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

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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0answers
55 views

Algorithm to numerically solve this system of three polynomial equations of degree $6$ [closed]

Mathematica Nsolve gave all $6$ solutions without an initial guess, whereas sympy Nsolve gave one solution closest to the ...
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1answer
47 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
19 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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23 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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18 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
49 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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19 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
13 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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11 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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41 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
30 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
28 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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0answers
30 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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42 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
33 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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7 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
99 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
31 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?
2
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1answer
39 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
14 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
49 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
30 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
49 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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16 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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20 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
22 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
38 views

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

$$ x_1 + x_2 + \cdots +x_n = c_1 $$ $$ \frac{x_1^2}{2} + \frac{x_2^2}{2} + \cdots +\frac{x_n^2}{2} = c_2 $$ $$ \vdots $$ $$ \frac{x_1^n}{n!} + \frac{x_2^n}{n!} + \cdots +\frac{x_n^n}{n!} = c_n $$ ...
2
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1answer
29 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
33 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
20 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 + ...
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0answers
60 views

Magnus series expansion

In the theory of the Magnus series expansion, it can be found that $$ \Omega(t) = \int_0^t A(\tau)d\tau - \frac{1}{2}\int_0^t \left[ \int_0^\tau A(\sigma)d\sigma, A(\tau) \right]d\tau + ...
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1answer
15 views

Composition of two functions in normed spaces

Let $\Omega_1, \Omega_2 \subset \mathbb{R^n}$ be bounded. The mapping $ F: \Omega_1 \rightarrow \Omega_2 $ shall be bijective, continuously differentiable and such that $||DF(x)||$ and ...
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1answer
32 views

Quadrature obtained from Simpson's rule, and its order of error

Express $Q$ as a weighted combination of the five function values $f(a)$ through $f(e)$ and establish that its order is six. (See section 6.2.) This is from Numerical Methods by Moler, ...
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2answers
51 views

Help with Runge-Kutta method for solving systems of differential equations

I am currently doing an investigation with SIR model for predicting the progress of an infectious disease. However, I am not very much familiar with systems of differential equations,so I would need ...
2
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1answer
36 views

Help find error bound of trapezoidal quadrature

I'm having trouble finding the error bound of this function. My professor says it's "trivial" and thus, he refused to offer me any help beyond a simple hint I kind of knew anyway. I am given this ...
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0answers
16 views

diffusion equation

Kindly give me suggestions on my following assignment of Simulations in Fluid Flow: Solve the following differential equation for transport of f(x,y,z,t) by MS Excel ∂f/∂t+Ux ∂f/∂x+Uy ∂f/∂z+Uz ...
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1answer
44 views

MATLAB, 1st order 2d hyperbolic equation, problem with convergence.

Follow up to my previous question: MATLAB: solving 1st order hyperbolic equation in 2 spacial dimensions The equation I'm solving has the form: $$f_t + A y f_x - B x f_y =0$$ I wrote the following ...
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2answers
47 views

Numeric methods?

I would please like to receive advice on the following: Prove that the given equation has a real solution, then find it, numerically; $2^x-x^2=20$ I don't know how to do this...any help?
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2answers
34 views

Does Runge Kutta need future state of system?

In order to use the RK methods, you need to know the state of the system at future time-steps which can be expensive to compute (e.g., in physics simulations). As a simple example I'll use RK-2: In ...
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1answer
18 views

Properties of carry in base $b$ multiplication

Consider $n$ bit numbers $A$ and $B$. Let they be represented in base $b$. When you multiply $A$ and $B$ using school multiplication: $(1)$ how many carry propagations can one expect? $(2)$ what ...
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how to numerically solve a problem in calculus of variations?

I am asked to solve a problem in calculus of variations via numerical methods as below in 2 states: I have no idea what to do? can any one help me urgently? or offer me some tutorials, books, or ...
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1answer
97 views

Determine $p$, $q$ and $r$ so that the order of the fixed point iteration for computing $a^{1/3}$ becomes as high as possible

So I'm given the following equation for computing $a^{1/3}$ $$x_{k+1}=px_k + \frac{qa}{x_k^2} + \frac{ra^2}{x_k^5}$$ and I have to find the p, q, and r so that this equation converges to $a^{1/3}$ as ...
2
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1answer
36 views

Weed out numerical artifacts from matrix inversion

I am working with the inverses to a set of large sparse matrices (in Matlab). A key indicator for my application is the number of non-zero entries in each row, and I recently discovered that I was ...
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0answers
15 views

linear differential operator 2d, order of error h^4?

I have to show that following discretization of a linear differential operator satisfies the equation $\Delta_h u(x) = \Delta u(x) + \frac{h^2}{12} \Delta^2 u(x) + O(h^4)$ $$\Delta_h u = ...
1
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1answer
29 views

Introduction to Newtons method

I'm supposed to come up with two ways to introduce Newtons method for the approximation of zeros for highschool students. (That is the method using tangents and with the formula $ x_{n+1} = x_{n} - ...
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1answer
19 views

Nearest-neighbor interpolation

I read in a book that the nearest-neighbor interpolation results in a function whose derivative is either zero or undefined. Can anyone explain what does it mean when the derivative of a function is ...
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1answer
12 views

Approximating the value of the limit of a sequence defined recurrently

Suppose I have a sequence defined by recurrence, i.e. $x_{n+1}=f(x_n)$ for some $f:\mathbb{R}\to \mathbb{R}$, and $x_0\in \mathbb{R}$. Suppose that $f$ is $K$-Lipschitz for some $K<1$. Then $f$ has ...
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0answers
17 views

Strictly diagonally dominant matrix -LU factorization

Let $A\in\mathbb{C^{n\times n}}$ be strictly diagonally dominant. I want to show that the LU factorizations with and without partial pivoting are the same for these matrices. For start, I created ...
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0answers
21 views

Transformation for two different boundary functions in Stefan problem

Peace be upon on all of you, I have one-dimensional Stefan problem. Let say we have two boundary conditions of $u(t,s_{1}(t))=g_{1}(t)$ and $u(t,s_{2}(t))=g_{2}(t)$, where $u$ is temperature, $t$ is ...