Numerical algorithms for problems in analysis and algebra, scientific computation

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Recommendations for binomial system solver

I am interested in solving binomial systems of the form $$ \begin{cases} a_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} + b_1 x_1^{d_{11}} x_2^{d_{12}} \cdots x_n^{d_{1n}} &= 0 \\\\ ...
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122 views

approximation methods in integral equations

Recently I was reading about integral equations and I am a beginner in it. There was a constant reference to the non-availability of methods to find the exact solutions and hence lot of approximation ...
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1answer
196 views

How are real-analytic functions encoded in computer algebra?

I would like to know how are encoded the real-analytic functions on the interval by the computers. When I think in a real-analytic function I just think in a composition of the ''typical'' analytic ...
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1answer
517 views

Matrix perturbation theory

I am having matrix $M_0$ with coresponding eigenvectors and 4 eigenvalues {0,0,a,-a}. Eigenvalue $\lambda=0$ is double degenerated. Now I am appliing small perturbation $\epsilon M_1$ and want to get ...
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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 ...
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0answers
169 views

Heat transfer coefficients - correlation equations for Nusselt number - vertical pipe

I need a heat transfer coefficient applicable to free convection of water INSIDE a vertical pipe. I have the correlation equation from Churchill and Chu, but it is only valid for D/L>=35/Gr^(1/4), ...
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1answer
91 views

Kronecker-structured matrix kernel

Let $A,B\in\mathbb{C}^{n\times 3n}$ be two matrices, and denote the Kronecker matrix product by $\otimes$. The matrix $$ M= \begin{bmatrix} A \otimes I_n \\\\ I_n \otimes B\end{bmatrix} $$ has size ...
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213 views

Approximation theory under $L_1$-error

Is there a reference for results in approximation theory of bounded functions of one (and multiple) variables under $L_1$-error? Formal definitions for functions of one variable are below. Let $C$ ...
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131 views

Radius of convergence to be proved more precisely (differential equation)

There is a differential equation in polar coordinates: $r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const. It is possible to get a solution which is a power series (see below). However, I am looking for an ...
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1answer
275 views

Lebesgue constant as condition number of polynomial interpolation

Let $T = \{ x_0,\ldots,x_n \}$ be a set of $n+1$ different points in the real interval $[a,b]$. Let $X_T$ be the associated interpolation operator on $C[a,b]$: it takes a function $f \in C[a,b]$ into ...
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1answer
788 views

Smoothing L1 norm, Huber vs Conjugate

I'm trying to minimize a convex (not necessarily strictly convex) function involving an L1 norm (similar to lasso), which makes it non-differentiable at some points. So I'd like to smooth it and treat ...
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47 views

Recovering Shared Eigenvector Set

Suppose we are given a set of $M$ pairs $\{(\vec{x}^{(i)},\vec{y}^{(i)})\}$, with $\vec{x}^{(i)}\in\mathbb{R}^N$, $\vec{y}^{(i)}\in\mathbb{R}^N$, $M\gg N$ such that $\vec{y}^{(i)} = Q^{(i)} ...
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2answers
691 views

Approximating erf by tanh

It appears to be well-known that $\tanh(x)\le \mathrm{erf}(x)$ on $[0,\infty)$. It's off-handedly mentioned here, for example. Where can I find a formal proof? On the one hand, it's hard to imagine ...
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121 views

integrate square of bessel 1st and 2nd kind

Has anyone ever solved the integral of square of Bessel and Neumann functions? This is for standing wave analysis on a cylindrical object. I need to integrate $(AJ(kr) + BY(kr))^2$ from $r=a$ to ...
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75 views

Integrating B-Spline composed with log

If $f$ is a real B-Spline and $a, b$ are real numbers, then is there a numerically stable way to evaluate the following expression? $\int_a^b f (\log x) \mathrm{d}x$
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3answers
840 views

Easy functions ?

Let $f$ be an analytic function, and suppose that we want to compute $f(x)$. The input consists of the digits of $x$ and the output of a rational number approximating $f(x)$. A function $f$ is called ...
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138 views

Numerical linear algebra: how to compute $B^TC^{−1}B$ efficiently

Hi, my question is similar to this one. I have to compute $B^TC^{−1}B$, where $C$ is a strictly positive definite $n\times n$ matrix and $B$ is $n\times m$. The matrix $C$ is huge ($n$ up to a ...
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2answers
296 views

Condition number of matrix after partial orthogonalization

I'm wondering about which bounds one can put on the condition number of a $n\times n$ square matrix which is obtained from another $n\times n$ square matrix by orthogonalizing the first $m < n$ ...
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137 views

Upper bound on integrals of Legendre polynomials

Hi, If $P_n(x) $ is unnormalized shifted Legendre polynomial, and $g_{n,m}(x) = \int_0^x P_n(x_1)x_1^m dx_1, n>m $ then what is the upper bound $ |g_{n,m} (x)|_{max} , x\in (0,1) $ as a function ...
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418 views

Parabolic cylinder functions - explicit estimates?

I need estimates for the parabolic cylinder functions $U(a,z)$ (first studied by Whittaker). Most work in the literature focuses on $a$ real. As it happens, I am interested in $U(a,z)$ on a strip in ...
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308 views

How to solve a system of linear equations without storing the matrix?

I have a procedurally defined Hermitian matrix $M$, i.e. I can get any matrix element by calling a black box function (e.g. a library function), and a vector $Y$. And I have to solve a system of ...
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4answers
775 views

When we use Bernstein polynomials in application

When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple ...
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99 views

Matrix Minimax problem

I have the equation $\Sigma_k(M_k{p_k})V=EV$, where the $M_k$ are n*n real Hermitian matrices, $V$ is a n*n eigenvector matrix, $E$ a dim-n energy eigenvector and the $p_k$ scalar parameters. The ...
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26 views

Problem with the convergence of a Nystrom algorithm

I programmed a Nystrom Algorithm specifically for my problem: This is the exact equation i want to solve: $y''=(w^2-e*cos(t))*sin(t)-b*y'$ And this is my algorithm ...
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1answer
149 views

An upper bound on a simple sum

Hi, I am trying to put a bound on a sum. Given $\omega=\exp(2\pi i/3)$ and $n$ positive real numbers $ 0=\tau_0 < \tau_1 < \tau_2 < ...\tau_{n-1}< \tau_n=1 $ such that ...
4
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1answer
310 views

Norm of inverse confluent Vandermonde matrix

Let $\{x_1,\dots,x_n\}$ be pairwise distinct complex numbers and $l_1+l_2+\dots+l_n=N$. The $N\times N$ confluent Vandermonde matrix is defined as $$V= \begin{bmatrix} ...
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135 views

Checking for error in conjugate gradient algorithm

What is a good way to check if the any numerical error is occured in conjugate gradient algorithm. Additionally why is it not suggested to check error by checking A-orthogonality of search direction ...
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138 views

How to interpolate in 3-D non-euclidean space?

Assume, one has a 3-D non-euclidean space of points $p_i = \left(x_i, y_i, z_i\right) \in \mathcal{R}^2 \times \mathcal{R}_{> 0}$ with the following "distance" function $d\left(p_1, p_2\right) = ...
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252 views
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479 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: ...
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149 views

Algorithm for numerically approximating the Prokhorov metric?

Question: What is known about algorithms for numerically computing/approximating the Prokhorov distance between two measures? Recall that the Prokhorov distance metrizes the topology of weak(-*) ...
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2k views

Any good books on numerical methods for ordinary differential equations?

I need to find some masters-level exercises about numerical methods for solving ODEs. Are there any good references?
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1answer
1k views

Upper bound on largest eigenvalue of a real symmetric n*n matrix with all main diagonal >0, everywhere else <=0

Is there a good analytic upper bound on the largest eigenvalue of a real symmetric n*n matrix with all main diagonal entries strictly positive, all other entries <=0 with typically many of them ...
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309 views

Condition Number related to Root finding problems

Suppose we want to find the root of the equation $f(x)=\phi(x) - d = 0$, where d is a real constant and $f$ is continuously differentiable function. The problem is well posed if the inverse ...
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2answers
253 views

Delauney triangulation in high (>20) dimensions

Hi all, I know that its very hard to find the Delauney triangulation of high dimensional spaces, especially if there are several thousand points that need to be triangulated. So I was wondering . . ...
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3answers
317 views

What software one needs to solve a big linear system on a small computer?

A time ago I was intrigued by the following remark: "... one ends up with a (non-sparse) system of equations in about 10000 real variables. One important practical point is that solving such systems ...
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1answer
216 views

Most orthogonal lattice basis

Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...
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250 views

Solving system of nonlinear equations

Dear all, Can anyone tell me all the algorithms that are available for finding all solutions of a system of nonlinear equations? I am particularly interested in solving problems of the form: ...
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394 views

Global Error Analysis of Euler's Method

I know that the local error at each step of Euler's method is O(t^2), where t is the time step. And since there are (b-a)/t steps, the order of the global error is O(t). However, I saw a derivation ...
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159 views

fast multipole method and geometric algebra

Hello, I just learned about fast multipole method(FMM) from this article http://math.nyu.edu/faculty/greengar/shortcourse_fmm.pdf and I really liked the use of complex numbers in 2d. But I didn't ...
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4answers
619 views

Applications of Hodge-de Rham Laplacian on p-forms ($p\neq 0,n$) in physics or engineering

The Hodge-de Rham Laplacian $L=(d+d^*)^2$, where $d$ is the boundary operator of the de Rham complex, is well-known in the math community. Recently, I tried very hard to search for examples of its use ...
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1answer
78 views

Is is preferable to use a difference formula of higher order of accuracy for spatial derivatives to solve this IVP problem ?

I want to numerically integrate the equation $\partial_t u= a(t) \partial_xu+b\partial_{xxx}u+c$ to get $u(t)$. Is is preferable to use a difference formula of higher order of accuracy for spatial ...
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55 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
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385 views

Numerical Methods for ODEs - History

Wikipedia presents a timeline of important developments in Numerical Methods for ODEs, namely: ...
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651 views

why do we need algorithms, and why is non-convex optimization difficult? [closed]

A simple question, but (I'm quite sure) not a superficial one: is the basic distinction between algorithms and much of the rest of math that algorithms try to tackle problems for which we lack global ...
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297 views

Gauss Legendre Method for Implicit Integration

Methods that are usually adopted for time integration in transport phenomena problems are either: Euler (explicit, first-order accurate) $\frac{dY}{dt}=f(t,Y)$ $Y^{n+1}=Y^n+\Delta t f(t,Y^n)$ ...
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376 views

Series acceleration for more complicated types of oscillating series

Question: It is well known that the greatest integer function has a Fourier series representation. Since the greatest integer function itself is not periodic, the representation is derived from the ...
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63 views

Is there a Krylov subspace method for solving D+epsilon*S where D is diagonal, epsilon small and S skew-symmetric

I'm working on a problem that gives a matrix system of the form D + epsilon*S, where S is a skew-symmetric matrix. I'm interested in finding if any work has been done to develop a conjugate gradient ...
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1answer
368 views

Valid use of Laplace's method?

I am trying to say something about the asymptotics of $$\int_{\mathbb{R}} e^{cx - x^{4/3}}dx$$ as $c \to +\infty$, and need a sanity check. As I understand it, Laplace's method is to write $$q(x) = ...
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224 views

Discrete Sobolev space of $R^n$ valued maps

Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say $f:\Omega ...