Rectangle Rule

Description

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 I_[a,b](f) be the integral of f(x) over the closed and bounded interval [a,b], and let R_h(f) be the result of applying the rectangle rule with n subintervals of length h, i.e.
R_h(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 I_[a,b](f) and R_h(f)
R_h(f) = I_[a,b](f) - (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 R_h(f) = I_[a,b](f) but rather what the theorem says is that
lim_h->0( R_h(f) - I_[a,b](f) ) / h^2p < M, where M > 0.

If f is at least twice differentiable on the interval [a,b], then applying the mean-value theorem to
R_h(f) - I_[a,b](f) = - (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
R_h(f) - I_[a,b](f) = - (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
R_0.1(f) = I_[a,b](f) - 0.00042 [ f'(b) - f'(a) ] + (0.00000012) [ f'''(b) - f'''(a) ] + ··· and if h = 0.01 then
R_0.01(f) = I_[a,b](f) - 0.0000042 [ f'(b) - f'(a) ] + (0.000000000012) [ f'''(b) - f'''(a) ] + ··· while if h = 10 then
R₁₀(f) = I_[a,b](f) - 4.1667 [ f'(b) - f'(a) ] + 12.15 [ f'''(b) - f'''(a) ] + ··· However, if the function f(x) is a 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 R_h(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 leftR_h(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 a non-negative function is increasing in the interval of integration, addition should be performed left to right and if the non-negative function is decreasing in the interval of integration, addition should be performed right to left.

Function List

Functions:

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 i^th element is the function evaluated at the midpoint of the i^th 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 i^th element is the function evaluated at the midpoint of the i^th 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

File: rectangle_rule.c

////////////////////////////////////////////////////////////////////////////////
// File: rectangle_rule.c                                                     //
// Routines:                                                                  //
//    Rectangle_Rule_Sum_LR                                                   //
//    Rectangle_Rule_Sum_RL                                                   //
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
//                                                                            //
//  Description:                                                              //
//     A Newton-Cotes formula is an interpolatory quadrature formula using    //
//     the values of the integrand (function being integrated) at equally     //
//     spaced nodes to approximate the integral of the function over a        //
//     closed and bounded interval.                                           //
//                                                                            //
//     The rectangle rule is an (open) Newton-Cotes formula with a single     //
//     node at the midpoint of the closed and bounded interval.  The          //
//     interpolating polynomial is a constant, the constant being the value   //
//     of the function evaluated at the node.                                 //
//     The estimation of the integral then is simply the function evaluated   //
//     at the mid-point times the length of the interval.                     //
//                                                                            //
//     In order to integrate a function using the rectangle rule over a       //
//     closed and bounded interval [a,b], divide the interval [a,b] into N    //
//     subintervals of length h = (b-a) / N.  The integral of the function    //
//     over the interval [a,b] is the sum of the integrals of the function    //
//     over the subintervals [a,a+h], [a+h, a+2h],...,[b-h,b].  The integral  //
//     is approximated then by applying the rectangle rule to each            //
//     subinterval.  By abuse of language, this composite rectangle rule      //
//     is simply also called the rectangle rule.                              //
//                                                                            //
//     Let I(f) be the approximation of the integral of f(x) over [a,b] then  //
//     the rectangle rule is:                                                 //
//       I(f) = h { f( [x[0] + x[1]] / 2) + ... + f( [x[n-1] + x[n]] / 2) }   //
//     where h is the step size, h = x[i] - x[i-1]; n is the number of        //
//     steps, n = abs(b - a) / h and x[0] = a, x[i] = x[i-1] + h for          //
//     i = 1,...,n-1 so that x[n] = b.                                        //
//                                                                            //
//     For a function f(x) which is at least be differentiable of class C2    //
//     on the interval of integration the truncation error estimate is        //
//                            n * h^3 * f''(xm) / 24                          //
//     where f''(xm) is the value of the second derivative evaluated at some  //
//     (unknown) point, a <= xm <= b.                                         //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
//  double Rectangle_Rule_Sum_LR( double a, double h, int n,                  //
//                                                    double (*f)(double) );  //
//                                                                            //
//  Description:                                                              //
//     This function integrates f(x) from a to a+nh using the rectangle       //
//     rule by summing from the left end of the interval to the right end.    //
//                                                                            //
//  Arguments:                                                                //
//     double a   The lower limit of the integration interval.                //
//     double h   The length of each subinterval, h > 0.                      //
//     int    n   The number of subintervals, the upper limit of integration  //
//                is a + n * h.                                               //
//     double *f  Pointer to the integrand, a function of a single variable   //
//                of type double.                                             //
//                                                                            //
//  Return Values:                                                            //
//     The integral of f(x) from a to (a + n * h).                            //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
double Rectangle_Rule_Sum_LR( double a, double h, int n, double (*f)(double) ) { 
   double x = a + 0.5 * h;
   double upper_limit = a + (n - 0.25) * h;
   double integral = (*f)(x);

   for (x += h; x < upper_limit; x += h) integral += (*f)(x);
 
   return h * integral;
}


////////////////////////////////////////////////////////////////////////////////
//  double Rectangle_Rule_Sum_RL( double a, double h, int n,                  //
//                                                    double (*f)(double) );  //
//                                                                            //
//  Description:                                                              //
//     This function integrates f(x) from a to a+nh using the rectangle       //
//     rule by summing from the right end of the interval to the left end.    //
//                                                                            //
//  Arguments:                                                                //
//     double a   The lower limit of the integration interval.                //
//     double h   The length of each subinterval, h > 0.                      //
//     int    n   The number of subintervals, the upper limit of integration  //
//                is a + n * h.                                               //
//     double *f  Pointer to the integrand, a function of a single variable   //
//                of type double.                                             //
//                                                                            //
//  Return Values:                                                            //
//     The integral of f(x) from a to (a + n * h).                            //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
double Rectangle_Rule_Sum_RL( double a, double h, int n, double (*f)(double) ) { 
   double x = a + (n - 0.5) * h;
   double integral = (*f)(x);

   for (x -= h; x > a; x -= h) integral += (*f)(x);
 
   return h * integral;
}

File: rectangle_rule_tab.c

////////////////////////////////////////////////////////////////////////////////
// File: rectangle_rule_tab.c                                                 //
// Routines:                                                                  //
//    Rectangle_Rule_Tab_Sum_LR                                               //
//    Rectangle_Rule_Tab_Sum_RL                                               //
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
//                                                                            //
//  Description:                                                              //
//     A Newton-Cotes formula is an interpolatory quadrature formula using    //
//     the values of the integrand (function being integrated) at equally     //
//     spaced nodes to approximate the integral of the function over a        //
//     closed and bounded interval.                                           //
//                                                                            //
//     The rectangle rule is an (open) Newton-Cotes formula with a single     //
//     node at the midpoint of the closed and bounded interval.  The          //
//     interpolating polynomial is a constant, the constant being the value   //
//     of the function evaluated at the node.                                 //
//     The estimation of the integral then is simply the function evaluated   //
//     at the mid-point times the length of the interval.                     //
//                                                                            //
//     In order to integrate a function using the rectangle rule over a       //
//     closed and bounded interval [a,b], divide the interval [a,b] into N    //
//     subintervals of length h = (b-a) / N.  The integral of the function    //
//     over the interval [a,b] is the sum of the integrals of the function    //
//     over the subintervals [a,a+h], [a+h, a+2h],...,[b-h,b].  The integral  //
//     is approximated then by applying the rectangle rule to each            //
//     subinterval.  By abuse of language, this composite rectangle rule      //
//     is simply also called the rectangle rule.                              //
//                                                                            //
//     Let I(f) be the approximation of the integral of f(x) over [a,b] then  //
//     the rectangle rule is:                                                 //
//       I(f) = h { f( [x[0] + x[1]] / 2) + ... + f( [x[n-1] + x[n]] / 2) }   //
//     where h is the step size, h = x[i] - x[i-1]; n is the number of        //
//     steps, n = abs(b - a) / h and x[0] = a, x[i] = x[i-1] + h for          //
//     i = 1,...,n-1 so that x[n] = b.                                        //
//                                                                            //
//     For a function f(x) which is at least be differentiable of class C2    //
//     on the interval of integration the truncation error estimate is        //
//                            n * h^3 * f''(xm) / 24                          //
//     where f''(xm) is the value of the second derivative evaluated at some  //
//     (unknown) point, a <= xm <= b.                                         //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
//  double Rectangle_Rule_Tab_Sum_LR( double h, int n, double f[] )           //
//                                                                            //
//  Description:                                                              //
//     Assuming that a is the lower limit of integration and the ith element  //
//     of the array f[] is the value of the function evaluated at the midpoint//
//     of the ith subinterval, i.e. f[i] = f( a + (i+0.5)*h ), this routine   //
//     integrates f(x) using the rectangle rule by summing from the left      //
//     end of the interval to the right end.                                  //
//                                                                            //
//  Arguments:                                                                //
//     double h   The length of each subinterval, h > 0.                      //
//     int    n   The number of subintervals, the upper limit of integration  //
//                is a + n * h, where a is the lower limit of integration.    //
//     double f[] Array of function values the ith element of which is the    //
//                function evaluated at a + (i + 0.5)*h.                      //
//                                                                            //
//  Return Values:                                                            //
//     The integral of f(x) from a to (a + n * h).                            //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
double Rectangle_Rule_Tab_Sum_LR( double h, int n, double f[] ) { 

   double integral = f[0];
   int i;

   for (i = 1; i < n; i++) integral += f[i];
 
   return h * integral;
}


////////////////////////////////////////////////////////////////////////////////
//  double Rectangle_Rule_Tab_Sum_RL( double h, int n, double f[] )           //
//                                                                            //
//  Description:                                                              //
//     Assuming that a is the lower limit of integration and the ith element  //
//     of the array f[] is the value of the function evaluated at the midpoint//
//     of the ith subinterval, i.e. f[i] = f( a + (i+0.5)*h ), this routine   //
//     integrates f(x) using the rectangle rule by summing from the right     //
//     end of the interval to the left end.                                   //
//                                                                            //
//  Arguments:                                                                //
//     double h   The length of each subinterval, h > 0.                      //
//     int    n   The number of subintervals, the upper limit of integration  //
//                is a + n * h.                                               //
//     double f[] Array of function values the ith element of which is the    //
//                function evaluated at a + (i + 0.5)*h.                      //
//                                                                            //
//  Return Values:                                                            //
//     The integral of f(x) from a to (a + n * h).                            //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
double Rectangle_Rule_Tab_Sum_RL( double h, int n, double f[] ) { 

   double integral = f[n-1];
   int i;

   for (i = n - 2; i >= 0; i--) integral += f[i];
 
   return h * integral;
}