# Questions tagged [numerical-methods]

Questions on numerical methods; methods for approximately solving various problems that often do not admit exact solutions. Such problems can be in various fields. Numerical methods provide a way to solve problems quickly and easily compared to analytic solutions.

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### Implementation of a simulation of an incompressible Newtonian fluid with uniform density

Let $d\in\left\{2,3\right\}$ and $\Omega\subseteq\mathbb R^d$ be a bounded domain. I want to simulate an incompressible Newtonian fluid with uniform density $\rho$ and viscosity $\nu$. The evolution ...
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### orthogonal functions with orthogonal first derivatives

Is there any set of functions $\phi_1(x) , \phi_2(x) , \ldots , \phi_n(x) , \ldots$ defined on $[a,b]$ such that \begin{eqnarray} &&\langle\phi_i, \phi_j\rangle = \int_a^b \phi_i(x)\phi_j(x)...
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### What’s the best way to cut an apple?

Take the apple in one hand, and the knife in the other. In the first cut, the apple is divided in two pieces: a small one that drops into the plate and a big one that is still hold with the hand. This ...
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### Nature of ODE $\dot x=x^2-\frac{t^2}{1+t^2}$

Discuss the equation $\dot{x}=x^2-\frac{t^2}{1+t^2}$. Make a numerical analysis. Show that there is a unique solution which asymptotically approaches the line $x=1$. Show that all solutions below ...
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### Evaluate $\int_0^2 \sqrt[3]{x^2 + 2x - 1} \, dx$

Calculate the value of the integral $$\int_0^2 \sqrt[3]{x^2 + 2x - 1} \,dx$$ with measurement uncertainty not larger than $10^{-3}$. I know we can evaluate integration using the "trapezoidal ...
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### Optimized way to compute L1 distance matrix

I'm computing distances between two groups of multi-dimensional points giving a matrix of distances pairwise between points. For the L2 (euclidean) distance I can use optimized matrix multiplication ...
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### Solving numerically the equation of motion of D7 brane perturbation

I want to solve this equation $$\partial_{\rho}^{2}\phi+\frac{3}{\rho}\partial_{\rho}\phi+\left(\frac{M^{2}}{(1+\rho^{2})^{2}}-\frac{l(l+2)}{\rho^{2}}\right)\phi=0$$ numerically. I know that ...
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### Approximating a function with a piecewise constant function

I have some distribution X of values (which I don't know exactly but I can sample many times). I also have a function $f : X \to Y$ which may be complicated. I want to approximate $f$ with a piecewise ...
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Is there a procedure/algorithm for calculating sums of the form $$\sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}}$$ where $$L_i(n_1,\ldots, n_r) :... 5 votes 0 answers 105 views ### Numerical integration of functions of bounded variation Let f: \mathbb{R} \rightarrow \mathbb{R} be such that f\in BV(\mathbb{R}) \cap L^1(\mathbb{R}). Now, since f\in BV(\mathbb{R}) pointwise values makes and consequently we can define numerical ... • 1,152 5 votes 0 answers 69 views ### Convergence of the Implicitly Restarted Arnoldi Method (IRAM) I know that the Implicitly Restarted Arnoldi Method (IRAM) consists of using the implicitly shifted QR algorithm to restart the Arnoldi process without losing to much information about the eigenspace. ... • 5,534 5 votes 1 answer 185 views ### Initial conditions for which these two recursive sequences converge The following problem is a generalization of an exercise that the professor give me and that I have already solved. In the initial statement \alpha=0 and given 0<a_0<1, the limit of the ... • 649 5 votes 0 answers 70 views ### Why is the average magnitude of the imaginary parts of the roots of these polynomials greater than that of the real part? Let f_n(x) be a polynomial obtained by replacing 10 with x in the base 10 expansion of n, n \ge 10. Eg. f_{98}(x) = 9x+8 and f_{2021}(x) = 2x^3 + 2x + 1. f_n(x) has exactly [k = \log_{... • 19.6k 5 votes 0 answers 131 views ### Approximate solution of \Gamma(x+a)=k\, \Gamma(x+1) for 0 \leq x \leq a As the title says, I am looking for good approximate solution of the equation$$\Gamma(x+a)=k\, \Gamma(x+1) \qquad \text{for} \qquad 0 \leq x \leq a$$for a given real and positive value of a and ... • 261k 5 votes 1 answer 212 views ### Discrete entropy inequality for scalar conservation laws Consider a scalar conservation law u_t+f(u)_x=0. A three point monotone scheme given by, \begin{eqnarray} u_i^{n+1}=u_i^{n}-\lambda (F(u_i^n,u_{i+1}^n)-F(u_{i-1}^n,u_i^n)) \end{eqnarray} where F(u,... • 1,015 5 votes 0 answers 95 views ### Are there any tricks to find an orthonormal basis for polynomials by hand? I am taking a Numerical Analysis class. On a previous exam, there is a question which involved finding an orthonormal basis for \mathbb{P}_2 w.r.t the inner product (u, v) = \int_0^2 u(t)v(t) dt. ... • 492 5 votes 0 answers 100 views ### "Solving" for n the equation {2n\brack n}=k (Stirling numbers of the first kind) Interested by this question, I wondered how easily we could "solve" for n the equation$${2n\brack n}=k$$where the left hand side is the unsigned Stirling number of the first kind. I ... • 261k 5 votes 0 answers 124 views ### Evaluating series \sum_{n=2}^\infty \left(n^{1/n}-1-\frac{\log n}{n} \right) This is similar to my recent question, but probably more interesting. What can we say about this slowly converging series:$$S=\sum_{n=2}^\infty \left(n^{1/n}-1-\frac{\log n}{n} \right)$$We can ... • 31.5k 5 votes 0 answers 594 views ### How can I numerically compute a stochastic integral? I am trying simulate a to solve a 2-dimensional stochastic process and Y_t^1 is a mean-reverting square root process which I simulated on a time grid using its known conditional distribution. I ... • 3,697 5 votes 0 answers 92 views ### Efficient method to find H given by H(x)=\int_0^x f(x-u) f(x-au) e^u \, du Question Let f:[0,\infty) \rightarrow [0,\infty) be some continuously differentiable function and a \in (0,1) then we define the function H:[0,\infty) \rightarrow [0,\infty) by letting:$$H(x)=\...
Briefly speaking, the Feynman-Kac formula gives a way for constructing $C^2$ functions satisfying certain PDEs in the classical sense (at least, it's how it is explained in Oksendal's book that I am ...