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

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56 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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63 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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136 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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107 views

Numerical integration of function with singularities

I am currently trying to solve a semi-infinite integral containing a set of singularities lying on the real axis numerically. The process I am using is breaking the integral into small steps $\Delta ...
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32 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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48 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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72 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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147 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 ...
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235 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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40 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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32 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 ...
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54 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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51 views

Approximating the Fourier transform with DFT/FFT

Suppose I have a continuous function $f(x)$, $x\in[-L/2,L/2]$. Its $L-$periodic Fourier coefficients are given by $$ \hat{f}[k]=\frac{1}{L}\int_{-L/2}^{L/2}f(x)\exp(-2\pi ikx/L)dx $$ If I apply ...
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58 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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48 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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38 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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25 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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53 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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37 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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109 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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37 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 ...
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243 views

Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
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33 views

Metastable solution for system of nonlinear equations

System of nonlinear equations: $$E_i=\epsilon_i+\sum_{j\neq i}^N \left(\frac{1}{1+\exp(E_j/T)}-\frac{1}{2}\right)\frac{e^2}{r_ij} \tag 1$$ where $T=0.05$, $r_{i,j}$ is given symmetric matrix with ...
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216 views

Runge-Kutta method accuracy

I got Runge-Kutta method here and I solve this system using it. So here's Runge-Kutta stuff $k_1 = f(t_n, y_n)$ $k_2 = f(t_n + h/2, y_n + hk_1/2) $ $k_3 = f(t_n+h, y_n - hk_1 + 2hk_2)$ $y_{n+1} ...
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85 views

Numerical Methods for estimating divergence over an improper integral

Problem given a function $f(x)$, defined on $[ \epsilon, \infty )$. Is there a way to "numerically estimate" whether the integral of the function diverges over the domain $[ \epsilon, \infty )$? ...
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24 views

Nontrivial Matrix-estimate

I try to proof the following estimate: \begin{align} h' W^{-1} H W^{-1} h \geq c h' H h \qquad c>0, \qquad\qquad (1) \end{align} where $h\in\mathbb{R}^{K-1}$ and ...
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119 views

Inverse of Sum of Matrix Inverses

Given $N$ positive-definite matrices $\Lambda_i$, I need to efficiently compute $\Gamma_N$, where $$ \Gamma_n = \left(\sum_{i=1}^n \Lambda_i^{-1}\right)^{-1}. $$ Applying the Woodbury matrix identity ...
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0answers
117 views

How to generate a random matrix which have given singular values?

I know one method: generate a random matrix, apply SVD decomposition, modify singular values, and then multiply those matrices back together. However, I'm wondering how random this method is. Since ...
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39 views

Is scalar product a well-conditioned operation?

I'm reading a course and one of the exercises is about establishing whether scalar product is a well-conditioned operation. Here's their solution. They disturb each element of the vector by ...
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48 views

Inequality in H-curl function space

Define a function space V , $$ V:=\{\mathbf{v} \in \mathbf{L}^{1+\alpha}(\Omega), \mathbf{curl}~\mathbf{v} \in \mathbf{L}^2(\Omega)\}, $$ equipped with graph norm $$ \|\mathbf{v}\|_{V} := ...
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79 views

how to choose point spacing to approximate a parametric curve using line segments?

Suppose I have a parametric equation for a curve $\vec{r} = f(t)$, which I wish to draw using line segments between some set of points at times $t_0, t_1, t_2,$ etc. If I want to achieve a given ...
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0answers
46 views

When solving PDEs is there an alternative to interpolation for out-of-grid point?

I'm numerically solving a PDE where the space domain is huge. So, I often need to interpolate to get out-of-grid points needed by the finite difference algorithm. As a result, I've a lot of numerical ...
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0answers
118 views

Solution of an implicit Fourier transform equation

How does one solve the following equation ($\hat{a}(k)$ denotes the Fourier transform of $a(x)$ and $q$ is real positive): $$\hat{a}(k)=f(k)\widehat{a^q}(k).$$ This equation appeared in some paper. ...
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754 views

Show that the averaged vector field one step method is well-defined

Let $\dot{\mathbf{y}} = \mathbf{f}(\mathbf{y}), \;\mathbf{f}: D \subset \mathbb{R}^d \to \mathbb{R}^d$ be an autonomous differential equation with $\mathbf{f}$ smooth. We define the averaged vector ...
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57 views

To calculate a derivative of a set of points, is it more correct to interpolate finite differences or to derivate the interpolation?

I have a series of points extracted from numerical simulations. I also recently discovered the amazing power of finite differences. Nevertheless, I was used to estimate my derivatives from the ...
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47 views

Finding $k$ unknowns given the sum of their first $k$ powers

Motivation: The motivation for this question came from a Computer Science problem of finding duplicates in a list in constant time and constant space. If the list of numbers was $i_1, i_2, \ldots, ...
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66 views

automatization of numerical derivation

I would like to know if there is an automized or fast way to numerically derivate a large number of tab-delimited files (derived from the program kaleidagraph) and to automatically extract some key ...
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144 views

monotonic smoothing fit to be implemented (in python or other language)

In a post that already exists, implementation-of-monotone-cubic-interpolation, there is a good method for fitting data which necessarily includes all of the given points. But, what if I need to ...
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0answers
84 views

van der pol and liapunov

i have attempted this question and done as much as i possibly could, any help regarding this question would be very helpful and appreciated. a) show that the second-order differential equation for ...
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0answers
294 views

Proving invertibility of matrices using banachs lemma

I'm studying for finals and trying to understand how you can possibly use banach's lemma for anything worthwhile, more particularly we have a bunch of sample questions that say it can be used to prove ...
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0answers
77 views

How to can I transform the 2D cuasi Laplace equation with variable coefficients to finite difference scheme?

I want to solve $$\frac{\partial}{\partial x}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial x}\right) + \frac{\partial}{\partial y}\left(\frac{1}{\rho(x,y)}\frac{\partial \Phi}{\partial ...
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0answers
51 views

What kind of numerical methods are best applicable to this?

I'm wondering: what would be the best numerical method for solving a nonlinear integral equation of the form $$f(x) = a(x) + \int_{-A}^{A} K(x, t, f(t)) dt$$ where $f$ is the unknown function, a ...
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0answers
137 views

Induction proof of polynomial interpolation theorem

show if $\phi (x) = f(x)g(x)$, this is valid: $\phi [x_0,x_1,...,x_n]=\sum\limits_{r=0}^n f[x_0,x_1,..,x_r]g[x_r,x_{r+1},...,x_n]$ by induction. I have tried to prove it by the divided differences ...
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42 views

maximal m-elements of the matrix inversion

Suppose the $n\times n$ matrix $A$ is invertible, and all its elements are between 0 and 1. The existing matrix inversion operation of $A^{-1}$ will take $O(n^3)$ time. Now I just want to find the ...
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159 views

Optimization problem about large matrices

I'd like to solve the following optimization problem: Find non-negative scalar $a$, $b$, $c$ to minimize $\| (D-(aA+bB+cC+D^{-1})^{-1})y\|^2+2\operatorname{trace}((aA+bB+cC+D^{-1})^{-1})$ where ...
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0answers
380 views

Consistency order of backward Euler method

How can I proof that backward Euler method has consistency order 1? Implicit function theorem states that for a sufficiently small $h$, $$ \vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1) $$ has a unique ...
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188 views

all eigenvalues of a large sparse symmetric matrix

my question is similar to how to diagonalize a large sparse symmetric matrix, to get the eigenvalues and eigenvectors however i wish to be more concrete and ask if one can, on a standard PC (e.g. a ...
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0answers
96 views

Suggestions for projects in mathematics of finance

I am looking for computational projects related to mathematics of finance suitable for a senior level independent study for a student who has seen the green light, (or the light of the green!). I had ...
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0answers
55 views

Solution to pertubed linear system

Suppose one has the following system of linear equations $$(A + \Delta A) x = b$$ where $A$ and $\Delta A$ are large sparse matrices and $\Delta A$ is "small" compared to $A$, furthermore vector $x$ ...
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30 views

Approximate a constant function with sequence of spline functions

Suppose that for a constant $c \in \mathbb{R}$ $$\sup_{t \in [a,b]}\Big|c- \sum_{l=1}^{m}a_{l}\ B_{l}(t;q)\Big|< \epsilon.$$ The $B_l$ form a B-spline basis of degree $q$ on the interval $[a,b]$ ...