Numerical algorithms for problems in analysis and algebra, scientific computation

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1answer
375 views

sign-flipping inverse

Consider this matrix: $Z=\begin{bmatrix}23.9 & -7 & -17 \\\\ -7 & 23.9 & -17 \\\\ -17 & -17 & 33.9 \end{bmatrix}$ Its inverse is entrywise negative (you can check...) and ...
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0answers
337 views

Fast inversion of a special kind of matrices - approximations are ok

Suppose I have a stochastic matrix $M$ (with thousands or millions of stochastic column vectors), which I split into two matrices: $D$ containing only the diagonal entries of $M$, and $R$ containing ...
2
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0answers
432 views

Update the inverse of sum of two symmetric matrices

There are two invertible symmetric matrices A and B, of which B is a block diagonal. A and B have the same dimensions. I need to iteratively calculate the inverse of M = s * A + B, where s is a ...
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13 views

Numerical evaluation of orthogonal polynomials [on hold]

I've written some Matlab procedures that evaluate orthogonal polynomials, and as a sanity check I was trying to ensure that their dot product would be zero. But, while I'm fairly sure there's not ...
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0answers
37 views

What's the advantage of majorization-minimization (MM) algorithm [on hold]

The majorization-minimization (MM) algorithm is a framework for convex and nonconvex optimization. When applied to nonconvex optimization, the MM algorithm solves a sequence of convex problems to ...
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2answers
1k views

How to solve a fifth degree polynomIal

Charles Hermite have created a method using elliptic functions to solve fifth degree polynomial, to get around the theory of Galois. Can someone explain me it and give a simple example? Tank you.
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9answers
2k views

What is… A Grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...
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2answers
373 views

Discrete Wavelet Transform and L2 Basis

Using the mother wavlet $phi$ one obtains an orthonormal basis $\phi_{j,k}(x):=2^{j/2}\,\phi(2^j\,x-k)$of L^2 (on the unit interval say). Given a function $f$ on can calculate the coefficients using ...
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7answers
6k views

There must be a good introductory numerical analysis course out there!

Background As a numerical analyst, I've frequently taught the 'Introductory Numerical Analysis' class. Such courses are found in many major universities; the audience typically consists of reluctant ...
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0answers
89 views

Evaluate a Function to Full Machine Precision [closed]

If we want to evaluate $$f(x)=\frac{e^x-1-x}{x^2}$$ then we have to observe its large relative error as $x\to 0$. My question is that how can we find a method so that we can compute $f(x)$ to full ...
10
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2answers
246 views

Why Householder reflection is better than Givens rotation in dense linear algebra?

It’s obvious that Givens rotation works better with sparse matrices. But I don’t know why Householder reflection is better for dense matrices. Does it require less computations? Or it’s numerically ...
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3answers
207 views

Numerical solution of SDEs with colored noise

I am trying to numerically solve an SDE with both white and colored noise that models a non-linear circuit: $$ dX_t = f(X_t) dt + \sigma_w dW + \sigma_c dC $$ where $W$ is a standard Brownian motion ...
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3answers
579 views

Numerical solution to Fisher-Kolmogorov equation

Can you use the Crank-Nicolson method to get a numerical approximation to the fisher-kolmogorov equation? If not what would be the easiest way to model the equation using matlab? Thanks and sorry its ...
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0answers
25 views

state-of-art numerical contour (complex) integration method when contour is square and available values are evenly spaced

What is current state-of-art for numerical contour integration method (for $f(z)$ with $z$ being complex number and $f$ complex-valued) when contour is square on complex plane, and one only has ...
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0answers
22 views

numerical differentiation of sum of one-dimensional sinusoids with angular frequency close to Nyquist one

Suppose that $f(t) = \sum_i C_i e^{i\omega_i t}$, and $f$ is sampled at certain sampling angular frequency $\omega_s$. All $\omega_i$s are very close to $\omega_s/2$, and thus standard finite ...
3
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1answer
49 views

Injectivity of vector functions: Numerical Verification

Problem Setup Let $f:A\rightarrow B$, be a continuous function, $A\subset\Re^{n}$,$B\subset\Re^{m}$, $m\geq n$ and $A, B$ compact. The function $f(\cdot)$ can only be evaluated numerically. ...
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1answer
103 views

Reference Request: Variational Problem

I want to solve approximately the following variational problem: Given a function $c:[-1,1]^2\rightarrow [0,1]$, constants $p_1...p_n\in \mathbb{R}^+$, $\alpha_1...\alpha_n\in \mathbb{R}$, and ...
2
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1answer
111 views

Solving a simple Schrödinger equation with Fast Fourier Transforms

While trying to solve a stochastic Gross-Pitaevskii equation I have found a problem that can be tracked down to something buggy occurring in the simplest Schrodinger equation possible: $$\partial_t ...
5
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2answers
174 views

Alternating binomial Dirichlet series

I have come across the following deceptively simple expression: $$ H_n^s=\sum_{j=1}^n(-1)^{j-1}\left(\begin{array}{c}n\\j\end{array}\right)j^{-s} $$ We have (using eg mathematica, though probably ...
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2answers
2k views

Finding the formula for Bezier curve ratios (hull/point : point/baseline)

Given a cubic Bezier curve defined by points $p_1$, $p_2$, $p_3$, and $p_4$, a point $B$ on that curve at some $t$ value (where $0 \leq t \leq 1$), a point $A$ on the line $(p_2 - p_3)$ at distance ...
21
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2answers
5k views

The unreasonable effectiveness of Pade approximation

I am trying to get an intuitive feel for why the Pade approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can't ...
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2answers
138 views

Roots of the Chebyshev polynomials of the second kind

It is known that the roots of Chebyshev polynomials of the second kind, denote it by $U_n(x)$, are in the interval $(-1,1)$. I have noticed that, by looking at the low values of $n$, the roots of ...
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1answer
278 views

Can I find the gap between the two least eigenvalues of this special matrix A(t)?‎

I am interested in finding the gap between the two least eigenvalues of $A(t)$, a Hermitian $N\times N$ sparse ‎matrix whose diagonal elements are $a_it+b_i\,(1\leq i\leq N)$, and all off-diagonal ...
3
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1answer
565 views

Claimed Quadrature Results seem Impossible

We've been preparing a preprint that shows that the convergence bounds proved for tanh-sinh quadrature for numerical integration, cannot possibly hold, and an error must exist - since they imply a P ...
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0answers
72 views

Numerical inversion involved confluent hypergeometric (1F1) (or Kummer function)

Edit: The question is solved !! The code is actually correct. There is not error in the codes. I miss-used it. Thank you for your attention : ) This problem arises when I tried to compute the valua ...
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1answer
327 views

Reduced eigenvalue problem

Given a symmetric $n\times n$ matrix matrix and $m$ of the eigenvalue/vector pairs, is there an efficient and numerically stable way to factor out the known structure such that only a smaller ...
3
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0answers
39 views

numerical stability of root identification via Newton-Raphson iteration of Stieltjes residue sums

I have asked several questions on math.SE in order to compute numerically the poles of high-degree Padé approximations for $e^{-x}$, because a computation directly from the polynomial ...
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1answer
43 views

Is spectral properties a general term for condition number?

I am reading an article about solving large sparse linear systems, in this paper it’s said that most of the iterative methods to solve $Ax = b$ are very much influenced by the spectral properties of ...
107
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2answers
10k views

What are the shapes of rational functions?

I would like to understand and compute the shapes of rational functions, that is, holomorphic maps of the Riemann sphere to itself, or equivalently, ratios of two polynomials, up to Moebius ...
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0answers
25 views

Numerical techniques for nonlinear, coupled integro-differential equations

The gist of the problem I have is I want to be able to find a numerical solution to these three coupled, rather unpleasant looking integro-differential equations (1): $$ \frac{d^2 x(t)}{dt^2} = ...
60
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19answers
21k views

Why were matrix determinants once such a big deal?

I have been told that the study of matrix determinants once comprised the bulk of linear algebra. Today, few textbooks spend more than a few pages to define it and use it to compute a matrix inverse. ...
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2answers
401 views

“Fractally self-similar” numbers

This is another question about visualization of Ford circles, the previous one being Confusion with practically implementing rational approximations. Here is an output of zooming into Ford circles at ...
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1answer
204 views

Confusion with practically implementing rational approximations

Writing a program visualizing Ford circles I've encountered a seemingly purely programmatic puzzle but then gradually realized there are some mathematical aspects of it which I don't understand. Let ...
4
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1answer
190 views

Existence of nonlinear equation

How can we prove that equation (1) has solutions for every $p$. I mean, is there an analytic method that can be used to show that there exist solutions for every $p$ for this nonlinear equation: ...
11
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1answer
4k views

Meaning of $\Subset$ notation

The symbol $\Subset$ (occurring in places where $\subseteq$ could occur syntactically) comes up frequently in a paper I'm reading. The paper lives at the intersection of a few areas of math, and I ...
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0answers
21 views

Monotonicity per dimension of multivariate scattered data

For my thesis, I am working on interpolation using the RBF method (Radial Basis Functions). Before interpolating, I want some a priori insight into the data, for example check in which dimensions it ...
5
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1answer
68 views

Difference stencils approximating Laplacian

Let $\Delta$ be the Laplace operator on the interval $[0,1]\subset \mathbb{R}$. Divide $[0,1]$ into small intervals of size $h$ to get an equidistant grid. One can approminate $-\Delta$ on this grid ...
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2answers
125 views

The condition number of a scaled Vandermonde matrix

Let $V(x_1,..,x_n)$ be the Vandermonde matrix induced by $x_1,..,x_n$, and let $\tilde{V} := V(\frac{x_1}{h},...,\frac{x_n}{h})$. My intuition says that the condition number should be invariant under ...
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1answer
130 views

Iterative Root Finding

Consider a function $f(x)=g(x,h(x))$, which we know has a unique root $x^*$. The functions $f$, $g$ and $h$ are all continuous in $x$ and behave nicely. Iteratively solving $g(x_{i+1}, h(x_i))=0$ with ...
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1answer
99 views

Does Gaussian Quadrature actually refer to Gauss-Legendre Quadrature?

When the term Gaussian Quadrature appears in most Literatures, does it actually refer to Gauss-Legendre Quadrature. In other words, do they implicitly admit that they use the Legendre orthogonal ...
4
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1answer
129 views

How to calculate log or exp of a value in GF(2^n) using log/exp table of GF((2^k)^m) where n=k*m?

Consider Galois fields $\mathbb{F}_{2^n}$ and $\mathbb{F}_{2^k}$, where $n=km$, and $\mathbb{F}_{2^k}$ is a ground field of $\mathbb{F}_{2^n}$. I’d appreciate pointers to papers or suggestions on: ...
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0answers
109 views

An alternative to the Euler--Maclaurin formula: Approximating sums by integrals only

The Euler--MacLaurin summation formula can be written as $$ \sum_{i=0}^{n-1} f(k)\approx \int^{n-1}_0f(x)\,dx + \frac{f(n-1) + f(0)}2 + \sum_{j=1}^m\frac{B_{2j}}{(2j)!}[f^{(2j - ...
2
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1answer
104 views

numerical methods for discontinuous ODEs

Greetings, what are state of art methods for numerical solution of ODEs with discontinuous right side? I'm mostly interested piecewise-smooth right side functions, e.g. sign.
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22 views

Numerical integration over a cube with non-product weight

Numerical integration over an interval with (well-behaved) weight functions is a research area that has received considerable attention in the past centuries. Any cubature formula over a interval ...
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1answer
60 views

The classical two phase Stefan problems

What is the most commonly used treatment method of the moving interface in the classical two phase Stefan problems with the finite element method. Here I mean the water-ice two phase problem under ...
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0answers
131 views

Is there a brute force method for determining irreducible representations?

Suppose I have some groups $G_1$, $G_2$, $G_3$, etc... Then the direct product is given by $G = G_1 \times G_2 \times G_3 \ldots$ I know that the sub-representations of a reducible representation ...
2
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0answers
60 views

Natural neighbor interpolation

Recently I am interested in Natural neighbor interpolation, that is : Given a function $P(x)$ and some interpolation points $\{x_i,P(x_i)\}_{i=1}^N$, we have the interpolation function ...
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0answers
29 views

On the computation of generalized eigenvalues of a low-rank approximation using SVD

I have trouble deriving an expression of the generalized eigenvalues of a matrix pair, found in http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4618700 . The setup is the following and can ...
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1answer
150 views

Stone-Weierstrass Theorem, polynomial interpolation, divided difference in complex plane

Setting: Let $\Gamma$ be a simple smooth($C^\infty$) curve in $\mathbb{C}$ parametrized by the injective map $\gamma:[0,1] \to \mathbb{C}$. Assume $f$ is a function defined on $\Gamma$ s.t. $f$ is ...
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2answers
598 views

Distribution of the Error term in GH Hardy's “curious result” $\sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$

In an early paper, GH Hardy talks about the distribution of "curious" sum: $$ \sum_{\nu \leq n } \{ \nu \theta \}^2 = \tfrac{1}{12} n + O(1)$$ where $\{x\}:=x-\left \lfloor x \right \rfloor -1/2$. ...