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

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26 views

Mean value theorem for sequences

This is a problem I am trying to solve. Given a sequence $x_n$ defined $x_{n+1}=F(x_n)$. Assume $\lim_{n \to \infty}x_n=x$ and $F'(x)=0$. Need to show that $$x_{n+2}-x_{n+1}=o(x_{n+1}-x_{n}).$$ ...
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66 views

Trapezoidal rule - Multivariable

If I wanted to integrate the function $f(x,y)$ over the region $[a,b]\times[c,d]$ with two segments, am I going about this the right way? $$I(f) = \int_a^b \int_c^d f(x,y)\ dy\,dx = \int_a^b g(x) \ ...
2
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33 views

What is the best method to calculate the square root when I know that the root is always an integer?

I have been through the wikipedia page, but wanted to know if there was a preferred (most efficient) method when there is an exact solution to find?
2
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52 views

Formula for $s_n = \sum_{i = 1}^n i^3$ Newton's Forward Difference Interpolation

Use Newton's Forward Difference formula to find an expression for $$ S_n = \sum_{i = 1}^{n} i^3$$ This is from an Introductory Numerical Analysis paper. I cannot figure out the connection ...
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85 views

Solving systems of polynomials with an oracle

I need to solve a system of polynomials. Let the variables be $x_1, \dots, x_n$, and let the polynomials be $f_1, \dots, f_n$ Let's say we have these conditions we can already assume: there are ...
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61 views

Convergence of Newton method under some assumptions

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 508 here) that if $x^\ast$ is the unique root of equation $$f(x)=0$$on interval $[a,b]$ and if the function has ...
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49 views

Why does this nonlinear ODE solution not work?

I am relatively new to Python and trying to use it to solve a second order nonlinear differential equation, specifically the Poisson-Boltzmann equation in an electrolyte. $$\phi''(r) + \frac2 ...
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66 views

Calculating gradient from finite difference results

I am solving the steady-state heat equation in two dimensions: $$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial ...
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38 views

Numerical Analysis, divided differences

This is what I have to prove: $$f[x_0, x_1, \dots, x_n] = \frac{(-1)^n}{(x_0+a)(x_1+a) \dots (x_n+a)}$$ where $f(x) = \frac{1}{x+a}$ and $f[x_0, \dots, x_n]$ is the divided difference of $f$ in ...
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35 views

Can the Lanczos algorithm converge very fast by taking a good initial guess?

Suppose I have the two lowest eigenvectors $v_1$, $v_2$ of a matrix $M$. If slightly change $M$ to $M'$. Can I use $v_1$ or $v_2$ as an initial guess for $M'$? If so, which one should be used, $v_1$ ...
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135 views

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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42 views

How does this equation hold (Secant method)?

Consider we are approxinating a root by the secant method. Then, the interation is given by $x_{n+1}=x_n - f(x_n)\frac{x_n - x_{n-1}}{f(x_n)-f(x_{n-1})}$. In my text (Atkinson), it's written that: ...
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52 views

Show that $\displaystyle\sum_{i=0}^{N-1}|\epsilon_i|\to0, N\to\infty$

Let $I_o=[t_0,t_0+T]\subset\mathbb R, T>0$, If $f\in C^0(I_0\times\mathbb R,\mathbb R)$ and satisfies the Lipschitz condition: $\forall t\in I_0, \forall y,y^{*}\in\mathbb ...
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105 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
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56 views

difference between runge kutta methods of same order

I recently read about runge kutta methods for solving differential equations. So far I understood the idea but up to know nobody could answer me following question: If we consider the explicit rk ...
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59 views

Numerically solve for maximum root

I am looking for an efficient algorithm that can numerically solve a piecewise function for its maximum zero root. The piecewise function will normally take the form of the plots below where by below ...
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99 views

Boundary Conditions for a Finite Difference Approximation of a Sixth Derivative

I am trying to use a finite difference scheme to numerically solve sixth order parabolic equations such as \begin{equation} u_t = u_{xxxxxx} \end{equation} with symmetry conditions \begin{equation} ...
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69 views

Moment Generating Function of a Beta random variable.

After getting some excellent help on this problem in the statistics SE, I am reformuluating my question. Let me know if I should just delete it and ask a new one. Let $V$ be a $Beta(\alpha,1)$ ...
2
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77 views

Constrained non-linear Optimization using Newton's method - Portfolio optimization

I want to solve following constrained optimization problem from portfolio optimization: The solution is supposed to be a modified risk parity portfolio: The optimization problem is: \begin{align} ...
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43 views

Multivariate Appoximation

I have a mathematical model for a complex system which I would like to approximate it. My idea is to run this complex model once and produce some outputs, and then fit a polynomial for these outputs. ...
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69 views

Reciprocal Non-linear Hammerstein integral equation

I came across a problem that looks like a non-linear Hammerstein equation: $$ \displaystyle y(t)= v(t)+\int_{0}^{\infty} \frac{e^{\iota ts}}{y(s)}\mathrm{d}s $$ I tried solving it by collocation ...
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34 views

Finding Volume of Monte Carlo Integration

Suppose $\mathbf{X}\in\mathrm{R}^n$ is an $n-$ dimensional random vector having joint Gaussian distribution i.e. $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol\mu,\boldsymbol\Sigma\right)$, where, ...
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46 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
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69 views

Numerically approximate the maximum of an element of a vector after a series of matrix multiplications.

Where S is a sigmoidal function, A_i is a matrix, and x is an input vector, and ...
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60 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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69 views

How to scale “probabilities” to a given mean?

I have a set of scores $x_i$, $i=1,\ldots,N$ (mimicking probabilities, $0\le x_i\le 1$) and I want to transform them so that the result has a given mean $m$, while remaining in the interval $[0;1]$. ...
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51 views

Solving systems of equations with trigonometric terms

I am trying to solve (or rather find the least squares solution for) a system of equations with trigonometric terms in them. The system consists of pairs of equations of the form $a_1 \cos\theta - ...
2
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63 views

solving singular linear system $Ax=0$

what are computational methods for solving square singular linear system $Ax=0$ for a nonzero $x$ with $A$ of large dimensions?
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14 views

Remainder of the minimax approximation polynomial - number of extrema

Recall some definitions. Let $f \in C [a,b]$. The minimax polynomial $p_n$ is the polynomial $p_n (x) $ of degree $\leq n$ that minimizes $||f-p_n||_\infty $. It can be proved that this polynomial ...
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94 views

How to solve this complicated differential equation?

I need to know how to solve this complicated differential equation in $z$ either analytically or numerically : \begin{eqnarray} \frac{dx_1}{dz} &=& -ib_1x_1 - ikx_2 \\ \frac{dx_2}{dz} ...
2
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167 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
2
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0answers
71 views

Effective computation of matrix commutator

Is there a faster way to compute the commutator of large (at least one of them sparse) matrices $[A,B]$ then to compute $AB$ ,$BA$ and subtract them?
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72 views

Are there high performance computing applications for symbolic integration?

Currently there are a number of applications for numerical integration in applied mathematics and physics. Many of these are integral transforms (often Fourier or Laplace), or solving definite ...
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158 views

Boundary integral method to solve Poisson equation

Suggest how to solve Poisson equation \begin{equation} σ ∇^2 V = - I δ(x-x_s) δ(y-y_s) δ(z-z_s) \nonumber \end{equation} by using the boundary integration method to calculate the potential ...
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0answers
40 views

Numerical Integration of Highly Oscillatory Integral with Misbehaving Derivatives

I'm attempting to numerically handle an equation of the following form: \begin{equation*}f: x \rightarrow \int_{0.00001}^{2} d\omega e^{i \omega x} f(\omega)\end{equation*} where $f(\omega) = ...
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58 views

how to prove this curious identity with the Chebyshev polinomials

we defined the Tm like this (where Tm are the Chebyshev polinomials) Then I showed this: And now I have no idea how to proove this: I also have to make the remark that I also proved that the ...
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0answers
86 views

(newbie) spectral derivative

I have data that form a scalar field on a 2D grid, evenly spaced. The grid has a finite size. There is no particular periodicity patern in my data. I want to calculate the value of the gradient at ...
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225 views

Standard symmetric tridiagonal matrix Eigenvalue decomposition algorithm?

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue ...
2
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0answers
348 views

Numerically calculate the boundary of a basin of attraction for a high dimensional dynamical system

I am looking for an efficient, non-exponential time algorithm to calculate the boundary of a basin of attraction for a stable fixed point in a high dimensional nonlinear dynamical system. The naive ...
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0answers
49 views

Interpolation Concept or Misconcept?

This question is regarding Interpolation, say we are given table of data ,$x_0 <x_1<\dotsb<x_k<\dotsb<\dotsb<x_n$ as well as $f(x_0),f(x_1),\dotsc,f(x_n)$ it is said that "we ...
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0answers
38 views

Time Discretization

I wonder why we work with constant discretization in Time Discretization of numerical approximation for numerical scheme if we take not necessarily constant Discretization Is the numerical scheme ...
2
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0answers
61 views

Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
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0answers
74 views

Lagrange's interpolation to solve for 0 of y(x)

I have the data composing of 7 elements x is from 0 → 3 incrementing by 0.5 y is from 1.8241 → -1.5427 I am supposed to use Lagrange's interpolation of three nearest neighbor data points. I am ...
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0answers
58 views

Numerically stable Lanczos process? I need to compute Elements of inverse in sparsity pattern of A

I have a large sparse symmetric positive definite matrix NxN matrix $A$. Let $s$ be the average number of non-zeros per row (i.e. $sN$ total non-zero elements). I would like to compute the elements ...
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0answers
39 views

Accurate computation of arcsec near branch points

The direct numerical implementations of the usual definitions of the complex $\mathrm{arcsec}(z)=\arccos(1/z)$ and similar for $\mathrm{arccsc}(z), \mathrm{arcsech}(z), $ etc are not accurate near ...
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27 views

Does eigenvalue theory remain in numerical applications?

If I have a symmetric matrix $A \in \mathbb{Q}^{n \times n}$ in matlab, then in theory it is guaranteed that $A$ has a orthogonal basis of eigenvectors and real eigenvalues. Does this remain in ...
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62 views

We are learning about LU Decomposition .. because?

I know what LU Decomposition is but I don't know why we have have to learn about it. What are we using it for? (What's the point to know about it?) Thanks.
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40 views

Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
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143 views

lagrange interpolation question here

We have the function : $f(x)=\cos(x) + \sin(x)$ and $x_0=0, x_1=0.25 , x_2=0.5, x_3=1$ a)Find Lagrange polynomial for this function. c)Find the real approximation error. d)Find the limit of the ...
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0answers
44 views

Is numerical stability preserved under basis transformation for pde

I'm using a Backward Time Central Space method to solve the heat equation in polar coordinates. In Cartesian coordinates, it is easy to show this is unconditionally stable (by assuming solution of the ...