Trapezoidal Rule

Description

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 I_[a,b](f) be the integral of f(x) over the closed and bounded interval [a,b], and let T_h(f) be the result of applying the trapezoidal rule with n subintervals of length h, i.e.
T_h(f)=h [ f(a)/2 + f(a+h) + ··· + f(b-h) + f(b)/2 ]. The Euler-Maclaurin summation formula relates I_[a,b](f) and T_h(f)
T_h(f) = I_[a,b](f) + (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 T_h(f) = I_[a,b](f) but rather what the theorem says is that
lim_h->0( T_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
T_h(f) - I_[a,b](f) = (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
T_h(f) - I_[a,b](f) = (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
T_0.1(f) = I_[a,b](f) + 0.00083 [ f'(b) - f'(a) ] - (0.00000014) [ f'''(b) - f'''(a) ] + ··· and if h = 0.01 then
T_0.01(f) = I_[a,b](f) + 0.0000083 [ f'(b) - f'(a) ] - (0.000000000014) [ f'''(b) - f'''(a) ] + ··· while if h = 10 then
T₁₀(f) = I_[a,b](f) + 8.3333 [ f'(b) - f'(a) ] - 13.89 [ 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 ) + 122.0 = 123.0 > 122.0 = 0.400 + ( 0.400 + ( 0.400 + 122.0 ) ). 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 T_h(f)=h [ (((···( ( f(a)/2 + f(a+h) ) + f(a+2h) ) + ··· + f(b-2h) ) + f(b)/2 ]and the other which adds right to leftT_h(f)=h [ f(a)/2 + ( f(a+h) + ( f(a+2h) + ··· + (f(b-h) + f(b) )···))) ].In general, if the function is increasing in the interval of integration, addition should be performed left to right and if the function is decreasing in the interval of integration, addition should be performed right to left.

Function List

Functions:

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 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 Trapezoidal_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: trapezoidal_rule.c

////////////////////////////////////////////////////////////////////////////////
// File: trapezoidal_rule.c                                                   //
// Routines:                                                                  //
//    Trapezoidal_Rule_Sum_LR                                                 //
//    Trapezoidal_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 trapezoidal rule is a (closed) Newton-Cotes formula for which the  //
//     interpolating polynomial is a line, the line joining the points        //
//     (x,f(x)) at the endpoints of the interval.                             //
//     The estimation of the integral then is simply the area of the trapezoid//
//     formed by the line segment joining the points (x, f(x)) at the         //
//     endpoints, the line segments joining (x,0) to (x,f(x)) at the endpoints//
//     and the interval of integration.                                       //
//                                                                            //
//     In order to integrate a function using the trapezoidal 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 trapezoidal rule to each          //
//     subinterval.  By abuse of language, this composite trapezoidal rule    //
//     is simply also called the trapezoidal rule.                            //
//                                                                            //
//     Let I(f) be the approximation of the integral of f(x) over [a,b] then  //
//     the trapezoidal rule is:                                               //
//       I(f) = h { f(x[0]) / 2 + f(x[1]) + ... + f(x[n-1]) + f(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.                                        //
//                                                                            //
//     The truncation error estimate for the trapezoidal rule is              //
//                           n * h^3   f''(xm) / 12                           //
//     where f''(xm) is the value of the second derivative evaluated at some  //
//     (unknown) point a <= xm <= b.                                          //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
//  double Trapezoidal_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 trapezoidal     //
//     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 Trapezoidal_Rule_Sum_LR( double a, double h, int n,
                                                       double (*f)(double) ) { 
   double bound = a + (n - .5) * h;
   double integral = 0.5 * (*f)(a);

   for (a += h; a < bound; a += h) integral += (*f)(a);

   return h * ( integral + 0.5 * (*f)(a) );
}


////////////////////////////////////////////////////////////////////////////////
//  double Trapezoidal_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 trapezoidal     //
//     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 Trapezoidal_Rule_Sum_RL( double a, double h, int n,
                                                       double (*f)(double) ) { 
   double x = a + n * h;
   double bound = a + 0.5 * h;
   double integral = 0.5 * (*f)(x);

   for (x -= h; x > bound; x -= h) integral += (*f)(x);

   return h * ( integral + 0.5 * (*f)(a) );
}

File: trapezoidal_rule_tab.c

////////////////////////////////////////////////////////////////////////////////
// File: trapezoidal_rule_tab.c                                               //
// Routines:                                                                  //
//    Trapezoidal_Rule_Tab_Sum_LR                                             //
//    Trapezoidal_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 trapezoidal rule is a (closed) Newton-Cotes formula for which the  //
//     interpolating polynomial is a line, the line joining the points        //
//     (x,f(x)) at the endpoints of the interval.                             //
//     The estimation of the integral then is simply the area of the trapezoid//
//     formed by the line segment joining the points (x, f(x)) at the         //
//     endpoints, the line segments joining (x,0) to (x,f(x)) at the endpoints//
//     and the interval of integration.                                       //
//                                                                            //
//     In order to integrate a function using the trapezoidal 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 trapezoidal rule to each          //
//     subinterval.  By abuse of language, this composite trapezoidal rule    //
//     is simply also called the trapezoidal rule.                            //
//                                                                            //
//     Let I(f) be the approximation of the integral of f(x) over [a,b] then  //
//     the trapezoidal rule is:                                               //
//       I(f) = h { f(x[0]) / 2 + f(x[1]) + ... + f(x[n-1]) + f(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.                                        //
//                                                                            //
//     The truncation error estimate for the trapezoidal rule is              //
//                           n * h^3   f''(xm) / 12                           //
//     where f''(xm) is the value of the second derivative evaluated at some  //
//     (unknown) point a <= xm <= b.                                          //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
//  double Trapezoidal_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 left    //
//     endpoint of the ith subinterval for 0 <= i < n and f[n] is the right   //
//     endpoint of the nth interval, i.e. f[i] = f( a + i*h ), i = 0,...,n,   //
//     then  this routine integrates f(x) using the trapezoidal rule by       //
//     summing from the left end of the interval to the right end.            //
//                                                                            //
//  Arguments:                                                                //
//     double h   The length of each subinterval, h > 0.                      //
//                a + n * h, if a is the lower limit of integration.          //
//     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    //
//                f[i] = f(a + i*h), i = 0,...,n, i.e. the dimension of f is  //
//                n + 1.                                                      //
//                                                                            //
//  Return Values:                                                            //
//     The integral of f(x) from a to (a + n * h).                            //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
//                                                                            //
double Trapezoidal_Rule_Tab_Sum_LR( double h, int n, double f[] ) { 

   double integral = 0.5 * f[0];
   int i;
   for (i = 1; i < n; i++) integral += f[i];

   return h * ( integral + 0.5 * f[n] );
}


////////////////////////////////////////////////////////////////////////////////
//  double Trapezoidal_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 left    //
//     endpoint of the ith subinterval for 0 <= i < n and f[n] is the right   //
//     endpoint of the nth interval, i.e. f[i] = f( a + i*h ), i = 0,...,n,   //
//     then  this routine integrates f(x) using the trapezoidal 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, where a is the lower limit of integration.    //
//     double f[] Array of function values the ith element of which is the    //
//                f[i] = f(a + i*h), i = 0,...,n, i.e. the dimension of f is  //
//                n + 1.                                                      //
//                                                                            //
//  Return Values:                                                            //
//     The integral of f(x) from a to (a + n * h).                            //
//                                                                            //
////////////////////////////////////////////////////////////////////////////////
//                                                                            //
double Trapezoidal_Rule_Tab_Sum_RL( double h, int n, double f[] ) { 

   double integral = 0.5 * f[n];
   int i;

   for (i = n - 1; i > 0; i--) integral += f[i];

   return h * ( integral + 0.5 * f[0] );
}