Hybrid Symbolic-Numeric Integration in 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Keith O. GeddesSymbolic Computation Group, School of Computer Science
University of Waterloo, Waterloo, ON, N2L 3G1, Canada
<Text-field style="Heading 1" layout="Heading 1">Abstract</Text-field>The Maple scientific computation environment supports both symbolic and numeric mathematical computation. In this presentation, the paradigm of hybrid symbolic-numeric computation is employed to achieve an enhanced level of computational power. The overall strategy is to apply an appropriate combination of symbolic mathematical analysis and numerical computation.The hybrid symbolic-numeric methods illustrated here are part of an automated numerical integration strategy [Geddes86, Geddes92] implemented in Maple for the numerical evaluation of definite integrals. In particular, symbolic analysis techniques are used to deal with integrand singularities.
<Text-field style="Heading 1" layout="Heading 1">Example 1: Finite and Infinite Interval of Integration</Text-field>SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==PkkiZkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCooOSQiIiMtSSNsbkdGJDYjRiwiIiItSSRleHBHRiQ2IywkKiRGLEYtISIiRjFGJEYkRiQ=LUklcGxvdEc2IjYkSSJmR0YkOyIiISIiJQ==Form the integral on 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 and attempt to evaluate in symbolic mode.PkklaW50Zkc2Ii1JJGludEdGJDYkLUkiZkdGJDYjSSJ4R0YkL0YrOyIiISIiJQ==The integral does not evaluate in symbolic mode. Apply direct numerical integration (at standard precision).LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJJWludGZHNiI=The above result was computed by invoking compiled NAG routines.Or at higher precision:LSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiNENiNJJWludGZHNiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnNotice from the graph that the integral on 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 will have approximately the same numerical value as the above result on LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtbkdGJDYkUSIwRicvRjNRJ25vcm1hbEYnLUkjbW9HRiQ2MFEiLEYnRj4vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGRi8lKnN5bW1ldHJpY0dGRi8lKGxhcmdlb3BHRkYvJS5tb3ZhYmxlbGltaXRzR0ZGLyUnYWNjZW50R0ZGLyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUSQwZW1GJy8lJ3JzcGFjZUdRM3Zlcnl0aGlja21hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUZBNjBRMSZJbnZpc2libGVUaW1lcztGJ0Y+RkRGR0ZJRktGTUZPRlFGU0ZWRllGZm5GaW4tRjs2JFEiNEYnRj5GPkYr.Curiously, for the infinite integral a closed form expression can be obtained!PkkpaW5lcnRJbnRHNiItSSRJbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JC1JImZHRiQ2I0kieEdGJC9GLjsiIiFJKWluZmluaXR5R0YoLUkmdmFsdWVHNiI2I0kpaW5lcnRJbnRHRiQ=LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMsKCokSSNQaUdGJCMiIiIiIiMjRioiIiUqJkYoRilJJmdhbW1hR0YkRiojISIiIiIpKiZGKEYpLUkjbG5HNiRGJEkoX3N5c2xpYkc2IjYjRitGKiNGMUYtOr, we may apply direct numerical integration to the infinite integral.LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJKWluZXJ0SW50RzYiNote: Numerical evaluation of the exact formula loses some precision. The result from numerical integration is correct to 10 significant digits.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">Example 2: Logarithmic singularity at left endpoint</Text-field>SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==PkkiZ0c2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCooLUkkc2luR0YkNiM5JCIiIi1JI2xuR0YkRi5GMC1JJGV4cEdGJDYjLCQqJEYvIiIkISIiRjBGJEYkRiQ=LUklcGxvdEc2IjYkSSJnR0YkOyIiISIiJA==Form the integral on 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, attempt symbolic evaluation, and apply numerical integration.PkklaW50Z0c2Ii1JJGludEdGJDYkLUkiZ0dGJDYjSSJ4R0YkL0YrOyIiISIiJA==LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiNJJWludGdHNiI=Evaluating numerically to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEjMjVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= digits yields the following result.LSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiNENiNJJWludGdHNiI=Note that the integrand has a logarthmic singularity at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= as the following series expansion shows.LUknc2VyaWVzRyUqcHJvdGVjdGVkRzYkLUkiZ0c2IjYjSSJ4R0YoL0YqIiIhThe hybrid symbolic-numeric technique includes the concept of term-by-term integration of such a series expansion.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">Example 3: Subtracting off a singularity</Text-field>SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==QyQ+SSJoRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkjbG5HRiU2IywmIiIiRjAtSSRjb3NHRiU2IywkOSQiIiMhIiJGJUYlRiVGNw==PkkpaW5lcnRJbnRHNiItSSRJbnRHRiQ2JC1JImhHRiQ2I0kieEdGJC9GKzsiIiEiIiI=Note that the graph of this integrand goes to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2MFEoJm1pbnVzO0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRkYvJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5RictSSNtbkdGJDYkUSgmaW5maW47RidGLw== at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=. However, this is an integrable singularity: it behaves like 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 near LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk=.LUklcGxvdEc2IjYkSSJoR0YkOyIiISIiIg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnNote: The technique of subtracting off a singularity illustrated below, takes place automatically within Maple's numerical integration routines.Suppose it is desired to compute the result to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEjMjVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= digits of accuracy. The generalized series expansion of 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 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= takes the following form.QyQ+SSdEaWdpdHNHNiIiI0QhIiI=LUknc2VyaWVzRyUqcHJvdGVjdGVkRzYlLUkiaEc2IjYjSSJ4R0YoL0YqIiIhIiIpThe non-regular part isPkkicUc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwkLUkjbG5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2IzkkIiIjRiRGJEYkThe new expressionPkklbmV3aEc2IiwmLUkiaEdGJDYjSSJ4R0YkIiIiLUkicUdGJEYoISIiis analytic on the interval 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. Thus it can be integrated easily by the default numerical integration method.PkkjcjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJEludEc2JEYnSShfc3lzbGliR0YkNiRJJW5ld2hHRiQvSSJ4R0YkOyIiISIiIg==Integrating 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 is easy because it has the indefinite integralLUkkaW50RzYiNiQtSSJxR0YkNiNJInhHRiRGKQ==and its definite integral can therefore be computed symbolically:PkkjcjJHNiItSSRpbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JC1JInFHRiQ2I0kieEdGJC9GLjsiIiEiIiI=Finally, summing the two values, we obtain the value for the original definite integration problem.Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiaEdGKDYjSSJ4R0YoL0YtOyIiISIiIiwmSSNyMUdGKEYxSSNyMkdGKEYxLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">Example 4: Algebraic transformation of variables</Text-field>SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JJXNxcnRHRiQ2Iy1JJHNpbkdGJDYjOSRGJEYkRiQ=PkkpaW5lcnRJbnRHNiItSSRJbnRHRiQ2JC1JIkZHRiQ2I0kieEdGJC9GKzsiIiEiIiM=Note: The change of variables illustrated below takes place automatically within Maple's numerical integration routines.The generalized series expansion of the integrand 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 at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRJmluZml4RicvJSdsc3BhY2VHUS90aGlja21hdGhzcGFjZUYnLyUncnNwYWNlR0ZPLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUkjbW5HRiQ2JFEiMEYnRjk= is of the formLUknc2VyaWVzRyUqcHJvdGVjdGVkRzYlLUkiRkc2IjYjSSJ4R0YoL0YqIiIhIiImApplying the change of variables 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 yieldsLUkkSW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQqJi1JIkZHRic2IyokSSJ0R0YnIiIjIiIiLUklZGlmZkdGJTYkRi1GLkYwL0YuOyIiIS1JJXNxcnRHRiQ2I0YvThe new integrand is analytic on the interval of integration. Therefore it can be integrated easily by the default numerical integration method, even at high accuracy. Suppose that the result is desired to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEjNTBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= digits of accuracy:LSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiNdNiMtSSRJbnRHNiRGJUkoX3N5c2xpYkc2IjYkLCQqJi1JJHNpbkdGKzYjKiRJInRHRi0iIiMjIiIiRjZGNUY4RjYvRjU7IiIhKiRGNkY3LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">Example 5: Integrating a series term-by-term</Text-field>SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==PkkiR0c2ImYqNiNJInZHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJComLUkkZXhwR0YkNiMsJjkkIiIiKiRGMCIiIyMhIiJGM0YxLCZGMUYxLUYtNiNGMCNGMUYzRjVGJEYkRiQ=PkkpaW5lcnRJbnRHNiItSSRJbnRHRiQ2JC1JIkdHRiQ2I0kidkdGJC9GKzsiIiFJKWluZmluaXR5RyUqcHJvdGVjdGVkRw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnNote: The development of the integrand in a generalized series expansion and its integration term-by-term as illustrated below takes place automatically within Maple's numerical integration routines.Suppose it is desired to compute the integral to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEjMjBGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRic= digits of accuracy. First, the interval of integration is split into 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and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEiMUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNjBRIy4uRidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSVmb3JtR1EocG9zdGZpeEYnLyUnbHNwYWNlR1EwbWVkaXVtbWF0aHNwYWNlRicvJSdyc3BhY2VHUSQwZW1GJy8lKG1pbnNpemVHRi4vJShtYXhzaXplR1EpaW5maW5pdHlGJy1GLDYkUSgmaW5maW47RidGLw==. For the finite interval, the default numerical integration method has no difficulty.PkknRGlnaXRzRzYiIiM/PkkkcjAxRzYiLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRJbnRHNiRGJ0koX3N5c2xpYkdGJDYkLUkiR0dGJDYjSSJ2R0YkL0YxOyIiISIiIg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnFor the infinite interval, the change of variables 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 transforms the problem into the following new integration problem.PkkmcjFpbmZHNiItSSRJbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2JCwkKiYtSSJHR0YkNiMqJEkieEdGJCEiIiIiIi1JJWRpZmZHRig2JEYwRjFGM0YyL0YxOyIiIUYzLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLet the integrand appearing here be named LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== . The first few terms of the generalized series expansion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== are as follows.QyQ+SSJnRzYiLUkjb3BHJSpwcm90ZWN0ZWRHNiQiIiJJJnIxaW5mR0YlISIiQyQ+SSJuRzYiIiInISIiPkkic0c2Ii1JNGV2YWxmL2ludC9nZW5zZXJpZXNHRiQ2JUkiZ0dGJEkieEdGJEkibkdGJA==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnNote that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== is a series in increasing powers of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUklbXN1cEdGJDYlLUkjbW9HRiQ2MFEvJkV4cG9uZW50aWFsRTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUlZm9ybUdRJ3ByZWZpeEYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHUTJ2ZXJ5dGhpbm1hdGhzcGFjZUYnLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYjNiUtRjk2MFEoJm1pbnVzO0YnRjxGPkZBRkNGRUZHRklGSy9GTlEmaW5maXhGJ0ZQL0ZURlJGVkZZLUkmbWZyYWNHRiQ2KC1JI21uR0YkNiRGWEY8LUYsNiVRInhGJ0YvRjIvJS5saW5ldGhpY2tuZXNzR1EiMUYnLyUrZGVub21hbGlnbkdRJ2NlbnRlckYnLyUpbnVtYWxpZ25HRlxwLyUpYmV2ZWxsZWRHRkBGKy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGKw== . Maple's symbolic integrator determines that the first two terms of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEic0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== have the following indefinite integrals.QyQ+SSZ0ZXJtc0c2IjcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLUkpc2ltcGxpZnlHNiRGKUkoX3N5c2xpYkdGJTYkLUkjb3BHRik2JEkiaUdGJUkic0dGJUkpc3ltYm9saWNHRiUvRjM7IiIiSSJuR0YlISIiLUkkaW50RzYiNiQmSSZ0ZXJtc0dGJDYjIiIiSSJ4R0YkLUkkaW50RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQmSSZ0ZXJtc0dGJzYjIiIjSSJ4R0YnLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnSimilarly, the integral of each term can be computed symbolically. If we compute the definite integral over [0, 0.25] of the successive terms of the generalized series expansion of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== , we find that the successive values are as follows. For the numerical evaluation of the symbolic formula for each term, and for the addition of the terms to get an accurate floating point result, it is generally necessary to set the working precision higher for this stage of the computation.QyQ+SSdEaWdpdHNHNiIsJEYkIiIjISIiQyQtSSRzZXFHJSpwcm90ZWN0ZWRHNiQtSSRpbnRHNiRGJUkoX3N5c2xpYkc2IjYkLUkjb3BHRiU2JEkiaUdGK0kmdGVybXNHRisvSSJ4R0YrOyIiISQiI0QhIiMvRjA7IiIiSSJuR0YrRjs=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnClearly, the series representation is converging reasonably quickly on this interval (the magnitude of successive terms is decreasing at a good rate).To obtain a numerical result accurate to 20 digits, it is necessary to sum more than six terms of the series as can be seen from the magnitude of the terms above. For this example, 20 terms is more than necessary.n := 20:s := `evalf/int/genseries`(g, x, n):terms := [seq(simplify(op(i,s),symbolic), i=1..n)]:The following loop adds successive terms until the magnitude of the terms becomes negligible for 20 digit accuracy.Digits := 2*Digits:epsilon := 1e-20:t := evalf( int(terms[1], x=0..0.25) ): val := t:for i from 2 to n while abs(t/val) > epsilon do
t := evalf( int(terms[i], x=0..0.25) );
val := val + t
end do:number_of_terms := i-1;We now have the following result for the integral of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== over [0, 0.25] .QyQ+SSdEaWdpdHNHNiIiIz8hIiI=PkkjcjFHNiItSSZldmFsZkclKnByb3RlY3RlZEc2I0kkdmFsR0YkFor the remaining interval [0.25, 1] , ordinary numerical integration methods encounter no difficulties because there are no nearby singularities.PkkjcjJHNiItSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJEludEc2JEYnSShfc3lzbGliR0YkNiRJImdHRiQvSSJ4R0YkOyQiI0QhIiMiIiI=Add these two values to get the value of the integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEmcjFpbmZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= .QyQ+SSZyMWluZkc2IiwmSSNyMUdGJSIiIkkjcjJHRiVGKEYoFinally, we have obtained the answer for the original integration problem.Ly1JJEludEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiR0dGKDYjSSJ2R0YoL0YtOyIiIUkpaW5maW5pdHlHRiYsJkkkcjAxR0YoIiIiSSZyMWluZkdGKEY0LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">Multidimensional Numerical Integration</Text-field>In Maple, one can always express a multiple integral using nested one-dimensional integrals. For example:SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==QyQ+SSJmRzYiZio2JUkieEdGJUkieUdGJUkiekdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiotSSRleHBHRiU2IywoOSQiIiI5JUY0OSZGNEY0LCZGMyIiJkY0RjQhIiIsJkY1IiM1IiIjRjRGOSwmRjYiIzoiIiRGNEY5RiVGJUYlRjk=LUkkSW50RzYiNiQtRiM2JC1GIzYkLUkiZkdGJDYlSSJ4R0YkSSJ5R0YkSSJ6R0YkL0YtOyIiISIiJS9GLjtGMiIiJC9GLztGMi1JJXNxcnRHRiQ2IyIiIw==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnIn older versions (prior to Maple 8), one could invoke the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEmZXZhbGZGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRic= command on a multiple integral such as above; however, the only solution method was to invoke, recursively, one-dimensional numerical integration methods. This was an inefficient approach to numerical multiple integration problems which would succeed only on the simplest problems.In more recent versions, specific methods for numerical multiple integration were introduced. The following command automatically invokes a compiled C routine to compute the multiple integral.LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRJbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUYnNiQtRic2JCoqLUkkZXhwR0YoNiMsKEkieEdGKiIiIkkieUdGKkY2SSJ6R0YqRjZGNiwmRjUiIiZGNkY2ISIiLCZGNyIjNSIiI0Y2RjssJkY4IiM6IiIkRjZGOy9GNTsiIiEiIiUvRjc7RkRGQS9GODtGRCokRj4jRjZGPg==LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnIt is also possible to express multiple integrals using a special list syntax, which is sometimes more convenient.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">Special (list) syntax for multiple integrals</Text-field>A numerical multiple integration problem can be specified in a natural way using nested one-dimensional integrals, as shown above. In 1-D input notation, this takes the general form: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where the integrand LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== depends on 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 . Such a problem can also be specified using the following special multiple integration notation with a list as the second argument: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 .Additional optional arguments can be stated just as in the case of one-dimensional integration. Also as in one-dimensional integration, the integrand LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JlEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUwZm9udF9zdHlsZV9uYW1lR1ErMkR+Q29tbWVudEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJw== can be specified as a procedure in which case the second argument must be a list of ranges: 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 . Whether a multiple integration problem is stated using nested integrals or using the list notation, the arguments are extracted so as to invoke the same numerical multiple integration routines. See the help page 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 for further details and examples.
<Text-field style="Heading 2" layout="Heading 2">Example 6</Text-field>Numerical evaluation of the triple integral given above, expressed in 1-D input notation, takes the following form (again using nested one-dimensional integrals):LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRJbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUYnNiQtRic2JC1JImZHRio2JUkieEdGKkkieUdGKkkiekdGKi9GMzsiIiEiIiUvRjQ7RjgiIiQvRjU7RjgtSSVzcXJ0R0YoNiMiIiM=Using the list syntax, this same computation can be expressed as follows:LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRJbnRHNiRGJEkoX3N5c2xpYkc2IjYkLUkiZkdGKjYlSSJ4R0YqSSJ5R0YqSSJ6R0YqNyUvRi87IiIhIiIlL0YwO0Y1IiIkL0YxO0Y1LUklc3FydEdGKDYjIiIjLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">Example 7</Text-field>SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==QyQ+SSJkRzYiLCYiIiJGJyoqSSJ3R0YlIiIjSSJ4R0YlRipJInlHRiVGKkkiekdGJUYqISIiRi4=QyQ+SSJnRzYiLCYqJkkiZEdGJSIiIi1JJGNvc0dGJTYjKipJIndHRiVGKUkieEdGJUYpSSJ5R0YlRilJInpHRiVGKUYpRikqLkYoRilGLkYpRi9GKUYwRilGMUYpLUkkc2luR0YlRixGKSEiIkYpLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRJbnRHNiI2JC1GJzYkLUYnNiQtRic2JEkiZ0dGKC9JIndHRig7IiIhIiIiL0kieEdGKEYzL0kieUdGKEYzL0kiekdGKEYzThe same computation can be expressed using the list syntax as follows.LUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtSSRJbnRHNiRGJEkoX3N5c2xpYkc2IjYkSSJnR0YqNyYvSSJ3R0YqOyIiISIiIi9JInhHRipGMC9JInlHRipGMC9JInpHRipGMA==It is important to note that expressing the computation in either form leads to the same numerical multiple integration routines being invoked.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">Example 8</Text-field>When low accuracy is sufficient, the Monte Carlo method may be used. For example, suppose that we wish to compute the following multiple integral.SShyZXN0YXJ0RyUqcHJvdGVjdGVkRw==QyQtSSRJbnRHNiI2JC1GJDYkLUYkNiQtRiQ2JC1GJDYkLUYkNiRJImhHRiUvJkkieEdGJTYjIiIiOyEiIkY2LyZGNDYjIiIjRjcvJkY0NiMiIiRGNy8mRjQ2IyIiJUY3LyZGNDYjIiImRjcvJkY0NiMiIidGN0Y4whereQyQ+SSJoRzYiKiQsJiIiIyIiIi1JJHNpbkdGJTYjKihJI1BpRyUqcHJvdGVjdGVkR0YpLUklc3FydEdGJTYjIiMoKUYpLC4mSSJ4R0YlNiNGKUYpJkY2NiNGKEYpJkY2NiMiIiRGKSZGNjYjIiIlRikmRjY2IyIiJkYpJkY2NiMiIidGKUYpRikhIiJGRg==The computation can be expressed as follows.QyQtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1JJEludEc2JEYlSShfc3lzbGliRzYiNiZJImhHRis3KC8mSSJ4R0YrNiMiIiI7ISIiRjMvJkYxNiMiIiNGNC8mRjE2IyIiJEY0LyZGMTYjIiIlRjQvJkYxNiMiIiZGNC8mRjE2IyIiJ0Y0L0knbWV0aG9kR0YrSSxfTW9udGVDYXJsb0dGKy8uSShlcHNpbG9uR0YrJEZFISIkRjU=Only trust about LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbW5HRiQ2JFEiM0YnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJw== digits because the requested accuracy was LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbW5HRiQ2JFEkMC41RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2MFEnJnNkb3Q7RidGLy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOC8lKXN0cmV0Y2h5R0Y4LyUqc3ltbWV0cmljR0Y4LyUobGFyZ2VvcEdGOC8lLm1vdmFibGVsaW1pdHNHRjgvJSdhY2NlbnRHRjgvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRJDBlbUYnLyUncnNwYWNlR0ZKLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUklbXN1cEdGJDYlLUYsNiRRIzEwRidGLy1GIzYkLUYzNjBRKCZtaW51cztGJ0YvRjZGOUY7Rj1GP0ZBRkNGRUZIRktGTUZQLUYsNiRRIjJGJ0YvLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJw== . (The Monte Carlo method is only appropriate for computing a few digits of accuracy.)LSZJJmV2YWxmRyUqcHJvdGVjdGVkRzYjIiIkNiNJIiVHNiI=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">References</Text-field>[Geddes86] K.O. Geddes, Numerical integration in a symbolic context. Proceedings of SYMSAC'86, B.W. Char (ed.), ACM Press, New York, 1986, pp. 185-191.[Geddes92] K.O. Geddes and G.J. Fee, Hybrid symbolic-numeric integration in Maple. Proceedings of ISSAC'92, P.S. Wang (ed.), ACM Press, New York, 1992, pp. 36-41.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn