In this problem we will investigate using adaptive quadrature to approximate a definite integral. The trapezoid rule can use used to approximate an integral I(f, a, b) = Så f(x) dx by I:(f,a, b) = (f(a) + f(b)) = Qab, which is called a quadrature rule. Adaptive quadrature on a given interval [a, b) proceeds as follows: %3D • calculate the midpoint c = t° • calculate Qab, the trapezoid rule approximation on [a, b] • calculate Qac, the trapezoid rule approximation on [a, c] • calculate Qeb, the trapezoid rule approximation on [c, b] approximate the error as |Qab – (Qac + Qcb)| • if the error approximation is less than ɛ, return the approximation Qac + Qcbi otherwise perform adaptive quadrature on [a, c] and also on [c, b] and return the sum of these two results as the approximation In this question we will study the function f(x) cos(x). %3D Write a program that defines a “math" (anonymous) function for f and another for the true integral of ƒ (where you can just do the integration by hand and hard code the result as, for example, If). The main program should then call a function get input which should • prompt the user for a value for a • prompt the user for a value for b (you may assume they enter a b such that a < b) • keep prompting the user and giving error messages until a positive value is entered for ɛ • return the values for a, b, and ɛ The main program should then call the function adaptive_quadrature which should • take in a function f, values for a and b, and a value for e • print an entry message • implement adaptive quadrature as described above, calling itself recursively when the error approximation is above ɛ • return the approximation of the integral of f from a to b The main program should print the approximation returned by the adaptive quadrature function. Finally, the main program should calculate and print the true definite integral (calculated using the Fundamental Theorem of Calculus) and the absolute error in the approximation. Use the format specifier %e for the error. Sample output:
In this problem we will investigate using adaptive quadrature to approximate a definite integral. The trapezoid rule can use used to approximate an integral I(f, a, b) = Så f(x) dx by I:(f,a, b) = (f(a) + f(b)) = Qab, which is called a quadrature rule. Adaptive quadrature on a given interval [a, b) proceeds as follows: %3D • calculate the midpoint c = t° • calculate Qab, the trapezoid rule approximation on [a, b] • calculate Qac, the trapezoid rule approximation on [a, c] • calculate Qeb, the trapezoid rule approximation on [c, b] approximate the error as |Qab – (Qac + Qcb)| • if the error approximation is less than ɛ, return the approximation Qac + Qcbi otherwise perform adaptive quadrature on [a, c] and also on [c, b] and return the sum of these two results as the approximation In this question we will study the function f(x) cos(x). %3D Write a program that defines a “math" (anonymous) function for f and another for the true integral of ƒ (where you can just do the integration by hand and hard code the result as, for example, If). The main program should then call a function get input which should • prompt the user for a value for a • prompt the user for a value for b (you may assume they enter a b such that a < b) • keep prompting the user and giving error messages until a positive value is entered for ɛ • return the values for a, b, and ɛ The main program should then call the function adaptive_quadrature which should • take in a function f, values for a and b, and a value for e • print an entry message • implement adaptive quadrature as described above, calling itself recursively when the error approximation is above ɛ • return the approximation of the integral of f from a to b The main program should print the approximation returned by the adaptive quadrature function. Finally, the main program should calculate and print the true definite integral (calculated using the Fundamental Theorem of Calculus) and the absolute error in the approximation. Use the format specifier %e for the error. Sample output:
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
Transcribed Image Text:Enter a value for a: 0
Enter a value for b: pi/2
Enter a value for epsilon: -2
Invalid epsilon -2.0000
Enter a value for epsilon: 0
Invalid epsilon 0.0000
Enter a value for epsilon: 1e-2
adaptive quadrature: Entering with a
adaptive quadrature: Entering with a
adaptive quadrature: Entering with a
adaptive quadrature: Entering with a
adaptive quadrature: Entering with a
Adaptive quadrature with a = 0.00 and b = 1.57 gives 0.993953
The true integral with a = 0.00 and b = 1.57 is 1.000000
= 0.000000 and b = 1.570796
= 0.000000 and b = 0.785398
= 0.000000 and b = 0.392699
= 0.392699 and b = 0.785398
= 0.785398 and b = 1.570796
The absolute error in the approximation is 6.046921e-03
![In this problem we will investigate using adaptive quadrature to approximate a definite
integral. The trapezoid rule can use used to approximate an integral I(f, a, b) = S f(x) dx
by I:(f, a, b)
quadrature on a given interval [a, b] proceeds as follows:
b-a
(f(a) + f(b)) = Qab, which is called a quadrature rule. Adaptive
• calculate the midpoint c =
a+b
• calculate Qab, the trapezoid rule approximation on [a, b]
calculate Qac, the trapezoid rule approximation on [a, c]
calculate Qeb, the trapezoid rule approximation on [c,b]
approximate the error as Qab - (Qac + Qcb)|
if the error approximation is less than ɛ, return the approximation Qac +Qcb;
otherwise perform adaptive quadrature on [a, c] and also on [c, b] and return the sum
of these two results as the approximation
In this question we will study the function f(x) = cos(x).
Write a program that defines a “math" (anonymous) function for f and another for the
true integral of f (where you can just do the integration by hand and hard code the result
as, for example, If). The main program should then call a function get_input which
should
prompt the user for a value for a
• prompt the user for a value for b (you may assume they enter a b such that a < b)
keep prompting the user and giving error messages until a positive value is entered
for e
• return the values for a, b, and ɛ
The main program should then call the function adaptive_quadrature which should
• take in a function f, values for a and b, and a value for e
print an entry message
implement adaptive quadrature as described above, calling itself recursively when
the error approximation is above e
return the approximation of the integral of ƒ from a to b
The main program should print the approximation returned by the adaptive quadrature
function. Finally, the main program should calculate and print the true definite integral
(calculated using the Fundamental Theorem of Calculus) and the absolute error in the
approximation. Use the format specifier %e for the error. Sample output:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa7502be5-dc00-4bdc-a983-1aa8633bc5af%2F58636772-8957-461f-8d41-25b711f4f1d8%2Ff0zmqi_processed.png&w=3840&q=75)
Transcribed Image Text:In this problem we will investigate using adaptive quadrature to approximate a definite
integral. The trapezoid rule can use used to approximate an integral I(f, a, b) = S f(x) dx
by I:(f, a, b)
quadrature on a given interval [a, b] proceeds as follows:
b-a
(f(a) + f(b)) = Qab, which is called a quadrature rule. Adaptive
• calculate the midpoint c =
a+b
• calculate Qab, the trapezoid rule approximation on [a, b]
calculate Qac, the trapezoid rule approximation on [a, c]
calculate Qeb, the trapezoid rule approximation on [c,b]
approximate the error as Qab - (Qac + Qcb)|
if the error approximation is less than ɛ, return the approximation Qac +Qcb;
otherwise perform adaptive quadrature on [a, c] and also on [c, b] and return the sum
of these two results as the approximation
In this question we will study the function f(x) = cos(x).
Write a program that defines a “math" (anonymous) function for f and another for the
true integral of f (where you can just do the integration by hand and hard code the result
as, for example, If). The main program should then call a function get_input which
should
prompt the user for a value for a
• prompt the user for a value for b (you may assume they enter a b such that a < b)
keep prompting the user and giving error messages until a positive value is entered
for e
• return the values for a, b, and ɛ
The main program should then call the function adaptive_quadrature which should
• take in a function f, values for a and b, and a value for e
print an entry message
implement adaptive quadrature as described above, calling itself recursively when
the error approximation is above e
return the approximation of the integral of ƒ from a to b
The main program should print the approximation returned by the adaptive quadrature
function. Finally, the main program should calculate and print the true definite integral
(calculated using the Fundamental Theorem of Calculus) and the absolute error in the
approximation. Use the format specifier %e for the error. Sample output:
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