Numerical algorithms for problems in analysis and algebra, scientific computation

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35 views

### Inner Products vs Discretizations of Functions

Let $f:(0,1)\rightarrow [0,1]$ be a $\mathcal{C}^{\infty}$ function.
We say that a function $D_r^f:\{1,...,r\}\rightarrow [0,1]$ is an $r$-discretization of $f$ if $$D_r^f(j) = \frac{1}{|a(j)-b(j)|}\...

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14 views

### Why the negative gradient gives the direction of the steepest decrease in the gradient descent algorithm? [migrated]

I understand that the gradient vector gives the direction of the maximum growth. What I don't get is why going the exact opposite direction is going to get the maximum decrease?
By sure that holds ...

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93 views

### Does the method below provide any advantages?

I completed course intro to numerical method and lately interested and trying to find special way to solve this ODE.I have try and ask in math exchange stack but do not get any respond so decided to ...

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80 views

### Orthogonal Procrustes problem for sub-spaces?

By Orthogonal Procrustes problem I mean given matrix $A$ and $B$ finding a orthogonal matrix $R$ which most closely maps $A$ to $B$, this has a solution as shown in https://en.wikipedia.org/wiki/...

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68 views

### Negative eigenvalue of Toeplitz Hermitian matrix?

I am working on estimation of a covariance matrix and I know that the matrix is Toeplitz. The desired matrix should not produce negative eigenvalues at all. However, sometime my estimation leads to a ...

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36 views

### Proving Fixed Point Algorithms

In Thomas Minka's paper on Estimating the Dirichlet Distribution (link here http://research.microsoft.com/en-us/um/people/minka/papers/dirichlet/minka-dirichlet.pdf), the author presents a fixed ...

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61 views

### Eigenvalues of approximations to product-convolution operators

Consider an operator $T: L^2 \mapsto L^2$ of the form $TA = g (h \ast A)$ where $g$ is and $h$ are bounded $C^\infty$ functions.
This operator $T$ can be shown to be Hilbert-Schmidt, hence compact. ...

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42 views

### Largest instance of highly nonlinear benchmark functions (e.g. Rastrigin function)

What is the largest instance size (number of variables) ever numerically solved for highly nonlinear (continuous, not combinatorial) optimization benchmarks functions, such as Rastrigin, Schwefel or ...

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92 views

### Connection between cardiac equations and untangling knots?

I was surprised to learn that there is (conjecturally) a connection between a cardiac muscle model known as the FitzHugh-Nagumo equations, and untangling knots:
Maucher, Fabian, and Paul Sutcliffe....

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23 views

### Dennis More' Superlinear Convergence_refrences request

Why in the proof of superlinear convergence of restricted broyden class (for the unconstrained optimization) we need the bounded deterioration condition for the approximation of all the true hessian ...

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**1**answer

92 views

### A centralised website for computational attempts in graph theory and metric geometry?

The set of questions below stems from this question.
1) does a website exist that contains (at least links to) code and data files, with the aim to centralise computational results in graph ...

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119 views

### Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...

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**1**answer

94 views

### Fast Fourier Transforms for non-trigonometric bases

The fast Fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other bases, e.g. orthogonal polynomial bases ...

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69 views

### Deriving Milne's predictor of order four from extrapolation polynomial [closed]

I am trying to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = y_{n-3}+\...

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24 views

### Adaptive refinement of integral domain

In electromagnetics we need to calculate the radiated power which is defined as something like
$P_r=\int_0^{2\pi}\int_0^{\pi}R(\theta,\phi)\sin{\theta}d{\theta}d\phi$
We already have $R(\theta,\phi)...

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**1**answer

95 views

### Efficient computation of matrix exponential of trace zero matrix [closed]

I am looking for identities that may help with numerical computation of the matrix exponential ${\rm exp}(A)$ where ${\rm tr}(A)=0$. I am already aware of general-purpose algorithms for computing the ...

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102 views

### How to rotate a covariance matrix which contains quaternion elements? [closed]

I am implementing a paper which recovers full-3d body pose from images.
It represents individual body parts as 7D vectors containing first the absolute 3D location [x y z] and then the unit ...

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**1**answer

249 views

### How to compute $\sum_{x \in \mathbb{Z}^n} e^{-x^TMx}$ efficiently

Let $M$ be a real symmetric integer valued positive definite matrix with $\det(M) \geq 1$. I would like write code to compute
$$S_M= \sum_{x \in \mathbb{Z}^n} e^{-x^TMx}.$$
One option is to simply ...

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vote

**1**answer

41 views

### Subquadratic multiplication of probability mass functions (with log-convolution?)

We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$.
That is, we are given two finite input vectors $x$ ...

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51 views

### Separable Least squares - is there a notion of conjugate directions?

I have a general question.
Suppose I have the following to optimize
$$\|Y-A(\mathbf{x})B(\mathbf{y})\|^2$$
where $Y$ is a vector, $A(\mathbf{x})$ is a matrix that depends on a vector $\mathbf{x}$ in a ...

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160 views

### Efficiently calculate the trace of the product of two large but symmetric matrices, one of which is an inverse

Sorry about the long title. I need to calculate the trace of $M(M+D)^{-1}$, where $M$ is a dense symmetric matrix, and $D$ is a diagonal matrix. The main issue is the dimension could be large (usually ...

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27 views

### Circulant Block Sparse Matrix Least Squares

I have what boils down to a least squares problem that I am trying to exploit structure in order to calculate efficiently.
Consider a NxN circulant matrix A. Where the first row is [1,2,3,4,0,0,0,...,...

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90 views

### What does the Von Neumann's stability analysis tell us about non-linear finite difference equations?

I've asked this question on computation science stackexchange, but it did not receive any answers so I have decided to ask it here as well.
I am reading a paper [1] where they solve the following non-...

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**2**answers

75 views

### Euler Schemes in Stochastic Differential Equations

So i am trying to understand what happens in Implicit (backward) and Explicit (forward) Euler in Stochastic Differential Equations
I ll start with explicit. Say i have the following SDE known as ...

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**1**answer

78 views

### Add a multiple of $I$ to a matrix to minimize its operator norm

Given $A\in\mathbb{C}^{n\times n}$, what is $s_* = \arg\min \|A-sI\|$?
Here $\|A\|$ is the operator norm, $\|A\|=\rho(A^*A)^{1/2}$, and $I$ is the identity.
The corresponding problem for the ...

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3k views

### Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black
holes one billion light-years away. One of the black holes had 36
times the mass of the sun, and the other 29. After the merging the
...

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46 views

### Find optimal value for a regularization parameter in generalized eigenvalue problem

Consider the generalized eigenvalue problem :
$ \Sigma_{XY} \Sigma_{YX} {W} = \lambda \Sigma_{XX} {W} $
where $\Sigma_{XX} $ and $\Sigma_{XY}$ are sample covariance matrices are of the matrices $X$...

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59 views

### What's the advantage of majorization-minimization (MM) algorithm [closed]

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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218 views

### An inequality which involves a sum of integrals

Please help me to prove
$$
\sum\limits_{j=2}^n \frac{1}{j^\alpha (j-1)^\alpha} \int\limits_{j-1}^j \frac{dx}{x^{1-\alpha}(n-x)^\alpha} \leq \int\limits_0^1 \frac{dx}{x^{1-\alpha}(n-x)^\alpha},\quad \...

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**1**answer

53 views

### Uniqueness and invariance of the LDLT decomposition

A real symmetric positive semi-definite matrix $A$ can be decomposed in the form
$A = P^TLDL^TP$,
where $P$ is a permutation matrix, $L$ is a lower unit triangular matrix and $D$ is a diagonal ...

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98 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 ...

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24 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 ...

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**1**answer

56 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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**1**answer

110 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 $\...

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votes

**1**answer

217 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 \...

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**0**answers

18 views

### How can I filter the effects of a variable from a correlation matrix?

I have a correlation matrix (it contains 500 columns and 500 rows) and I would like to make an other correlation matrix in which one variable (and its influences) is filtered from the initial matrix. ...

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15 views

### Maximization of the difference of a monotone submodular function and a linear function with a cardinality constraint

Maximizing a monotone submodular function with a cardinality constraint can be solved by using a simple greedy heuristic. However, if the submodular function is non-monotone, the greedy heuristic can ...

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**2**answers

207 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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283 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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**2**answers

164 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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**1**answer

572 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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50 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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**1**answer

53 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 ...

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3k 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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84 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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42 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} = \frac{...

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**2**answers

436 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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**1**answer

209 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:
...