Numerical Integration - Numerical Methods in Engineering with MATLAB - page 204

Recursive Trapezoidal Rule

Let I k be the integralevaluatedwith the composite trapezoidal rule using 2 k − 1 panels.
Note that if k is increasedbyone, the number of panels is doubled. Using the notation

H = b − a

= ,

weobtain fromEq. (6.5) the following results for k 1 2 and 3.

k = 1 (1 panel):

f ( b )] H
2

I 1 = [ f ( a ) + (6.8)

k = 2 (2 panels):

f ( a )

2 f a f ( b ) H

4

f a

H
2

H
2

1
2 I 1 +

H
2

I 2 = + + + = +

k = 3 (4 panels):

f ( a )

2 f a 2 f a 2 f a f ( b ) H

8

H
4

H
2

3 H
4

I 3 =

+ + + + + + +

f a

f a

H
4

1
2 I 2 +

H
4

3 H
4

= + + +

Wecan now see thatfor arbitrary k > 1wehave

f a

, k

2 k − 2

1
2 I k − 1 +

H
2 k − 1

(2 i 1) H

2 k − 1

−

I k = + = 2 , 3 ,... (6.9a)

i = 1

which is the recursive trapezoidal rule . Observethat the summation containsonly
the newnodes that werecreatedwhen the number of panels was doubled. Therefore,
the computation of the sequence I 1 ,

I k fromEqs. (6.8) and (6.9) involves the
same amountofalgebra as the calculation of I k directly fromEq. (6.5). The advantage
of using the recursivetrapezoidal rule isthat it allows ustomonitor convergence and
terminate the process when the difference between I k − 1 and I k becomes sufficiently
small.Aform of Eq. (6.9a)that iseasier to rememberis

I 2 , I 3 ,...,

h f ( x new )

1
2 I (2 h )

I ( h ) = + (6.9b)

where h = H / ( n − 1) is the width of each panel.

trapezoid

The function trapezoid computes I ( h ), given I (2 h )fromEqs. (6.8) and (6.9).We
can compute a

= , ,...

f ( x ) dx by calling trapezoid repeatedlywith k 1 2 until the

desiredprecisionis attained.