum of a series, and I'd like to either find its values, or approximate them analytically: $F(x) = \frac{1}{w} ...
-1
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2answers
56 views

Aprroximate graph to function

there is a set of points which set a graph that is not linear. Is there any method to approximate a function that is close enough to this graph? I've read some articles and got to know approximation ...
1
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1answer
117 views

Need numerical approximation for Fourier{max(0,f(x,y))} given Fourier{f(x,y)}

Given $\mathscr{F}\{f(x,y)\}$ is there a way to numerically approximate $\mathscr{F}\{max(0,f(x,y))\}$ ? I am not necessarily looking for a closed formula. Even some iterative method would be ...
3
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1answer
665 views

Upper and lower bounds of a ratio involving vector norms

I'm working on a signal processing problem and need to analyze the following expression $$ G = \frac{n}{\sum\limits_{i=1}^n |w_i|} \frac{ \sum\limits_{i=1}^n g_i w_i^2}{\sum\limits_{i=1}^n g_i |w_i|} ...
5
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3answers
283 views

“$O$” notation in Stirling approximation

In the Stirling approximation the formula as typically used in applications is $$\ln n! = n\ln n - n +O(\ln(n))$$ I'm confused about the last term "$O$" . What does it mean actually, and when do we ...
5
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0answers
154 views

Watson's Lemma Extension

We all know that Watson's Lemma is used to approximate the integral $$ F\left( s \right)=\int_0^\infty {{e^{ - st}}f\left( t \right)dt} $$ for large $s$. However, for arbitrary $s$, are there any ...
5
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1answer
233 views

Approximating a weird sum

How can I approximate the sum$$\sum_{k=1}^n \left(\frac{2k}{n} \left\lceil \frac{n}{k} \right\rceil \left\{ \frac{n}{k} \right\}-1\right)$$ where $\{x\}$ is the fractional part function, and $\lceil ...
5
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5answers
7k views

Software to find a function for data approximation

I've got some y(x) 2D data set. I would like to find a function fitting this data: Is there any open source or free software to find a function to approximate a data sequence like the above? Here ...
2
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1answer
54 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
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1answer
115 views

Integral approximation.

Can you help me to show that $$\int^{\pi/2}_{0}{d\theta\over (1-m^2\cos^2\theta)^2} \approx {(2-m^2)\pi\over4(1-m^2)^{3/2}}$$ to first order, such that $0 \lt m \lt 1$
1
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1answer
259 views

Knot placement for a natural cubic spline

I am trying to approximate a function via a natural cubic spline. Suppose I sample the function on a grid i.e. I know the value of the function at a fixed number of equidistant points, say on 200 ...
3
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0answers
66 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
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1answer
153 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
544 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
181 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
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1answer
129 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
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1answer
419 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
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1answer
41 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 ...
3
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1answer
1k 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
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1answer
129 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
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1answer
308 views

Show that if m/n is a good approximation of $\sqrt{2}$ than $(m+2n)/(m+n)$ is better

Claim: If $m/n$ is a good approximation of $\sqrt{2}$ then $(m+2n)/(m+n)$ is better. My attempt at the proof: Let d be the distance between $\sqrt{2}$ and some estimate, s. So we have ...
3
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1answer
92 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
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1answer
538 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
177 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
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1answer
95 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
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1answer
103 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 ...
14
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1answer
284 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 ...
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0answers
693 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
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1answer
46 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
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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
369 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
145 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
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5answers
1k 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)\\ ...
9
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3answers
389 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
559 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
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1answer
120 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
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1answer
546 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: $$ ...
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0answers
213 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
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2answers
298 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
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0answers
308 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
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2answers
132 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
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1answer
324 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
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1answer
360 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 ...
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0answers
218 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 ...
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0answers
65 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
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3answers
322 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 ...
3
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3answers
2k 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
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2answers
679 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
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1answer
132 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 ...