Numerical methods for definite integrals

Question 1

How do you approximate an integral ?

Answer 1

Find v values by integrating x^j-1 between the given interval

Produce a vadermole matrix with nodes, u.

Use V^T and V to find W values.

Î(f)=w₁f(u₁)+w₂f(u₂)+w₃f(u₃)….

Question 2

What is a canonical interval?

Answer 2

Something like [0,1] or [-1,1]. Often is is easier to map an interval to one of these and perform the integration between these limits.

They are denoted by E

Question 3

If you have a set of Legendre or Chebyshev polynomials, how do you find appropriate nodes for an aproximation?

Answer 3

The roots of P_n

Question 4

What is an inner product?

Answer 4

Question 5

What are the rules govening polynomials used in Gauss aproximation?

Answer 5

P₀(x)=1

The degree of P_i(x)=i

The roots of P_i(x) are real and distinct

The polynomials are othogona with respect to the inner product. (the inner product of any 2 is 0)

Question 6

What is the degree of exactness of n point Gauss rules?

Answer 6

d=2n-1

Question 7

What are quadrature rules for?

Answer 7

Approximating the values of integrals.

Question 8

What is the general form of a quadature rule?

Answer 8

The w numbers are called weights and the points, x, are called nodes

Question 9

What is the algorthm to find produce a Lagrange interpolatory quadrature rule?

Answer 9

Question 10

Suppose you are given Legendre polynomials and are asked to construct an N point gauss-legrandre rule, what do you do?

Answer 10

The roots of the polynomial of degree N are what you will used for nodes.

Proced in the normal way with a vandermole matrix and weights.

Question 11

If you have produced an interpolatory quadrature rule, how do you map it to a cannonical interval?

Answer 11

Produce the map t(x)

Change the nodes with the map

The new weights, w*

w*=dt/dx w

Question 12

What are Gauss quadrature rules for?

Answer 12

They are for aproximating integrals of the form

∫μ(x)f(x) dx between a and b

μ(x) is a weight function