Questions tagged [computational-mathematics]

This tag concerns computational problems central to mathematical and scientific computing. The scope includes algorithms, numerical analysis, optimization, and linear algebra, computational topology, computational geometry, symbolic methods, and inverse problems.

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Investigating the numerical accuracy of a truncated Legendre polynomial expansion of an unknown function

I have an integral equation involving an unknown function $f(x)$, of the most basic form $$ \int_{-1}^{1} e^{iω(t) x} f(x) \ dx = g(t) $$ I am solving for an approximation of $f(x)$ by substituting in ...
Silver Pages's user avatar
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algorithm guaranteed to converge for convex function

I have a multi-variate convex function, and I want to find its global minimum. We know there is one and only one minimum. Is there any algorithm which is guaranteed to converge to the minimum starting ...
poisson's user avatar
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what software are used for signed barcodes, fibered barcode, graded Betti numbers and greyscale. Please help.

what software are used for signed barcodes, fibered barcodes, graded Betti numbers, and greyscale. Please help. where can I learn this software? I am a fetus in coding. Help would be appreciated. :) &...
Rabia Sagheer's user avatar
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Need help with proving energy stability of forward Euler for heat equation

I have the 1D heat equation $u_t= \alpha u_{xx}$ on $x \in [0,1]$. It has homogeneous Dirichlet boundary conditions. I intend to use the forward Euler numerical scheme. $$\frac{u^{n+1}-u^{n}}{k} = \...
laplacian18's user avatar
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Is there a way to find if there relationship of numbers

I have a challenge. This may be little tricky or even not possible but wanted to check if anyone has any thoughts on this? PS : This question is in general and not related to only to R. May be I can ...
manu p's user avatar
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verifying Ramanujan constant

The famous Ramanujan constant $ e^{\pi \sqrt{163}} $ is a near-integer. see the link here. I tried to calculate this number with matlab and failed. Matlab cannot even deliver the first 9 apparently ...
S. Kohn's user avatar
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2 votes
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Is it possible to find something better than binary search for this problem?

Let's say we have $n$ urns (numbered $1$ through $n$) and the first $k$ urns have a ball in them (for some $k$ unknown to us) and the remaining urns are empty. Our goal is to determine $k$ by looking ...
user23571113's user avatar
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How would you Illustrate the Symplectic Euler method for a general initial value problem of the second order to a person who has never heard of it? [closed]

Consider a person who has never heard of symplectic Euler but knows about Initial Value Problems, forward Euler, backward Euler, and Taylor series. Say, you are required to give them an idea of what ...
rayyan's user avatar
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Fast solvers for saddle-point problem (linear system)

I would like to speed up my multi body simulator. There, in every time step a linear system, often referred to as saddle-point problem, has to be solved. The system looks like this $$ \begin{bmatrix} ...
freddy90's user avatar
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Finding Maximum Value of Variable Using Only Its Fourier Transform?

Assume I have a variable u, which is an $m\times n$ array filled with exclusively real values. I want to find the maximum value within this ...
Jacob Ivanov's user avatar
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How to stop a negative exponential from rounding to zero?

I'm doing materials homework and calculating vacancy density, which has some large constants (Na and k), and have to do the function. the right side of the density equation is exp(-Q/(kT)), which ...
Erwin Davinky's user avatar
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What is the time complexity of multiplying two matrices over an arbitrary ring?

I know that the time complexity of matrix multiplication over a field is well studied (multiplying two $n \times n$ matrices can be done in $n^\omega$ field operations, where $\omega$ is the matrix ...
GHPR's user avatar
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Showing that an explicit s-stage RK method with its order of accuracy higher than s

I was asked to show an s-stage explicit Runge-Kutta Method cannot obtain accuracy higher than $s$. It suffices to consider autonomous system $y' = f(y)$ (otherwise by introducing a new variable $t$, ...
Stack_Underflow's user avatar
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Which 2nd root finding method does the HP32E use to compute $Q^{-1}$?

The ancient pocket calculator HP32E computes $Q$, areas under the normal curve, with the "Algorithm 39" (same in FORTRAN) and the inverse $Q^{-1}$, quantile, using two different root finder. ...
m-stgt's user avatar
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generalized implicitly restarted lanczos method

I am looking for references on how to solve the generalized eigen-problem : $$Ax = \lambda Bx \tag{1}$$ Where $A$ is a symmetric matrix and $B$ is symmetric positive definite. I know a standard ...
Alexandre Hoffmann's user avatar
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2 answers
249 views

When does $(a+b\sqrt n)^3+(a-b\sqrt n)^3=c^3$ have integer solutions $(a,b,c,n)$?

From this post Where Fermat's Last Theorem fails, we find the nice, $$(18+17\sqrt2)^3+(18-17\sqrt2)^3=42^3$$ Using this initial solution, an infinite more can be generated using P. Tait's identity, $$\...
Tito Piezas III's user avatar
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Is there a way for me to change the order of multiplying these matricies?

Given an equation of this form: \begin{equation} \vec{X}=\sigma(t) B^{-1} \label{MasterEq} \end{equation} where $\vec{X}$ has the following components: \begin{equation} x_i= \sum_{\alpha = ...
TheEndernaut Infinity's user avatar
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How do computers calculate irrational numbers? like how do they compute root 2? [duplicate]

I do know how to like get root 2 but it's the long division method, and from just seeing it I am genuinely confused how they would implement it. Then there is the prime factorization method but it ...
Tejas Agrawal's user avatar
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Given base $b$ what is the expected value of round off error in rounding to $d$ digits?

Given base $b$, what is the expected value of round off error (relative, not absolute) when we round $x$ to $d$ digits, where $x$ is a random variable? Assume $x$ is drawn from a uniform distribution ...
SRobertJames's user avatar
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A special type of one symbol escape huffman code. Which source(s) will it be optimal for?

Background / Context: The other day I implemented a special case of Huffman coding constructed with $2^{N}-1$ valid code words and 1 "escape" into a set of longer length codes. As there is ...
mathreadler's user avatar
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Compute integral closure using a computer

Suppose given an inclusion $A\subset B$ of finitely-presented commutative algebras over a field. Is there a CAS which can decide whether $B$ is a finite $A$-module? What if instead of f.p. k-algebras, ...
Tomo's user avatar
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MVUE, Minimum variance unbiased estimator [duplicate]

Let $(X_{1},X_{2},\ldots X_{n})$ be a random sample from the distribution with density $$f(x)=\begin{cases} e^{-(x-\delta)},&\text{if x > $\delta$}\\ 0, &\text{otherwise} \end{cases} $$ ...
maths and chess's user avatar
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A lower bound of the distance between 2 distinct roots of 2 distinct polynomials

Here is one of my homework in the computer algebra class: Let $f(x),g(x)\in \mathbb{Z}[x]$ be of positive degree $m,n$, and $f(\alpha)=g(\beta)=0$, where $\alpha\neq \beta$ are real. Show that $$ |\...
Zoudelong's user avatar
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Navier Stokes FDM: Iterative Solvers (Component Form vs Matrix Form)

I hope you are all enjoying a pleasant holiday season, if you happen to be celebrating. Currently, I am engrossed in a university assignment that involves conducting a benchmark with various solvers ...
natsukashi_heiwa's user avatar
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1 answer
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Compare magnitude of integers with limited computation

I have a polynomial $f:\mathbf{Z}\to\mathbf{Z}$ and very large integers $n, x$ such that $f(n)\approx x$ (specifically, I can say that $f(n)$ is within some relatively small value $\epsilon$ of $x$). ...
Ethan Chapman's user avatar
1 vote
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Creating nonuniform grids for FDM with multiple points of concentration

If I am creating a grid in the $S_i$ direction with $N_S+1$ grid points. If I want more steps around some $K$, I can use: $$ S_i=K+c \sinh \left(\xi_i\right), \quad i=0,1, \ldots, N_S $$ where $c=\...
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How can I use algebraical properties of group operations for generating subsets of blocks of binary numbers of same population count?

Considering blocks of binary digits, are from all binary numbers $\in \{0,\cdots,2^{N}-1\}$ Which operations can I define which will preserve the number of 1-bits in any given number? Is there some ...
mathreadler's user avatar
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4 votes
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Evaluating $\lim_{n \to \infty} \frac{1}{n}\sum_{t = 1}^{n}e^{-k\cos^2(\omega t)}$, where $k>0$ and $0<\omega<\pi$

Need to evalute a closed form expression of the following limit: $$\lim_{n \to \infty} \frac{1}{n}\sum_{t = 1}^{n}e^{-k\cos^2(\omega t)}$$ where $k>0$ and $0<\omega<\pi$. Empirically, I have ...
Suryasis Jana's user avatar
2 votes
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Why does using $f(x)=x^2$ not work in Pollard's rho algorithm?

In Pollard's rho algorithm for integer factorization, we use pseudorandom sequences of the form $x_{i+1}=f(x_i)$ and look at them$\mod{n}$ until we get a cycle. Utilizing the birthday paradox, we can ...
shp's user avatar
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How to simulate from $dY_i=Y_i( \mu dt$ $+ \sigma_{(2)}( \alpha dB^{(1)}_i + \sqrt{1- \alpha ^2} dB^{(2)}_i))$ for $\mu, \sigma>0, \alpha \in [-1,1]$?

I am studying numerical methods from the textbook Monte Carlo Methods in Financial Engineering by Paul Glasserman and have encountered the following exercise: I want to simulate from the stochastic ...
FD_bfa's user avatar
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Fast expression to rotate vector by the same rotation that orients one vector to another

The title is a bit convoluted but I don't know a good way to express it. I am wondering whether this expression that I've worked out for my specific task has been discussed in the past since I have ...
Matrefeytontias's user avatar
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Model on NetLogo: Color Fractions

I have the following code in NetLogo: patches-own [ column-number ] to setup clear-all ask patches [ set plabel-color black set column-number pxcor + max-pxcor ] go reset-ticks end to go let n ...
José Carlos Pérez Garrido's user avatar
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How can one do fast bit-wise inverse derivative / gradient over $Z^2$ or $GF(2)$?

In computer language what I have is a "bit-stream" in 1D or a "bit-map" in 2D. With boolean algebra "xor" corresponds to difference or sum in $Z_2$ or GF(2), whereas &...
mathreadler's user avatar
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1 vote
1 answer
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How would I generate a Pade Approximant using the coefficients of a Taylor Series?

I would like to find an effective way to make a Pade Approximant using the coefficients of a Taylor Series. I've heard of Wynn's epsilon algorithm and using the Extended Euclidean Algorithm, but what ...
Deniz Kirbiyik's user avatar
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0 answers
25 views

Finite Difference method, ADI Scheme of Douglas and Rachford

I am trying to implement the ADI scheme of Douglas and Rachford. For $p(X,Z,t)$, there is: $$ \begin{gathered} A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\...
THAT'S MY QUANT MY QUANTITATIV's user avatar
1 vote
0 answers
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Efficiently mapping a number between two bases given a particular condition on the digits

I am currently faced with the following problem. Suppose I have a number $N$ in base $a$, and let $N_0, N_1, \cdots, N_n$ be the digits of said number in that base. It is also given that $N_i < b \...
GPU'njoyer's user avatar
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1 answer
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Could spectral methods solve first-order differential equations? [closed]

I've been learning spectral methods mainly through reading the book "spectral mehtods in matlab" by Trefethen. I have a simple question: Could spectral methods solve first-order differential ...
user35734's user avatar
1 vote
2 answers
393 views

Calculate roll length based on inner/outer diameter and number of turns.

I have seen in this link that there is a way to calculate the rolled material length and the number of turns based on: the material thickness inner diameter of the roll external diameter of the roll ...
Marinos An's user avatar
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0 answers
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Can projection of 3D affine motion onto a pin hole camera be described by a 2D 2nd order polynomial?

Question : will a second order polynomial be rich enough to be able to express motion vector fields stemming from affine transformations in 3 dimensions as seen through a pin-hole camera? $$\hat v_x = ...
mathreadler's user avatar
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Multiple summation for 1D Ising model

I am trying to perform multiple summation in a Matlab code, but I don't know how to write a code to perform multiple for's . $$\sum_{S_1=\pm1}...\sum_{S_N=\pm1}(\Pi_{i=1}^{N-1}e^{S_iS_{i+1}})=\sum_{...
Pedro Ye's user avatar
1 vote
0 answers
25 views

Is the weak form on this book about a level set FSI problem wrong?

I'm trying to repeat a level set FSI problem on the book :Level Set Methods for Fluid-Structure Interaction, on the page 89, the provided freefem code define a weak form of the discretized equation ...
吴yuer's user avatar
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3 votes
3 answers
291 views

$a^3+b^3+c^3=(c+1)^3=A^3+B^3+c^3$ solutions.

(Note: Positive integers throughout, except where specified.) I. Parameterization Following on from this excellent question from Tito Piezas III, "Sums of Cubes of form $a^3+b^3+c^3=(c+1)^3$"...
Old Peter's user avatar
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4 votes
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What are the family of geodesics/curves between point weights/balls on an elastic sheet?

ASSUMPTIONS: no deformation is separate, all deformations touch with at least another points deformation, i.e., no deformation is by itself. point force preferred, but small dense balls okay too. The ...
Teg Louis's user avatar
5 votes
0 answers
136 views

How fast does the coprime probability converge to $6/\pi^2$?

It is known that the probability that two positive integers are coprime is $6/\pi^2$. This is an amazing result. I wanted to see experimentally how the probability converges to $6/\pi^2$, but I found ...
Martin Brandenburg's user avatar
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31 views

Uniqueness of quotients when reducing with Gröbner basis

Let $K$ be a field, and let $G = (g_1, \ldots, g_m)$ be a Gröbner basis in $K[x_1, \ldots, x_n]$ (i.e. $G$ is a Gröbner basis for the ideal it generates). By Adams, Loustaunau - An Introduction to ...
Adelhart's user avatar
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1 vote
0 answers
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Claim that the class of context-free languages is closed under $f$ [closed]

Given an alphabet $\Sigma$, consider the operation $f : \Sigma^* → \Sigma^*$, defined as $$ f(w) = w_n \circ w_{n−1} \circ w_{n−2} \circ \cdots \circ w_1, $$ where $w = w_1 \circ w_2 \circ w_3 \circ \...
Hammad Nadeem's user avatar
0 votes
1 answer
68 views

Fitted $1/x$ by a linear combination of $e^x, \sin(x)$ and gamma function $\Gamma(x)$.

I am working on my homework, which is exercise 11.2b) from "Numerical Linear Algebra" by Trefethen. The exercise asks me to fit a function $f(x) = \dfrac{1}{x}$ by a linear combination of ...
hxllearnmath's user avatar
2 votes
0 answers
41 views

Are there practical applications for a method of finding n! using the constant resulting in the differences between exponents of consecutive numbers

I have been tinkering with a variation of Pascal's triangle using the differences between the exponents of consecutive numbers to derive a factorial. Generally, the bottom row starts with 1^n , 2^n , ...
elvexo's user avatar
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3 votes
1 answer
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How to approximate rational exponents?

On a recent exam, in an integer-type question in Chemistry, I calculated my final answer to be 10^0.6, I thought I had calculated the answer incorrectly as there was no way that would be the answer. I ...
OneChannelEverything's user avatar
0 votes
1 answer
20 views

Finding the Maximum Step Size for a IVP using various methods.

I'm stuck in a problem where I must simulate the trajectory of a spherical pendulum given the initial conditions. I will start with the things i have already done, then comes the question. To ...
Zamukerz's user avatar

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