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# Example.

The Gaussian integral

was computed with 14 accurate digits.

The results are given below.

 SINGLE PRECISION Tol Trapezoidal Simpson error n error n 0.1E+01 -.2E-01 0.5E-03 0.1E+00 -.2E-01 0.4E-04 0.1E-01 -.1E-02 0.4E-04 0.1E-02 -.3E-03 0.4E-04 0.1E-03 -.2E-04 0.3E-05 0.1E-04 -.1E-05 0.2E-06 0.1E-05 -.2E-06 -.8E-07 0.1E-06 -.2E-06 0.0E+00 0.1E-07 -.1E-04 0.8E-07 0.1E-08 -.1E-04 0.8E-07 0.1E-09 -.1E-04 0.8E-07 0.1E-10 -.1E-04 0.8E-07 0.1E-11 -.1E-04 0.8E-07

 DOUBLE PRECISION TOL Trapezoidal Simpson error n error n 0.1E+01 -.2E-01 0.5E-03 0.1E+00 -.2E-01 0.4E-04 0.1E-01 -.1E-02 0.4E-04 0.1E-02 -.3E-03 0.4E-04 0.1E-03 -.2E-04 0.3E-05 0.1E-04 -.1E-05 0.2E-06 0.1E-05 -.3E-06 0.1E-07 0.1E-06 -.2E-07 0.7E-09 0.1E-07 -.1E-08 0.7E-09 0.1E-08 -.3E-09 0.4E-10 0.1E-09 -.2E-10 0.3E-11 0.1E-10 -.1E-11 0.2E-12 0.1E-11 -.3E-12 0.6E-14

Aside from this presentation, we compare the results above with the results obtained via Gaussian Quadrature integration formula. This method gives excellent results with only several node points; it is therefore much more efficient than both trapezoidal and Simpson rules.

 Gauss Quadrature error n -3.0E-4 2 -1.2E-5 3 4.4E-7 4 -8.0E-9 5

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