Convergence analysis of integration methods on segment

Evaluation of relative quadrature error with the following parameters:

fun_names = [
    "exp",
]

methods = [
    'Gauss',
    'Radau_I',
    'Radau_II',
    'Lobatto_III',
]

# all_nsteps = range(2,11)
all_nsteps = [5]
refinement_lvl = np.array(range(1,100))
# refinement_lvl = np.array([2**i for i in range(18)])

The following plots give the measured relative error as a function of the number of quadrature subintervals

plt.show()

The following plots give the measured convergence rate as a function of the number of quadrature subintervals. The dotted lines are theoretical convergence rates.

plt.show()

We can see 3 distinct phases on these plots:

• A first pre-convergence phase, where the convergence rate is growing towards its theoretical value. the end of the pre-convergence phase occurs for a number of sub-intervals roughtly independant of the convergence order of the quadrature method.

• A steady convergence phase where the convergence remains close to the theoretical value

• A final phase, where the relative error stagnates arround 1e-15. The value of the integral is computed with maximal accuracy given floating point precision. The approximation of the convergence rate is dominated by seemingly random floating point errors.