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Trapezoidal Rule

Trapezoidal Rule

The trapezoidal rule approximates the integral of a function f(x) on the closed and bounded interval [a, a+h] of length h > 0 by the (signed) area of the trapezoid formed by the line segments joining (a, 0) to (a+h, 0), (a+h, 0) to (a+h, f(a+h)), (a+h, f(a+h)) to (a, f(a)) and
(a, f(a)) to (a, 0). The composite trapezoidal rule is used to approximate the integral of a function f(x) over a closed and bounded interval [a, b] where a < b, by decomposing the interval [a, b] into n > 1 subintervals of equal length h = (b - a) / n, then adding the results of applying the trapezoidal rule to each subinterval. By abuse of language both the composite trapezoidal rule and the trapezoidal rule sometimes are referred to simply as the trapezoidal rule.
Let abf( x ) dx be the integral of f(x) over the closed and bounded interval [a,b], and let Th(f) be the result of applying the trapezoidal rule with n subintervals of length h, i.e.

Th(f)=h [ f(a)/2 + f(a+h) + ··· + f(b-h) + f(b)/2 ].

The Euler-Maclaurin summation formula relates abf( x ) dx and Th(f)

Th(f) = abf( x ) dx + (h2/12) [ f'(b) - f'(a) ] - (h4/720) [ f'''(b) - f'''(a) ]
+ ··· + K h 2p-2 [f(2p-3)(b) - f(2p-3)(a) ] + O(h 2p)
,

where f', f''', and f(2p-3) are the first, third and (p-3)rd derivatives of f and K is a constant.

The last term, O(h 2p) is important. Given an infinitely differentiable function in which the first 2p-3 derivatives vanish at both endpoints of the interval of integration, it is not true that
Th(f) = abf( x ) dx but rather what the theorem says is that
limh→0 | ( Th(f) - abf( x ) dx ) / h2p | < M,
where M > 0.

If f is at least twice differentiable on the interval [a,b], then applying the mean-value theorem to
Th(f) - abf( x ) dx = (h2/12) [ f'(b) - f'(a) ] - (h4/720) [ f'''(b) - f'''(a) ]
+ ··· + K h 2p-2 [f(2p-3)(b) - f(2p-3)(a) ] + O(h 2p)

yields the standard truncation error expression

Th(f) - abf( x ) dx = (h2/12) (b - a) f''(c), for some point c where a ≤ c ≤ b.

A corollary of which is that if f''(x) = 0 for all x in [a,b], i.e. if f(x) is linear, then the trapezoidal rule is exact.

The Euler-Maclaurin summation formula also shows that usually n should be chosen large enough so that h = (b - a) / n < 1. For example, if h = 0.1 then
T0.1(f) = abf( x ) dx + 0.00083 [ f'(b) - f'(a) ] - (0.00000014) [ f'''(b) - f'''(a) ] + ···
and if h = 0.01 then
T0.01(f) = abf( x ) dx + 0.0000083 [ f'(b) - f'(a) ] - (0.000000000014) [ f'''(b) - f'''(a) ] + ···
while if h = 10 then
T10(f) = abf( x ) dx + 8.3333 [ f'(b) - f'(a) ] - 13.89 [ f'''(b) - f'''(a) ] + ···
However, if the function f(x) is linear, then n may be chosen to be 1.

Besides the truncation error described above, the algorithm is also subject to the usual round-off errors and errors due to number of significant bits in the machine representation of floating point numbers.

The source code below is programmed in double precision, but the number of significant bits can easily be extended to extended precision by changing double to long double and by affixing an L to any floating point number e.g. 0.5 to 0.5L.

Mathematically addition is associative, (a+b)+c = a + (b+c), but if intermediate results are rounded to machine precision, machine addition is not necessarily associative. E.g. consider a decimal machine with 3 significant digits, then
( ( 0.400 + 0.400 ) + 0.400 ) + 122 = 123 > 122 = 0.400 + ( 0.400 + ( 0.400 + 122 ) ).
In order to reduce the effect of intermediate result round-off errors when adding a series of floating point numbers, there are two versions of the rectangular rule, one which adds left to right
Th(f)=h [ (((···( ( f(a)/2 + f(a+h) ) + f(a+2h) ) + ··· + f(b-2h) ) + f(b)/2 ]
and the other which adds right to left
Th(f)=h [ f(a)/2 + ( f(a+h) + ( f(a+2h) + ··· + (f(b-h) + f(b) )···))) ].
In general, if the magnitude of a function is increasing in the interval of integration, addition should be performed left to right and if the magnitude of a function is decreasing in the interval of integration, addition should be performed right to left.

Function List

  • double Trapezoidal_Rule_Sum_LR( double a, double h, int n, double (*f)(double) )

    Integrate the user supplied function (*f)(x) from a to a + nh where a is the lower limit of integration, h > 0 is the length of each subinterval, and n > 0 is the number of subintervals. The sum is performed from left to right.

  • double Trapezoidal_Rule_Sum_RL( double a, double h, int n, double (*f)(double) )

    Integrate the user supplied function (*f)(x) from a to a + nh where a is the lower limit of integration, h > 0 is the length of each subinterval, and n > 0 is the number of subintervals. The sum is performed from right to left.

  • double Trapezoidal_Rule_Tab_Sum_LR( double h, int n, double f[ ] )

    Integrate the function f[] given as an array of dimension n whose ith element is the function evaluated at the midpoint of the ith subinterval, where h > 0 is the length of each subinterval, and n > 0 is the number of subintervals. The sum is performed from left to right.

  • double Trapezoidal_Rule_Tab_Sum_RL( double h, int n, double f[ ] )

    Integrate the function f[ ] given as an array of dimension n whose ith element is the function evaluated at the midpoint of the ith subinterval, where h > 0 is the length of each subinterval h, and n > 0 is the number of subintervals. The sum is performed from right to left.

C Source Code

  • The file, trapezoidal_rule.c, contains the versions of Trapezoidal_Rule_Sum_LR( ) and Trapezoidal_Rule_Sum_RL( ) written in C.

  • The file, trapezoidal_rule_tab.c, contains the versions of Trapezoidal_Rule_Tab_Sum_LR( ) and Trapezoidal_Rule_Tab_Sum_RL( ) written in C.