In [62, 2.4] the measure d λ(t)=(-t^2-sint) d t on ℝ is considered as an "exotic choice" for ill

Step 1: Understand the problemu000AWe need to evaluate the integral $I u003D u005Cint_{u005Cmathbb{R}} u005Csin^2(10

Question

In $[62, \S 2.4]$ the measure $\mathrm{d} \lambda(t)=\exp \left(-t^2-\sin t\right) \mathrm{d} t$ on $\mathbb{R}$ is considered as an "exotic choice" for illustrating the Riemann-Hilbert approach to Gaussian quadrature. The specific example given is $I=\int_{\mathbb{R}} f(t) \mathrm{d} \lambda(t)$ with $f(t)=\sin ^2 10 t$. (a) Use the SOPQ routines sr_hermite.m and sgauss.m to obtain I accurate to 24 decimal digits. (b) Use the QPQ routine modis.m to genenate the first $N$ recurrence coefficients $\alpha_k, \beta_k$, $k=0,1,2, \ldots, N-1$, for the measure $\mathrm{d} \lambda$. (c) Reproduce Fig. 5(b) of [62] showing the absolute error of n-point Gaussian quadrature, $n=1,2, \ldots, 120$, for the integral $I$.

   In $[62, \S 2.4]$ the measure $\mathrm{d} \lambda(t)=\exp \left(-t^2-\sin t\right) \mathrm{d} t$ on $\mathbb{R}$ is considered as an "exotic choice" for illustrating the Riemann-Hilbert approach to Gaussian quadrature. The specific example given is $I=\int_{\mathbb{R}} f(t) \mathrm{d} \lambda(t)$ with $f(t)=\sin ^2 10 t$.
(a) Use the SOPQ routines sr_hermite.m and sgauss.m to obtain I accurate to 24 decimal digits.
(b) Use the QPQ routine modis.m to genenate the first $N$ recurrence coefficients $\alpha_k, \beta_k$, $k=0,1,2, \ldots, N-1$, for the measure $\mathrm{d} \lambda$.
(c) Reproduce Fig. 5(b) of [62] showing the absolute error of n-point Gaussian quadrature, $n=1,2, \ldots, 120$, for the integral $I$.

Orthogonal Polynomials in MATLAB : Exercises and Solutions

Walter Gautschi 1st Edition

Chapter 2, Problem 34

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Step 1

The measure is $d\lambda(t) = \exp(-t^2-\sin t)dt$. Step 2: Part (a) - Using SOPQ routines For part (a), we need to use sr_hermite.m and sgauss.m to compute the integral accurately to 24 decimal digits. These routines are specialized orthogonal polynomial Show more…

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In $[62, \S 2.4]$ the measure $\mathrm{d} \lambda(t)=\exp \left(-t^2-\sin t\right) \mathrm{d} t$ on $\mathbb{R}$ is considered as an "exotic choice" for illustrating the Riemann-Hilbert approach to Gaussian quadrature. The specific example given is $I=\int_{\mathbb{R}} f(t) \mathrm{d} \lambda(t)$ with $f(t)=\sin ^2 10 t$. (a) Use the SOPQ routines sr_hermite.m and sgauss.m to obtain I accurate to 24 decimal digits. (b) Use the QPQ routine modis.m to genenate the first $N$ recurrence coefficients $\alpha_k, \beta_k$, $k=0,1,2, \ldots, N-1$, for the measure $\mathrm{d} \lambda$. (c) Reproduce Fig. 5(b) of [62] showing the absolute error of n-point Gaussian quadrature, $n=1,2, \ldots, 120$, for the integral $I$.

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