# Tagged Questions

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### Discrete Poincare Inequality

For the sake of this question, let $\Omega \subset \mathbb{R^2}$ be a regular domain. In variational problems involving the Sobolev space $W^{1,1}(\Omega)$ (or $BV(\Omega)$) one often uses the Sobolev ...
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### Numerical analysis- Runge Kutta

I have: $$y'(x)= \sin(y); y(0)=1$$ I need to calculate the function values with Runge-Kutta. The range is [0,1]. My problem is that I need to choose the h (=dx) such that the error will be in order ...
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Given $f: [a,b] \to R$ and $K: [a,b]$ x $[a,b]$ $\to R$, we want to find a solution $\varphi:[a,b] \to R$ to the Fredholm integral equation: $$\varphi(x) = f(x)+\int _{ a }^{ b }{ K( x,t)\varphi ... 2answers 44 views ### Firstly what is an O(h^3) formula? Also I am not quite sure how to answer the question? The forward-difference formula can be expressed as$$f'(x_0)=\frac{1}{h}(f(x_0 +h)- f(x_0))-\frac{h}{2}f''(x_0) - \frac{h^2}{6}f'''(x_0) + O(h^3).Use Richardson's extrapolation to derive an ... 1answer 50 views ### Checking if the Hessian is the derivative of the gradient Suppose f: \Bbb R^n \to \Bbb R. I have a code that computes the gradient of f. I have another code that computes the Hessian of f times a vector. Now I want to check if they are correct. ... 1answer 9 views ### Change of variables from intinite to bounded support. I may be missing something simple, but I am stuck. My question: I am solving a system of partial differential equations numerically, but one of the variables can take on any value, ie x \in ... 3answers 101 views ### Intuitive Numerical Analysis Texts Steven Strogatz has a great informal textbook on Nonlinear Dynamics and Chaos. I have found it to be incredibly helpful to get an intuitive sense of what is going on and has been a great supplement ... 1answer 49 views ### Euler's Numerical Method Let \eta(x;h) be the approximate solution furnished by Euler's method for the initial-value problem y'=y, y(0)=1. I proved that: i) \eta(x;h)=(1+h)^{x/h}; ii) \eta(x;h) has the expansion ... 0answers 32 views ### Sequence and intermediate value theorem For part (a), By intermediate value theorem, there exist c between 0 and 1 such that f(c) = 0 Now, I supposed that there also exist d between 0 and 1 such that  f(d) = 0  and  c \neq d  I am ... 1answer 219 views ### Solve non-linear equations of 3 variables using Newton-Raphson Method iterms of c,s and q. The three non-linear equations are given by $$c[(6.7 * 10^8) + (1.2 * 10^8)s+(1-q)(2.6*10^8)]-0.00114532=0$$ s[2.001 *c + 835(1-q)]-2.001*c =0 ... 1answer 32 views ### How to write continued fraction as polynomial? I have \begin{align} r(x)= 1 + \frac{x}{\frac{1}{2}+\frac{x-1}{-1+\frac{x+1}{1+\frac{x-1}{-1}}}} \end{align} for an interpolation problem, and I need to write r(x) such that nominator and ... 1answer 48 views ### Quadrature Rule “is exact for polynomials of degree n” Could someone kindly explain what "a quadrature rule is exact for polynomials of degree n" means? Here is what I understand about numerical (Newton-Cotes) quadrature rules: Suppose we want to ... 1answer 20 views ### Calculating the limit of an analytic function which gives a log answer. I am trying to read through a paper and have gotten stuck at the following calculation several times. I've left if for a few days and came back to try it again four different times, but still no luck. ... 1answer 51 views ### Show that the iteration x_{n+1} = x_n - 2\frac{f(x_n)}{f'(x_n)} converges quadratically to x_* provided x_0 is sufficiently close to x_* We have the following conditions for the above slightly-modified Newton's method iteration: f is a real function of one real variable f'' is Lipschitz continuous f(x_*) = f'(x_*) = 0 I also ... 0answers 35 views ### how to prove this curious identity with the Chebyshev polinomials we defined the Tm like this (where Tm are the Chebyshev polinomials) Then I showed this: And now I have no idea how to proove this: I also have to make the remark that I also proved that the ... 0answers 29 views ### Hermite interpolation with interior points I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ... 1answer 55 views ### Find the rate of convergence? Given is the iteration x_{k+1}=\frac{1}{11}(1-\cos(x_{k})) with x_{0}\in (-\frac{\pi }{2},\frac{\pi }{2}) without 0. Check if the sequence converges to x^{*}=0 and find its convergence rate. ... 1answer 100 views ### Gaussian Quadrature -Deriving a Formula- eThe following is an exercise in the problem section of the Gaussian Quadrature chapter. The theorem: Derive a formula of the form\int_{a}^{b} f(x)dx \approx w_0f(x_0) + w_1f(x_1) + w_2f'(x_2) ...
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For the Gaussian Quadrature Thrm., why are we letting$c_0$ and $c_1$ be $0$ in the example in the picture?
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### 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 ...
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### Method of undetermined coefficients for finite difference approximations

I'm reading over my text, and the first mention of deriving the coefficients states: "Suppose we want a one-sided approximation to $u'(x)$ based on $u(x), u(x-h)$, and $u(x-2h)$ of the form: ...
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### Question on numerical quadrature and precision

I am studying numerical analysis and I came across this question: Let $P_{n+1}(x)\in\Pi_{n+1}$ be orthogonal to $\Pi_n$ relative to a weight function $w(x)\geq 0$ on $[a,b].$ Denote by ...
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### Richardson's Extrapolation

Use Richardson's extrapolation to find a 3 point 2nd order approximation of f '(x). I'm not sure how to go about to start this, i'm not the best when using richardson's extrapolation.
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