Numerical algorithms for problems in analysis and algebra, scientific computation

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votes

**1**answer

51 views

### Machine learning and Linear programming [on hold]

Her's the problem I'm trying to solve :
I want to find $n$ function $f_i : \mathbb{R} \rightarrow\mathbb{R}$ and the vector $X=(x_1,...x_n)$ that minimizes :
$\displaystyle{\sum_{i=1}^{n} ...

**4**

votes

**1**answer

491 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:
...

**0**

votes

**0**answers

97 views

### numerical method (implicit) for nonlinear pde [on hold]

$\newcommand{\lbar}{\underline{\lambda}}$
I need a numerical method (implicit , backward difference or forward difference) for estimate $A$ in this nonlinear PDE:
$$ A_t + \mu(\lambda -\lbar ) ...

**2**

votes

**1**answer

78 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 ...

**-2**

votes

**0**answers

35 views

### A Function $ g[0,\infty) \mapsto [0,1)$ Sharply Changing on Both Ends [on hold]

I need a function $ g[0,\infty) \mapsto [0,1)$ that sharply decreases near 0 and sharply increases near 1. Preferably, it wouldn't be defined in a piecewise manner.
Can anyone provide an example, ...

**1**

vote

**1**answer

154 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 ...

**0**

votes

**0**answers

29 views

### Numerical analysis- Runge Kutta [closed]

i have:
y'(x)= sin(y); y(0)=1
i need to calculate the function values with runge kutta.
my problem is that i need to choose the h (=dx) such that the error will be in order of 0.0001.
how i choose ...

**1**

vote

**1**answer

54 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 ...

**4**

votes

**4**answers

198 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

46 views

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

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 ...

**1**

vote

**3**answers

147 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} ...

**0**

votes

**1**answer

123 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

116 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 ...

**2**

votes

**2**answers

151 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

**0**answers

146 views

### open problems in Numerical Analysis [closed]

There are plenty of open problems in Graph Theory, Number Theory, and Combinatorics, according to the Open Problem Garden: http://www.openproblemgarden.org/. Yet there are no open problems in ...

**1**

vote

**1**answer

61 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 $n$ ...

**1**

vote

**2**answers

757 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 ...

**8**

votes

**1**answer

329 views

### Gaps between roots of trigonometric polynomials

[Cross-posted from Math.SE because I got no responses there.]
Given a polynomial in $e^{\mathrm{i}k t}$ of the form
$$ p(t) = \sum_{-n\leq k\leq n} c_k e^{\mathrm{i}k t} $$
with $\bar c_{-k} = c_k$, ...

**3**

votes

**1**answer

72 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 ...

**7**

votes

**2**answers

830 views

### Why are there so few zero-dimensional polynomial system solvers and is this because there is no real market for them?

My questions involve the quotes below from wikipedia regarding solving polynomial systems, which given the size of the market for Big Data & Predictive Analysis applications I find puzzling:
...

**2**

votes

**2**answers

304 views

### Integral Fredholm equation of the second type

There is an equation
$$
w(x) = g(x)+\int\limits_0^M w(y)f(x-y)\,dy
$$
where $f\geq 0$, $f\in C^\infty(\mathbb R\setminus\{c\})$ for some point $c$ and $\int\limits_{-\infty}^\infty f(t)\,dt\leq 1$. ...

**1**

vote

**0**answers

90 views

### Asymptotic expansion of an integral, related to Maass forms

I am trying to compute the asymptotic expansion of the integral
$I(t) = \int_{C} e^{\sqrt{1+u}(\frac{1}{t}+\frac{t^2}{\sqrt{u}})}\frac{u^\eta}{\sqrt{1+u}}du$
as $t$ is real and $t\rightarrow +\infty$, ...

**1**

vote

**1**answer

53 views

### Finite differencing scheme for Hamilton's equation with planar linkages

I am trying to simulate the movement of a planar linkage in the plane whose position and momentum obey Hamilton's equations, which is to say that $${{dq}\over{dt}} = {{dH}\over{dp}}$$ and ...

**8**

votes

**4**answers

249 views

### Software tools for medium-scale systems of polynomial equations

I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real ...

**4**

votes

**1**answer

69 views

### Sensitivity of the range of a matrix

The distance between two subspaces $\mathcal{U}$ and $\widetilde{\mathcal{U}}$ is classically defined as $d(\mathcal{U},\tilde{\mathcal{U}}):=\|P-\tilde{P}\|$, where $P$ and $\tilde{P}$ are orthogonal ...

**4**

votes

**2**answers

2k views

### Computational complexity of calculating the nth root of a real number

Several sources state that the computational or time complexity of square rooting is the same as that of multiplication (or division). See for example:
Jean-Michel Muller, "Elementary Functions: ...

**2**

votes

**2**answers

98 views

### Boundedness of ratio of linear functions

Consider the function
\begin{eqnarray}
f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i},
\end{eqnarray}
over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...

**1**

vote

**1**answer

157 views

### Dragilev method

What are the advantages and disadvantages of the Dragilev ( http://www.mathnet.ru/php/person.phtml?&personid=32359&option_lang=eng
and ...

**2**

votes

**0**answers

118 views

### Are there workable numerical approaches for the pentagon equation?

Warning: this post is the "numerical" analog of
Are there workable algebraic geometry approaches for the pentagon equation?
I've replaced "algebraic geometry" by "numerical" in its content,
...

**7**

votes

**1**answer

221 views

### Compute an arbitrary decimal place of $\pi$

Is there a method to find the value of the $n$-th decimal place of $\pi$ which is more efficient than having to compute all decimal places before as well?

**2**

votes

**1**answer

106 views

### Projection onto $\ell^{2,1}$ ball

Does anyone have an idea how to project onto the $\ell^{2,1}$ ball efficiently, i.e. how to solve
$$ u = \arg \min_u \|u-f \|^2 \text{such that } \left(\sum_i \big(\sum_j |u_{i,j}|\big)^2 ...

**1**

vote

**1**answer

91 views

### Are there some algorithms to solve the diagonal matrix $X$ to the following matrix equation?

Suppose $X$ is an unknown $m \times m$ diagonal matrix. Given a scalar $0<c<1$, and a matrix $A$ of $m \times m$ size whose entries $0<A_{i,j}<1$. Are there some algorithms to find the ...

**2**

votes

**0**answers

196 views

### Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...

**6**

votes

**3**answers

1k views

### Eigenvalues of non-symmetric matrix and its transpose

What more can be said about the eigenvalues (especially the spectrum) of the $N \times N$ matrix ${\bf M} = {\bf A} + {\bf A}^T$ in terms of $\bf A$ if $\bf A$ is not symmetric and its eigenvalues are ...

**2**

votes

**0**answers

123 views

### A question on discrete numerical simulation on fluids mechanics

I read the paper "Stable, circulation-preseving simplicial fuids" by Elcott, et al: http://www.cs.jhu.edu/~misha/Fall09/Elcott07.pdf. It gives a structure preseving discretization of fluids. I have ...

**4**

votes

**1**answer

73 views

### 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

**1**answer

97 views

### Roots of modified polynomials

Consider the following two polynomials:
$$
g=x^3 - x^2 - (c + 2)x + c
$$
and
$$
h=x^3 - x^2 - cx + c
$$
The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, ...

**6**

votes

**4**answers

559 views

### solving Lyapunov-like equation

The following matrix equation might be a Lyapunov-like equation, but it seems hard for me to develop a simpler way to solve it. From the computation effort, I need some help for solving the special ...

**6**

votes

**1**answer

128 views

### Estimates of Hausdorff dimension (and its derivatives)

For example, the cookie cutter maps, say $T:I_1 \cup I_2 \subset [0,1] \to [0,1] $ is a $C^2$ map such that $|T'|>1$ and provided $I_1$ and $I_2$ are disjoint closed intervals and $T(I_i)=[0,1]$. ...

**0**

votes

**0**answers

25 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^1( \mathbb{R} ^d ) $
...

**5**

votes

**2**answers

371 views

### How to estimate the Haar measure on $G_2$

I have a sequence of real numbers. I want to know whether this sequence looks like the traces in the standard representation of a random sequence of elements of $G_2$. (Here random is according to the ...

**1**

vote

**0**answers

43 views

### interpolation and approximation [closed]

Given a function $f$ in C^k[a,b], can we always construct function $g \neq f$ such that $g(x) \ge f(x)$ for all $x \in [a,b]$, $f^{(m)}(a)=g^{(m)}(a)$ and $f^{(m)}(b)=g^{(m)}(b)$ for $m=0,1,\dots, k$ ...

**11**

votes

**13**answers

3k views

### Basic software libraries for numerical analysis using modern programming languages?

I'm looking for a software library with a scope similar to "numerical recipes", but implemented in a modern programming language. "Modern" in this context means to me: object oriented (not C or ...

**3**

votes

**0**answers

101 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 ...

**1**

vote

**0**answers

78 views

### Bounds for the infinity norm of the inverse for certain diagonaly dominant matrices

I m trying to analyse the stability against perturbations for a specific system of linear equations $Ax=b$.
For this, i use the standard condition number $||A||_{\infty}||A^{-1}||_{\infty}$.
Here ...

**1**

vote

**0**answers

43 views

### Using FFT to aproximate a fuction [closed]

I am trying to use the FFT to approximate a given function. So i have 10 points xk that are given for example, if i use the FFT that will give me Xk. So now using the inverse FFT we can get the ...

**3**

votes

**3**answers

119 views

### Quadrature formula max accuracy

I'm looking for a maximum accuracy quadrature formula:
$$
\int_{-1} ^{1} \sqrt{\frac {1-x}{1+x}} f(x)dx = A_1f(x_1)+A_2f(x_2)+R(f)
$$
I don't know exactly if it's Trapezoidal rule which has the ...

**0**

votes

**1**answer

72 views

### Decompositions of sparse symmetric matrices and methods for solving large linear equations

I am writing code for solving linear equations of the form
$$A_{n\times n}\cdot x=1_n$$
where $n$ is on the order of $10^6$ and $A$ is a symmetric matrix with approx $10^3$ nonzero entries in each ...

**1**

vote

**1**answer

153 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 ...

**-1**

votes

**1**answer

80 views

### Convergence for symmetric, positive semi-definite operator

Assume $u$ is a vector in the Euclidean space $\mathbb{R}^N$, $\|u\|=\sqrt{\langle u, u\rangle}$, where $\langle u, v\rangle = \sum_{i=1}^N u_i v_i$.
I have that $\|u^{k+1}-u\|\leq \|I - c ...