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

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11 views

What is the difference between zero and absolute stabilty?

In numerical ODEs, what is the difference between these two types of stability in multistep methods? From what I can gather, zero stability places a bound on how much future approximations can ...
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1answer
53 views

Recursive formula for integration by parts of given functions

I need to find, if it actually exists, a recursive formula for the following evaluations of indefinite integrals: \begin{align} I_{1,n}(x,R) &= \underset{n \,\text{terms}}{\underbrace{\int dx ...
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24 views

Least Square Apply Non-Linear Function

I am study numerical methods and I see that question. Considering that $f(1)=0.6065, f(1.5)=0.8825, f(2)=1, f(2.5)=0.8825, f(3)=0.6065.$ Utilizing the method for least squares, approximate the ...
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1answer
23 views

A Theorem about Interpolation Method?

I have a question about interpolation. I think that question is a theorem, but I don´t find nothing about that. Anyone can help me? Show that, if $g$ is the polynomial of degree $m<n$ that ...
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1answer
25 views

Abitrary derivatives of lagrange basis functions

The lagrange basis functions are given by \begin{align} \phi_k(x) =\prod_{j\not = k} \frac{x-x_j}{x_k-x_j} \end{align} I try to reproduce the numerical results of a paper. In this paper, the ...
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0answers
15 views

How to work with difference-of-elements penalty in optimization

I am trying to solve the optimization problem $$\min_{H,S>0} \|W(H+S)-X\|^2_F+Q(H)+\eta\|S G\|_F^2$$ where $X\in\mathbf{R}_+^{m\times T}$, $W\in\mathbf{R}_+^{m\times k}$, ...
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1answer
40 views

Not able to use fzero function in Matlab

I am new to Matlab. I am trying to solve a non-linear equation using this inbuilt Matlab function called fzero() but it's not giving me the results. The main file ...
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56 views

SOLVED Implementing Euler's Method step

Given: $$ \frac{dy}{dx} = -2y + 4e^{-x} $$ $$ \frac{dz}{dx} = -\frac{yz^{2}}{3} $$ $$ y(0) = 2, z(0) = 4, x\in[0,1],h = 0.2 $$ Do I have to implement the step of h = 0.2 any differently for z'? That ...
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11 views

Proving if the graph of F is over it's tangent line at all points then Newton Raphson converges.

Here's the problem: Let $F : \mathbb{R} \rightarrow \mathbb{R} \,, F \in \mathbb{C}^1$ so that $F'(x) < 0 \, \forall \, x$ and there's a unique $r$ so that $F(r)=0$ and let $L_{x_0} (x)$ be the ...
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25 views

A lower bound for the condition number of Hilbert Matrix

I am trying to prove that $\operatorname{cond}_{\infty}(H_n)\ge n^2$ for all natural $n$, where $H_n$ is the Hilbert matrix in $\mathbb R^{n\times n}$, i.e. $$(H_n)_{i,j} = 1/(i+j-1)$$ I am trying ...
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35 views

Approximate solutions for quintic equation

The other day I asked a question in here about deriving the equations $$u^2\left(\left(1-s_1\right)+3u+3u^2+u^3\right) =\alpha\left(s_0+2s_0u+\left(1+s_0-s_1\right)u^2+2u^3+u^4\right),$$ where ...
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0answers
19 views

Integrate for 3 step method

I need to integrate $$\int_0^{3h}\frac{(t-t_{i-1})(t-t_{i-2})}{2h^2}f(t_i,y(t_i))- \frac{(t-t_i)(t-t_{i-2})}{h^2}f(t_{i-1},y(t_{i-1}))- \frac{(t-t_{i-1})(t-t_i)}{2h^2}f(t_{i-2},y(t_{i-2})) dt$$ where ...
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1answer
102 views

Finding minimum point of banana function using Newton's Method

I am using the banana function $F(x_1,x_2)=(1-x_1)^2+100(x_2-x_1^2)^2$ over $x_1,x_1 \in \Re$. I am using $f_1(x_1,x_2)=0, f_2(x_1,x_2)=0$ to find the minimum point of $F$ which it is (1,1). Now I ...
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52 views

Matlab project - Jacobi method for tridiagonal matrices…

I have to do a project in Matlab to my University and I don't quite understand what I should do. I was given script that solves systems of equations with Jacobi's method with given tolerance and ...
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3answers
77 views

Prove that $x^3 -3x^2 +6 = 0$ has only one real root

I know that if I take the derivative of $$x^3 -3x^2 +6 = 0$$ and prove it is always greater than zero, I'll find that this functions is always increasing, and therefore if I find an interval where ...
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1answer
23 views

Integral over the unit ball in $\mathbb{R}^n$

Let $f(x)=|x|^r$ on $B_1(0)$ real valued function.Where $B_1(0)$ is the unit ball in $\mathbb{R}^n$. I am trying to show that if $r>1-n$ f has a weak derivative. ATTEMPT: I know from the ...
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1answer
70 views

Newton's method convergence implementation

How can I solwe this problem: Experimentaly examine convergence Newton's method for conformation: \begin{align} 2x^3-y^2-1=0 \\ xy^3-y-4=0 \end{align} for various loaded inputs with start points ...
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1answer
22 views

Large error after factoring h from fourth-order Runge–Kutta method

Consider $$\frac{dy}{dx} = 2x-y, \qquad y(0)=1$$ That has an exact solution $y(t) = 2t+3 e^{-t}-2$. But I want a numerical solution. So I decided to use forth-order Runge–Kutta method ...
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1answer
42 views

Solve a viscous Burgers' equation with a Newton-GMRes method

I implemented a preconditioner for a GMRes method. To test this preconditioner I want to solve this one dimensional viscous Burgers' equation $$\partial_t u(x,t) + u(x,t) \partial_x ...
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0answers
26 views

3 step ODE method using interpolating polynomial

I am trying to find a 3-step method of the form $y_{i+1}=y_{i-2}+b_0f_{i-2}+b_1f_{i-1}+b_2f_i$ to solve the ODE $y'=f(t,y)$ by using an interpolating polynomial and then finding $\int_0^{3h} ...
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1answer
14 views

Galerkin Method: Why Set the Residuals to Zero?

I don't understand why the Galerkin method weighs the residual by the shape functions and sets it equal to zero. I'd like to know the reason why. Any intuitive explanation would be greatly ...
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1answer
57 views

Two dimensional Numerical integration

If I am numerically integrating my function $f(x,y)$ on a two dimensional cartesian grid, say $[0,1]\times[0,1]$ with $\Delta x=\Delta y$ using the values at the center of each cell, what is the order ...
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1answer
25 views

Discretization of differential equations, solving boundary value problem

For $\Omega = (0,1)^2 \subseteq \mathbb R^2$ and $f \in C(\Omega)$ consider the boundary value problem: $- \Delta u(x,y) = f(x,y)~ \forall (x,y) \in \Omega,~ u(x,y)=0 ~ \forall (x,y) \in \partial ...
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0answers
26 views

$f\in\mathcal{C([a,b])}^{m+1}$root multiplicty $m\ge2$ in $x_*$ Newton method linear convergence

Let $f:[a,b]\rightarrow \mathbb{R}$, $f\in\mathcal{C([a,b])}^{m+1}$, $f$ has root multiplicty $m\ge2$ in $x_*\in(a,b)$. Thus it is: $$f(x_*)=f'(x_*)=\cdots=f^{m-1}(x_*)=0; f^{m}(x_*)\neq0$$ Proof ...
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0answers
21 views

Find all local minimums of polinomial function of two variables

I am intrigued by the task of numerically finding all local minimums of a polynomial $f(x,y) = a_1 + a_2x + a_3y + a_4xy + a_5x^2 + a_6y^2 + a_7x^2y + \ldots$ in the interval $[0,1]\times [0,1]$ ...
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1answer
52 views

Bifurcation Example Using Newton's Method

I am studying dynamical systems as part of a research project. I have been using Newton's Method and studying the dynamic properties. Does anyone know where I could find a relatively simple example ...
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0answers
20 views

Conjugate Gradient

Why sometime our conjugate gradient routine never reaches the stop condition even if the result is correct? As stop condition we use the following: $$\delta > \epsilon^2 \delta_0$$ To avoid ...
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0answers
17 views

extension of trigonometric functions as basis functions to higher dimensions

Trigonometric functions forms an orthonormal basis functions for $L^2[a,b]$, with corresponding normalization coefficients. I want to know if this result can be extended to higher dimensions. For ...
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13 views

Error terms for Composite Newton-Cotes formulas

There are four general error formulas for Newton-Cotes rules (according to closed/open formulas and odd/even points). My question is: are there four general error formulas for composite Newton-Cotes ...
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44 views

Solution of inhomogenous ODE (4th order)

Hello stackexchangers, I have an inhomogenous ODE in 4th order. This ODE is the constitutive law to describe a material by using the "Wiechert model" (p. 15) which is given by $p_0\sigma + ...
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1answer
31 views

How many fixed points does a function have?

How many fixed points does $$f(x) = \sin\left(\sin\left(\dfrac{x^3}{11} + \dfrac{x^2}{7} + \dfrac{x}{3}\right)\right) $$ have in the range $[0,1]$? How does one come to solve this? I thought ...
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2answers
32 views

Finite difference method

I wanted to ask something regarding the finite difference approximation. I used the finite difference to calculate the numerical derivatives of my function. The finite difference is given by the ...
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33 views

QR Algorithm without Shifts (Trefethen and Bau)

A real symmetric matrix $A$ has eigenvalue 1 of multiplicity 8, while all the rest of the eigenvalues are $\leq 0.1$ in absolute value. Describe an algorithm for finding an orthonormal basis of the ...
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0answers
56 views

Lyapunov function

How to do this problem? Find a Lyapunov function for $(0,0)$ in the system: $$x˙=3xy^2−11x^2$$ $$y˙=11x^3−4y^3$$ I know there is no formula for finding Lyapunov functions for a system, so how do I ...
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1answer
35 views

Distributed Newton methods for large scale problems

I am keen to know about the literature landscape for distributed convex optimization methods which use second order information like the Newton step. This is as such a less evolved area compared to ...
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0answers
9 views

Numerical Optimization non concave function

Supposed I have a function $f:R^{m\times n} \rightarrow R$ that is not concave. Suppose that for each $x \ in R^m$, the function $f(x,\cdot)$ is strictly concave. When optimizing $f$ over $R^{m\times ...
2
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1answer
111 views

Relating convergence theorem for Newton-Raphson method to Newton fractal

I have created a Newton fractal (below) using the Newton-Raphson method to find the five solutions of f = (z^5-1) The convergence theorem of Newtons method say ...
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0answers
32 views

A-stability of Runge-Kutta methods

I am studying Runge-Kutta methods, but I can't understand why explicit Runge-Kutta methods are not A-stable. Someone can explain it to me?
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1answer
44 views

Higher order numerical PDE schemes near boundaries, implementation in MATLAB

Followup to my previous question. The first order scheme proved unstable for my pde: $$f_t + A y f_x - B x f_y =0$$ So I'm looking to implement a higher order scheme (using these tables). I was ...
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0answers
17 views

Estimate accuracy of inaccurate fast function having exact values of slow one

Let’s say we have functions $F$ and $H$ to calculate a series $S$ of integers and that: $S_{i} = H(x_{i}) = F(x_{i}) + e_{i}$ Being $e_{i}$ the error of $F(x_{i})$ to estimate $S_{i}$ The problem ...
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2answers
57 views

Looking for numerical methods for finding roots of convex vector function ${\bf f}({\bf x})={\bf 0}$

Consider the function ${\bf f}:\mathbb{R}^n\to\mathbb{R}^m$ defined as ${\bf f} = (f_1,f_2,\ldots,f_m)$ where each $f_i:\mathbb{R}^n\to\mathbb{R}$ is twice-continuously differentiable convex in ${\bf ...
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1answer
32 views

Understanding what exactly an upper bound on an error is in numerical analysis

I think the hardest part of numerical analysis for me is understand what constitutes an "upper bound", and this has caused me alot of strife because often times my answer differs from the book, but ...
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0answers
61 views

calculate the second derivative using `ode45`

I have a second order differential equation. I am using ode45 to solve the problem. ode45 converts the equations to the first ...
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0answers
18 views

Derivation of the Duckworth-Lewis method G50 table

I have got a question about the way the Duckworth-Lewis method G50 table is derived. The table is the following: So, how are the percentages inside the table calculated? Thanks!
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0answers
26 views

What does instability mean and examples, boundary condition

The Upwind-Scheme for the numerical solution of first order PDE's (homogenous case) of the form $u_t + cu_x = 0$ is given by $$ u_j^{n+1} = \left\{ \begin{array}{ll} u_j^n - \frac{c\Delta t}{\Delta ...
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0answers
23 views

Best way to fit an equation for the given graph

I have 450 pair $(x,y)$ of data. The plot is like this: I need to fit an equation: $y=f(x)$ for the given data, and to find out values of $y$ when $x=500$. Now, my question is: What kind of ...
3
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2answers
42 views

Algorithm to solve the system $\sum_{i=1}^nx_i^k = k!c_k$, $k=1,2,\ldots,n$ efficiently

$$ x_1 + x_2 + \cdots +x_n = c_1 $$ $$ \frac{x_1^2}{2} + \frac{x_2^2}{2} + \cdots +\frac{x_n^2}{2} = c_2 $$ $$ \vdots $$ $$ \frac{x_1^n}{n!} + \frac{x_2^n}{n!} + \cdots +\frac{x_n^n}{n!} = c_n $$ ...
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1answer
30 views

A question on a error bound with trigonometric functions

I have a link to a paper on a solution below http://math.berkeley.edu/~zworski/128/psol07.pdf This is related to my other question on the same problem. For problem 7, the author achieves a second ...
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2answers
36 views

Notation laplace operator squared $\Delta^2$

I have the following expression (in a numerical context) $$\Delta_h u(x) = \Delta u(x) + \frac{h^2}{12} \Delta^2 u(x) + O(h^4)$$ The $\Delta$ is the Laplace operator so $\Delta u = u_{xx}+u_{yy}$. ...
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1answer
21 views

Confusion about trigonometric error bounds in numerical analysis

I have a link to a paper on a solution below http://math.berkeley.edu/~zworski/128/psol07.pdf For problem 7, the author of the paper does something like so: $$f''(\xi) = -5e^{2\xi}sin3\xi + ...