This is especially problematic for quad2d and integral2 with the 'tiled' method. The underlying quadrature rule gives the best results when the integrand is smooth, and a discontinuity where the integrand drops precipitously to zero will cause the adaptive quadrature algorithm to subdivide and further subdivide in that area until the error estimates across the discontinuity are sufficiently small. Conversely, masking the integrand has devastating effects on performance and accuracy. Telling the integrator where the boundary is gives it an enormous advantage. For example, to integrate F over a triangle with the vertices (0,0),(1,0), and (1,1), you would write F = <= x),0,1,0,1) Because dblquad would only integrate over rectangles, one would "mask" the input function to make it zero inside of a rectangular region.
As an example of this, consider the way integrating over non-rectangular regions was formerly done in MATLAB. Usually the tiled approach is the faster of the two, but it has its weaknesses. So, why have two methods? Each has different strengths. The 'iterated' method, as the name suggests, performs the integration as iterated one-dimensional integrals, so its "divide-and-conquer" strategy is to tackle one dimension first and then move on to the other.
If the error condition on a rectangle is not met, the rectangle is divided into quadrants and so forth. The 'tiled' method is based on quad2d's approach of dividing the region into quadrants and approximating the integral over each quadrant by a 2-D quadrature rule. Integral2 has, at present, two different integration methods, 'tiled' and 'iterated', not counting the 'auto' method that chooses between them.Įach integration method employs a type of "divide-and-conquer" approach to double integration but in very different ways.
Can You Take Advantage of these New Integration Routines?.
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