Fri Mar 6 09:42:39 2026 line_nco_rule_test(): python version: 3.10.12 numpy version: 1.26.4 Test line_nco_rule(). line_nco_rule_test01(): line_nco_rule() computes the open Newton-Cotes rules using N equally spaced points for an interval [A,B]. Open Newton-Cotes Rule 1 I X(I) W(I) 0 0 2 Sum(|W)|) = 2.0 Open Newton-Cotes Rule 2 I X(I) W(I) 0 -0.333333 1 1 0.333333 1 Sum(|W)|) = 2.0 Open Newton-Cotes Rule 3 I X(I) W(I) 0 -0.5 1.33333 1 0 -0.666667 2 0.5 1.33333 Sum(|W)|) = 3.3333333333333326 Open Newton-Cotes Rule 4 I X(I) W(I) 0 -0.6 0.916667 1 -0.2 0.0833333 2 0.2 0.0833333 3 0.6 0.916667 Sum(|W)|) = 1.9999999999999993 Open Newton-Cotes Rule 5 I X(I) W(I) 0 -0.666667 1.1 1 -0.333333 -1.4 2 0 2.6 3 0.333333 -1.4 4 0.666667 1.1 Sum(|W)|) = 7.6000000000000085 Open Newton-Cotes Rule 6 I X(I) W(I) 0 -0.714286 0.848611 1 -0.428571 -0.629167 2 -0.142857 0.780556 3 0.142857 0.780556 4 0.428571 -0.629167 5 0.714286 0.848611 Sum(|W)|) = 4.516666666666671 Open Newton-Cotes Rule 7 I X(I) W(I) 0 -0.75 0.973545 1 -0.5 -2.01905 2 -0.25 4.64762 3 0 -5.20423 4 0.25 4.64762 5 0.5 -2.01905 6 0.75 0.973545 Sum(|W)|) = 20.484656084656073 Open Newton-Cotes Rule 8 I X(I) W(I) 0 -0.777778 0.797768 1 -0.555556 -1.25134 2 -0.333333 2.21741 3 -0.111111 -0.763839 4 0.111111 -0.763839 5 0.333333 2.21741 6 0.555556 -1.25134 7 0.777778 0.797768 Sum(|W)|) = 10.060714285714287 Open Newton-Cotes Rule 9 I X(I) W(I) 0 -0.8 0.891755 1 -0.6 -2.57716 2 -0.4 7.35009 3 -0.2 -12.1407 4 0 14.9519 5 0.2 -12.1407 6 0.4 7.35009 7 0.6 -2.57716 8 0.8 0.891755 Sum(|W)|) = 60.87125220458518 Open Newton-Cotes Rule 10 I X(I) W(I) 0 -0.818182 0.758509 1 -0.636364 -1.81966 2 -0.454545 4.3193 3 -0.272727 -4.70834 4 -0.0909091 2.45019 5 0.0909091 2.45019 6 0.272727 -4.70834 7 0.454545 4.3193 8 0.636364 -1.81966 9 0.818182 0.758509 Sum(|W)|) = 28.112009479718452 Open Newton-Cotes Rule 11 I X(I) W(I) 0 -0.833333 0.83342 1 -0.666667 -3.09706 2 -0.5 10.6544 3 -0.333333 -23.0561 4 -0.166667 37.0525 5 0 -42.7742 6 0.166667 37.0525 7 0.333333 -23.0561 8 0.5 10.6544 9 0.666667 -3.09706 10 0.833333 0.83342 Sum(|W)|) = 192.16103896103863 Open Newton-Cotes Rule 12 I X(I) W(I) 0 -0.846154 0.727116 1 -0.692308 -2.35157 2 -0.538462 7.03889 3 -0.384615 -11.9379 4 -0.230769 13.1062 5 -0.0769231 -5.58266 6 0.0769231 -5.58266 7 0.230769 13.1062 8 0.384615 -11.9379 9 0.538462 7.03889 10 0.692308 -2.35157 11 0.846154 0.727116 Sum(|W)|) = 81.48870916505062 line_nco_rule_test02(): Use a sequence of Open Newton Cotes rules for an estimate Q of the integral: I = integral ( 0 <= x <= 1 ) exp(x) dx. The exact value is: I = 1.718281828459045 N Q |Q-I| 1 1.64872 0.0695606 2 1.67167 0.0466086 3 1.71778 0.000505296 4 1.71793 0.00035166 5 1.71828 1.73578e-06 6 1.71828 1.22681e-06 7 1.71828 3.44711e-09 8 1.71828 2.45358e-09 9 1.71828 1.10678e-11 10 1.71828 4.87962e-10 11 1.71828 5.10751e-11 12 1.71828 9.31833e-10 13 1.71828 8.06306e-08 14 1.71828 1.92466e-06 15 1.71829 3.18399e-06 16 1.71832 3.72229e-05 17 1.71724 0.00103932 18 1.71795 0.000327102 19 1.7471 0.0288222 20 1.68564 0.0326399 21 1.70743 0.0108509 22 8.47767 6.75939 line_nco_rule_test(): Normal end of execution. Fri Mar 6 09:42:39 2026