Exercise · ZW-GAUSSIAN-QUADRATURE-008 · ★★☆ Intermediate
If the \(n\)-point rule is exact up to degree \(2n-1\), up to what degree is it exact with \(n=3\)?
Choose one option.
Show answer
- B. \(5\)
Explanation
With \(n=3\) the degree is \(2\cdot3-1=5\). Taking \(n\) as the degree or using \(2n-2\) are common errors.
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
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- 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-009 Which statement about the 2-node Gaussian quadrature on [-1,1] is false?