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Chapter 4: Problem 14

State the degree of precision of the closed Newton-Cotes formula on 5 nodes, Bode's Rule.

Short Answer

Expert verified
Bode's Rule has a degree of precision of 7.

Step by step solution

01

Understanding the Closed Newton-Cotes Formula

The closed Newton-Cotes formulas are a group of numerical integration methods that approximate the integral of a function. They use equally spaced points, or nodes, within the integration interval. On 5 nodes, this integration rule is referred to as Bode's Rule.
02

Identifying the Degree of the Polynomial

The degree of precision of a numerical integration formula like Bode's rule is defined as the highest degree of a polynomial that the formula can integrate exactly. For the closed Newton-Cotes formula with n+1 nodes, the degree of precision is at most n, where n is the number of subintervals or nodes minus one.
03

Calculating the Degree of Precision for 5 Nodes

Since Bode's Rule involves 5 nodes, it corresponds to dividing the interval into 4 subintervals. Thus, the polynomial degree that Bode's Rule can exactly integrate is 2n-1, where n=4. Therefore, the degree of precision is 2(4)-1 = 7.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Newton-Cotes formulas
Newton-Cotes formulas are essential tools in numerical integration, designed to approximate the integral of functions using polynomial interpolants. These formulas come in both open and closed varieties, determined by whether they include the endpoints of the interval of integration.
  • In the closed form, the function is evaluated at the interval's endpoints.
  • These formulas utilize evenly spaced nodes within the integration interval.
  • The number of nodes determines the complexity and potential accuracy of the approximation.
A few of the common closed Newton-Cotes formulas include the Trapezoidal Rule, Simpson's Rule, and Bode's Rule. Closed Newton-Cotes formulas are popular due to their straightforward application and the ability to approximate integrals with satisfying accuracy.
Degree of Precision
The degree of precision of a numerical integration method refers to the highest degree polynomial it can integrate exactly. In other words, it tells us the method's accuracy in terms of polynomial functions.For closed Newton-Cotes formulas:
  • The degree of precision is influenced by the number of nodes, particularly as it relates to the intervals formed.
  • If a formula uses \(n+1\) nodes, it can theoretically integrate polynomials up to degree \(2n - 1\) exactly.
This is because the integration formula works by constructing an interpolating polynomial through the given nodes and integrates it. Knowing the degree of precision helps choose the right integration method for a problem to ensure the desired accuracy level.
Bode's Rule
Bode's Rule is a specialized member of the closed Newton-Cotes formulas family which uses 5 equally spaced nodes to perform the integration.
  • It is employed for better accuracy when dealing with more complicated functions, compared to simpler methods like Simpson's Rule.
  • By utilizing 5 nodes, Bode's Rule divides the integral interval into 4 subintervals.
  • The degree of precision for Bode's Rule is 7, since it can integrate polynomials up to degree 7 exactly.
This makes Bode's Rule a valuable option for accurately approximating integrals of higher-order polynomial functions while still maintaining manageable computational complexity.

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Most popular questions from this chapter

Let \(I=\int_{0}^{2} x^{2} \ln \left(x^{2}+1\right) d x\). (a) Approximate \(I\) using the Midpoint rule. (b) Use your answer to (a) to estimate the number of subintervals needed to approximate \(I\) to within $$ 10^{-4} \text {. NOTE: } I=\frac{24 \ln (5)-6 \tan ^{-1}(2)-4}{9} \text { . } $$

Suppose that \(N(h)\) is an approximation of \(M\) for every \(h>0\) and that $$ M-N(h)=K_{1} h+K_{2} h^{2}+K_{3} h^{3}+\cdots $$ for some constants \(K_{1}, K_{2}, K_{3}, \ldots .\) Use the values \(N(h), N(h / 3),\) and \(N(h / 9)\) to produce an \(O\left(h^{3}\right)\) approximation of \(M .[\Lambda]\)

Find the degree of precision of the quadrature formula $$ \int_{-1}^{1} f(x) d x=f\left(-\frac{\sqrt{3}}{3}\right)+f\left(\frac{\sqrt{3}}{3}\right) $$

Derive an \(O\left(h^{3}\right)\) formula for approximating \(M\) that uses \(N(h), N\left(\frac{h}{2}\right),\) and \(N\left(\frac{h}{3}\right),\) and is based on the assumption that $$ M=N(h)+K_{1} h+K_{2} h^{2}+K_{3} h^{3}+\cdots $$

State the degree of precision of the five point formula. $\$$ $$ \begin{array}{r} f^{\prime}\left(x_{0}\right)=\frac{1}{12 h}\left[-25 f\left(x_{0}\right)+48 f\left(x_{0}+h\right)-36 f\left(x_{0}+2 h\right)\right. \\ \left.\quad+16 f\left(x_{0}+3 h\right)-3 f\left(x_{0}+4 h\right)\right]+\frac{h^{4}}{5} f^{(5)}(\xi) \end{array} $$
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