Gaussian Quadrature September 25, 2011 Interpolation Approximation - - PowerPoint PPT Presentation

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Interpolation Gaussian Quadrature September 25, 2011 Interpolation Approximation of integrals Approximation of integrals by quadrature Many definite integrals cannot be computed in closed form, and must be approximated numerically.

SLIDE 1

Interpolation

Gaussian Quadrature

September 25, 2011

SLIDE 2

Interpolation Approximation of integrals

Approximation of integrals by quadrature

◮ Many definite integrals cannot be computed in closed form,

and must be approximated numerically.

◮ Basic building block

1

−1

u(ξ)w(ξ) dξ ≈

N

u(ξi)ωi (1)

◮ ξi are the quadrature points ◮ ωi are the quadrature weights ◮ w(ξ) > 0 is a weight function

◮ Integration can be done with function evaluations at ξi ◮ How do I pick ξi and ωi to minimum errors and/or maximize

efficiency

SLIDE 3

Interpolation Approximation of integrals

Example Quadrature: Trapezoidal integration

◮ w = 1 ◮ Quadrature points are equally spaced:ξi = 2i N − 1 ◮ Quadrature weights are

ωi =

2N

for i = 0, N

2 N

for 1 ≤ i < N (2)

◮ Formula takes the form, with ui = u(ξi),

1

−1

u(ξ) dξ ≈ 2 N u0 2 + u1 + · · · + uN−1 + uN 2

◮ Can we do better?

SLIDE 4

Interpolation Approximation of integrals

Possible route

◮ Choose ξi and vary ωi to maximize order of exact

integration.

◮ I can impose (N + 1) constraints on ωi, e.g. maximize the

degree of polynomials that quadrature approximates exactly:

ξk

i ωk =

1

−1

ξk dξ, 0 ≤ k ≤ N (4)

◮ Gauss quadrature adjust ξi so that formula above applies

for k > (N + 1). Relies on properties of orthogonal polynomials.

SLIDE 5

Interpolation Approximation of integrals

Gauss quadrature

Let pm(x) be the set of orthogonal polynomials on the interval a ≤ x ≤ b w.r.t. weight function w(x) and of degree m b

a

pm(x)pn(x)w(x) dx = δm,npm2 (5) Let the collocation points be the roots of pN+1(xi) = 0 and the weights:

xk

i ωk =

b

a

xkw(x) dx, 0 ≤ k ≤ N (6) then...

SLIDE 6

Interpolation Approximation of integrals

Gauss quadrature

◮ The weights are all positive ωi > 0. ◮ Gauss quadrature is exact for all polynomials, q, of degree

less or equal to 2N + 1

q(xi)ωk = b

a

q(x)w(x) dx (7)

◮ It is not possible to find a xi, ωi combination where the

integration is exact for polynomials of degree 2N + 2.

SLIDE 7

Interpolation Approximation of integrals

Gauss quadrature flavors

Gauss quadrature comes in different flavors depending on whether the quadrature points include none, one, or both end points.

◮ Gauss quadrature: all quadrature points are strictly inside

the interval: a < xi < b

◮ Gauss-Lobatto quadrature: x0 = a and xN = b.

Useful for imposing BC as we will see later.

◮ Gauss-Radau quadrature: x0 = a or xN = b but not both.

Useful for integrals or PDEs on semi-infinite intervals

SLIDE 8

Interpolation Approximation of integrals

Gauss-Lobatto quadrature

◮ Formula is exact for polynomials of degree ≤ 2N − 1. ◮ For Jacobi type polynomials, quadrature points are the

roots of (1 − x2)p′

N(x) = 0

Jacobi polynomials include Chebyshev, Legendre and associated Legendre polynomials.

SLIDE 9

Interpolation Examples

Example: Chebyshev Gauss Quadrature

◮ Roots are known analytically since TN+1(x) = cos(N + 1)θ,

with x = cos θ.

◮ The roots are then

ξi = cos θi = (2i + 1)π 2(N + 1) , 0 ≤ i ≤ N (8)

◮ Quadrature weights are

ωi = π N + 1 (9)

◮ The formula is exact for polynomial of degree 2N + 1:

1

−1

q(x) √ 1 − x2 dx =

N

q(xi)ωi (10)

SLIDE 10

Interpolation Examples

Example: Chebyshev Gauss-Lobatto Quadrature

◮ Roots are known analytically since T ′ N(x) = N sin Nθ sin θ ◮ The roots of (1 − x2)p′ N(x) = sin θ sin Nθ

ξi = iπ N , 0 ≤ i ≤ N (11)

◮ Quadrature weights are

ωi = π 2N for i = 0, N, and ωi = π N , 1 ≤ i < N (12)

◮ The formula is exact for polynomial of degree 2N − 1:

1

−1

q(x) √ 1 − x2 dx =

N

q(xi)ωi (13)

SLIDE 11

Interpolation Examples

Example: Legendre Gauss Quadrature

◮ Roots must be computed numerically LN+1(xi) = 0 ◮ Quadrature weights are

ωi = 2 (1 − xi)2L′

N+1(xi)

(14)

◮ The formula is exact for polynomial of degree 2N + 1:

1

−1

q(x) dx =

N

q(xi)ωi (15)

SLIDE 12

Interpolation Examples

Example: Legendre Gauss-Lobatto Quadrature

◮ Roots must be computed numerically (1 − x2)L′ N(x) = 0 ◮ Quadrature weights are

ωi = 2 N(N + 1)[LN(ξ)]2 (16)

◮ The formula is exact for polynomial of degree 2N − 1:

1

−1

q(x) dx =

N

q(xi)ωi (17)

SLIDE 13

Interpolation Some Math Proofs

Some mathematical proofs for Gaussian Quadrature

◮ How do we know that the orthogonal polynomial pN+1 has

N + 1 roots?

◮ How can we prove that formula is exact for polynomials of

degree 2N + 1?

◮ Starting point is that pm is the set of orthogonal

polynomials: b

a

pm(x) pn(x) w(x) dx = δm,npm2 (18)

SLIDE 14

Interpolation Some Math Proofs

Multiplicity and location of roots

◮ Assume pN has only k roots in [a, b]: pN can be written as

pN = (x − r1)(x − r2) . . . (x − rk)h(x) (19) where hk is a polynomial of degree N − k that must be single-signed in [a, b]. If it changes sign, then there must be another root rk+1

◮ Consider the following polynomial:

z(x) = (x − r1)(x − r2) . . . (x − rk)pN(x) (20) z(x) = [(x − r1)(x − r2) . . . (x − rk)]2 h(x) (21)

◮ z(x) must also be single-signed.

SLIDE 15

Interpolation Some Math Proofs

Multiplicity and location of roots

pN is orthogonal to all polynomials of degree k < N then b

a

(x − r1)(x − r2) . . . (x − rk)

polynomial of degree k

pN(x)w(x) dx = δN,k b

a

[(x − r1)(x − r2) . . . (x − rk)]2 h(x)w(x) dx = δN,k Since z is single signed this is possible only if k = N, then pN has N roots in [a, b] Since pN is of degree N and has N roots, the roots must be isolated. Likewise p′

N(x) must have N − 1 roots corresponding to the

extrema of pN(x).

SLIDE 16

Interpolation Some Math Proofs

Exactness of quadrature for polynomials

Let f(x) be a polynomial of degree 2N + 1, it can always be written as f(x) = pN+1(x)q(x) + r(x) (22) where q(x), and r(x) are polynomials of degree at most N. b