Numeric Integration and Error Bounds One of the main themes of this chapter has been estimating the theoretical error bounds for various methods of numeric integration. As we have seen in class, the problems that make numeric integration worth doing also get very messy for checking the computations. This worksheet will be used to work through how to use Maple to do some of these problems, We will work with a slight modification of our favorite example from the last worksheet. We will be trying to find NiMtJSRpbnRHNiQtJSRzaW5HNiMqJiUieEciIiMqJiIiJSIiIiUjUGlHRi4hIiIvRio7IiIhRiw= . to an accuracy of .001. Once again, we start by reading in the student package. with(Student[Calculus1]); 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 We next want to define the function that we will be working with. Falling into the usual rut of math classes we will call the function f. f:= sin(x^2/(4*Pi)); NiM+SSJmRzYiLUkkc2luRzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiU2IywkKiZJInhHRiUiIiNJI1BpR0YpISIiIyIiIiIiJQ== For the various error bound, we need limits on the magniftude of the first, second and fourth derivatives of f. We compute them below, and name them f1, f2, and f4 respectively. f1:= diff(f,x); NiM+SSNmMUc2IiwkKigtSSRjb3NHNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjLCQqJkkieEdGJSIiI0kjUGlHRishIiIjIiIiIiIlRjVGMEY1RjJGMyNGNUYx f2:= diff(f,x,x); NiM+SSNmMkc2IiwmKigtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjLCQqJkkieEdGJSIiI0kjUGlHRishIiIjIiIiIiIlRjVGMEYxRjIhIiMjRjNGNiomLUkkY29zR0YqRi1GNUYyRjMjRjVGMQ== f4:= diff(f,x,x,x,x); NiM+SSNmNEc2IiwoKigtSSRzaW5HNiRJKnByb3RlY3RlZEdGK0koX3N5c2xpYkdGJTYjLCQqJkkieEdGJSIiI0kjUGlHRishIiIjIiIiIiIlRjVGMEY2RjIhIiUjRjUiIzsqKC1JJGNvc0dGKkYtRjVGMEYxRjIhIiQjRj1GNiomRihGNUYyISIjRj4= Although the derivatives are straightforward, we would not want to do them by hand. We could, of course, find the maximum of each derivative, by taking its derivative, setting it equal to zero, solving, and then evaluating the original derivative at that point. That is far to much work. Instead we will graph and find a bound visually. We start by graphing the first derivative between 0 and NiMqJiIiJSIiIiUjUGlHRiU= plot (f1, x=0..4*Pi); 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 From looking at the graph it is clear that we can use NiMvJiUiS0c2IyIiIiIiIw== as a bound on the first derivative. We repeat the process with the second derivative. plot (f2, x=0..4*Pi); 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 We see from the graph that we can use NiMvJiUiS0c2IyIiIyIiJQ== as a bound on the secon derivative. Now we bound the fourth derivative. plot(f4, x=0..4*Pi); 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 From the graph we see that we can let NiMvJiUiS0c2IyIiJSIjOQ== . (Recall that we want a bound on the absolute value of the derivative rather than on the derivative itself.) We are now ready to look at the error bounds. The book tells us that: NiMxLSUkYWJzRzYjLCYlIklHIiIiJiUiTEc2IyUibkchIiIqKCYlIktHNiNGKUYpKiQsJiUiYkdGKSUiYUdGLiIiI0YpKiZGN0YpRi1GKUYu and NiMxLSUkYWJzRzYjLCYlIklHIiIiJiUiUkc2IyUibkchIiIqKCYlIktHNiNGKUYpKiQsJiUiYkdGKSUiYUdGLiIiI0YpKiZGN0YpRi1GKUYu . Thus in our case if we want the error to be no more than .001, we need to have NiMxKigiIiMiIiIqJCwmKiYiIiVGJiUjUGlHRiZGJiIiISEiIkYlRiYqJkYlRiYlIm5HRiZGLSRGJiEiJA== . Equivalently, NiMxKioiIiMiIiIqJComIiIlRiYlI1BpR0YmRiVGJiIlKzVGJkYlISIiJSJuRw== . num[left] := evalf(2*1000*(4*Pi)^2/2); NiM+JkkkbnVtRzYiNiNJJWxlZnRHRiYkIiswbjh6OiEiJQ== Thus we need almost 160,000 intervals to get the theoretical error under .001. The case of the midpoint and trapezoid rules are somewhat better. Then the book tells us that: NiMxLSUkYWJzRzYjLCYlIklHIiIiJiUiTUc2IyUibkchIiIqKCYlIktHNiMiIiNGKSokLCYlImJHRiklImFHRi4iIiRGKSomIiNDRikqJEYtRjNGKUYu and NiMxLSUkYWJzRzYjLCYlIklHIiIiJiUiVEc2IyUibkchIiIqKCYlIktHNiMiIiNGKSokLCYlImJHRiklImFHRi4iIiRGKSomIiM3RikqJEYtRjNGKUYu . Thus in our case, if we want the error to be no more than .001, we need to have NiMxKigiIiUiIiIqJCwmKiZGJUYmJSNQaUdGJkYmIiIhISIiIiIkRiYqJiIjQ0YmKiQlIm5HIiIjRiZGLCRGJiEiJA== and NiMxKigiIiUiIiIqJCwmKiZGJUYmJSNQaUdGJkYmIiIhISIiIiIkRiYqJiIjN0YmKiQlIm5HIiIjRiZGLCRGJiEiJA== respectively. Equivalently, NiMxKioiIiUiIiIqJComRiVGJiUjUGlHRiYiIiRGJiIlKzVGJiIjQyEiIiokJSJuRyIiIw== and NiMxKioiIiUiIiIqJComRiVGJiUjUGlHRiYiIiRGJiIlKzVGJiIjNyEiIiokJSJuRyIiIw== respectively. num[midpoint] :=evalf(sqrt(1000*4*(4*Pi - 0)^3/24));
num[trap] := evalf(sqrt(1000*4*(4*Pi - 0)^3/12)); NiM+JkkkbnVtRzYiNiNJKW1pZHBvaW50R0YmJCIrRlclNHYmISIo NiM+JkkkbnVtRzYiNiNJJXRyYXBHRiYkIisxTzFMIikhIig= So we only need about 580 and 820 subintervals respectively with these two rules. We are now ready for Simpson's rule which has an error bound of NiMxLSUkYWJzRzYjLCYlIklHIiIiJiUiU0c2IyUibkchIiIqKCYlIktHNiMiIiVGKSokLCYlImJHRiklImFHRi4iIiZGKSomIiQhPUYpKiRGLUYzRilGLg== . We showed above that we can set NiMvJiUiS0c2IyIiJSIjOQ== . Thus to get an error of .0001, we need NiMxKigiIzkiIiIqJCwmKiYiIiVGJiUjUGlHRiZGJiIiISEiIiIiJkYmKiYiJCE9RiYqJCUibkdGKkYmRi0kRiYhIiQ= . Equivalently NiMxKioiIzkiIiIqJComIiIlRiYlI1BpR0YmIiImRiYiJSs1RiYiJCE9ISIiKiQlIm5HRik= . num[simpson] := evalf((14*(4*Pi)^5*1000/180)^(1/4)); NiM+JkkkbnVtRzYiNiNJKHNpbXBzb25HRiYkIisnSCNIRXEhIik= So we only need 72 subintervals if we use Simpson's rule. <Text-field layout="Heading 2" style="Heading 2"><Font family="Times New Roman">Exercises</Font></Text-field> 1) For the integral, NiMtJSRpbnRHNiQtJSRzaW5HNiMqJiUieEciIiMqJiIiJSIiIiUjUGlHRi4hIiIvRio7IiIhRiw= , above, How many intervals do we need with each rule for an accuracy of .0001? This exercise is like the problem done above. It simply changes the desired degree of accuracy. We know from above that k1 = 2, k2 = 4 and k4 = 14. Plugging into the error formulas we see that for the right and left rules we need NiMxKioiIiMiIiIqJComIiIlRiYlI1BpR0YmRiVGJiIlKzVGJkYlISIiJSJuRw== , for the midpoint and trapezoid rules we need NiMxKioiIiUiIiIqJComRiVGJiUjUGlHRiYiIiRGJiImKysiRiYiI0MhIiIqJCUibkciIiM= and NiMxKioiIiUiIiIqJComRiVGJiUjUGlHRiYiIiRGJiImKysiRiYiIzchIiIqJCUibkciIiM= respectively, while for Simpson's rule we need NiMxKigiIzkiIiIqJCwmKiYiIiVGJiUjUGlHRiZGJiIiISEiIiIiJkYmKiYiJCE9RiYqJCUibkdGKkYmRi0kRiYhIiU= . Solving gives: num[left] := evalf(2*10000*(4*Pi)^2/2);
num[midpoint] :=evalf(sqrt(10000*4*(4*Pi - 0)^3/24));
num[trap] := evalf(sqrt(10000*4*(4*Pi - 0)^3/12));
num[simpson] := evalf((14*(4*Pi)^5*10000/180)^(1/4)); NiM+JkkkbnVtRzYiNiNJJWxlZnRHRiYkIiswbjh6OiEiJA== NiM+JkkkbnVtRzYiNiNJKW1pZHBvaW50R0YmJCIrNCQzJz09ISIn NiM+JkkkbnVtRzYiNiNJJXRyYXBHRiYkIitPMCE+ZCMhIic= NiM+JkkkbnVtRzYiNiNJKHNpbXBzb25HRiYkIisjNHIlXDchIig= So the needed numbers are 1, 579, 137 for the right and left rules, 1819 for the midpoint rule, 2572 for the traezoid rule, and 126 for simpson's rule. 2) Use your favorite numeric integration technique to find the integral, NiMtJSRpbnRHNiQtJSRzaW5HNiMqJiUieEciIiMqJiIiJSIiIiUjUGlHRi4hIiIvRio7IiIhRiw= , with an accuracy of .001? evalf(simpson(sin(x^2/(4*Pi)), x=0..4*Pi, 72));
evalf(int(sin(x^2/(4*Pi)), x=0..4*Pi));
NiMtSShzaW1wc29uRzYiNiUtSSRzaW5HNiRJKnByb3RlY3RlZEdGKkkoX3N5c2xpYkdGJTYjLCQqJkkieEdGJSIiI0kjUGlHRiohIiIjIiIiIiIlL0YvOyIiISwkRjFGNSIjcw== NiMkIis9M3FCPCEiKg== 3) For the integral, NiMtJSRpbnRHNiQtJSRzaW5HNiMqJiUieEciIiMqJiIiJSIiIiUjUGlHRi4hIiIvRio7IiIhKiYiIidGLkYvRi4= , how many intervals do we need with each rule for an accuracy of .001? This problem modifies the interval rather than the accuracy. Thus we need to recompute the bounds on the derivatives. plot(f1, x=0..6*Pi);
plot(f2, x=0..6*Pi);
plot(f4, x=0..6*Pi); 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 Thus we set k1=3, k2=9, k4=80. We now plug into the formulas. num[left] :=evalf(3*1000*(6*Pi)^2/2);
num[midpoint] := evalf(sqrt((9*1000*(6*Pi)^3))/24);
num[trap] := evalf(sqrt((9*1000*(6*Pi)^3))/12);
num[simpsons] := evalf((80*1000*(6*Pi)^5/180)^(1/4)); NiM+JkkkbnVtRzYiNiNJJWxlZnRHRiYkIit5amVIYCEiJQ== NiM+JkkkbnVtRzYiNiNJKW1pZHBvaW50R0YmJCIrU2khXEIkISIo NiM+JkkkbnVtRzYiNiNJJXRyYXBHRiYkIishWzcpcGshIig= NiM+JkkkbnVtRzYiNiNJKXNpbXBzb25zR0YmJCIrOkpOLj0hIig= 4) Use your favorite numeric integration technique to find the integral, NiMtJSRpbnRHNiQtJSRzaW5HNiMqJiUieEciIiMqJiIiJSIiIiUjUGlHRi4hIiIvRio7IiIhKiYiIidGLkYvRi4= , with an accuracy of .001? evalf(ApproximateInt(sin(x^2/(4*Pi)), x=0..6*Pi, method=simpson, partition=180));
evalf(int(sin(x^2/(4*Pi)), x=0..6*Pi)); NiMkIitObFlhRCEiKg== NiMkIiswYVlhRCEiKg== 5) For the integral, NiMtJSRpbnRHNiQtJSRleHBHNiMsJCokJSJ4RyIiIyEiIi9GKzsiIiEiIiQ= ,, how many intervals do we need with each rule for an accuracy of .001? f:= exp(-x^2);
f1:= diff(f,x);
f2:= diff(f,x,x);
f4:= diff(f,x,x,x,x); NiM+SSJmRzYiLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRilJKF9zeXNsaWJHRiU2IywkKiRJInhHRiUiIiMhIiI= NiM+SSNmMUc2IiwkKiZJInhHRiUiIiItSSRleHBHNiRJKnByb3RlY3RlZEdGLUkoX3N5c2xpYkdGJTYjLCQqJEYoIiIjISIiRikhIiM= NiM+SSNmMkc2IiwmLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkKiRJInhHRiUiIiMhIiIhIiMqJkYvRjBGJyIiIiIiJQ== NiM+SSNmNEc2IiwoLUkkZXhwRzYkSSpwcm90ZWN0ZWRHRipJKF9zeXNsaWJHRiU2IywkKiRJInhHRiUiIiMhIiIiIzcqJkYvRjBGJyIiIiEjWyomRi8iIiVGJ0Y0IiM7 plot(f1, x=0..3);
plot(f2, x=0..3);
plot(f4, x=0..3); 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 Thus we set k1=1, k2=2, k4=12. We now plug into the formulas. num[left] :=evalf(1*1000*3^2);
num[midpoint] := evalf(sqrt((2*1000*3^3)/24));
num[trap] := evalf(sqrt((2*1000*3^3)/12));
num[simpsons] := evalf((12*1000*3^5/180)^(1/4)); NiM+JkkkbnVtRzYiNiNJJWxlZnRHRiYkIiUrISoiIiE= NiM+JkkkbnVtRzYiNiNJKW1pZHBvaW50R0YmJCIrIVw7TXUlISIp NiM+JkkkbnVtRzYiNiNJJXRyYXBHRiYkIitKUj8zbiEiKQ== NiM+JkkkbnVtRzYiNiNJKXNpbXBzb25zR0YmJCIrRzQ9RzYhIik= 6) Use your favorite numeric integration technique to find the integral, NiMtJSRpbnRHNiQtJSRleHBHNiMsJCokJSJ4RyIiIyEiIi9GKzsiIiEiIiQ= ,, with an accuracy of .001? evalf(ApproximateInt(exp(-x^2), x=0..3, method=simpson, partition=12));
evalf(int(exp(-x^2), x=0..3));
evalf(ApproximateInt (exp(-x^2), x=0..3, method=midpoint, partition= 50)); NiMkIisnPnQ/JykpISM1 NiMkIismW3Q/JykpISM1 NiMkIisoZXU/JykpISM1