Exercise · ZW-GAUSSIAN-QUADRATURE-002 · ★☆☆ Basic
What are the nodes \(x_i\) of an \(n\)-point Gaussian quadrature?
Choose one option.
Show answer
- D. Roots of the orthogonal polynomial of degree \(n\)
Explanation
The nodes are roots of an orthogonal polynomial of degree \(n\), not fixed points of the interval. Equally spaced points belong to Newton-Cotes rules.
Review the concept: Gaussian quadrature
Numerical integration method that chooses optimal nodes \(x_i\) and weights \(w_i\) to approximate an integral with the highest possible accuracy for a fixed number of function evaluations.
\[\int_{a}^{b} f(x)\,dx \approx \sum_{i=1}^{n} w_i\, f(x_i)\]
More practice problems: gaussian quadrature
- ZW-GAUSSIAN-QUADRATURE-001 What is the name of the integral approximation that chooses nodes x_i and weights w_i to gain accur…
- ZW-GAUSSIAN-QUADRATURE-003 In the formula ∑ w_i f(x_i), what do the w_i represent?
- ZW-GAUSSIAN-QUADRATURE-004 On which standard interval are the Gauss-Legendre nodes defined?
- ZW-GAUSSIAN-QUADRATURE-005 Which family of orthogonal polynomials generates the Gauss-Legendre nodes?
- ZW-GAUSSIAN-QUADRATURE-006 Apply the 2-point rule (nodes pm 1/√3, weights 1) to ∫_-1^1 x^2 dx. What do you get?
- ZW-GAUSSIAN-QUADRATURE-007 To map [a,b] onto [-1,1], which change of variable is used?
- ZW-GAUSSIAN-QUADRATURE-008 If the n-point rule is exact up to degree 2n-1, up to what degree is it exact with n=3?
- ZW-GAUSSIAN-QUADRATURE-009 Which statement about the 2-node Gaussian quadrature on [-1,1] is false?