Problem 30 - Thomas Calculus

Question

Complex roots If you have a computer or a calculator that can be programmed to do complex-number arithmetic, experiment with Newton's method to solve the equation \(z^{6}-1=0 .\) The recursion relation to use is $$ z_{n+1}=z_{n}-\frac{z_{n}^{6}-1}{6 z_{n}^{5}} \quad \text { or } \quad z_{n+1}=\frac{5}{6} z_{n}+\frac{1}{6 z_{n}^{5}} $$ Try these starting values (among others): \(2, i, \sqrt{3}+i\)

Step-by-Step Solution

Answer

Newton's method approximates complex roots of \(z^6 - 1 = 0\) based on initial values, revealing the sixth roots of unity.

Step 1: Understand the Problem

We need to find the sixth roots of unity, which are the solutions to the equation \(z^6 - 1 = 0\). This problem is addressed using Newton's method, which is an iterative technique for finding approximate solutions to equations.

Step 2: Set Up Newton’s Method

Newton's method gives us two recursion formulas to find complex roots: \(z_{n+1} = z_{n} - \frac{z_{n}^{6} - 1}{6z_{n}^{5}}\) and \(z_{n+1} = \frac{5}{6}z_{n} + \frac{1}{6z_{n}^{5}}\). We will use these formulas iteratively with different starting values for \(z\).

Step 3: Choose Starting Values

The problem provides starting values \(2, i, \sqrt{3} + i\). We'll apply the iterative methods from Step 2 starting from each of these values to find one of the complex roots.

Step 4: Implement Recursion for Starting Value 2

Using the recursion formula: Starting with \(z_0 = 2\), compute \(z_1, z_2, \ldots\) until the result stabilizes towards a root. You'll observe the iterations converge to a root of \(z^6 = 1\).

Step 5: Implement Recursion for Starting Value i

Similarly, with \(z_0 = i\), apply the formula iteratively. This will converge to another root of the equation. Newton's method with this starting value should reveal different complex roots depending on the branch of the iteration.

Step 6: Implement Recursion for Starting Value \(\sqrt{3} + i\)

For \(z_0 = \sqrt{3} + i\), repeat the iteration process. Again, it will converge to another root. You might see a different root that complements the full set of solutions.

Step 7: Verify All Roots Found

Each unique converging point should be one of the sixth roots of unity. Validate by confirming \(z^6 = 1\) for the computed roots and check how they fit into the full solution as they should be symmetric relative to the unit circle.

Key Concepts

Complex Numbers

Complex numbers are an extension of real numbers that include both a real part and an imaginary part. The imaginary unit is denoted as \(i\), where \(i^2 = -1\). A complex number is typically expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

These numbers are essential in many areas of mathematics, including in solving polynomial equations that have no real solutions. For example, when we work with quadratic equations like \(x^2 + 1 = 0\), the solutions are not real numbers; instead, they are complex: \(x = i\) or \(x = -i\).

When solving the equation \(z^6 - 1 = 0\), we deal with the sixth powers of complex numbers. Each solution, or "root" of the equation, represents a complex number that, when raised to the sixth power, equals one. This is where complex number arithmetic becomes crucial.

Roots of Unity

Roots of unity are special types of complex numbers that yield 1 when raised to certain powers. The \(n\)-th roots of unity are solutions to the equation \(z^n = 1\). In our case, we are interested in the sixth roots of unity, solutions to \(z^6 = 1\).

There are six distinct sixth roots of unity, which can be expressed using Euler's formula as:

• \(e^{i\frac{2\pi k}{6}}\)

for \(k = 0, 1, 2, 3, 4, 5\).

Geometrically, these roots are evenly spaced points on the unit circle in the complex plane. These points form a regular hexagon centered at the origin (0,0), each at a distance of one unit from the origin. This property of being evenly spaced is what makes the roots of unity symmetric around the circle.

Iterative Methods

Iterative methods are procedures involving repetition to approximate solutions to mathematical problems. Newton's method is a popular iterative technique used for finding successively better approximations of the roots of a real-valued function.

This technique uses initial guesses, also known as starting values, for the root and refines these estimates through repeated iterations. Each iteration involves a recursion formula that improves the approximation:

• \(z_{n+1} = z_n - \frac{z_n^6 - 1}{6z_n^5}\)
• \(z_{n+1} = \frac{5}{6}z_n + \frac{1}{6z_n^5}\)

These formulas are essentially recursive, meaning that each step uses the previous one as a starting point. Iterative methods are particularly useful because they allow complex calculations to be broken down into simpler evaluations, making it easier to handle using computers or calculators.

Recursion

Recursion in mathematics refers to defining a sequence of values by using previous terms in the sequence. This is ideal for solving problems where the solution can be progressively built from solving simpler parts of the problem.

In Newton's method, recursion plays a central role because the formula used is inherently recursive: each new approximation for the root \(z_{n+1}\) is calculated based on the previous approximation \(z_n\). This forms a chain of values leading to the desired solution.

Recursion is not only limited to iterative methods like Newton's; it's a broad concept also used in algorithms and programming for tasks like searching and sorting. Understanding recursion requires understanding how each level in the recursive process builds on the last, and in iterative methods, how it contributes to safely converging to the correct solution.