MATH 5520
Numerical Integration 2.
Dmitriy Leykekhman
Spring 2009
IGauss Quadrature.
IComposite Quadrature Formulas.
IMATLAB’s Functions.
D. Leykekhman - MATH 5520 Finite Element Methods 1 Numerical Integration 2 – 1
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Gauss Quadrature and Composite Quadrature Formulas in Numerical Integration - Prof. Dmitri, Study notes of Mathematics
Gauss quadrature, a method for numerical integration that chooses nodes and weights to make the formula exact for a polynomial of maximum degree. The document also covers composite quadrature formulas, which approximate integrals by approximating each integral with a low degree quadrature formula. Examples and explanations of the gauss-legendre quadrature formulas, composite midpoint rule, composite trapezoidal rule, and composite simpson's rule. Matlab's functions for numerical integration, quad and trapz, are also mentioned.
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MATH 5520
Numerical Integration 2.
Dmitriy Leykekhman
Spring 2009
I (^) Gauss Quadrature.
I (^) Composite Quadrature Formulas.
I (^) MATLAB’s Functions.
Gauss Quadrature.
I (^) Idea of the Gauss Quadrature is to choose nodes x 0 ,... , xn and the weights w 0 ,... , wn such that the formula ∫ (^) b
a
p(x) dx ≈
∑^ n
i=
wip(xi).
is exact for a polynomial of maximum degree.
I Lemma
There is no choice of nodes x 0 ,... , xn and weights w 0 ,... , wn such that ∫ (^) b
a
pN (x) dx ≈
∑^ n
i=
wipN (xi).
for all polynomials pN of degree less or equal to N if N > 2 n + 1. I (^) The above lemma give an upper bound on the maximum degree.
Gauss Quadrature.
Example
Let’s force the formula to be exact for 1 , x, x^2 , and x^3. This gives us
w 1 + w 2 =
− 1
1 dx = 2
w 1 x 1 + w 2 x 2 =
− 1
x dx = 0
w 1 x^21 + w 2 x^22 =
− 1
x^2 dx =
w 1 x^31 + w 2 x^32 =
− 1
x^3 dx = 0.
a nonlinear system of 4 equations with for unknowns. Usually we need a nonlinear solver to solve nonlinear systems, but in this example we can solve it analytically to obtain
w 1 = w 2 = 1, x 1 = −
, x 2 =
Table of Gauss-Legendre Quadrature Formulas.
The weights and nodes for the first 3 Gauss-Legendre formulas on [− 1 , 1]. xi wi exact for pN ,
− √^13 , √^13 1,1 N = 3
3 5 ,^0 ,
3 5
5 9 ,^
8 9 ,^
5 9 N^ = 5
-0. -0.
N = 7
Composite Midpoint Rule.
Example
∫ (^) b
a
f (x)dx ≈
n∑− 1
i=
(xi+1 − xi)f
xi+1 + xi 2
Composite Trapezoidal Rule.
Example
∫ (^) b
a
f (x)dx ≈
n∑− 1
i=
xi+1 − xi 2
(f (xi+1) + f (xi))
The function values f (x 1 ), f (x 2 ),... , f (xn− 1 ) appear twice in the summation. This has to be utilized in the implementation of the composite Trapezoidal rule: ∫ (^) b
a
f (x)dx ≈ x 1 − x 0 2
f (x 0 )
n∑− 1
i=
xi − xi− 1 2
xi+1 − xi 2
f (xi)
xn − xn− 1 2
f (xn)
MATLAB’s quad, trapz
MATLAB has several build in functions for numerical integration. We will mention a couple quad and trapz. You can get more information by typing
help quad help trapz
MATLAB’s quad
The syntax for quad
QUAD Numerically evaluate integral, adaptive Simpson quadrature.
Q = QUAD(FUN,A,B) tries to approximate the integral of scalar-valued function FUN from A to B to within an error of 1.e-6 using recursive adaptive Simpson quadrature.
FUN is a function handle.
The function Y=FUN(X) should accept a vector argument X and return a vector result Y, the integrand evaluated at each element of X.