# Questions tagged [numerical-calculus]

This tag is for various question on numerical calculus / numerical analysis which concerned with all aspects of the numerical solution of a problem, from the theoretical development and understanding of numerical methods to their practical implementation as reliable and efficient computer programs.

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### Zeros of Legendre Polynomial and Quadrature rule

Comsider the quadrature rule $I_{n}(f)=w_{1}f(x_{1})+w_{2}f(x_{2})+\ldots +w_{n}f(x_{n})$ for approximating $I(f)=\int_{-1}^{1}f(x)dx$ a)Suppose $x_{1},x_{2},\ldots x_{n}$ are the zeros of the ...
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### given $x_0,...,x_{n}\in\mathbb{R}$ and $y_0,...,y_n\in\mathbb{R_{+}}$ then there exists a nonnegative polynomial with deg $\leq 2n$ s.t. $p(x_i)=y_i$

I have the following question: let $n \in \mathbb{N}$ and $x_0,x_1,\dots,x_n \in \mathbb{R}$ different points and $y_0,\dots,y_n \in \mathbb{R}$ nonnegative points show that there exists non-negative ...
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Consider a numerical integration rule of the form $\int_{0}^{1} f(t)dt \approx af(0)+bf(c)$ a)Find a,b,c such that this quadrature rule has highest order of precision. b)The quadrature rule above has ...
1 vote
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### Stability of multistep methods that contain second derivative

Consider the numerical method $y_{n+1}=y_{n}+\frac{h}{2}[\frac{dy_{n}}{dt}+\frac{dy_{n+1}}{dt}]+\frac{h^{2}}{12}[\frac{d^{2}y_{n}}{dt^{2}}-\frac{d^{2}y_{n+1}}{dt^{2}}]$ where $\frac{d}{dt}$ is the ...
1 vote
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### Local approximation of geodesic distance

Consider a surface $S\subset \mathbb{R}^3$ which is the graph of a function $Z(x,y)$ and two points $p,q\in S$ which are "very close to each other". Is there a simple (and computationally ...
• 381
553 views

### Numerically compute and clear divergence of discrete vector field

I have a fluid simulation that represents velocity as a vector field in a grid of cells. The cells all have the same width and the same height, but the height is not necessarily equal to the width. I ...
• 409
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### Numerical Inverse of an integral

Let $f: \mathbb{R} \to (0, \infty)$ be a continuous function. Define $F: \mathbb{R} \to (0, \infty)$ by: \begin{equation*} F(x) = \int_{-\infty}^x f(t)dt \end{equation*} Clearly, $F$ is strictly ...
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1 vote
56 views

### Double integrals: strange accuracy behavior

In double integrals: when x-y region is rectangular as in $$\int_0^3\int_0^3(x^2+y^2) \, dy \, dx$$ accuracy of Simpson 1/3 rule is better than that of Trapezoidal rule which in turn is better than ...
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