# Tagged Questions

Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

13k views

### “The Egg:” Bizarre behavior of the roots of a family of polynomials.

In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ ...
11k views

### Proof that ${\left(\pi^\pi\right)}^{\pi^\pi}$ (and now $\pi^{\left(\pi^{\pi^\pi}\right)}$) is a noninteger

Conor McBride asks for a fast proof that $$x = {\left(\pi^\pi\right)}^{\pi^\pi}$$ is not an integer. It would be sufficient to calculate a very rough approximation, to a precision of less than 1, and ...
14k views

### Why do we still do symbolic math?

I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one. Numerical computations, to my understanding, never ...
1k views

### How much does symbolic integration mean to mathematics?

(Before reading, I apologize for my poor English ability.) I have enjoyed calculating some symbolic integrals as a hobby, and this has been one of the main source of my interest towards the vast ...
3k views

### What is the fastest/most efficient algorithm for estimating Euler's Constant $\gamma$?

What is the fastest algorithm for estimating Euler's Constant $\gamma \approx0.57721$? Using the definition: $$\lim_{n\to\infty} \sum_{x=1}^{n}\frac{1}{x}-\log n=\gamma$$ I finally get $2$ decimal ...
17k views

### What algorithm is used by computers to calculate logarithms?

I would like to know how are logarithms calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that ...
691 views

### How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$

I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
831 views

### What's the most efficient way to mow a lawn?

For $S\subseteq\Bbb R^2$ and $x\in\Bbb R$, define $E_x(S)=\{y\in\Bbb R^2:d(y,S)<x\}$. ($E_x(S)$ represents the expansion of $S$ by $x$.) Given a path $\gamma:[0,1]\to\Bbb R^2$, denote its length as ...
5k views

### How do you calculate the decimal expansion of an irrational number?

Just curious, how do you calculate an irrational number? Take $\pi$ for example. Computers have calculated $\pi$ to the millionth digit and beyond. What formula/method do they use to figure this out? ...
2k views

### Why does my calculator show $2^{-329} = 0?$

On my calculator, I usually get a $0$ when I divide something by $2$, a lot of times if that makes sense, but I was just wondering why does $2^{-329} = 0?$
17k views

### Simple numerical methods for calculating the digits of $\pi$

Are there any simple methods for calculating the digits of $\pi$? Computers are able to calculate billions of digits, so there must be an algorithm for computing them. Is there a simple algorithm that ...
915 views

678 views

### Wiggly polynomials

I'd like to be able to construct polynomials $p$ whose graphs look like this: We can assume that the interval of interest is $[-1, 1]$. The requirements on $p$ are: (1) Equi-oscillation (or ...
8k views

### Determining whether a symmetric matrix is positive-definite (algorithm)

I'm trying to create a program, that will decompose a matrix using the Cholesky decomposition. The decomposition itself isn't a difficult algorithm, but a matrix, to be eligible for Cholesky ...
2k views

### Detecting perfect squares faster than by extracting square root

Given the radix-$r$ representation of a integer $n$, and a small integer constant $k$, there is an $O(\log n)$ algorithm for detecting whether $n$ is a multiple of $k$, namely, division, which ...
879 views

### In what sense is the derivative the “best” linear approximation?

I am familiar with the definition of the Frechet derivative and it's uniqueness if it exists. I would however like to know, how the derivative is the "best" linear approximation. What does this mean ...
225 views

### Plotting $\left(1+\frac{1}{x^n}\right)^{x^n}$.

When I plot the following function, the graph behaves strangely: $$f(x) = \left(1+\frac{1}{x^{16}}\right)^{x^{16}}$$ While $\lim_{x\to +\infty} f(x) = e$ the graph starts to fade at $x \approx 6$. ...
9k views

I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits. By pen and paper that is. I'm doing this old school. What first came to mind was to use $\log(ab) = \log(a)... 2answers 4k views ### Why does Newton's method work? I find many sites explaining how to use Newton's method, but none explaining why it works. Could someone give me the intuition behind it? Thanks. 2answers 411 views ### how to compare$\sin(19^{2013}) $and$\cos(19^{2013})$how to compare$ \sin(19^{2013})$and$\cos (19^{2013})$or even find their value range with normal calculator? I can take$2\pi k= 19^{2013} \to \ln(k)= 2013 \ln(19)- \ln(2 \pi)=5925.32 \to k= 2....
9k views

In order to find the local minima of a scalar function $p(x), x\in \mathbb{R}^3$, I know we can use the gradient descent method: $$x_{k+1}=x_k-\alpha_k \nabla_xp(x)$$ where $\alpha_k$ is the step size ...
26k views

### What is difference between Finite Different Method, Finite Element Method and Finite Volume Method for PDE?

Can you help me explain the basic difference between FDM, FEM and FVM? What is the best and why? Advantage and disadvantage of them?
284 views

### Fastest curve from $p_0$ to $p_1$

I'm trying to solve a problem in path planning: Given points $p_0$ and $p_1$ and vectors $v_0$ and $v_1$, find a function $p(t)$ st. $p(0) = p_0$, $p(T) = p_1$, $p'(0) = v_0$ and $p'(T) = v_1$...
13k views

### How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
302 views

### Did Feynman mentally compute $\sqrt[3]{1729.03}$ by linear approximation?

In the biopic infinity'' about Feynman. (11:48~15:50) Feynman compute $\sqrt[3]{1729.03}$ by a mental calculation. I guess that he use the linear approximation. That is, he observe that $1728=12^3$....
1k views

1k views

### Fast inverse square root trick

I found what appears to be an intriguing method for calculating $$\frac{1}{\sqrt x}$$ extremely fast on this website, with more explanation here. However, the computer-science lingo and "floating"-...
### How to calculate $2^{\sqrt{2}}$ by hand efficiently?
I've been trying to calculate $2^{\sqrt{2}}$ by hand efficiently, but whatever I've tried to do so far fails at some point because I need to use many decimals of $\sqrt{2}$ or $\log(2)$ to get a ...