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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Firstly what does relative tolerance mean?

Apply the Jacobi iteration method to the system Ax=b with $$ A= \begin{pmatrix} 3 & -1 & 1 \\ 3 & 6 & 2 \\ 3 & 3 & 7 \\ ...
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1answer
62 views

Is this possible or hopeless to try to prove?

If I have $x_1, ..., x_k=o(n)$ and $j=O(1)$. Is it possible to prove something like: $$\sum_{i=1}^k {n \choose j} \left(\frac{x_i}{n}\right)^j \left(1-\frac{x_i}{n}\right)^{k-j} \sim {n \choose j} ...
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1answer
371 views

Approximating the exponential function

I have found experimentally something that seems graphically like an approximation of the exponential function. However, it is totally experimental and I have no idea whether it really converges ...
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0answers
61 views

Rational numbers imply reals?

I was solving an inequality today and I proved it for rational numbers (it was easier because I was able to "strengthen" by doing things like "$\frac{p}{q}>\frac{r}{s}\implies ps\ge qr+1$ since ...
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1answer
34 views

Can someone explain what destructive cancellation is as well as how to answer the question?

Let $f(x)= \sqrt{x^2 + 1} - 1$ (taking the positive real square root, as usual). When $a = 10^{−3}$, compute $f(a)$, working to $5$ significant figures at every stage of the calculation. Also it can ...
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1answer
82 views

Why is $\tan 54^\circ\approx \frac{\sin24^\circ}{1-\sqrt{3}\sin24^\circ}$

This question was asked as an equality on MSE and I am quite surprised to find that its strictly false However I would like to see why is their difference of the order $10^{-15}$? $$\tan ...
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4answers
3k views

How much gas does a car use to carry its own gas?

I have always been curious about this one. Since the gas has some weight, the car will have to burn some extra gas to carry it's own fuel around. How can I calculate how much that extra gas is? ...
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0answers
23 views

How to get a nice approximation of $f(N,s)=\sum_{k=0}^{N}{N \choose k}{k \choose s-k+N}$ when $N>>1$ and $|s|<<N$?

I need to approximate the above sum in order to calculate $\mathbb{E}(s^2)$, which is the expectation value determined by the probability density function $f$ and the position $s$. Any idea?
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14 views

Approximating the action of the U(N) exponential map

Let's say that I have a curve in $\mathbb{C}^N$ given by the action of the unitary group: $$x(t) = e^{Ht}x_0,~ H \in \mathfrak{u}(N),~ ||x_0||=1$$ I can approximate this to first order as: $$\tilde ...
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1answer
34 views

I am not sure how to use the secant method formula without a function being given?

Calculate an approximation value for $4^{\frac34}$ using four steps of the secant method with the starting values of $x_0=3$ and $x_1=2$.
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1answer
137 views

Upper and lower bound of an approximate computation

I'm currently working on an exercise in The Art Of Computer Programming, Vol. 1 relating the algorithm for computing $\log_{10} x$ presented in the Mathematical Preliminaries. The algorithm is ...
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1answer
37 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
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1answer
86 views

Approximation for lower incomplete gamma function

Does any one know approximations for the lower incomplete gamma function $\gamma(a,bx)$. The problem is this: I want to find the quantile function for the CDF of the gamma distribution. The CDF of the ...
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2answers
44 views

Firstly what is an $O(h^3)$ formula? Also I am not quite sure how to answer the question?

The forward-difference formula can be expressed as $$f'(x_0)=\frac{1}{h}(f(x_0 +h)- f(x_0))-\frac{h}{2}f''(x_0) - \frac{h^2}{6}f'''(x_0) + O(h^3).$$ Use Richardson's extrapolation to derive an ...
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2answers
65 views

show that $\lim_{k\to\infty}\frac{ k!} { k^k}= 0$ using stirling's formula [closed]

Prove that $\lim _{k\to \infty \:}\left(\frac{k!}{k^k}\right)$ = 0:
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2answers
36 views

What does “symbolically tractable” mean?

What does "symbolically tractable" mean in the following quote? "Traditional treatments of mechanics concentrate most of their effort on the extremely small class of symbolically tractable dynamical ...
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0answers
26 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
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1answer
93 views

How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is accurate to within 0.0001?

I'm very lost on the following problem and will appreciate your help very much. How large should n be to guarantee that the Simpson's Rule approximation on the Integral (from 0 to 1) 19e^x^2 dx is ...
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0answers
45 views

Which (approximative) methods are there to compute the inverse of a complicated function?

I have a complicated function $f(x)$ for which I want to compute the inverse $f^{-1}$ over a certain range $R(f): a \leq f(x) \leq b$. The only way to find the inverse I can think of is power series ...
2
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1answer
51 views

approximation of $\log(1+z)=z$ as $z\to 0$

This is new to me and I have not done any asymptotic approximation. I don't understand how they get that $\frac{n}{N}$ stays close to $\frac{2}{3}$ as N goes to infinity. Also how do they do get ...
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0answers
25 views

How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
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2answers
49 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
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138 views

What domains of $\mathbb{R}^n$ have the property that $H^1(\Omega)=H^1_0(\Omega)$?

i wonder what are sufficient conditions on an unbounded domain of $R^n$ called $\Omega$ to get : $C_c^\infty (\Omega)$ dense in $H^1 (\Omega)$ ? where $C_c^\infty$ stands for the set of functions with ...
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0answers
29 views

solve equation involving digamma function

I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast ...
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1answer
20 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that ...
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3answers
260 views

How could I improve this approximation?

In a computer application, I need to solve trillions of times an equation which can be reduced to $$f(x)=\sin(x)-a x=0$$ Newton methods (quadratic and higher orders) are used for the solution. ...
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2answers
39 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
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1answer
51 views

Newton-Cotes Quadrature formula

Im trying to find more information about numerical integration methods. When is a Newton-Cotes Quadrature formula on n nodes exact?
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1answer
43 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
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0answers
24 views

Spatial Basis Functions…any practical hints on choosing them?

I am not coming from a pure mathematical background, and now in my daily life as an engineering student, I am getting faced to spatial basis functions to approximates variables. The problem is: Are ...
3
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1answer
28 views

Find a maximum triangle that lies on a polyline (with constraints)

If there's a polyline (a GPS track, actually) with a lot of points (could be several thousand), that looks like this 1) How can I find such a triangle with the biggest possible perimeter, that its ...
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3answers
899 views

Is it possible to find square root using only rational numbers and elementary arithmetic operators

Suppose I have a number a How can I find it's square root using only +, -, /, ...
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1answer
27 views

Looking for a approximation/solution to my mortgage calculator function

I'm working on a little function, $t(A,y,r)$ that calculates the monthly payment of a fixed-rate mortgage, where $A$ is the amount borrowed, $y$ is the number of years over which the loan will be ...
0
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1answer
51 views

How obtain a (accurate) function from this graph with these points?

I need obtain the function from 0 to 20 from this graph: I have the even numbers in the {x, f(x)} format: {0, 0}, {2, 1.8}, {4, 2}, {6, 4}, {8,4}, {10,6}, {12,4}, {14,3.6},{16,3.4}, {18,2.8}, ...
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1answer
39 views

Estimating the absolute error of the function $f(x)=4x^2$

I have to estimate the value of $f(x)=4x^2$ for $x\in [1,2]$, and $x$ is unknown. the approximated value for $x$ is $\tilde x$, which is also in $[1,2]$. What is the maximum absolute error of $x$, ...
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0answers
39 views

Truncation error in Padè approximants

Suppose only the following data are known about a rational function $R(x)=P(x)/Q(x)$ (for $P,Q$ polynomials): (a) the degree of $P$ is $\leq m$ and the degree of $Q$ is $\leq n$; (b) the first $k$ ...
2
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1answer
17 views

lower bound of modified Besselfunction

i'm looking for an lower bound for the modified Bessel function of the first kind $I_\nu(x)$ of a +ive real argument. There should be one of the form $$ce^{x^\alpha} \le I_\nu(x)$$.
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1answer
51 views

Is this interpolation, does it have a name?

I was waching Signle Variable Calculus MIT lectures (I stop on 9 about linear approximation) I was also learn interpolation at my university and I thought that I'll create my own equation for ...
2
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1answer
52 views

How is equivalent to approximation property?

I'm trying to prove Lemma 4.8 of [1] online reading: Notation: $\tau_C(X)$ means the topology of uniform convergence on the compact subsets of $X$. Lemma 4.8. For a Banach space $X$ the following ...
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2answers
270 views

Difference between “≈”, “≃”, and “≅”

In mathematical notation, what are the usage differences between the various approximately-equal signs "≈", "≃", and "≅"? The Unicode standard lists all of them inside the Mathematical Operators ...
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138 views
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1answer
65 views

Function Approximation

I need to solve the following equation $$-\frac{\partial S(x,y,t)}{\partial t}=ax^2+bx\frac{\partial S(x,y,t)}{\partial x}+c\Big[\frac{\partial S(x,y,t)}{\partial ...
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0answers
41 views

Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube

I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ...
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2answers
137 views

What is the sum that the square root button on calculator does so I can do it without the calculator button [duplicate]

I am not very good when it comes to Maths but the current work I am doing means I need to get better and quick. I have been teaching myself about areas, diagonals and square roots. However I am ...
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1answer
48 views

Understanding approximation $\sqrt{\frac{z}{z-2h}} \approx 1 + \frac{h}{z}$, $h>0, |z| \gg h.$

I am having troubles showing the above step regarding the approximation $$ \sqrt{\frac{z}{z-2h}} - \sqrt{\frac{z}{z+2h}} \approx \frac{2h}{z}, \quad h>0, |z| \gg h$$ given in an old exam. My ...
2
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2answers
182 views

Monotonically approximate $L^p$ function by step function

It is a classical fact that a $L^p(R^d)$ ($1\leq p<+\infty$) function can be approximated by step functions with compact support, but my question will be, can we require that the step function is ...
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1answer
71 views

Meaning of $\sim$?

I often read $f(x) \sim g(x)$ and I wonder what the Standard Interpretation of this $\sim$ is. It seems to mean something like asymptotically equally distributed, something like $f(x)=g(x)(1+o(1))$. ...
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82 views

How to approximate large sum of exponential variables

Is there any way to approximate the following sum: $$ \sum_{i_1=1}^N \sum_{i_2=1}^N \cdots \sum_{i_k=1}^N \cdots \sum_{i_N=1}^N \exp(-r_{i_1} - r_{i_{k+1}} - r_{i_{2k+1}} - r_{i_{3k+1}} \cdots - ...
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2answers
99 views

Newton Raphson Method for double roots

I am currently working on Newton Raphson Method. I am kind of facing a problem how Newton Raphson Method work on more than second order quadratic functions with double roots. I have googled and ...