Questions on numerical analysis/numerical methods; methods for approximately solving various problems that often do not admit exact solutions.

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31 views

Can be a weight function from Favard's theorem determined?

Favard's theorem on orthogonal polynomials essentially states that if a system of polynomials satisfies a certain three-term recurrence, then it forms an orthogonal system with respect to some weight ...
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97 views

Tridiagonal system problem with Gaussian elimination and partial pivoting

What happens to the tridiagonal system (shown below) if Gaussian elimination with partial pivoting is used to solve it? In general, what happens to a banded system? $$ \pmatrix{d_1 & c_1 & ...
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28 views

Polynomials of the form $g(x)=f[x,x_1,x_2,…,x_m]$.

Can anybody point me to some materials about polynomials of the form $g(x)=f[x,x_1,x_2,...,x_m]$, meaning that for given $x$ they will give back a leading coefficient of function $f$ interpolated at ...
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40 views

overdetermined system

I'm trying to solve an overdetermined system with Second-order differential equations. Without noise everything works fine. When I add Gaussian noise the solution is not stable anymore and I've got a ...
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26 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 ...
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54 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 ...
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18 views

Interpolation of a function given two sets

$P$ and $Q$ are sets of $k+1$ points. $P\bigcap Q$ has $k$ points. $p(x)$ in $\mathbb{P}_k$ interpolations $f(x)$ at points of $P$. $q(x)$ in $\mathbb{P}_k$ interpolations $f(x)$ at points of $Q$. Let ...
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49 views

Euler's method versus Backward Euler method

What's the difference between a standard Euler's method and backward Euler method? Are there any good examples to help me understand this difference?
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63 views

Financial mathematic with Feynman-Kac

I have a really big task in financial mathematics and a small part of it (to set up the problem), I need to write a PIDE (the Feynman-kac) where we estimate options with jumps. It is derived from the ...
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304 views

Exact inversion of matrix complexity (by Gaussian elimination)

I would like to check if what I have done is correct. Please, any input is appreciated. Problem statement: Consider a non-singular matrix $A_{nxn}$. Construct an algorithm using Gaussian elimination ...
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23 views

How to extract projected points from Factor Analysis?

I am currently comparing different projection methods to derive 2D screen coordinates from high-dimensional point clouds. This is often done with PCA, MDS, LLE, SNE, etc. and I want to compare them ...
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22 views

$\ell^{\infty}$ vs. Finite $\ell^{p}$ Norms for Estimating Rates of Convergence of ODE Methods

Suppose we have a numerical metod $\Phi(h)$ which estimates a solution to some ODE IVP at time $T$ using stepsize $h$ ($N=\frac{T-t_{0}}{h}$), and that we have an asymptotic error expansion of the ...
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58 views

This system is contractive?

I have a system which has a form of find point problem, described as following $$p_i=h_i(\mathbf{p})$$ where $$p_i\in[0,1]$$ is the $i$-th components of the $n$-dimensional column vector ...
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43 views

Numerically solving first order system of ODES

I am currently trying to numerically solve a first order system of ODES, of the form: $$\frac{d\rho}{ds}(s)=f_1(\rho,\theta,k_1,k_2)$$ $$\frac{d\theta}{ds}(s)=f_2(\rho,\theta,k_1,k_2)$$ ...
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29 views

Prove $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{-m} \cdot \|{v}\|_{L^{q}(T)}$

My professor asked me to derive this inverse estimate: $\|{v}\|_{W^{m,p}(T)} \le C|T|^{1/p - 1/q} \cdot h_{T}^{l-m} \cdot \|{v}\|_{W^{l,q}(T)}$, for $l \le m$ So I divided the problem into 2 steps: ...
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95 views

Condition number for multiplication and division identical

Consider the maps $F:(x,y) \mapsto xy$ and $G:(x,y) \mapsto x/y$ taking $\mathbb R^2_+ \to \mathbb R$. (Let $\mathbb x := (x,y)$) Because they are differentiable, one may take $$ Kf(\mathbb x) = ...
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104 views

Numerical Integration of a differential equation of fisrt-order with constant and variable sampling rate

Example 1 A non-homogeneous differential equation of first-order is given and the integral of it need to be calculated. The differential equation is shown in Figure 1: $ {y}'(t) = Asin(\omega t + ...
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49 views

I need help about a Jacobian Matrix for Newton's Method

Let f (x) = $x^3$, with $x \in $. Given three points $x_{i-1} <x_i <x_{i +1}$ central difference: $$f[x_{i-1}; x_{i+1}] := \dfrac{f(x_{i+1}) - f(x_{i-1})} {x_{i+1} - x_{i+1}}$$ $f_0$ ...
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52 views

Derivation of weights for estimating second derivative from Butcher's tableau of Runge-Kutta methods

Given an explicit Runge-Kutta method (following this notation): $y_{n+1} = y_n + h \sum_{i=1}^s b_i k_i$ $k_i = f\left(t_n + c_i h, y_n + h \sum_{j = 1}^{s} a_{ij} k_j\right)$ How would one derive ...
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33 views

How to generate random matrices when it's singular values are given?

Consider matrix S as nxn diagonal matrix with singular values populated across the diagonals in non-increasing order. I want to know how to create random matrix A whose singular values with be the ...
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37 views

What is the error when approximating $L^2([0,1])$ by a finite dimensional space?

Let $X \subset L^2([0,1])$ such that $f([0,1]) \subset [-M,M]$, for some constant $M$, and any $f \in X$. By choosing a finite dimensional basis $V=\left(v_i\right)_{i=0}^n$, where each $v_i \in X$, ...
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33 views

Check for the value of x for function f is ill conditioned

I have two examples: $$ i) \ f(x) = \sqrt{1 - x^2} $$ $$ ii) \ f(x) = \sqrt{x^2 + 1} - x $$ I must check the value of x for which the calculation of the values ​​of the function $ f $ is ill ...
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106 views

Discretization of a 2nd Order Differential Equation

Good Day to ALL! I want to UNDERSTAND the method of discretizing a 2nd order differential equation. I have tried the same, but my matlab model was a bust (it computes but the result is incoherent). I ...
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33 views

Convergence of $xe^x - R$

Basing my question on one of the previous questions I have passed before Root of the function $f(x)=xe^x-R$, I was wondering why does $xe^x - R$ always converge? I was told that the function will ...
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47 views

Show convergence by Cauchy sequence

\begin{equation} x(k+1)=\arccos\bigg( -\frac{1}{2(Dr^{\frac {|\sin(2x(k)+\theta)|}{M\sin x(k)\sqrt{A+2B\cos(2x(k)+\theta)}}}+1)} \bigg) \nonumber \end{equation} $A,B,D,r,M,\theta$ are constants. ...
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36 views

Finding zero of nonlinear function

I have a function $g(x)$ and would like to numerically find all the zeros of the function for $x$ in some interval $[a ,b]$. My function $g$ is nonlinear and discontinuous. I can compute the ...
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56 views

Richardson's Extrapolation problem

The forward-difference formula can be expressed as $f'(x_0) = 1/h [f(x_0) + h) - f(x_0)] - h/2f''(x_0) - h^2/6 f'''(x_0) + O(h^3)$. Use extrapolation to derive $O(h^3)$ formula for $f'(x_0)$. I ...
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124 views

Newton's Interpolation Polynomial Question

The approximate population of the U.S. was 150.7 million in 1950, 179.3 million in 1960, 203.3 million in 1970, 226.5 million in 1980, and 249.6 million in 1990. Using Newton's interpolation ...
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128 views

Is my form of Lorentz and Gaussian equations suitable for curve fitting?

I am attempting to curve fit gas chromatography spectra. These spectra can be Lorentz or Gaussian or a combination of both. I am using the following code for these equations. ...
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18 views

Why the Hermite Polynomial $H_{2n+1}$ of $f \in C^{2n+2}[a,b]$, it's g'(t) has (2n + 2) distinct zeros in [a,b]?

$g(x) = f(t) - H_{2n+1}(t) - \frac{(t-x_0)^2 \cdots (t-x_n)^2}{(x-x_0)^2 \cdots (x-x_n)^2}[f(x) - H_{2m+1}]$. Thank you.
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38 views

Approximating the speed of an object from a finite list of approximate positions

An observer watches an object and notes that, for $1 \leq i \leq n$, the object was seen approximately at point $p_i$ in the plane at time $t_i$. For the purpose of this problem, suppose that there ...
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46 views

Secant Method Convergence

Prove that if the iterates in Secant's method converge to a point $r$ for which $f'(r)\ne0$, then $f(r)=0$. Hint: For the secant method, the mean-value theorem is useful. This is the case $n=0$ in ...
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197 views

Newton's Method and Secant Method Convergence proof

Show that if the iterates in Newton's method converge to a point $r$ for which $f'(r)\ne0$, then $f(r)=0$. Also establish the same assertion for the secant method. Hint: For the secant method, the ...
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23 views

Determining that a difference scheme is “dissipative of order 2r” in practice

First off, a difference scheme is dissipative of order 2r if there is a constant $\delta > 0$ such that for all the associated frozen coefficient schemes the eigenvalues of the amplification matrix ...
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77 views

What are some examples of difficult sums?

I'm looking for sums such that evaluating $f(x)$ is easy and fast, but evaluating $$\sum_a^{a+n}{f(x)}$$ is slow and hard. To be more scientific, evaluating $f(x)$ takes time $O(n)$, but evaluating ...
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540 views

Finite difference approximation of heat equation with source term

I am using the implicit finite difference method to discretize the 1-D transient heat diffusion equation for solid spherical and cylindrical shapes. The general equation is: $$ ...
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56 views

Bisection method, involving inequalities of roots and its intervals

If the bisection method generates intervals $[a_0, b_0], [a_1, b_1]$, and so on, which of these inequalities are true for the root r that is being calculated. Give proofs or counterexamples. a. $|r ...
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48 views

Please help me about conjugate gradient method

As I know, the error function of neural network is the sum of difference between actual output and the target value. But in conjugate gradient method, they use quadratic function: $E(w) = \frac{1}{2} ...
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68 views

Numerical Error in Computation - What Are the Students' Expected to Know?

I'm going to conduct an educational research about math undergraduates' conceptions about "numerical error." So I'm making a list of items I think someone with a BSc in mathematics is expected to know ...
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110 views

If $f$ is a differentiable function and $f'$ is bounded show that there exists a unique fixed point

let $f$ be a continuously differentiable function from $\mathbb{R}$ to $\mathbb{R}$ and such that $|f'(x)| \leq 4/5$ for all $x \in \mathbb{R}$. Show that there exists a unique $x$ in $\mathbb{R}$ ...
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50 views

Approximating $x - y$ when $x$ and $y$ are really close to each other.

We have this formula $ A = \sqrt {x + \frac {2}{x}} - \sqrt{x - \frac {2}{x}}$ and also $x\gg1$ as you know in these situations we simply multiply $A$ with $ \sqrt {x + \frac {2}{x}} + \sqrt{x - ...
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17 views

Do primitive recursive languages suffice for numerical simulations?

According to the Wikipedia article, a primitive recursive programming language can't do while-loops (but it can do for-loops), but the upshot is that primitive recursive functions are not partial ...
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71 views

Solve: $2div(G(\nabla H_{u,v}\nabla H_{u,v}^\top)\nabla H_{u,v})$

From the work: Algorithms for 3D Shape Scanning with a Depth Camera. Specifically on reducing the following equation in terms of the variable $X\in\Re^{k\times t} $, to be able to program it in ...
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52 views

Numbers immediately to the right and left of $2^k$

What are machine numbers immediately to the right and left of $2^k$ ? How far is each from $2^k$? Please, help!
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140 views

FFT for Fourier Integrals of analytic Functions

So, I'm trying to implement a fourier-transform of an analytic function through DFT or FFT as I'm only interested in a certain frequency range. So far I've tested Numerical Recipes' FFT algorithm ...
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129 views

Numerical Approximation of Black-Scholes, too big for MATLAB?

Note, this is a homework problem. Okay, so the problem is to approximate the Black-Scholes model using a semi-discretization with forward Euler in time and the usual spatial finite difference for ...
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72 views

Small symbols behind parantheses

I am currently reading the following paper where the author uses constantly small symbols after parantheses, but I do not know what this means. I am particularly interested in equation (23), so you ...
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51 views

Comparison of upper bounds of operator norms

I am working with a $C^{1}$ function $F: D \rightarrow \mathbb{R}^{n}$ with Frechet derivative $F'$, and assume the following condition (1) $||F'(z)^{-1}\left(F'(y) - F'(x))(y-x)\right)|| \leq ...
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191 views

Solve Master Equation (Differential Equation) Numerically

any ideas on how to solve the following master equation numerically? $$ \frac {\partial P(x,y,t)} {\partial t} = rP(x-1,y,t) + (1-2r)P(x,y-1,t) + rP(x+1,y-2,t)+\gamma P(x,y+1,t) - (1 + \gamma)P(x,y,t) ...
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42 views

Appropriate numerical scheme for eigen value problem

I am looking for a numerical scheme which can easily handle the following eigen value problem I already had the analytical results of this problem, now I want to know how to treat this problem ...