Numerical algorithms for problems in analysis and algebra, scientific computation

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192 views

Is there an example where the error of Gauss-Laguerre quadrature does not vanish?

The $n$th Gauss-Laguerre quadrature aims to approximate integral $$\int_{\mathbb{R}_+} f(x) \exp(-x)$$ by the sum $$\sum_{i=1}^n f(x_i) w_i$$ where $x_1,...,x_n$ are the roots of the $n$th Laguerre ...
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95 views

Stationary Distribution for Markov-like system?

Let \begin{equation} A= \begin{pmatrix} 0 & a_{1,2} & a_{1,3} \\ a_{2,1} & 0 & a_{2,3} \\ a_{3,1} & a_{3,2} & 0 \end{pmatrix}, \end{equation} \begin{equation} B= ...
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1answer
110 views

Trace of multiplied positive definite matrices

I have to compute $Tr(K^{-1}\Sigma)$ where both $K$ and $\Sigma$ are symmetric positive definite matrices. Question is considering that I have computed the Cholesky, $L_1$ of $K$ previously, is there ...
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2answers
119 views

Numerical calculation of Fourier transform with a nice error bound

I'd like to have an algorithm for a numerical calculation of Fourier transform with a nice error bound. To be precise, if $f$ is a function from $L_1(R)$, $F[f]$ is it's exact Fourier transform and ...
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125 views

solving a system of algebraic equations

I'm trying to solve the following system of nonlinear algebraic equations for $q_{e}\in\mathbb{R}^{4}$ $$ ...
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81 views

What do we know about the generalized eigenvalue problem involving a projector?

Consider a matrix $A\in\mathbb{R}^{n\times n}$ and a projector $P\in\mathbb{R}^{n\times n}$. Are there results regarding the generalized eigenpairs $(v,\lambda)$ of the generalized eigenproblem ...
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0answers
147 views

Algorithms to compute largest gap between smallest nonzero eigenvalues of sparse symmetric matrix

I am looking mainly for implementations but also for theoretical algorithms to compute gaps between smallest positive eigenvalues of symmetric, singular matrix or real numbers. To be precise, I want ...
2
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2answers
124 views

Seeking a class of functions for which sums approximate integrals well

Is there a "natural" class of integrable functions $f: {\mathbb R} \rightarrow {\mathbb R}$ for which it is true (and, preferably, not too hard to prove!) that $\sup_{0 \leq a < h} |h S(a,h) - I|$ ...
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101 views

Error of midpoint method for differentiable functions

Is it the case that for every differentiable function $f$ on $[0,1]$ (with finite one-sided derivatives at the endpoints), the midpoint method of estimating $\int_0^1 f(x) \: dx$ has error $o(1/n)$? ...
3
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1answer
200 views

Error of midpoint method for functions that are not twice-differentiable

All of the bounds I've seen for the error of the midpoint method of integration are expressed in terms of the second derivative of the function. What bounds are available when the function is not ...
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1answer
163 views

Is there any geometric and intuitive interpretation of Newton-like iterative steps in numerical optimization?

Are the iterative steps in optimization affected by the intrinsic and extrinsic curvatures of the objective functions ? and How? Is there any geometric and intuitive demo show illustrating the ...
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1answer
391 views

How to determine the distance between two matrices under the meaning of a matrix function? [closed]

Suppose a nonlinear infinitely continous differentiable function $f:\mathbb{D}\mapsto \mathbb{R^+}$, where $\mathbb{D}\subset\left\{X|\text{rank}{X}=2,X\in\mathbb{R}^{3\times 3}\right\}$ is a ...
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1answer
142 views

Delay Differential Equations Numerical methods

I have a general question about delay differential equations. I know that even simple ones hardly have analytic solutions and mine clearly doesn't have any as it is a system of non-linear delay ...
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80 views

unique positive real root fast computation

What is the fastest way to compute the value of the unique positive real root corresponding to the following polynom: :p(x) = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x - f = 0 where a, b, c, d, e, f are ...
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0answers
394 views

Linearizing and solving a nonlinear PDE numerically

Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
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2answers
226 views

Local positivity of solutions to linear differential inequalities (Chaplygin's theorem)

According to the entry "Differential inequality" of the Encyclopedia of Mathematics http://www.encyclopediaofmath.org/index.php/Differential_inequality the following result is due to Chaplygin ...
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1answer
87 views

Estimating the vector potential

My question is, that given a vector field only numerically discrete in space, is there a way to estimate its vector potential? Theoretically, I see this which requires the vector field over all of ...
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2answers
1k views

Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...
6
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1answer
190 views

Approximating an iteratively defined function

Let $f_0,f_1,\ldots$ be a sequence of functions $f_n : [0,1] \rightarrow R$ defined as follows: $$f_0(x) =1+2x$$ $$f_{n}(x) := \left\{\frac{5+t}{2} : \text{ where t solves } ...
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1answer
190 views

Approximation of the sum involving binary entropy function

Given the following sum: $S(n) = \sum_{i=1}^{n} \frac{1}{(1-\operatorname{H}(p))^i}$ where $H$ is the binary entropy function defined as: $\operatorname{H}(p) = -p\log p - (1-p)\log (1-p) $. Let ...
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2answers
151 views

dense lattices in high dimensions

I want a collection of points $\{ x_1, \dots, x_m\}$ to sample a unit cube $[0,1]^n$ with $n >>1 $ in high dimensions so that summing over these points is approximate the integral over that ...
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1answer
292 views

The geometric-mean factorial

Think of the factorial as $f(n) = n \odot (n-1) \odot \cdots \odot 2 \odot 1$, where $\odot$ is the binary operator for multiplication, $\cdot$. This suggests exploring replacing $\odot$ with other ...
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2answers
152 views

Does an implicit Runge Kutta scheme applied on a nonlinear ODE give a nonlinear set of equations to solve in each step?

We want to approximately solve an ODE $$\frac{dy}{dt} = f(y,t)$$ with the Runge Kutta method $$y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$$ $$k_i = f\left(y_n + h \sum_{j=1}^s a_{ij} k_j,\,t_n + c_i ...
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1answer
161 views

Accurate bounds for derivatives of Legendre polynomials

Let $P_n(x)$ denote the $n$th Legendre polynomial. What bounds can one give for $d_{n,m}(x) = |\frac{d^m}{dt^m}P_n(t)|_{t=x}$ assuming that $|x| \le 1$? Clearly $$d_{n,m}(x) \le d_{n,m}(1) = ...
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3answers
544 views

Best known bounds on tensor rank of matrix multiplication of 3×3 matrices

Years ago I attended a conference where they taught us that matrix multiplication can be represented by a tensor. The rank of the tensor is important, because putting it into minimal rank form ...
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1answer
63 views

Augmenting orthonormal system into complete orthonormal system in a numerically stable way

Let us suppose we have a, say, 10 dimensional real space with 3 orthogonal unit vectors given. How do I complete this orthonormal system with 7 additional vectors into a complete ONS in a way that is ...
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0answers
69 views

How to solve a divergent linear system using iterative methods?

I have a matrix A which is symmetric and non-diagonal dominant. I tried to use Jacobi/Gauss-Seidel/SOR to solve it but it diverges. Is there any mechanism to condition the matrix for convergence using ...
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0answers
162 views

Simplify this expression with modified Bessel functions of the second kind

I'm interested in the function $$\frac{K^{(0,2)}(0,x)}{K(0,x)}$$ where the numerator is the modified Bessel function of the second kind twice differentiated with respect to $\alpha$ and taken at ...
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64 views

Variational problem for optimal weight function leading to shorter intervals with many primes

The motivation for the following problem stems from the recent preprint by James Maynard, see also Proposition 5 of the recent blogpost by Terrence Tao. The solution of this problem could give better ...
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1answer
209 views

Does this algorithm terminate in all scenarios?

Let $x \in \mathbb{R}^p$ denote a $p$-dimensional data point (a vector). I have two sets $A = \{x_1, \dots, x_n\}$ and $B = \{x_{n+1}, \dots, x_{n+m}\}$, so $|A| = n$, and $|B| = m$. Given $k \in ...
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1answer
164 views

Literature on root finding of convex Functions

I am interested in using a result about Newton's method, which basically states that if f is convex on $[a,b]$ and it holds $f(a)<0$ and $f(b)>0$, then the Newton iteration converges to ...
2
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1answer
65 views

Fast numerical approximation of Lauricella series of the fourth kind for real variables and real parameters

I'm looking for a method to efficiently compute a numerical approximation of $$F^n_D(x_1,\ldots,x_n) = \sum_{m=0}^{\infty} \sum_{i_1 +\ldots+i_n=m}\frac{(a)_{m}(b_1)_{i_1}\ldots ...
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0answers
55 views

Approximate Laplace (forward) transforms

Given f(t) let's write Lf(s) = INT_0^inf f(t) exp(-ts)dt for its Laplace transform. I lately find myself looking at expansions of the form SUM_i a_i exp(-t_i s) that converge to Lf of some given ...
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1answer
77 views

Quadrature rules exact for given functions

Many quadrature schemes are defined by the degree of polynomials for which they are exact. For example, we say that the rule SUM_i a_i f(x_i) is order n when SUM_i a_i p(x_i) = INT p(x) dx for all ...
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45 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 ...
2
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1answer
187 views

Convergence of fixed point iteration algorithm

It is obvious that for any given $f(x)$ there exists $g(x)$ such that $f(x)=0 \Leftrightarrow g(x)=x$. We could use this fact to solve any root finding problem using fixed point iteration method, only ...
1
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1answer
138 views

Eigenvalue problem with quadratic constraints

$\circ$ Consider the following eigenvalue problem : $$Ax=\lambda x \hspace{0.5cm} (1)$$ where matrice $A \in \mathbb{R}_{n \times n}$ is a positive semi-definite with eigenvectors $x = ...
2
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1answer
100 views

Updating $LU$ decomposition after adding a sparse matrix

How many elements of $LU$ decomposition of a symmetric matrix change after adding a sparse symmetric matrix? Is it more efficient to recompute $LU$ decomposition after adding a sparse matrix comparing ...
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0answers
231 views

Computing a 2D Fourier transform on a disk

I am looking for a reliable and fast way of evaluating an integral like $$ F(r, \phi)= \int_0^1 \int_0^{2\pi} f(\rho, \theta) e^{2\pi i \rho r \cos(\theta - \phi)}\rho\, d\theta\,d\rho, $$ where $f$ ...
4
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1answer
125 views

Do interpolation nodes have to be dense?

Let $f(x) = \exp(x)$ and $(\xi_i)_{i=0}^\infty, \, \xi_i \in (0,1)$ be a sequence of points from the unit interval. For $n \in \mathbb{N}$ let $P_n$ be a polynomial of degree $n$ that interpolates ...
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0answers
73 views

Looking for tighter bounds

I have to solve an equation which is $$\sum_{i=1}^N x_i = \sum_{i=1}^N y_i,$$ where $$x_i = \frac{z_i}{1 + (K_i - 1) w}$$ and $$y_i = \frac{K_i z_i}{1 + (K_i - 1) w}.$$ The $z_i$ are all positive ...
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131 views

About arithmetic-geometric mean

It's well known that if we set $a_0=x \geq 0, \ g_0=y \geq 0$, and $$ a_{n+1}=\dfrac{1}{2}(a_n+g_n), \ g_{n+1}=\sqrt{a_n g_n} ,$$ then both $\{a_n\}$ and $\{g_n\}$ will converge to $AGM(x,y)$. ...
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227 views

Eigenvalue computation using inverse iteration

I have a positive definite matrix $A$. I need to compute the max eigen value of $A$ using inverse iteration. The problem is that there are duplicate maximum eigen values and so inverse iteration ...
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84 views

Solutions of Fredholm integral equation of the fisrt kind with asymmetric kernel

Given the integral equation: $$\int_0^{\lambda_0}K(\lambda,T)S(\lambda)d\lambda=f(T)$$ with $S(\lambda)$ unknown function and the kernel: $$K(\lambda,T)=\frac{1}{\lambda^5(\exp(k/\lambda T )-1)}$$ ...
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134 views

When is it possible to split a non-linear operator into a composition of a linear and local one?

Let $A: L^2(R^n)\to L^2(R^n)$ be a non-linear operator. Is it known when it's possible to split $A$ into a composition of a linear operator $B: L^2(R^n)\to (L^2(R^n))^k$ and a local operator $C: ...
11
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2answers
524 views

How to project a vector onto a very large, non-orthogonal subspace

I have a difficult problem. I have a very large, non-orthogonal matrix $A$ and need to project the vector $y$ onto the subspace spanning the columns of $A$. If this were a small matrix, I would use ...
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3answers
267 views

What are the basis functions for a product space?

Let $X=L^1\left([0,1]^3\right)$, for numerical purpose, what are the possible basis function for $X$? In finite element method, the basis functions are tooth functions, or polynomial functions. Is ...
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1answer
267 views

Is this inverted integral transform valid?

I have the following transform: $$F(y) = \int_{0}^{\infty} y\exp{\left[-\frac{1}{2}(y^2 + x^2)\right]} I_0\left(xy\right)f(x)\;\mathrm{d}x$$ with the following conditions: $f(x)$ and $F(y)$ must ...
10
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1answer
359 views

Who is Petrov of the Petrov-Galerkin method?

I was not able to find the origin of the name Petrov in the Petrov-Galerkin method for the numerical approximation of PDEs. Wikipedia refers to a certain Alexander G. Petrov, but it is still not ...
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136 views

The rationale of QR algorithm for computing eigenvectors

For a symmetric matrix $A$, the rationale for the success of applying QR to compute the spectral decomposition of $A = UDU^T$ is, for large $k$, the QR factorization of $A^k = Q_kR_k$ obeys, ...