Numerical algorithms for problems in analysis and algebra, scientific computation

**3**

votes

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41 views

### Determining Nullspace Basis such that only one column is deleted or added as row is added or deleted, and remaining columns of basis stay the same

I would like to compute, in MATLAB, the basis Z for the nullspace of an m by n matrix A, such that if one row of A is added (resulting in A_a), the basis for A_a is n-m-1 of the n-m columns of Z, ...

**3**

votes

**0**answers

81 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 ...

**1**

vote

**1**answer

69 views

### Convergence of sequence of polynomials defined by boundary conditions

I'm sorry if my question sounds trivial, but analysis is not my field.
Consider the interval $[a,b]\subset \mathbb{R}$. On $[a,b]$, for every $n\in\mathbb{N}$, $n\ge 3$, I define the polynomials ...

**2**

votes

**1**answer

61 views

### Numerical solution of singular ODE

Consider the singular ODE
$y''+\frac{y'}{r}+p(r)y=0 \ \ with \ \ y(0)=1 \ \ and \ \ y'(0)=0$.
Theoretically such solution exists and is unique if $p$ is nice. Is there a method to numerically ...

**2**

votes

**2**answers

413 views

### Who is currently researching topics concerning applying algebraic topology and/or differential geometry to numerical methods? [closed]

I am interested in pursuing a PhD in mathematics from a top ranked university with a faculty member researching something akin to the following description: applications of algebraic topology and/or ...

**0**

votes

**1**answer

25 views

### Solving sparse linear least squares or a positive definite 5-band matrix system fast

I want to quickly solve linear least squares problem for $x \in \mathbb{R}^n$
$$min_x \left\| A x - b \right\|_2^2$$
with a special sparse structure where each row in $A$ has only up to 4 ...

**0**

votes

**1**answer

49 views

### Finding the distribution of a random variable numerically with sample data? [closed]

Just a thought that I had recently. Suppose given discrete data points for a random variable, could one numerically generate the probability function values at these discrete values? I tried looking ...

**2**

votes

**0**answers

39 views

### Scale vector in scaled pivoting (numerical methods)

I'm teaching students about several numeric methods, including scaled pivoting. There's a small section in this subject that I could never find a clear explanation to, either as intuition, or a more ...

**1**

vote

**1**answer

132 views

### Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?

I apologize if this is not a research level question (already tried asking http://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange with ...

**4**

votes

**1**answer

87 views

### Numerical equality testing

I am working on developing an online homework system.
One thing I would like to have is something which compares a student's answer (like $2\sin(x)\cos(x)$) with the intended answer (maybe ...

**2**

votes

**0**answers

29 views

### Finding an explicit constant in finite element error estimates

Background: In a finite element approximation to the solution of a linear PDEs, estimates on the order of convergence of the approximation to the solution rely on a theorem of Bramble and Hilbert ...

**2**

votes

**1**answer

320 views

### How to find representatives of $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$

While reading about the Teichmuller flow, I am reading about the space of lattices $SL(2,\mathbb{R})/SL(2, \mathbb{Z})$.
I could not a find a good way of computing the Teichmuller flow on this ...

**5**

votes

**0**answers

158 views

### Degree of Chebyshev polynomial necessary

In general, given $0<a<1$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[1-\frac{a}2,1+\frac{a}2]$ at every $x\in[1-a,1+a]$ and $f(0)=0$. What is minimum degree that is ...

**8**

votes

**3**answers

464 views

### Degree necessary of a polynomial?

Given $-1<a<b<0$, I want to find a polynomial $f(x)\in\Bbb R[x]$ such that $f(x)\in[a,b]$ at every $x\in[b^2,a^2]$ and $f(0)=0$. What is minimum degree that is needed and maximum degree that ...

**2**

votes

**0**answers

51 views

### Compute the smoothing of functions

Given a function $g:R^d\rightarrow R$, which is not necessarily continuous, I want to compute the "smoothing" of $g$, i.e.,
$G(\vec{y})=\int_{R^n} g(\vec{x}) f_{\vec{y}, \sigma}(\vec{x}) d\vec{x} $
...

**0**

votes

**0**answers

96 views

### Generalized arithmetic progressions contained in Bohr sets

Recall that generalized arithmetic progression of dimension $d$ is by definition a set of the form $P = P_1+\dots+P_d$, where $P_j = \{lp_j\ \mid \ |l|\leq l_j\}\subset \mathbb Z$ is an ordinary ...

**9**

votes

**2**answers

215 views

### Computing Gauss Legendre Quadrature for Large N

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscisas and weights $\{ x_j, w^j \} _j$ for large N. My question is how to do it, and why should it ...

**1**

vote

**0**answers

28 views

### LU growth factor applied to LDL of a Positive Semidefinite matrix [closed]

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...

**4**

votes

**0**answers

63 views

### Numerical integration error bounds on the unit sphere

A sequence of points $x_1,x_2,\dots$ on the unit sphere $S^{D-1}$ is said to be uniformly distributed if
\begin{align}
\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{j=1}^N f(x_j) = \int_{x \in ...

**0**

votes

**0**answers

37 views

### Global optimisation of the real part of impedance

I have the following global optimisation problem:
$$
\underset{\omega}{\min}-c^{T}\left(\omega^{2}\mathbf{1}+A^{2}\right)^{-1}b
$$
where $A$ is a $n \times n$ real matrix, $c$ and $d$ are ...

**2**

votes

**1**answer

83 views

### The bubble function

In the finite element method and more precisely the MINI element method in two dimensions, they use a function called the "bubble function" which is related to a triangle K of the space meshing and is ...

**0**

votes

**1**answer

74 views

### Upper bound for a ratio of modified Bessel functions

I am looking for an upper bound for the ratio of Bessel I functions $\dfrac{|I_\nu'(z)|}{|I_\nu(z)|}$ where $\nu$ is complex, and $z$ is a positive real number. Do you know any results about it? Thank ...

**9**

votes

**1**answer

218 views

### What happens to continuous spectrum upon discretization?

Excuse me for a bit of an vague question, but I haven't been able to find a definite answer for this for quite some time. My question is regarding (mostly non-normal )linear operators and their ...

**2**

votes

**1**answer

98 views

### Relation between Chebyshev Interpolation and Expansion

I am seeking connections between pointwise Lagrange interpolation (using Chebyshev-Gauss nodes) and generalized series approximation approach using Chebyshev polynomials.
Pointwise Lagrange ...

**2**

votes

**0**answers

39 views

### Estimating overshoot in spline interpolation

Say I have a spline space $\mathcal S$ of dimension $n$ with a set of unisolvent points $(\xi_i)_{i=1}^n$, i.e., points at which I can unambiguously interpolate within the spline space. So, given ...

**0**

votes

**0**answers

83 views

### inflow/outflow Boundary Conditions for flow in pipe

I have a question about boundary condition of solving Navier-Stokes equation through pipe.
When I simulate the flow in pipe using periodic boundary condition, it works good. But when I tried to change ...

**3**

votes

**2**answers

166 views

### What are interesting heuristics of determining how far given matrix is from a singular one?

The condition number and volume of matrix (defined as absolute value of its determinant) are things which come to mind. Is there more?
I think that over the years numerical folks (who are faced with ...

**5**

votes

**0**answers

103 views

### Hyperbolic toral automorphisms, and maximizing over orbits the minimum along an orbit of a function

Setup:
Let $\phi\colon T^2 \to T^2$ be a hyperbolic toral automorphism. Let $f\colon T^2 \to \mathbb{R}$ be a continuous function.
For $x \in T^2$, let $\underline{f}(x) = \inf_{n \in \mathbb{Z}} ...

**9**

votes

**4**answers

545 views

### Rounding errors in images of Julia sets

One typically computes Julia sets by iterating a complex function, such as a polynomial or rational function.
How do rounding errors affect the results?
I'm looking for references on this issue, ...

**2**

votes

**1**answer

106 views

### cohomology algebra of submanifold in euclidean space

If we write a manifold or CW-complex $X$ as a subset of $\mathbb{R}^n$, in expression of coordinates, for example, \begin{multline}
F(S^2,k+1)=\{(x_1,x_2,x_3,\cdots, ...

**6**

votes

**4**answers

380 views

### Expected value of a function over random sets

I am doing an analysis on the complexity of some set-related algorithm where the input is a random set. One of my setbacks can be formulated as follows:
Pick $k$ distinct numbers out of numbers ...

**0**

votes

**0**answers

49 views

### volterra equation of the first kind with K(0,t)=0

A standard assumption both in theory and numerical methods
for the Volterra equation of the first kind
$$ g(t) = \int_0^t K(s,t) f(s) ds$$
is that $K(s,t) \neq 0$. One can show existence of the ...

**4**

votes

**2**answers

129 views

### Accuracy of the formulas for angles between almost colinear vectors

Assume $x$ and $y$ are two vectors in $\mathbb{R}^3$ and we want to compute the acute angle $\alpha\in(0,\pi/2]$ between these two (noncolinear) vectors. There are (at least) two possibilities:
In ...

**4**

votes

**1**answer

66 views

### Numerical approximation to the Wasserstein metric?

Are there numerical methods for approximating/calculating the Wasserstein metric in particular cases?
Suppose that $f$ and $g$ are two density functions with the same support. How can I calculate the ...

**0**

votes

**0**answers

49 views

### Orthogonalization technique after cosparse dictionary update

I'm trying to adapt the cosparse dictionary learning (DL) approach described in Analysis K-SVD to a DL method that creates the dictionary as a union of orthonormal blocks (UONB).
For this I apply the ...

**0**

votes

**0**answers

42 views

### Smoothness of linear equation

Suppose $\beta$ is a solution to some linear equation $ M \beta = f$ where
$M$ is lower triangular with all negative entries except for the diagonal and first sub-diagonal where all entries are ...

**0**

votes

**0**answers

51 views

### Bits of precision matrix reconstruction

We have a real rank $r$ matrix $M\in\{0,1\}^{n\times n}$.
Suppose we have diagonalized using $LMR=D$.
I want to recover a real matrix $\widetilde{M}$ such that maximum absolute entry of ...

**3**

votes

**2**answers

194 views

### Monte Carlo integration of Gaussian integrals

I was doing a physical problem, and then it comes to this Gaussian integral. The dimension of the integral is very large (dimension = 300~600), and it is difficult to find the maximum of the ...

**1**

vote

**1**answer

100 views

### What is a one-parameter Newton's method?

The Newton's method that I know is defined as follows:
$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$
However, I've recently encountered a paper that talks about a one-parameter family of Newton's method ...

**0**

votes

**1**answer

109 views

### Generating random variables from the Cantor Distribution [closed]

I am looking for a method (exact, if possible, but at least asymptotically correct) for generating random variates from a Cantor Distribution? It seems like its abstract definition prevents this. In ...

**0**

votes

**0**answers

24 views

### Column Subset Selection implementations

Are there readily available implementations of algorithms for the CSSP - Column Subset Selection Problem?

**0**

votes

**0**answers

38 views

### Comparing Calculation Error in Divergent Numerical Methods

I'm not an expert in numerical methods, but I'm doing a simulation based on non-linear differential equations (General Relativity), there solutions has singularities, thus at some points numerical ...

**0**

votes

**1**answer

81 views

### What is exponentially fitted osculating straight line?

While reading an article about iterative methods for solving nonlinear equations I can't understand what is exponentially fitted osculating straight line. Could someone please briefly explain this ...

**3**

votes

**1**answer

235 views

### Analytic Solution to SDEs

Are there any example of SDEs with constant diffusion terms, other than the Ornstein Uhlenbeck process, which have exact solutions? I'm thinking of something of the form:
\begin{equation}
dX_t = ...

**-2**

votes

**1**answer

254 views

### BDF2 and TR-BDF2: what is better? [closed]

What method of numerical solving ODEs is better? BDF2 or TR-BDF2?
Namely, what advantages has TR-BDF2 over BDF2? Is TR-BDF2 more accurate? Does it damp "ringing" better?
The BDF2 method requires the ...

**0**

votes

**0**answers

28 views

### non-coherent estimation problem

I have the following signals
$$\left[\begin{array}{c} y_{mn} \\ y_{nm}\end{array}\right] =\left[\begin{array}{c} x_{n} \\ x_{m}\end{array}\right]h_{nm} +\left[\begin{array}{c} e_{mn} \\ ...

**7**

votes

**1**answer

202 views

### Approximation theory on the disc

Chebyshev theory provides a very effective method for approximating continuous real valued functions on the unit interval. Is there something similar for continuous real valued functions on the ...

**0**

votes

**0**answers

78 views

### Eigenvalue problem (finite difference operator)

Consider an arbitrary elliptic (perhaps, degenerate) finite difference operator
$$L_{i,j,k}=-\Delta_{i,j,k}+\alpha_{i,j,k}\frac{\partial}{\partial x}_{i,j,k}+\beta_{i,j,k}\frac{\partial}{\partial ...

**0**

votes

**0**answers

47 views

### Shooting method for non-oscillatory solutions

I am having some trouble with using the shooting method :
I am given a system of ode's representing initial value problem, and I know I want one of the unknowns (=u) to vanish at infinity (which I ...

**1**

vote

**1**answer

140 views

### Divergence of the Lagrange interpolation on the Chebyshev nodes

Faber theorem states that for every $\lbrace x_k^{(n)} \rbrace$ there exists a continuous $f$ function such that $\| f - L_n \|_{\infty} \not\rightarrow 0$, where $L_n$ is interpolation polynomial on ...