Hybrid Symbolic-Numeric Integration in Multiple Dimensions via Tensor-Product Series 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 Orlando A. Carvajal Frederick W. Chapman Keith O. Geddes Symbolic Computation Group, School of Computer Science University of Waterloo, Waterloo, ON, N2L 3G1, Canada

We present a new hybrid symbolic-numeric method for the fast and accurate evaluation of definite integrals in multiple dimensions. This method is well-suited for two classes of problems: (1) analytic integrands over general regions in two dimensions, and (2) families of analytic integrands with special algebraic structure over hyperrectangular regions in higher dimensions. The algebraic theory of multivariate interpolation via natural tensor-product series was developed in the doctoral thesis by Chapman, who named this broad new scheme of bilinear series expansions \342\200\235Geddes series\342\200\235 in honour of his thesis supervisor. This paper describes an efficient adaptive algorithm for generating bilinear series of Geddes-Newton type and explores applications of this algorithm to multiple integration. We will present test results demonstrating that our new adaptive integration algorithm is effective both in high dimensions and with high accuracy. For example, our Maple implementation of the algorithm has successfully computed nontrivial integrals with hundreds of dimensions to 10-digit accuracy, each in under 3 minutes on a desktop computer. Current numerical multiple integration methods either become very slow or yield only low accuracy in high dimensions, due to the necessity to sample the integrand at a very large number of points. Our approach overcomes this difficulty by using a Geddes-Newton series with a modest number of terms to construct an accurate tensor-product approximation of the integrand. The partial separation of variables achieved in this way reduces the original integral to a manageable bilinear combination of integrals of essentially half the original dimension. We continue halving the dimensions recursively until obtaining one-dimensional integrals, which are then computed by standard numeric or symbolic techniques.

The hybrid symbolic-numeric method presented in this paper is well-suited for two classes of integration problems: . analytic integrands over general regions in two dimensions, and . families of analytic integrands with special algebraic structure over hyperrectangular regions in higher dimensions.

The method is based on recursive dimension-halving via separation of variables. Multidimensional integrals are reduced to one-dimensional integrals using Geddes-Newton series expansions to generate accurate tensor-product approximations of the integrand. Test results show that our new adaptive integration algorithm is effective both with high accuracy and in high dimensions. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

A tensor product is a finite sum of terms where each term is a product of univariate functions: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn The rank of a tensor product is the minimum number of terms among all the equivalent representations. It is a natural tensor product when the factors of each term can be derived from the original function by a finite linear combination of linear functionals. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

For a bivariate function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnRjU=and a point 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, the splitting operator 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 defined by 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. The point 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 is called a splitting point. Note: 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 splits LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== into a rank-one tensor product. 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 interpolates 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 on the lines LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYsNiVRImFGJ0YvRjI= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYsNiVRImJGJ0YvRjI= , which means that 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 and 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. These properties allow us to generate a natural tensor-product series simply by iterating the splitting operator while varying the choice of splitting point.

Not every bivariate function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== can be expressed as a tensor product of finite rank. For an approximation in 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 where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is continuous, we can uniformly approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== by a rank-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYntensor product: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== 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

Ly0mSSJzRzYiNiNJIm5HRiY2JEkieEdGJkkieUdGJi1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkKiYtJkkiZ0dGJjYjSSJpR0YmNiNGKiIiIi0mSSJoR0YmRjY2I0YrRjkvRjc7RjlGKA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Define the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRIm5GJ0YvRjItRiw2JVEjdGhGJ0YvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRis= remainder : 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 . The following simple algorithm generates a natural tensor-product series expansion LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= of a given function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== .

. Set 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 and initialize LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiaUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtbkdGJDYkRlRGOQ== . . 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 , iterate the following: ) Choose 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 with 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 . ) Let 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 and 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 . . Let 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 and return 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 to approximate LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== with uniform error 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 on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiUkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== .

Refining terminology introduced by Chapman in his doctoral thesis, we call LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= the Geddes-Newton series expansion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== terms with respect to the splitting points 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 .

Consider the following function on 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 : LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNitGKy1GIzYlLUYsNiVRImZGJ0YvRjItSSNtb0dGJDYwUTAmQXBwbHlGdW5jdGlvbjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkQvJSlzdHJldGNoeUdGRC8lKnN5bW1ldHJpY0dGRC8lKGxhcmdlb3BHRkQvJS5tb3ZhYmxlbGltaXRzR0ZELyUnYWNjZW50R0ZELyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdGVi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInhGJ0YvRjItRj02MFEiLEYnRkBGQi9GRkYxRkdGSUZLRk1GT0ZRRlQvRlhRM3Zlcnl0aGlja21hdGhzcGFjZUYnRllGZm4tRiw2JVEieUYnRi9GMkZALUY9NjBRMSZJbnZpc2libGVUaW1lcztGJ0ZARkJGRUZHRklGS0ZNRk9GUS9GVVEvdGhpY2ttYXRoc3BhY2VGJy9GWEZecEZZRmZuLUY9NjBRIj1GJ0ZARkJGRUZHRklGS0ZNRk9GUUZdcEZfcEZZRmZuLUY9NjBGXHBGQEZCRkVGR0ZJRktGTUZPRlEvRlVRMG1lZGl1bW1hdGhzcGFjZUYnL0ZYRmZwRllGZm4tRiM2JS1GLDYlUSRleHBGJ0YvRjJGPC1Gam42JC1GIzYnRistSSVtc3VwR0YkNiVGXm8tRiM2Iy1JI21uR0YkNiRRIjJGJ0ZALyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJ0ZjcC1GYnE2JUZnb0ZmcUZqcUYrRkBGY3AtRiM2JS1GLDYlUSRjb3NGJ0YvRjJGPC1Gam42JC1GIzYlRl5vLUY9NjBRJyZwbHVzO0YnRkBGQkZFRkdGSUZLRk1GT0ZRRmVwRmdwRllGZm5GZ29GQEYrRis=

Ly1JImZHNiI2JEkieEdGJUkieUdGJSomLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJilGJyIiIyIiIilGKEYyRjNGMy1JJGNvc0dGLDYjLCZGJ0YzRihGM0Yz

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn The Geddes-Newton series to 3 terms is as follows, where the splitting points are 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 for 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 : 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

Ly0mSSJzRzYiNiMiIiQ2JEkieEdGJkkieUdGJi1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkKigmSSJjR0YmNiNJImlHRiYiIiItJkkiZ0dGJkY1NiNGKkY3LSZJImhHRiZGNTYjRitGNy9GNjtGN0Yo

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The splitting points in Example 1 are all located on the diagonal LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYsNiVRInhGJ0YvRjI= of the unit square, as illustrated in the following figure. 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 a := [1.0, 0.0, 0.616336]: n := nops(a): points := {plot([seq([a[i],a[i]], i=1..n)], style=POINT, symbol=BOX, symbolsize=11, axes=BOXED, color=BLACK)}: for i from 1 to n do horizontal[i] := plot([[0,a[i]], [1,a[i]]], style=LINE, axes=BOXED, color=BLACK); vertical[i] := plot([[a[i],0], [a[i],1]], style=LINE, axes=BOXED, color=BLACK); end do: lines := `union`(seq({horizontal[i], vertical[i]}, i=1..n)): diag := {plot([[0,0], [1,1]], style=LINE, linestyle=DASH, axes=BOXED, color=BLACK)}: plots[display](points union lines union diag);

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

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Note: 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 agrees exactly with 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 at all points on the horizontal and vertical lines through the three splitting points! With this strong interpolating property, even with only three splitting points we find that the maximum error over the unit square satisfies: 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 . If we request a series accurate to 15 digits for the function 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 defined in Example 1, we find that only 12 terms are required (i.e., 12 splitting points). In other words, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2MFE0JkRvdWJsZVZlcnRpY2FsQmFyO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRiw2JVEiZkYnRi9GMi1GNjYwUSgmbWludXM7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRkovRk5RJDBlbUYnL0ZRUTN2ZXJ5dGhpY2ttYXRoc3BhY2VGJ0ZSRlUtSSVtc3ViR0YkNiUtRiw2JVEic0YnRi9GMi1JI21uR0YkNiRRIzEyRidGOS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistRl5vNiVGNS1GLDYlUSgmaW5maW47RidGL0YyRmdvRistRjY2MFElJmx0O0YnRjlGO0Y+RkBGQkZERkZGSEZKRk1GUEZSRlUtRmRvNiRRJDUuMEYnRjktRjY2MFEoJnRpbWVzO0YnRjlGO0Y+RkBGQkZERkZGSEZKL0ZOUTBtZWRpdW1tYXRoc3BhY2VGJy9GUUZpcEZSRlUtSSVtc3VwR0YkNiUtRmRvNiRRIzEwRidGOS1GIzYkLUY2NjBGZ25GOUY7Rj5GQEZCRkRGRkZIRkpGTUZQRlJGVS1GZG82JFEjMTVGJ0Y5LyUxc3VwZXJzY3JpcHRzaGlmdEdGaW9GKw== . The distribution of the 12 splitting points for this particular function is illustrated in the following figure. restart; LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yna := [1.0, 0.0, 0.625, 0.3125, 0.8125, 0.15625, 0.90625, 0.46875, 0.078125, 0.953125, 0.71875, 0.0390625]: n := nops(a): points := {plot([seq([a[i],a[i]], i=1..n)], style=POINT, symbol=BOX, symbolsize=11, axes=BOXED, color=BLACK)}: for i from 1 to n do horizontal[i] := plot([[0,a[i]], [1,a[i]]], style=LINE, axes=BOXED, color=BLACK); vertical[i] := plot([[a[i],0], [a[i],1]], style=LINE, axes=BOXED, color=BLACK); end do: lines := `union`(seq({horizontal[i], vertical[i]}, i=1..n)): diag := {plot([[0,0], [1,1]], style=LINE, linestyle=DASH, axes=BOXED, color=BLACK)}: plots[display](points union lines union diag);

6?-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3++++++](=(!#=$""!""!7$$"3++++++](=(!#=$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6&7.7$$"""""!$"""""!7$$""!""!$""!""!7$$"3+++++++]i!#=$"3+++++++]i!#=7$$"3+++++++DJ!#=$"3+++++++DJ!#=7$$"3+++++++D")!#=$"3+++++++D")!#=7$$"3++++++]i:!#=$"3++++++]i:!#=7$$"3++++++]i!*!#=$"3++++++]i!*!#=7$$"3++++++](o%!#=$"3++++++](o%!#=7$$"3++++++]7y!#>$"3++++++]7y!#>7$$"3++++++DJ&*!#=$"3++++++DJ&*!#=7$$"3++++++](=(!#=$"3++++++](=(!#=7$$"3++++++D1R!#>$"3++++++D1R!#>-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I&POINTG6$%*protectedGI(_syslibG6"-I'SYMBOLG6$%*protectedGI(_syslibG6"6$I$BOXG6$%*protectedGI(_syslibG6""#6-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3++++++](o%!#=$""!""!7$$"3++++++](o%!#=$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"""""!7$$"""""!$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"""""!$""!""!7$$"""""!$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6&7$7$$""!""!$""!""!7$$"""""!$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I*LINESTYLEG6"6#""$-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$""!""!7$$"""""!$""!""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$""!""!7$$""!""!$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3++++++DJ&*!#=$""!""!7$$"3++++++DJ&*!#=$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3++++++D1R!#>$""!""!7$$"3++++++D1R!#>$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"3+++++++]i!#=7$$"""""!$"3+++++++]i!#=-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"3++++++DJ&*!#=7$$"""""!$"3++++++DJ&*!#=-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"3++++++D1R!#>7$$"""""!$"3++++++D1R!#>-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3+++++++]i!#=$""!""!7$$"3+++++++]i!#=$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"3+++++++DJ!#=7$$"""""!$"3+++++++DJ!#=-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3+++++++DJ!#=$""!""!7$$"3+++++++DJ!#=$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"3+++++++D")!#=7$$"""""!$"3+++++++D")!#=-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$"3+++++++D")!#=$""!""!7$$"3+++++++D")!#=$"""""!-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!""!-I&STYLEG6$%*protectedGI(_syslibG6"6#I%LINEG6$%*protectedGI(_syslibG6"-I'CURVESG6$%*protectedGI(_syslibG6"6%7$7$$""!""!$"3++++++]i:!#=7$$"""""!$"3++++++]i:!#=-I'COLOURG6$%*protectedGI(_syslibG6"6&I$RGBG6$%*protectedGI(_syslibG6"$""!""!$""!""!$""!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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Note similarities with the case of univariate polynomial interpolation via the Newton interpolation series. In both cases, addition of a new term in the series corresponds to the addition of a new interpolation point. However, in the case of Geddes-Newton series, the interpolation property holds not only at the new point but at all points on two lines passing through that point. Intuitively, for any function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== which is continuous in the unit square LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==, if the splitting points are dense along the diagonal then it is reasonable to expect that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= might converge uniformly to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiVUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRLSZSaWdodEFycm93O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtbkdGJDYkUSgmaW5maW47RidGOQ==. In other words, by choosing well distributed splitting points on the diagonal, we can cover the whole unit square with small grid cells. Recall: The remainder LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInJGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= vanishes on the boundary of each cell . Formal convergence results will appear in a separate paper. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The easiest case is when LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is analytic in a sufficiently large region. The convergence theorem and its proof are similar to the classical case where 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 is a function of a single complex variable and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ6RidGL0YyL0YzUSdub3JtYWxGJ0Yr denotes the interpolating polynomial. Recall the univariate case (Ref: P.J. Davis, Interpolation and Approximation), as summarized below. Basic tool: A contour integral representation of 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 . LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== be analytic on 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 and suppose that the interpolation points satisfy 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 . Let LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistSShtZmVuY2VkR0YkNiQtRiM2Iy1GLDYlUSJ6RidGL0YyL0YzUSdub3JtYWxGJ0Yr be the polynomial interpolating 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 at points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYmLUYjNiVGKy1JJW1zdWJHRiQ2JS1GLDYlUSJhRidGL0YyLUYsNiVRImlGJ0YvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJ0YrL0YzUSdub3JtYWxGJy8lJW9wZW5HUSJ8ZnJGJy8lJmNsb3NlR1EifGhyRidGKw==. 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, 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. Then for all LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRKiZFbGVtZW50O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRiw2JVEiQUYnRi9GMg== ,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 where 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 . Conclusion: If 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 then 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 . cVRGX1NIU1NGa1NHdXRQXXNGXUNFXUNIaVNRcVRGRE1QQE1WXFxtaGhvUj5hZkZgcz5gV1dla15gc15fa0ZhbWZlVUdgb15gZkZgc2ZgV1dFXFxNUEBdcGZlVUdAbUNFXWtVcG9nbmBxXl9sPmBrZmVVR2BxZmBmTmBqXmBXV2VrQkhTY0ZbU0d1dFBcXG1SbExTPG1VcG9nckZTY0ZbO0ZhZk5gakZhV1dla0Zhb15fbEZga2ZlVU9gaj5gZk5ga1ZgV1dlbGJGU2NGXTteYGZOYGteYFdXZWxyR1NjRl07QG1SbExTQG1UcG9nRF1tXl9sRmBwZmVVT0BlQ0VfO1JIU2NGXUNIdXRQX2NGY0NFX2tWcG9nRF1zXl9sTmBrZmVVc0ZhQ0VfY0ZhU1FxXFxvXl9sTkBKVmxMU0RNVHBvZ15gal5fbE5gb2ZlVWtSbExTRD0xOlwiXHtcfQ==cWRGZXNGU1NGbUNHdXRQX1NHaUNFXVNISlVATVBAbVZIbWhoT1NeYGZGYHNOYFdXZWxiR1NTRm07RmFmRmBzOmlzRlNTRjptQ0VdOz5ha15fazpWYGZGQFpwXl9rOnZgZkZAXVNRcWRGbUNGU1NGbUNGdXRQX2tSbGxSXFxtVnBvZ0RdbF5faz5hcmZlVU9AY0NFXUNIWnJeX2s+YXFmZVVPQG1DRV1DSGdTUXF0RltjRlNTRlBtaGhvU1ZgZkZgcl5gV1dlbVJHU1NGazt2YGZGYHI6XFxNUEBdbWZlVVdgaz5gZkZgcjpETVBAPUpUbGxSOlRNUEBNVjprQ0VdS1Nwb2dITVNMTVBAPWpWbGxSRmBXV2VtVmBqXl9rNjpcIlx7XH0=cXRGX3NGU1NGY1NIdXRQYWNGOjo0OlwiXHtcfQ==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cVRGZUNHU2NGX0NHdXRQXVNHZUNFX2NGZVNRcVRGWE1QRE1TWG1oMjpcIlx7XH0=cVRGZXNHU2NGX3NHdXRQXVNHOjo0OlwiXHtcfQ==cVRGZXNHU2NGX3NHdXRQXVNHbUNFX2NGbVNRcVRGOjI6XCJce1x9cVRGZVNIU2NGX1NIdXRQXWNHXUNFX3NGW1NRcVRGOjI6XCJce1x9cVRGZ1NHU2NGYXNGdXRQXWNHOmJHU2NGYTY7XCJce1x9cVRGaWNGU2NGYWNHdXRQXXNHOjo0OlwiXHtcfQ==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cXRGX3NGU1NGY3NGdXRQYWNGY0NFXUNHY1NRcXRGWnBmZVVXQEpUOnJGMjpcIlx7XH0=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cWRGbWNHU1NGZXNHdXRQX1NIY0NFXVNHWmteX2tmYHBmZVVPYHI+YWZGYG86WnNmZVVPQGlDRV1jR1tTUXFkRmBNUEBNVUxtaGhPU2BNUmxsUlRtVHBvZ0Q9ckd1dFBGQFRNVnBvZ0RdbF5fa25gc2ZlVWtTbGxSWE1ScG9nQkd2YGxmZVVPQGVDRV1zR2NTUXFcXHBqVVBtaGg/YTtrU1FxXFxsalVgbWhoP1tbcTprQ0g+YWpmZVVLVWxsUlxcPUJHU1NGOmVDRV1rVXBvZzpQPW5AWE1UWnJqVUQ9RkFYbVJwb2dEbVZAPWk7VkA+QDo0OlwiXHtcfQ==cVRGbUNIU1NGa1NIdXRQXVNIZ0NFXVNIW1NRcVRGWmxmZVVHQGNDRV1TSGNTUXFURkhNUEBtVlRtaGhvUk5gZkZgc3ZgV1dla1JGU1NGbUNIdXRQXUNIbUNFXVNIbVNRcVxccl5fa0ZhcjpjQ0VdS1VabV5fa1JHdXRQXUtTbGxSVmBXV2VrQkc6ZmBmRmBzTmBXW3FeX2tGQWpWOkZBOk5gZmpTWm1eP0w9XmBmalRab15fa2JHSlVsPGk7WE1QQF1yOmtDRUZBRE1SPE1QRE1SPG1oaE9TTmBmTmBqOjoyOlwiXHtcfQ==cVRGaVNGU2NGYVNHdXRQXXNHW0NFX3NGY1NRcVRGZ0NIU2NGYXNGdXRQXWtVbExTSE1TcG9nQF1wXl9sVmBrZmVVR0BjQ0Vfc0ZbU1FxVEZETVBEbVM6XUNFX2NGbVNRcVRGWnJmNkVcIlx7XH0=cVRGaUNIU2NGYUNGdXRQXXNHYkZrU1FxVEZgTVBETVNYbWhob1I6ZVNRcVRGWm1mZVVHYHI+YGZOYGxOYFdXZWtSRlNjRl9TRnV0UF07QG1WcG9nQF1sXl9sRmByZmVVa1NsTFNAbVVwb2dCR1NjRl1jR3V0UF07PkFSR1NjRl1TSDpEPWJHU2NGX2NGOl5gV1dla3JHU2NGX1NHSlZsTFNuYFdXRVpyOm1DRV9rVnBvZzphU0Z1dFBgTVJsTFNIPTE6XCJce1x9cVRGaUNIU2NGX1NHdXRQXXNHbUNFX2NGZ1NRcVRGa0NGU2NGX1NHSlNsTFNEPXJGU2NGX2NHalRsTFNEPXJHU2NGXztGYWZOQDo2OlwiXHtcfQ==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cWRGYUNHU2NHW1NRcWRGYWNGU2NHWzs8TVBUTVJKU1hNUFRNU3BvZ0Q9XmBXV2VsQkhTY0dlU1FxZEZqVXBvZ0Rdc15fcD5hV1dlbFZARmFXV2VsYkZGQUJHRkFSR1NjR2s7VE1QVE1VcG9nRF1xXl9wOkpTWnBqUnBvZ0Rdb1JHbVNRcVxcblJHaVNRcVxcbV5fb25gV1dFSlRwb2diRlBtU3BvZ0Rda15fb05gV1dFX1tvRmBXV0VcXF1vTkByR1BtU1pwXl9vVkAxOlwiXHtcfQ==cWRGZXNGU1NHX1NRcWRGZWNGU1NHXVNRcWRGQE1QUE1ScG9nRF1qXl9uRmFXV2VsXmByXl9uRkFiR1NDR207UE1QUE1SWm5eX29OQDpjU1FxZEZQPWVTUXFkRlhNUFBtVFpyUkdnU1FxZEZlW29uQFJGdmBXV2VsYkY+YVdXZWxacD5gV1dFSF1wRmBXV0VqU3BvZ0RdbEpUcG9nUkZUbVRwb2dCRlRNVXBvZ15AVE1VWnFiR2lTUXFcXHBiR2c7TF1wbkA6MzpcIlx7XH0= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn For our case, we consider a bivariate function 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 of two complex variables and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ3RidGL0YyLUkjbW9HRiQ2MFEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR1EzdmVyeXRoaWNrbWF0aHNwYWNlRicvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRiw2JVEiekYnRi9GMkZNRis= denotes the Geddes-Newton series expansion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== terms with respect to the splitting points LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYjNiVGKy1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2Ji1GIzYlRistRj42JC1GIzYpRistRjY2JS1GLDYlUSJhRidGL0YyLUYsNiVRImlGJ0YvRjIvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGWS8lKnN5bW1ldHJpY0dGWS8lKGxhcmdlb3BHRlkvJS5tb3ZhYmxlbGltaXRzR0ZZLyUnYWNjZW50R0ZZLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUZSNjBRIixGJ0ZVRldGWkZmbkZobkZqbkZcb0Zeb0Zgb0Zjb0Zmb0Zpb0ZccC1GUjYwRlRGVUZXL0ZlbkZZRmZuRmhuRmpuRlxvRl5vRmBvL0Zkb1EvdGhpY2ttYXRoc3BhY2VGJy9GZ29GZnBGaW9GXHAtRjY2JS1GLDYlUSJiRidGL0YyRktGTkYrRlVGK0ZVLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrLyUxc3VwZXJzY3JpcHRzaGlmdEdGUEYrRitGTkYr . Notation: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiekYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRKiZFbGVtZW50O0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRiw2JVEiVEYnRi9GMg== and 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 . We derive a contour integration formula for the remainder 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 : 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 Here, 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 and 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 . Then we can conclude that 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 where the constant LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== depends on the geometry, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYsNiVRKCZnYW1tYTtGJy9GMFEmZmFsc2VGJy9GM1Enbm9ybWFsRic= depends on separation constants (like before). If 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 and if the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRIm5GJ0YvRjItRiw2JVEjdGhGJ0YvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRis= boundary constant 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 as 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 then 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 . LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The given integration problem is 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 . Approximate the integrand 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 and then integrate: 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 . We replace the calculation of one integral in two dimensions by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21pR0YkNiVRIm5GJy8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRic= integrals each in one dimension: 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 The effectiveness of this scheme relies on the following facts: we have good methods to compute one-dimensional integrals, either in numerical mode or (advantageous in certain situations) symbolic mode the number of terms LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is of modest size (for many integrands of interest) due to the power of this particular interpolation scheme we exploit symmetry and apply efficient linear algebra computations, as discussed next LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

An integral with non-constant limits can be converted to a new integral with constant limits via a change of variables. Specifically, 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 using the change of variables: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYvLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUY2NjBRIj1GJ0Y5RjtGPkZARkJGREZGRkhGSkZNRlBGUkZVLUY2NjBGOEY5RjtGPkZARkJGREZGRkhGSi9GTlEwbWVkaXVtbWF0aHNwYWNlRicvRlFGaG5GUkZVLUYsNiVRImFGJ0YvRjItRjY2MFEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRkpGZ25GaW5GUkZVLUkobWZlbmNlZEdGJDYkLUYjNiUtSSNtbkdGJDYkRlRGOS1GNjYwUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIRkpGZ25GaW5GUkZVLUYsNiVRInNGJ0YvRjJGOUZlbi1GNjYwUScmcGx1cztGJ0Y5RjtGPkZARkJGREZGRkhGSkZnbkZpbkZSRlVGNS1GLDYlUSJiRidGL0YyLUY2NjBGX29GOUY7Rj5GQEZCRkRGRkZIRkpGTUZQRlJGVUZbcA== ; 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 Therefore limit attention to integrals on the unit square. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Making the integrand symmetric: For an integral on the unit square, we can always make the integrand symmetric, meaning 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. Namely, 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 (symmetric part + anti-symmetric part) where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRImZGJ0YvRjItRiw2JVEiU0YnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRistSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSJ4RidGL0YyLUkjbW9HRiQ2MFEiLEYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZRLyUqc3ltbWV0cmljR0ZRLyUobGFyZ2VvcEdGUS8lLm1vdmFibGVsaW1pdHNHRlEvJSdhY2NlbnRHRlEvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR1EzdmVyeXRoaWNrbWF0aHNwYWNlRicvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRiw2JVEieUYnRi9GMkZNLUZKNjBRMSZJbnZpc2libGVUaW1lcztGJ0ZNRk8vRlNGUUZURlZGWEZaRmZuRmhuL0Zcb1EvdGhpY2ttYXRoc3BhY2VGJy9GX29GX3BGYW9GZG8tRko2MFEiPUYnRk1GT0ZdcEZURlZGWEZaRmZuRmhuRl5wRmBwRmFvRmRvLUZKNjBGXHBGTUZPRlJGVEZWRlhGWkZmbkZobkZbb0Zeb0Zhb0Zkby1JJm1mcmFjR0YkNigtRiM2KEYrRistRiM2JUY4LUZKNjBRMCZBcHBseUZ1bmN0aW9uO0YnRk1GT0ZdcEZURlZGWEZaRmZuRmhuRltvL0Zfb0Zdb0Zhb0Zkb0ZBLUZKNjBRJyZwbHVzO0YnRk1GT0ZdcEZURlZGWEZaRmZuRmhuL0Zcb1EwbWVkaXVtbWF0aHNwYWNlRicvRl9vRmVxRmFvRmRvLUYjNiVGOEZdcS1GQjYkLUYjNiVGZ29GSUZGRk1GKy1JI21uR0YkNiRRIjJGJ0ZNLyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zmci8lKWJldmVsbGVkR0ZRRis= ; 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 Integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRImZGJ0YvRjItRiw2JVEiQUYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= over the unit square is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEiMEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw==, which gives us: 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 . Therefore we always have a symmetric integrand on the unit square. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

Consider the double integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== 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

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JComLUkkZXhwR0YkNiMqJilJInhHRiciIiMiIiIpSSJ5R0YnRjJGM0YzLUkkY29zR0YkNiMsJkYxRjNGNUYzRjMvRjU7IiIhLCZGM0YzRjEhIiIvRjE7RjxGMw==

Applying the change of variables to transform to the unit square, and then applying symmetrization, yields the new problem: 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

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JC1JIkZHRic2JEkic0dGJ0kidEdGJy9GLzsiIiEiIiIvRi5GMQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn where 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

Ly1JIkZHNiI2JEkic0dGJUkidEdGJSwkKigjIiIiIiIjRiwtSSRjb3NHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IywoRichIiJGKEY1KiZGJ0YsRihGLEYsRiwsJiomLUkkZXhwR0YwNiMqKClGJ0YtRiwpLCZGLEYsRidGNUYtRiwpRihGLUYsRixGP0YsRiwqJi1GOjYjKihGPUYsKSwmRixGLEYoRjVGLUYsRkBGLEYsRkZGLEYsRixGLA==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Result of computation: Geddes-Newton series for 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 computed to 3 terms The series was integrated (applying one-dimensional quadratures) Computed result: 0.385433 Result is correct to 5 significant digits (with only 3 terms) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

An efficient implementation is achieved by exploiting symmetry. The series resulting from our algorithm has the algebraic form: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== 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

Ly0mSSJzRzYiNiNJIm5HRiY2JEkieEdGJkkieUdGJi1JJFN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYkKigmSSJjR0YmNiNJImlHRiYiIiItRi02JComJkkia0dGJjYkRjZJImpHRiZGNy1JImZHRiY2JEYqJkkiYkdGJjYjRj5GNy9GPjtGN0Y2RjctRi02JComJkkibEdGJkY9RjctRkA2JCZJImFHRiZGREYrRjdGRUY3L0Y2O0Y3Rig=

where 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 are real-valued constants, and 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 is the splitting point used in the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUYsNiVRImlGJ0YvRjItRiw2JVEjdGhGJ0YvRjIvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRis= iteration. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Note that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYjNiVGKy1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2Ji1GIzYmRitGKy1GNjYlLUYsNiVRImFGJ0YvRjItRiw2JVEiaUYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRisvRjNRJ25vcm1hbEYnLyUlb3BlbkdRInxmckYnLyUmY2xvc2VHUSJ8aHJGJ0YrLyUxc3VwZXJzY3JpcHRzaGlmdEdGTEYrRitGSi1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJ0ZNLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Znbi8lKXN0cmV0Y2h5R0Znbi8lKnN5bW1ldHJpY0dGZ24vJShsYXJnZW9wR0Znbi8lLm1vdmFibGVsaW1pdHNHRmduLyUnYWNjZW50R0Znbi8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EvdGhpY2ttYXRoc3BhY2VGJy8lJ3JzcGFjZUdGaW8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictRlg2MFEiPUYnRk1GZW5GaG5Gam5GXG9GXm9GYG9GYm9GZG9GZ29Gam9GXHBGX3AtRlg2MEZaRk1GZW4vRmluRjFGam5GXG9GXm9GYG9GYm9GZG8vRmhvUSQwZW1GJy9GW3BRM3Zlcnl0aGlja21hdGhzcGFjZUYnRlxwRl9wLUYjNiVGKy1GNjYlLUYjNiVGKy1GOzYlLUY+NiYtRiM2JkYrRistRjY2JS1GLDYlUSJiRidGL0YyRkdGSkYrRk1GT0ZSRitGVUYrRitGSkYrRis= . Represent the series using matrices and vectors as 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 where 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 is the column vector of dimension LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== with elements 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiREYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRKCZ0aW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUTBtZWRpdW1tYXRoc3BhY2VGJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Yr diagonal matrix with elements 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiUEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnis a permutation matrix (identity matrix if splitting points all on diagonal) 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 is an LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRKCZ0aW1lcztGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUTBtZWRpdW1tYXRoc3BhY2VGJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJ0Yr unit lower triangular matrix. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Benefits of the matrix-vector representation: The cost of evaluating LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInJGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= is reduced from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEiT0YnRi9GMi1JI21vR0YkNjBRMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZULyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkobWZlbmNlZEdGJDYkLUYjNiVGKy1JJW1zdXBHRiQ2JS1GLDYlUSJuRidGL0YyLUkjbW5HRiQ2JFEiMkYnRj4vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRitGPkYr to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtRiw2JVEiT0YnRi9GMi1JI21vR0YkNjBRMCZBcHBseUZ1bmN0aW9uO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZULyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkobWZlbmNlZEdGJDYkLUYjNiMtRiw2JVEibkYnRi9GMkY+Ris= evaluations of the original function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== . The factors can be computed by exploiting efficient linear algebra operations for matrix-vector multiplication and inner product. We only need to perform LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== one-dimensional integrations of cross-sections of the original function: 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 for 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 . The net effect: Nearly all the processing time is spent evaluating the integrand, which is as desired. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The results in the following table are for the 2-D integral: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== LUkkSW50RzYiNiQtRiM2JC1JJHNpbkdGJDYjLCQqMCIiKSIiIkkjUGlHJSpwcm90ZWN0ZWRHRi5JInhHRiRGLiwmRi5GLkYxISIiRi5JInlHRiRGLiwmRi5GLkY0RjNGLiksJkYxRi5GNEYzIiIjRi5GLi9GNDsiIiFGLi9GMUY6

LUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtRiM2JC1JJHNpbkdGJDYjLCQqMCIiKSIiIkkjUGlHRiVGMUkieEdGJ0YxLCZGMUYxRjMhIiJGMUkieUdGJ0YxLCZGMUYxRjZGNUYxKSwmRjNGMUY2RjUiIiNGMUYxL0Y2OyIiIUYxL0YzRjw=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn Tol Time Result Pts RelErr 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 0.3 0.069551393139 15 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 3.7 0.06955139313890799 21 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 5.3 0.0695513931389079901727 26 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 7.2 0.069551393138907990172712980 33 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW5HRiQ2JFEiNUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRKCZ0aW1lcztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EwbWVkaXVtbWF0aHNwYWNlRicvJSdyc3BhY2VHRkovJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSVtc3VwR0YkNiUtRiw2JFEjMTBGJ0YvLUYjNiQtRjM2MFEoJm1pbnVzO0YnRi9GNkY5RjtGPUY/RkFGQy9GRlEhRicvRklRJDBlbUYnL0ZMRltvRk1GUC1GLDYkUSMzMEYnRi8vJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUkjbWlHRiQ2JUZpbi8lJ2l0YWxpY0dRJXRydWVGJy9GMFEnaXRhbGljRic= 14.2 0.06955139313890799017271297964487 38 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 Tol = requested accuracy tolerance Time = CPU time in seconds Result = computed value of the integral (to the number of digits requested) Pts = number of splitting points RelErr = relative error, compared with an accurate reference value LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

We have adapted our two-dimensional method to the case of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-dimensional integrands LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== which fit either of the following function patterns: 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 . Can generate tensor-product approximations of such functions over the unit hypercube 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 We call the method the Deconstruction / Approximation / Reconstruction Technique (DART). LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

. Deconstruction: Deconstruct LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== to obtain a bivariate function 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 or 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 . . Approximation: Approximate the symmetric function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== by a rank-LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== tensor-product series LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= in two variables on the unit square 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 . . Reconstruction: By a change of variables, reconstruct the original function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== from LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== and produce a LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-variable tensor-product approximation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIlNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== 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 from the bivariate approximation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRInNGJ0YvRjItRiw2JVEibkYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== on 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 .

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

The results below are for the following families of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-dimensional integrands: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn LyZJIkZHNiI2IyIiJiokKS1JJHN1bUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYkKiYmSSJiR0YlNiNJImlHRiUiIiImSSJ4R0YlRjNGNS9GNDtGNUkiZEdGJSIiI0Y1 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIkZGJ0YvRjItSSNtbkdGJDYkUSMxN0YnL0YzUSdub3JtYWxGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2MFEiPUYnRj8vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkovJSlzdHJldGNoeUdGSi8lKnN5bW1ldHJpY0dGSi8lKGxhcmdlb3BHRkovJS5tb3ZhYmxlbGltaXRzR0ZKLyUnYWNjZW50R0ZKLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0Zmbi8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1GIzYlLUYsNiVRI2xuRicvRjBGSkY/LUZFNjBRMCZBcHBseUZ1bmN0aW9uO0YnRj9GSEZLRk1GT0ZRRlNGVUZXL0ZlblEkMGVtRicvRmhuRmlvRmluRlxvLUkobWZlbmNlZEdGJDYkLUYjNiktRjw2JEZbb0Y/LUZFNjBRJyZwbHVzO0YnRj9GSEZLRk1GT0ZRRlNGVUZXL0ZlblEwbWVkaXVtbWF0aHNwYWNlRicvRmhuRmZwRmluRlxvRistSSttdW5kZXJvdmVyR0YkNictRkU2MFEmJnN1bTtGJ0Y/RkhGSy9GTkYxRk8vRlJGMS9GVEYxRlUvRlhRJ3ByZWZpeEYnRmhvL0ZoblEudGhpbm1hdGhzcGFjZUYnRmluRlxvLUYjNiUtRiw2JVEiaUYnRi9GMkZERmBwLUYsNiVRImRGJ0YvRjIvJSdhY2NlbnRHRkovJSxhY2NlbnR1bmRlckdGSkZlby1GNjYlLUYsNiVRInhGJ0YvRjJGZ3FGQUYrRj9GKy1GIzYjRitGKw== 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 Note: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIkZGJ0YvRjItSSNtbkdGJDYkUSI1RicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRIkZGJ0YvRjItSSNtbkdGJDYkUSI2RicvRjNRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGKw== contain coefficients LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1YkdGJDYlLUYsNiVRImJGJ0YvRjItRiw2JVEiaUYnRi9GMi8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnRis= which were assigned non-zero integer values in the range 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 Here, the region of integration is the unit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw==-cube: 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== was chosen based on the criterion that for the given integrand, dimension LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEiMkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRMSZJbnZpc2libGVUaW1lcztGJ0YvLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y4LyUpc3RyZXRjaHlHRjgvJSpzeW1tZXRyaWNHRjgvJShsYXJnZW9wR0Y4LyUubW92YWJsZWxpbWl0c0dGOC8lJ2FjY2VudEdGOC8lJWZvcm1HUSFGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZKLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbWlHRiQ2JVEiZEYnLyUnaXRhbGljR1EldHJ1ZUYnL0YwUSdpdGFsaWNGJw== would require > 3 min Time = CPU time in seconds Result = computed value of the integral (to the number of digits requested) RelErr = relative error, compared with an accurate reference value 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O.A. Carvajal, A New Hybrid Symbolic-Numeric Method for Multiple Integration Based on Tensor-Product Series. Master's thesis, University of Waterloo, Waterloo, Ontario, Canada, 2004. O.A. Carvajal, F.W. Chapman and K.O. Geddes, Hybrid symbolic-numeric integration in multiple dimensions via tensor-product series. Proceedings of ISSAC'05, Manuel Kauers (ed.), ACM Press, New York, 2005, pp. 84-91. F.W. Chapman, Generalized Orthogonal Series for Natural Tensor Product Interpolation. Ph.D thesis, University of Waterloo, Waterloo, Ontario, Canada, 2003. P.J. Davis, Interpolation and Approximation. Blaisdell, New York, 1963. D.H. Thomas, A natural tensor product interpolation formula and the pseudoinverse of a matrix. Linear Algebra and Its Applications, 13 (3), 1976, pp. 239-250. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn