# Tagged Questions

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### Gaussian Quadrature Confusion

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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I'm trying to understand this exercise: Well, my teacher told me that I need to suppose $q$ has $k<n$ different roots in $(a,b)$. So we have the roots $y_1,...,y_k$ of $q$. Then if I set $p(x)= ... 0answers 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 â€‹â€‹of the function$ f $is ill ... 1answer 58 views ###$a + b = a$in machine precision [closed] I have the following statement: "If$a + b = a$, then$b = 0$" may not true with the floating point operations. Actually, if$|y| â€Ž< (\varepsilon / B) |x|$, then$fl(x+y) = x$, where ... 1answer 73 views ### Positive real number has a finite number of binary when is in form$ m/2^n $Prove that positive real number$ ( x \in \mathbb{R} \ x > 0) $has a finite number of binary if and only if when is in form$ \frac{m}{2^n} $, where$ m, n \in \mathbb{N} $I found this solution: ... 1answer 97 views ### contraction point? This is an interesting question I saw in a book online: Suppose that$g:\mathbb R \to $$\mathbb R is a contraction. Then g has a unique fixed point c and that for any number x_0, the sequence ... 1answer 54 views ### Convergence and Constant sequence? Suppose that g:\mathbb R \to$$\mathbb R$is a contraction. Then$g$has a unique fixed point$c$and that for any number$x_0$, the sequence$x_0 x_1 x_2...$given by$x_n = g(x_{n-1})$. converges ... 1answer 191 views ### Prove that$ \ln[e(2/e)] $is a fast way to calculate$ \ln2 $Consider formula$ (*) \ln(x) = \sum_{k=1}^{\infty} (-1)^{k-1}\cdot \frac{(x-1)^k}{k}$. If you calculate$ \ln2 $with error less then$ \frac{1}{2} \cdot 10^{-6} $we need more than two milion ... 0answers 36 views ### What are the weights of the quadrate formula with weight function$x\mapsto (1-x^2)^{-1/2}$I'm trying to solve this numerical analysis exercise: I was able to prove everything until the part marked in red. I think I need to use this: So we get an exact result for$T_0$: ... 1answer 53 views ### Show that$\varphi_{j+1}(x)-C_j x \varphi_j (x) = \sum_{k=0}^j \alpha_{jk} \varphi_k (x)$where$\{\varphi_j \}$is a syst. of orth. polynom. This is a homework exercise. I'm only asking for hints, please don't give a full solution. This is the exercise: This is my attempt to solve this problem: If$C_j$is chosen to be equal to the ... 1answer 33 views ### Taking the derivative of$n$products I'm reading my numerical analysis book, but I don't understand this step: I'm assuming that this$l'$must be$l_0'$, as there is no$l$defined anywhere. If you want, you can read the text above ... 0answers 72 views ### Small symbols behind parantheses I am currently reading the following paper where the author uses constantly small symbols after parantheses, but I do not know what this means. I am particularly interested in equation (23), so you ... 2answers 138 views ### Solving$v_{t}+v(x,t)v_{x}=0$with initial condition This problem comes from an undergraduate course in PDE. The first question of the problem was to solve the following PDE:$v_{t}+v(x,t)v_{x}=0$with the following initial condition:$v(x,0)=5x\$ ...
We know that solutions exist for equations of the following variety: $$ye^y=x \iff y=W(x)$$ Where W is the Lambert W function. We can augment the problem slightly, and ask if there exist solutions ...