3) Integration and Quadrature

Question 1

What is the Trapezium Rule

Answer 1

Question 2

What is the error estimate for the Trapezium Rule

Answer 2

Question 3

What is a Quadrature Rule

Answer 3

Question 4

Give an example of a Qudrature Rule

Answer 4

The Trapezium Rule

Question 5

Show that quadrature rules of the form
I(f) = ∑ wk f(xk) are linear

Answer 5

Question 6

What is Simpson’s Rule

Answer 6

Question 7

What is a Newton-Cotes scheme of order n

Answer 7

• A type of quadrature rule that approximates the integral of a function using Lagrange basis polynomials.
• It constructs weights based on equally spaced nodes within the interval [a,b]

Question 8

Give an example of a Newton-Cortes Scheme

Answer 8

Simpson’s Rule is a Newton-Cotes scheme of order 2

Question 9

What is the Error Estimate of Simson’s Rule

Answer 9

Question 10

What is the Composite Trapezium Rule

Answer 10

Question 11

What is the bound of the absolute error for the composite trapezium rule

Answer 11

Question 12

What is the Composite Simpson’s Rule

Answer 12

Question 13

What is the bound of the absolute error for the Composite Simpson’s Rule

Answer 13

Question 14

What is the error bound of the four integration schemes

Answer 14

Question 15

What determines the degree of precision of a quadrature rule

Answer 15

A quadrature rule I(f) has a degree of precision k if it can exactly evaluate integrals of polynomials of degree up to k, but fails for a polynomial of degree k+1

Question 16

What is the degree of precision of the Trazpezium Rule, Simpson’s Rule and Newton’s Cortes

Answer 16

• The Trapezium rule has degree of precision 1
• Simpson’s rule has degree of precision 3
• In general, Newton-Cotes quadrature of degree n has degree of precision n, if n is odd, and n +1 if n is even