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## Rectangle Rule

The rectangle 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 rectangle with length h and height the value of the function f(x) evaluated at the midpoint of the interval, f(a+h/2). The composite rectangle 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 and adding the results of applying the rectangle rule to each subinterval. By abuse of language both the composite rectangle rule and the rectangle rule sometimes are referred to simply as the rectangle rule.
Let abf( x ) dx be the integral of f(x) over the closed and bounded interval [a,b], and let Rh(f) be the result of applying the rectangle rule with n subintervals of length h, i.e.

Rh(f)=h [ f(a+h/2) + f(a+3h/2) + ··· + f(b-h/2) ].

An immediate consequence of the Euler-Maclaurin summation formula yields the following equation relating abf( x ) dx and Rh(f)

Rh(f) = abf( x ) dx - (h2/24) [ f'(b) - f'(a) ] + (7h4/5760) [ 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 (2p-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
Rh(f) = abf( x ) dx but rather what the theorem says is that
limh→0 | ( Rh(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
Rh(f) - abf( x ) dx = - (h2/24) [ f'(b) - f'(a) ] + (7h4/5760) [ 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

Rh(f) - abf( x ) dx = - (h2/24) (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 rectangle 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
R0.1(f) = abf( x ) dx - 0.00042 [ f'(b) - f'(a) ] + (0.00000012) [ f'''(b) - f'''(a) ] + ···
and if h = 0.01 then
R0.01(f) = abf( x ) dx - 0.0000042 [ f'(b) - f'(a) ] + (0.000000000012) [ f'''(b) - f'''(a) ] + ···
while if h = 10 then
R10(f) = abf( x ) dx - 4.1667 [ f'(b) - f'(a) ] + 12.15 [ 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 rectangle rule, one which adds left to right
Rh(f)=h [ (((···( ( f(a+h/2) + f(a+3h/2) ) + f(a+5h/2) ) + ··· + f(b-3h/2) ) + f(b-h/2) ]
and the other which adds right to left
Rh(f)=h [ f(a+h/2) + ( f(a+3h/2) + ( f(a+5h/2) + ··· + (f(b-3h/2) + f(b-h/2) )···))) ].
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 Rectangle_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 Rectangle_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 Rectangle_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 Rectangle_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, rectangle_rule.c, contains the versions of Rectangle_Rule_Sum_LR( ) and Rectangle_Rule_Sum_RL( ) written in C.

• The file, rectangle_rule_tab.c, contains the versions of Rectangle_Rule_Tab_Sum_LR( ) and Rectangle_Rule_Tab_Sum_RL( ) written in C.