1 Numerical Integration

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NumericalIntegration

LagrangePolynomial

___(1)

(1) Integrate 2

Error

n = 1 x0 x1

------ (2)

(2)

n = 2 x0 x1 x2

Error O(h 4 ) Taylorseries

Integrate 2

Degreeofaccuracy precision quadratureformula n formula

x k , i = 1, 2,..., n

CompositeNumericalintegrationExample:Simpsonsruleh=2

CompositeSimpsonsruleh=1

CompositeSimpsonsruleh=1/2

Error

CompositeSimpsonsrule

RombergIntegration compositeTrapezoidal

k+1

GaussianQuadrature

x k k 2n1 ( k {1, 2,..., 2n 1} ) n1 n=2

f(x) 4 4

GussianQuadrature [1,1]

n 2 n=3 6 6 x k k 2n1 n=3 [a,b]

n=2

n 2 nodes