Numerical Integration Methods
When an integral is evaluated numerically, PTC Mathcad uses an adaptive quadrature method. You may want to change TOL, the endpoints, or the integrand to improve your results:
• Decreasing TOL may improve your results, but at some point the integral fails to converge. A good working range is 10-4 to 10-6.
• Setting large-valued endpoints to infinity and using the infinite endpoint algorithm may yield better answers.
• Sharply peaked integrands, or functions whose shape is not readily characterized by a single length scale, do not evaluate accurately. You may obtain better results by breaking an integral into pieces and separately integrating the peak from the rest of the plot.
• PTC Mathcad generally cannot integrate functions that have singularities in the interval of integration. Functions such as step and sawtooth functions with many finite discontinuities may also lead to nonconverging integrals. If you know the location of singularities in the integrand, you can often obtain a correct numerical evaluation by splitting the integral into a sum of integrals with these points as limits. To find potential singularities or discontinuities, plot the integrand.
Additional Information
Applying the adaptive method to an improper integral will likely produce an incorrectnumerical result.The Adaptive Integration algorithmrequires the function to be approximated by a polynomial in each subinterval division so the Gauss-Quadrature method can be used. Failing to meet the continuityrequirement on the integrand can lead to inaccurate results or failure to converge.