Numerical Analysis Chapter 4.2: Richardson's Extrapolation

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What is the formula for the (n+1)-point open Newton-Cotes formula?

hn+3 f (n+2) (ξ)

n X i=0 ai f (xi )

ni=0 ai f (xi ) (correct)

b Z a f (x) dx

What is the condition for the theorem to hold for even n in numerical integration?

f ∈ C n+3 [a, b]

f ∈ C n+2 [a, b] (correct)

f ∈ C n+1 [a, b]

f ∈ C n [a, b]

What is the error term for the Composite Simpson’s rule?

O(h6)

O(h4) (correct)

O(h5)

O(h3)

How is the interval [a, b] divided in the Composite Simpson’s rule?

into n subintervals Signup and view all the answers

What is the purpose of Algorithm 4.1?

To approximate the integral I = ∫b a f (x) dx Signup and view all the answers

What is the difference between the standard Simpson’s rule and the Composite Simpson’s rule?

The error term of the standard Simpson’s rule is O(h5), while the Composite Simpson’s rule has an error term of O(h4) Signup and view all the answers

What is the value of h in the Composite Simpson’s rule?

h = (b - a)/n Signup and view all the answers

Why can the error terms of the standard Simpson’s rule and the Composite Simpson’s rule not be compared directly?

Because the value of h is fixed in the standard Simpson’s rule, but not in the Composite Simpson’s rule Signup and view all the answers

What is the purpose of Step 3 in the algorithm?

To calculate the values of d1 and c1 Signup and view all the answers

What is the condition for the improper integral to converge with a singularity at the left endpoint?

0 Signup and view all the answers

What is the substitution used to change the improper integral with a singularity at the right endpoint?

z = -x Signup and view all the answers

What is the purpose of Step 9 in the algorithm?

To calculate the value of J Signup and view all the answers

What is the value of the integral when the integrand has a singularity at the left endpoint?

$(b-a)^{1-p}/(1-p)$ Signup and view all the answers

What is the purpose of Step 2 in the algorithm?

To iterate over the values of i from 1 to m Signup and view all the answers

What is the purpose of Step 6 in the algorithm?

To calculate the value of Q Signup and view all the answers

What is the purpose of Step 8 in the algorithm?

To calculate the value of J Signup and view all the answers

What is the purpose of setting h = (b − a)/n in the OUTPUT approximation XI to I?

To divide the interval into equal subintervals Signup and view all the answers

What is the main difference between the Composite Trapezoidal rule and the Composite Midpoint rule?

The way the function is evaluated at the nodes Signup and view all the answers

What is the form of the truncation error for the Composite Trapezoidal rule?

m−1 ∑ Kj h^2j + O(h^2m) Signup and view all the answers

What is the purpose of Richardson extrapolation?

To increase the accuracy of the approximation Signup and view all the answers

What is the result of applying Richardson extrapolation to the Composite Trapezoidal rule?

Romberg integration Signup and view all the answers

What is the primary challenge when integrating a function on an interval with both large and small functional variation?

Predicting the amount of functional variation and adapting the step size Signup and view all the answers

What is the condition on the function f for the application of the Composite Trapezoidal rule?

f ∈ C^2[a, b] Signup and view all the answers

What is the purpose of the tolerance TOL in Algorithm 4.3?

To specify the desired accuracy of the approximation Signup and view all the answers

What is the role of the variable X in the OUTPUT approximation XI to I?

It is used to evaluate the function at the nodes Signup and view all the answers

What is the role of the variable Li in Algorithm 4.3?

To count the number of levels Signup and view all the answers

What is the purpose of the variable XI0 in the OUTPUT approximation XI to I?

It is used to calculate the sum of the function values at the nodes Signup and view all the answers

What happens when the condition |S1 + S2 − v7| < v6 is met in Algorithm 4.3?

The algorithm stops and outputs the approximation Signup and view all the answers

What is the purpose of the variable v7 in Algorithm 4.3?

To keep track of the previous level's approximation Signup and view all the answers

What happens when the limit N is exceeded in Algorithm 4.3?

The algorithm outputs an error message Signup and view all the answers

How does Algorithm 4.3 adapt to regions of high functional variation?

By decreasing the step size Signup and view all the answers

What is the main advantage of using Adaptive Quadrature methods?

They can adapt to regions of high functional variation Signup and view all the answers

What is the potential drawback of using a small step size h in numerical calculations?

Dominance of round-off error Signup and view all the answers

What is the purpose of Richardson's extrapolation?

To generate high-accuracy results using low-order formulas Signup and view all the answers

What is the term used to describe the method of approximating a definite integral?

Numerical quadrature Signup and view all the answers

What is the name of the method that approximates the definite integral by dividing the area into trapezoids?

Trapezoidal rule Signup and view all the answers

What is the error term in the Trapezoidal rule?

-h/12 Signup and view all the answers

What is the purpose of Simpson's rule?

To integrate a function over [a, b] Signup and view all the answers

What is the condition for applying Richardson's extrapolation?

The approximation technique has an error term with a predictable form Signup and view all the answers

What is the role of the YouTube video by Mathispower4u in the context of the Trapezoidal rule?

It illustrates the application of the Trapezoidal rule Signup and view all the answers