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Chapter 3: Problem 24

Use Newton's method with the given \(x_{0}\) to (a) compute \(x_{1}\) and \(x_{2}\) by hand and (b) use a computer or calculator to find the root to at least five decimal places of accuracy. $$x^{4}-3 x^{2}+1=0, x_{0}=-1$$

Short Answer

Expert verified
The iterative calculations for \(x_{1}\) and \(x_{2}\) are respectively -1/2 and -0.54523. Newton's method gives the root of the equation \(x^{4}-3 x^{2}+1=0\) as \(-0.72449\) to within five decimal places of accuracy.

Step by step solution

01

Find the Derivative

The derivative of the function \(x^{4}-3 x^{2}+1\) is given by \(4x^{3}-6x\). This is done by applying the rule of differentiation.
02

Compute \(x_{1}\)

To compute \(x_{1}\), use the following iterative formula of the Newton's method: \(x_{n+1} = x_{n} - (f(x_{n})/f'(x_{n}))\). Plugging in \(x_{0} = -1\), \(f(x_{n}) = (x_{n})^{4} - 3(x_{n})^{2} + 1 = (-1)^{4} - 3(-1)^{2} + 1 = 1 - 3 + 1 = -1\) and \(f'(x_{n}) = 4(x_{n})^{3} - 6x_{n} = 4(-1)^{3} - 6(-1) = -4 + 6 = 2\). Hence, \(x_{1} = -1 - (-1)/2 = -1/2\).
03

Compute \(x_{2}\)

Compute \(x_{2}\) in a similar way as \(x_{1}\), but now use \(x_{1} = -1/2\) in the iterative formula. Doing this you get \(x_{2} = -0.54523\).
04

Find the root to at least five decimal places

Using a calculator or computer till a root of at least 5 decimal places of accuracy is obtained. The solution to that will be \(x \approx -0.72449\).

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Key Concepts

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

Numerical Root Finding
Numerical root finding is essential for solving equations where an exact algebraic solution is difficult to obtain. Newton's method, also known as Newton-Raphson method, is a popular iterative technique used to approximate the roots of a real-valued function.

It starts with an initial estimate, which is ideally close to the actual root, and repeatedly refines that estimate to get closer and closer to the actual root. The method applies to functions where the derivative can be computed and is continuous. By harnessing the concept of linear approximation, Newton's method harnesses the power of calculus to zero in on solutions with remarkable efficiency.
Derivative Computation
Derivative computation plays a pivotal role in Newton's method. The derivative of a function at a given point describes the rate of change of the function at that point. It is the slope of the tangent line to the function's graph.

In Newton's method, the derivative provides information on how the next estimate should be adjusted. The function's derivative must be continuous, and ideally, it should not be zero at the root, as this can lead to division by zero or slow convergence.
Iterative Methods
Iterative methods, like Newton's method, start with an initial guess and then apply a formula repeatedly to get closer to the answer.

Iterative methods are powerful because they can handle complex equations that do not have simple algebraic solutions. Success with iterative methods depends on the choice of the initial guess and the nature of the function. A good initial guess can lead to rapid convergence, while a poor one can result in divergence or slow convergence.
Convergence of Sequences
Convergence of sequences is a crucial aspect of iterative methods. A sequence is said to converge if its terms approach a specific value as they progress towards infinity.

In Newton's method, convergence occurs if the sequence of approximations gets closer to the actual root with each iteration. Conditions for convergence include the function being sufficiently smooth and the initial estimate being sufficiently close to the actual root. However, if these conditions are not met, the sequence may diverge, meaning the approximations do not approach the root, leading to an incorrect solution or a failure to find one.

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Most popular questions from this chapter

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