2) Numerical Integration Flashcards
(20 cards)
What is a quadrature rule in numerical integration
In a quadrature rule, what are the weights and nodes, and how do we distinguish between open and closed rules
What does it mean for a quadrature rule to have degree of precision d
How can interpolatory quadrature rules be constructed to guarantee a degree of precision at least n, and how are the weights determined
What is the midpoint rule in numerical integration, and what is its degree of precision
What are Newton–Cotes rules
Newton–Cotes rules are interpolatory quadrature rules based on equally spaced nodes: xk =a+kh,where h= (b−a)/n
What is the Newton–Cotes rule for n=1, and its degree of precision
What is Simpson’s rule and its degree of precision
What is the degree of precision of a quadrature rule based on the zeros of orthogonal polynomials
What are key properties of Gauss quadrature rules
- The nodes xk (zeros of Φn+1) are real, distinct, and lie in (a,b)
- Gauss rules are open (nodes lie strictly inside (a,b))
- All weights are positive, which makes the rule less sensitive to errors in f
- In general, the nodes and weights are irrational and often lack explicit formulas
What is the Gauss–Chebyshev quadrature rule for w(x)=(1−x^2)^−1/2 on [−1,1]
What is the idea behind composite quadrature rules
What is the composite midpoint rule
What is the error of the composite midpoint rule
What is the composite trapezium rule
What is the error of the composite trapezium rule
How can extrapolation improve quadrature approximations, and how does it lead to Simpson’s rule
What is the Romberg scheme, and how does it improve numerical integration accuracy
Romberg scheme improves accuracy by recursively eliminating error terms using extrapolation, based on the trapezium rule.
What is an adaptive quadrature algorithm
- Adaptive quadrature chooses where to sample based on the behavior of f(x), refining more in complex regions
- It uses two rules (like Simpson’s and Trapezoidal) to estimate error.
- The difference ∣R1−R2 ∣ tells us if we’re accurate enough. If not, split the interval and try again
- The algorithm ends when the total error is within the desired tolerance
Why can we use ∣R1 − R2 ∣ as our tolerence in adapative quadrature
