# Questions tagged [na.numerical-analysis]

Numerical algorithms for problems in analysis and algebra, scientific computation

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I am interested in the following general double sums, for integers $a\geq 1$ and $b\geq 2$, $$Z(a,b) = \sum_{k,\ell \geq 0} \frac{2k+3}{\binom{k+2}{2}^a} \frac{2\ell+3}{(\binom{k+2}{2}+\binom{\ell+2}{... 1 vote 0 answers 21 views ### A general question about spectral methods vs finite element methods According to this Wikipedia article: Spectral methods can be computationally less expensive and easier to implement than finite element methods; they shine best when high accuracy is sought in simple ... 0 votes 0 answers 20 views ### A simple procedure to simulate multifractional Brownian motion paths In a paper by Peltier and Vehel the multifractional Brownian motion (mBm) was defined for the first time, and they also give a procedure to simulate mBm sample paths. Briefly, mBm generalizes the ... 0 votes 0 answers 38 views ### Approximation of inverse trigonometric functions [closed] I would like to implement algorithms from scratch for inverse trigonometric functions (inverse of sine, cosine, and tangent). I found on Wikipedia that those functions are approximated by the Taylor ... 0 votes 1 answer 72 views ### FEM based solution to parabolic problem Consider the problem$$ \begin{cases} u_t - \Delta u = 0 &\text{ on } \Omega\times (0,T)\\u=0 &\text{ on } \partial \Omega\times (0,T) \\ u(x,0)=g(x) &\text{ on } \Omega \end{cases} ... 1 vote 0 answers 31 views ### Is there a more efficient computer algebra system to solve the system of nonlinear equations in N-R method or other numerical methods? Consider the system of infinite series \begin{align} &F=x+\frac{y^{3^2}}{3}+\frac{x^{3^5}}{3^2}+\frac{y^{3^7}}{3^3}+\frac{x^{3^{10}}}{3^4}+\frac{y^{3^{12}}}{3^5}+\cdots=0 \\ &G=y+\frac{x^{3^3}}... 0 votes 0 answers 25 views ### What are the convergence requirements for Inverse Power Method? I'm struggling to find the convergence requirements for the Inverse Power Method. I implemented this method in MATLAB as shown below. ... 0 votes 0 answers 45 views ### Fourier spectral methods for an elliptic equation I would like to study a linear elliptic problem on the torus \mathbb{T}^n (i.e. periodic boundary conditions) which is of the following form: -\Delta u + b^i \partial_i u + c u = f $$where b^i,... 3 votes 0 answers 58 views ### Tight bounds on the iterates of the Clenshaw algorithm for Chebyshev polynomials I'm trying to bound the iterates of the Clenshaw algorithm when applied to the Chebyshev series, related to a question I'm running into related to the stability of this algorithm. Recall that, for p(... 1 vote 0 answers 26 views ### Complexity of singular value decomposition using matrix multiplication oracles Suppose I have an n\times m real matrix A, n\ll m with full row rank (\mathrm{rank}(A) = n). I have an oracle that can compute Ax or A^T y for any x\in \mathbb{R}^m, y\in \mathbb{R}^n. ... 0 votes 1 answer 77 views ### Correct way to conduct equilibrium scaling of linear/integer/MIP program I would like to scale my linear/integer program and also mixed-integer program using the equilibrium scaling method. I have worked on two research papers and one research book. However, they did the ... 6 votes 1 answer 151 views ### Reporting inconclusive experimental searches In many areas of mathematics it is informative to conduct numerical experiments. But, it not uncommon that the searches do not lead to the examples or data one was hoping for. Since the numerical ... 0 votes 0 answers 49 views ### 3D interpolation function I've got a 3D figure created using around 30k points and has different regions colored in an specific way according to some unrelated variables that come from a project I'm creating. Taking in ... 0 votes 0 answers 29 views ### Finding the maximum of a certain trigonometric series Let a_0,_1,a_2,\dots,a_n be a finite sequnce of n  real numbers and consider the function$$f(t)=\sum_{j,k=0}^{n}a_j a_k \cos\left((j-k)t\right)$$for t \in [0,2 \pi] .If we want to maximise ... 1 vote 0 answers 81 views ### Generating Hermite polynomial with coefficient recurrance relation algorithm I am writing a math paper for my numerical analysis class about orthogonal Hermite polynomials. I want to implement the algorithm for generating the "probabilist's Hermite polynomials":$$ \...
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This might be an easy question but i am sorry for asking this. Let $f(x)\in\mathbb{Z}_p[x].$ Is it always true that $$f(x+y)=f(x)+f'(x)y+f''(x)\frac{y^2}{2}+zy^3$$ for some $z\in\mathbb{Z}_p.$ if it ...
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### Iterative method of finding root

I don't know much about numerical analysis. I need the following for help with my research in number theory. Is there a simple(not multistep) iterative method of finding the root of a real-valued ...
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### Singular value decomposition of truncated discrete Fourier transform matrix

Let $\mathbf{F}$ be a discrete Fourier transform (DFT) matrix such that \begin{align} F_{m,n}=e^{-j2\pi(m-1)(n-1)/N},\quad m,n=1,\ldots,N. \end{align} What we can say about the singular value ...
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### References for discrete calculus of variations

What would be good references for discrete calculus of variations? For applications such as minimizing a functional not on a $[0,1]\times[0,1]\to \mathbb{R}$ but on a bitmap image that approximates ...
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For Conjugate Gradient (to solve a linear system $Ax=b$) there is a theorem: "if $m$ is the number of distinct eigenvalues of matrix $A$, then the Conjugate Gradient method converges at the ...
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### How can I numerically solve the Laplace equation with cohomological data?

Consider the problem of solving for $u$ where $-\Delta u = f$, $[u] = [g]$ where $[\cdot]$ denotes cohomology class and $u, f, g$ are $p$-forms on a Riemannian manifold $M$. If $g$ instead was ...
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### closest equidistant point to N points in M dimensions

Is there a formula/algorithm/etc. to find the closest equidistant point (assuming it exists) to a set of points, allowing that the number of dimensions of the space is independent of the number of ...
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### Padé–Hermite approximants of the exponential of type II

I found the explicit expression of the Padé–Hermite approximants of the exponential function, of type I by complex integrals (see for instance Khémira - Approximants de Hermite–Padé, déterminants d’...
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### Determining polynomial approximations of piecewise constant functions

Let $t_1 < t_2 < \cdots <t_m$ be real, and $X = \cup_{i=1}^{m-1} (t_i, t_{i+1})$ be a union of real open intervals. Let $f:X \rightarrow \{-1, 1\}$ be any piecewise constant function of form ...
1 vote
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### How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
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1 vote
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### Typo in error a-priori estimate in a discontinuous Galerkin paper?

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
When solving a system of ODEs: $$\dot{x} = A x,$$ we call the system unstable when the eigenvalues of A have a positive real part. However, the stability region of the explicit Runge-Kutta 4 method ...
Let P be a given multilinear polynomial in $\mathbb{C}[z_1,\dots,z_n]$ and $D\subset \mathbb{C}$ be a given disc in the complex plane. Does there exist an efficient method for checking that $P$ has a ...