grandes-ecoles 2021 Q3

grandes-ecoles · France · centrale-maths2__pc Numerical integration Quadrature Formula Construction and Order Determination

In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.
Determine the coefficients $\lambda_0, \lambda_1, \lambda_2$ so that the formula $I_2(f) = \lambda_0 f(0) + \lambda_1 f(1/2) + \lambda_2 f(1)$ is exact on $\mathbb{R}_2[X]$. Is this quadrature formula of order 2?

In the case $I = [0,1]$ and $\forall x \in I, w(x) = 1$, we seek to approximate $\int_0^1 f(x)\,\mathrm{d}x$ when $f$ is a continuous function from $[0,1]$ to $\mathbb{R}$.

Determine the coefficients $\lambda_0, \lambda_1, \lambda_2$ so that the formula $I_2(f) = \lambda_0 f(0) + \lambda_1 f(1/2) + \lambda_2 f(1)$ is exact on $\mathbb{R}_2[X]$. Is this quadrature formula of order 2?

Paper Questions

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