Numerical algorithms for problems in analysis and algebra, scientific computation

**4**

votes

**1**answer

129 views

### Resolvent of a triangular matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$?
Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...

**0**

votes

**0**answers

34 views

### Bound on change of function given bound on Hessian

Suppose I have very some smooth function $F(x)$, and let $x_0 = \text{argmin}_x F(x)$. I would like to bound $F(x) - F(x_0)$ from above, in terms of the gradient $\nabla f(x)$ and the Hessian matrix ...

**-2**

votes

**2**answers

40 views

### Systems of ODEs that fulfill a matrix relationship at steady state [on hold]

It is well known that for a system of linear ODE $$x'(t) = A(t) \cdot x(t) + b(t)$$
with initial condition $x(t_0) = x_0$, that for a solution at any other time point $t_1$, $x(t_1) = (z_1, \ldots, ...

**6**

votes

**2**answers

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 ...

**2**

votes

**0**answers

39 views

### Estimating polynomial approximation error in high dimension

Question
Let $x \in [-1, 1]^d \subset \mathbb{R}^d$ be a $d$-dimensional variable and assume that -- given $n$ -- I have a way of computing a polynomial $p_n(x)$ of degree $n$ that approximates a ...

**0**

votes

**0**answers

27 views

### Probability of close approach for multivariate normal variables

The following problem comes from a physical model of two groups of particles in three dimensions. I need to know the probability that the two groups of particles approach each other within some ...

**0**

votes

**0**answers

37 views

### Initial Value for an ODE Problem [closed]

I have the following ODE
$\mathbf{A}\dfrac{d\vec{y}}{dt}+\mathbf{B}\vec{y}=\vec{x}$,
where $\vec{y}$ and $\vec{x}$ are $n\times1$ vectors and are functions of $t$, and $\mathbf{A}$ and $\mathbf{B}$ ...

**-5**

votes

**0**answers

35 views

### Diffusion Equation [closed]

Kindly give me suggestions on my following assignment of Simulations in Fluid Flow:
Solve the following differential equation for transport of f(x,y,z,t) by MS Excel
∂f/∂t+Ux ∂f/∂x+Uy ∂f/∂z+Uz ...

**2**

votes

**0**answers

58 views

### What is the computational complexity to compute the integral numerically?

Given $$\int_{\Delta}\frac{P_1(x_1,x_2,\dots,x_n)}{P_2(x_1,x_2,\dots,x_n)}$$ where $P_i$ is polynomial(that is $P_1(x_1,x_2,\dots,x_n), P_2(x_1,x_2,\dots,x_n)$ are polynomial) whose coefficients are ...

**4**

votes

**0**answers

119 views

### Rank 1 Approximation of Elementwise Inverse Matrix

I'm wondering whether there is a good way to solve the following optimisation problem.
Given a strictly positive quadratic matrix $A$, find two diagonal matrices $D_1$ and $D_2$ so that
$$ \| D_1 A ...

**4**

votes

**1**answer

508 views

### Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello!
Let's say I have
$(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$
with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$).
Now separate the Amplitude and Phase of the solution:
...

**1**

vote

**1**answer

31 views

### Condition Number and CFL Condition in Finite difference Methods

when applying a Finite Difference scheme for an IVP, two factors come to mind when considering stability:
One factor would be the condition number of the approximation operator. The other factor ...

**3**

votes

**1**answer

57 views

### Selecting Rays for Simulated Radon Transform

I have the task of determining approximations of a 2D function $f: (x,y)\in \mathbb{R}^2\mapsto\mathbb{R}$ from integrals along lines, i.e. from its Radon transform $R(\phi,\tau)[f(x,y)]$ and, because ...

**2**

votes

**1**answer

115 views

### books on very large scale linear optimization

Recently in my material science research, I have encountered problems of very large scale linear optimization. I read the introductory book "Introduction to Linear Optimization (Athena Scientific ...

**1**

vote

**2**answers

142 views

### iterative solution better than analytic solution? [closed]

My supervisor and I were discussing a specific optimisation problem this afternoon.
To be simple: solve for $R$ in the equation $Rx=y$, where $x$, $y$ are made of samples in two difference ...

**1**

vote

**0**answers

45 views

### Frozen coefficient method (von Neumann stability analysis)

Earlier it was considered that frozen coefficients method for Neumann stability analysis for finite difference scheme is more heuristic than rigorous. But I have read some information in a book by ...

**7**

votes

**1**answer

157 views

### Problems where Conjugate gradient works much better than GMRES

I am interested in cases where Conjugate gradient works much better than GMRES method.
In general, CG is preferable choice in many cases of SPD because it requires less storage and theoretical bound ...

**8**

votes

**1**answer

216 views

### Algorithm to produce random number with a gamma distribution

I'd like to produce pseudo-random numbers with different distributions for a Monte Carlo simulation.
I've got the poisson distribution working nicely with an algorithm from Knuth. I'm having trouble ...

**2**

votes

**1**answer

77 views

### Special Function, Series Expansion, or Simpler Form of a Certain Infinite Product?

$\prod _{n=1}^{\infty } \left(1+a (c+n)^b\right)$ where a > 0, b < -1, and c >= 0
Is there a special function, series expansion, or other simpler (or maybe just interesting) representation of ...

**6**

votes

**1**answer

132 views

### Computing certain integrals over high-dimensional polyhedra

Let $\delta>0$ be a small real number and consider the $k$-dimensional region consisting of points for which
$$\delta\leq x_1\leq x_2\leq\ldots \leq x_k$$
and
$$x_1+\ldots+x_k\leq 1.$$
I am ...

**0**

votes

**1**answer

209 views

### Improving Newton's Inequalities using the Taylor Theorem

Newton's inequalities say that if $f(x) = \sum \binom{n}{k} a_k x^k$ is a polynomial with all real roots then $ a_k^2 > a_{k-1}a_{k+1}$.
The proof this result uses that if $f(x)$ has all real ...

**1**

vote

**0**answers

35 views

### General reparameterization of a b-spline

Say I have a bspline function (or curve) of order $k_1$, defined over some knot vector
$\mathbf{t} = \{ t_i\}_1^{n_1}$, i.e. $$f(x) = \sum_i a^i B_{i,k_1}(x).$$
Do you know of a process of finding ...

**3**

votes

**1**answer

57 views

### The link and equivalence between variant definition of computation model and computational complexity over reals

To unify the numerical computation and classic computability theory, or to pave a foundation for the numerical computation, mathematicians present variant computation model and computational ...

**1**

vote

**3**answers

176 views

### What are some good sanity checks for simulating BNLS?

After doing some googling, I failed to find any explicit solution for the Biharmonic Nonlinear Schrodinger Equation, which states:
$$
i\psi (x,t) _t - \Delta ^2 \psi (x,t) + |\psi (x,t) | ^{2 \sigma} ...

**7**

votes

**1**answer

153 views

### Compute only selected components of an eigenvector

I am wondering whether it is possible to compute portions of the eigenvectors of a given (possibly very big) matrix. More formally, consider the eigenvalue problem $\mathbf{Ax} = \lambda \mathbf{x}$, ...

**7**

votes

**4**answers

202 views

### Numerical integration of legendre polynomials

I hope that numerical questions are also permitted here.
I want to expand a smooth functions $f \in C^{\infty}$in terms of Legendre polynomials. Thus I need to calculate integrals of the form ...

**2**

votes

**0**answers

33 views

### Conditions for convergence of Euler's method

It is know that a sufficient and necessary condition for
$$\dot y(t) = f(y(t), t), \quad t > 0, \quad y(0) = y_0$$
assumes a unique solution of $f$ is Lipschitz in $y$ and continuous in $t$. ...

**5**

votes

**4**answers

676 views

### Quanitative de Moivre–Laplace theorem (reference request)

The classical de Moivre-Laplace theorem states that we can approximate the normal distribution by discrete binomial distribution:
$${n \choose k} p^k q^{n-k} \simeq \frac{1}{\sqrt{2 \pi ...

**1**

vote

**1**answer

187 views

### Solving a matrix equation $X=c \cdot AXA' +I$ with a diagonal corrections

I am now struggling to solve the matrix $X \in R^{n \times n}$ in the following equation:
$X=c \cdot AXA' - diag(c \cdot AXA')+ I$,
where
(1) $A \in R^{n \times n}$ is a given matrix whose element ...

**2**

votes

**0**answers

56 views

### Numerical Methods for stochastic PDE, from rough paths to backward equations

this question is about some literary references regarding the state of the art in terms of numerical methods for SPDE's. In particular,
Have the numerical implications, if any, of the results in ...

**0**

votes

**0**answers

39 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 ...

**3**

votes

**0**answers

774 views

### Eigenvalue Problems with Linear Constraints

The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is ...

**0**

votes

**1**answer

59 views

### Could somebody recomends a good book or article about numerical methods for Stochastic Partial Differential Equations

Could somebody recomend a good book or article about numerical methods for Stochastic Partial Differential Equations. I'm looking for a good introductory material thanks.

**1**

vote

**1**answer

44 views

### Partial Constraint of Low Rank Matrix

Suppose $X \in \mathbb{R}^{m \times n}$ is a rank $r$ matrix. Let $\Omega$ be a generic subset of $\in \{1, \ldots, m\} \times \{1, \ldots, n\}$ of cardinality $r(m + n - r)$. Denote by $X_{\Omega}$ ...

**0**

votes

**0**answers

34 views

### Numerical method for self-consistency of one-dimensional probability density function

I have an integral equation for self-consistency of one-dimensional probability density function, like this
$$\rho_x(x) = \frac{1}{|a|}\int \int \rho_x\left(\frac{s-b}{a}\right) \rho_P(p) ...

**2**

votes

**1**answer

37 views

### IVP accuracy - scheme accuracy Vs. derivative accuracy?

General Question: If I have an IVP with periodic and continuous initial condition, which rules the accuracy of the scheme - the manner in which we approximate spatial derivative or the acuuracy of the ...

**7**

votes

**3**answers

282 views

### Multiprecision numerical evaluation of integral: Sage vs. PARI/GP vs. mpmath

I am trying to compute thousands of integrals of the below type, that comes up in a conformal mapping problem, to as many accurate digits as possible (preferably 50+):
$$
\int_{-1}^1\textrm{d}t ...

**0**

votes

**0**answers

48 views

### Why Does a quadratic phase term in BNLS causes collapse?

I've heard a couple of times that in the Biharmonic Nonlinear Schrodinger Equation,
$i\psi_z + \Delta ^2 \psi + |\psi | ^{2\sigma } \psi =0 $, $\psi (x, 0) = \psi _0 (x) \in H^2( \mathbb{R} ^d ) $
...

**6**

votes

**1**answer

154 views

### Are there any explicit probability conserving solvers for Pauli equation?

I know that there exist probability conserving explicit solvers for time-dependent Schrödinger's equation, for example, Visscher's one.
But when I tried to add spin into account in this scheme, it ...

**2**

votes

**0**answers

119 views

### numerical method (implicit , backward difference or forward difference) for nonlinear pde

$\newcommand{\lbar}{\underline{\lambda}}$
In this linear PDE:
\begin{cases}
B_t+b^Q(r,t)B_r+\frac{1}{2}d^2(r,t)B_{rr}+(\mu(\lambda,t)+\alpha \sigma (t))(\lambda -\lbar)B_{\lambda} \\ ...

**5**

votes

**2**answers

117 views

### Discrete gradient on point clouds

I am interested to know some ways to approximate discrete gradient if you have a function on point clouds in 2D or 3D.
If you have a function defined on a grid, it well known that you can use a ...

**4**

votes

**2**answers

108 views

### Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

I would like to numerically solve a hyperbolic PDE of the form
$\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t ...

**3**

votes

**2**answers

91 views

### Real solutions for systems of monomial equations

I have a $K$ equations of the form $x_1^{a_{i1}} \cdots x_n^{a_{in}}=c_i$ where $a_{ij}$ are non-negative integer constants and $c_i$ are real constants -- i.e. each equation is a monomial in $n$ ...

**-2**

votes

**1**answer

72 views

### reducing an n-order differential equation to a first order system of equations using either sagemath or sympy [closed]

I want to reduce a n-order ordinary differential equation into a first order system of equations. This is in preparation for numerical analysis. I use both Sympy and Sagemath for Computer Algebra, but ...

**2**

votes

**1**answer

62 views

### Monotonicity of Trapezoid Approximations

Here's a numerical analysis question which may not be very important, especially in practice, but has been bugging me.
Suppose $f$ is a continuous function on an interval $[a,b]$. Let $T_n(f)$ be ...

**5**

votes

**3**answers

237 views

### Estimating the Variance of a Discrete Normal Distribution

Let $f(x; \sigma) = \frac{1}{\sigma\sqrt{2\pi}}\cdot e^{-\frac{x^2}{2\sigma^2}}$ be the probability density function of a normal distribution $\mathcal{N}(0, \sigma^2)$. We consider a discrete normal ...

**0**

votes

**1**answer

135 views

### Proving that the eigenvalues of a certain matrix product are positive

Let $A$ be an $m \times n$ matrix, and define:
\begin{align*}
U &= {\rm diag} \{ \frac{1}{\beta_j} \}, \beta_j = \sum_{k=1}^m |a_{kj}|, j = 1 \dots n \\
V &= {\rm diag} \{ \frac{1}{\alpha_i} ...

**1**

vote

**1**answer

123 views

### Estimating the volume of a union of balls

Let $\{ B_i \}_{i=1}^n$ be a set of $n$ ball in the unit cube $C$ of dimension $d$.
If I want to estimate
$$
\frac{ \lambda \left( \cup B_i \right) }{\lambda\left( C \right) }, \tag{1}
$$
where ...

**1**

vote

**2**answers

175 views

### What is an extragradient method?

I've searched Google, but it seems that only research journal papers appear in search results, where some new, improved, or specialized extragradient method is discussed. I've also searched Wikipedia ...

**1**

vote

**2**answers

800 views

### What is the right citation for the power iteration method to find eigenvalues?

What is the right citation for the power iteration method to find eigenvalues, if I want to cite the method in a paper? I've seen some Google PageRank references in this context. But Brin and Page ...