Trapezoidal Method — SepalSolver

Trapezoidal Integration in SepalSolver

The Trapezoidal Rule is a numerical method used to approximate the definiintegral of a function. It works by approximating the region under the graph of the function :math: f(x) as a trapezoid and calculating its area.

Mathematical Definition

To compute the integral over the interval $[a, b]$ , we partition the interval into $n$ sub-intervals, each of width $h = \cfrac{b-a}{n}$ . The composite trapezoidal rule is defined as:

\int_{a}^{b}f(x) , dx \approx \cfrac{h}{2} \left[f(x_0) + 2 \sum_{i=1}^{n-1} f(x_i) + f(x_n) \right]

SepalSolver Implementation: Manual Approach Writing the algorithm manually allows for a deeper understanding of how the weighting of the endpoints :math:x_0 and :math:x_n differs from the interior points.

Example 1C#

// Define the function and interval
double a = 0;                      // Lower bound
double b = pi;                     // Upper bound
int n = 50;                        // Number of segments
double h = (b - a) / n;            // Step size
// Generate points
ColVec x = Linspace(a, b, n+1); 
ColVec y = Sin(x);
// The first and last points are multiplied by 1
// The internal points are multiplied by 2
double integral_result = (h/2) * (y[0] + 2*y[1..^2].Sum() + y[^1]);
Console.WriteLine($"The approximate integral is: \n{integral_result}");

OutputFrom the book

The approximate integral is: 
1.9953967383788942

Error Analysis

The error in the trapezoidal rule, often denoted as $E_t$ , is proportional to the square of the step size $h$ . Specifically, for a function that is twice continuously differentiable, the error is:

E_t = -\frac{(b-a)}{12}h^2 f''(\xi)

where $\xi$ is some number in the interval $[a, b]$ . This indicates that the method is :math: O(h^2), meaning that halving the step size $h$ will approximately reduce the error by a factor of four.

Method	Accuracy Order	Weighting Logic
Left Riemann	$O(h)$	Uses $f(x_{ i-1})$
Trapezoidal	$O(h^2)$	Average of endpoints
Simpson's	$O(h^4)$	Parabolic fit (1-4-1)

Example 2C#

// Define the function to be integrated
Func<double, double> f = x => Sin(x);
// Perform the integration using the trapezoidal rule
double result = Trapz(f, 0, pi);
// Print the result
Console.WriteLine($"The integral of sin(x) from {0} to pi is approximately {result}");

OutputFrom the book

The integral of sin(x) from 0 to pi is approximately 2.000000000000283