4
votes
2answers
63 views

Newton's Method for Roots of Polynomials

The standard way to use Newton's Method for finding a root of a polynomial $p(x)$ is to use the iteration formula $$x_{n+1}=x_n-{p(x)\over p'(x)}$$ I recently thought of a new way of finding the ...
0
votes
1answer
26 views

How to evaluate a condition number for a function of several variables.

I'm trying to get the condition number of a function $f(a,b,c)$ to see if it is stable. It is multivariate. I am reading the information here ...
0
votes
0answers
51 views

Infinite Order Fourier Series and Infinite Series with Piecewise Function

Given $$ f(x) = \left\{ \begin{array}{ll} 4\pi^2 & \quad x = 0 \\ x^2 & \quad 0<x\le2\pi \end{array} \right. $$ First, compute the infinite ...
1
vote
1answer
27 views

calculate Jacobian matrix without closed form or analytical form

The question is probably clear in the title. In many of my applications mostly computer vision, I might not have the closed-form or analytical form of f (a multivariable function). It's calculated ...
0
votes
2answers
54 views

Numerical root finding of function with unknown parameters

I have a multivariate function of which I want to find one of (or all) its roots. However, besides the variables, it also depends on a bunch of parameters. Now I only want to find roots which are ...
0
votes
0answers
40 views

How to find an error estimate for integral of curvilinear surface triangle when using quadrature

I would like to find a way to estimate the error due to the calculation of the normal when one tries to find the volume of a volume composed of quadratic surface triangles using numerical gauss ...
0
votes
0answers
21 views

Numerically solving partial differential equations

I am working on a project which involves solving Kramer's equation: $$ \frac{\partial p(x,v,t)}{\partial t} + v \frac{\partial p(x,v,t)}{\partial x} + (\frac{F(x)}{m}-\gamma v) \frac{\partial ...
0
votes
0answers
44 views

Calculating the Jacobian for sampled data

I am reading about the Jacobian matrix which I interpret as a generalized gradient. I would like to take my investigations a bit further, so I have constructed some sample data which I would like to ...
2
votes
1answer
148 views

Question about integral equations

Consider the equation $$g(t) = \int_a^b K(t,s)f(s) ds $$ where $g$ and the kernel $K$ are known and $f$ is to be determined. Suppose that the equation has a solution. Under what conditions on the ...
1
vote
0answers
86 views

Differentiating a multivariable function

In numerical mathematics, we looked at the "Taylor series method" to construct one-step methods. Let $\mathbf{\dot{y}} = \mathbf{f}(\mathbf{y})$ be a system of differential equations. We define a ...
0
votes
1answer
60 views

Approximating a complicated multi-variable function over an interval?

Consider $$ F(\mathbf{r})=F(x,y,z) = \frac{2z^2 - x^2 - y^2}{(x^2+y^2+z^2)^{5/2}} $$ where $x,y,$ and $z, $ are all $n^{\text{th}}$ order polynomial functions of a parameter $t$ with arbitrary ...
3
votes
1answer
255 views

How to show that the Hessian matrix of $G$ is positive definite?

Let $\{g_i:X\subset\mathbb{R}\rightarrow\mathbb{R};\;i=1,...,m\}$ be a linerly independet set of real functions. Given $n$ points $(x_1,y_1),...,(x_n,y_n)\in X$, consider the following function ...
2
votes
1answer
64 views

How to compute the second derivatives?

Motivation: In isogeometric analysis, state variables(e.g. displacement) are defined in the parametric domain, which can be mapped to the physical domain by $\boldsymbol{\xi}\mapsto \boldsymbol{x}$ ...
2
votes
1answer
97 views

Formulate optimization problem

My research area has "nothing to do with mathematics" but I still find it full of optimization problems. Therefore, I would like to learn to formulate and solve such problems, even though I am not ...
1
vote
1answer
51 views

Derivative of solution of ODE

I have a set of nonlinear differential equations with parameters. $$\dot{\vec{x}} = F(\vec{x},\vec{\beta}) $$ where $\vec{x} \in \mathbb{R}^p$ and $\vec{\beta} \in \mathbb{R}^q$ ($p,q \in ...
3
votes
1answer
210 views

Solving laplace's equation for an inviscid and incompresible fluid

Background I'm working on a 2D inviscid, incompressible fluid sim using vortex methods (that is, treating vortex as discrete particles), and I'm trying to (numerically) solve the no-through boundary ...
3
votes
2answers
76 views

Discrete approximations of $\nabla^2{\bf v}$

I am writing a Navier Stokes solver. The vector field is represented as a grid with integer coordinates I am looking at other people's computer code. I don't entirely understand the vector calculus, ...
2
votes
1answer
273 views

Derivation of weak form for variational problem

My question is about understanding the derivation of the weak form of a variational problem (to be used for the solution via the finite element method). The problem is as follows (it is an image ...
1
vote
1answer
775 views

Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
2
votes
1answer
155 views

Optimization, Gradients, and Multivariate Data

I would like to learn gradient based optimization for multivariate data. For example, assume the data I have is $X = (x_0, ..., x_n)$ where $x_i$ are some random variables and $f$ a function ...
0
votes
1answer
515 views

Separating equation into real and imaginary parts and eliminating integral

I have the following equation: $\tilde U(\tau ,\omega ) = \frac{1}{{\Lambda (\tau ,\omega )}}\exp \left[ {i\int\limits_0^\tau {\left( {{{\left( {\frac{\omega }{{{\omega _h}}}} ...
1
vote
0answers
160 views

Integration methods for functions with Delta distributions

Which Monte-Carlo methods are available for computing a multidimensional integral with Delta distributions (in case one cannot sample them explicitly)? PS: I also asked a similar question at ...
2
votes
1answer
225 views

Is the Hessian of the 'generalized least square function' positive semi-definite?

Let $F:\mathbb{R}^n\rightarrow \mathbb{R}$ a scalar field which has a quadratic from, $$F(\mathbf{x}) = \frac{1}{2}\mathbf{d}(\mathbf{x})^\top\Lambda\mathbf{d}(\mathbf{x})$$ with ...
0
votes
0answers
171 views

Double integration involving polynomial functions and sinc function

I encountered a problem which I can't seem to simplify/solve. I was wondering if any mathematicians or specialists knows how to approach this problem? $$\int^{0.5}_{-0.5} \int^{0.5}_{-0.5} \; ...
3
votes
1answer
303 views

Steepest Descent Algorithm for Solving Linear Systems

I am trying to implement the steepest descent algorithm for linear systems. The equation is below: $\begin{align*} Ax &= b\\ x_0 &= [0]*[m,n]\\ x_k &= x_{k-1} + ...
2
votes
1answer
274 views

Setting up and solving differential equation with The Euler Method

I recently started this question and it gave me some insight into the world of differential equations. However the solution was not fit for my goals as I wanted a general method for calculating the ...
0
votes
1answer
323 views

Discretizing gradient in upwind differencing, hamilton-jacobi equation

In the presentation here about Level set methods http://www.cs.au.dk/~bang/smokeandwater2006/Lecture9_IntroToWaterAndLS.ppt the author constructs a linear PDE $\frac{\partial \phi}{\partial t} + ...