Question

Compute a square root \(g \in Q(x)\) of \(f=1+4 x \in Q(x)\) modulo \(x^{8}\) such that \(g(0)=1\), using Newton iteration.

Short answer

The square root function \(g(x)\) is approximately \(1 + 2x + \cdots\) calculated iteratively.

Step-by-step solution

Initial Guess

Begin with an initial guess for the square root of the function. Since we want a square root \(g(x)\) such that \(g(0) = 1\), start with the initial polynomial estimate \(g_0(x) = 1\).

Applying Newton's Iteration Formula

Newton's method for approximating the square root involves iterating with the formula: \[ g_{k+1}(x) = \frac{1}{2} \left( g_k(x) + \frac{f(x)}{g_k(x)} \right) \]Using \(f(x) = 1 + 4x\), substitute this into the formula. For the starting iteration, \(g_0(x) = 1\), compute \(g_1(x).\)

First Iteration

Substitute into the formula from Step 2:\[ g_1(x) = \frac{1}{2} \left( 1 + \frac{1 + 4x}{1} \right) = \frac{1}{2} (1 + 4x + 1) = 1 + 2x \] This is the first approximation of \(g(x).\)

Second Iteration

Use the result of the first iteration to compute the next approximation:\[ g_2(x) = \frac{1}{2} \left(1 + 2x + \frac{1+4x}{1 + 2x}\right) \]Now expand and simplify the right-hand side to get the polynomial approximation up to the order \(x^8\).

Perform Polynomial Long Division

Perform polynomial division of \(1 + 4x\) by \(1 + 2x\) and continue simplifying the expression. This involves expanding the series and ensuring that terms higher than \(x^8\) are truncated.

Simplify and Iterate

Once polynomial division is complete, update \(g_2(x)\) using the simplified terms. Proceed by calculating:\[ \text{Simplify the expression and ignore higher order terms beyond } x^8. \] Each iteration refines \(g(x)\) closer to its actual square root modulo \(x^8\).

Further Iterations

Continue more iterations as necessary, evaluating \(g_3(x), g_4(x),\) and so on, maintaining the modulo condition. With each iteration, the approximation \(g_k(x)\) becomes more accurate.

Key concepts

Polynomial Long Division

Polynomial long division is a method similar to numerical long division but applied to polynomials. This technique is essential when simplifying expressions such as fractions with polynomials in the numerator and the denominator. In our exercise, we use it when dividing the polynomial \(1+4x\) by \(1+2x\) during the iterative process of Newton's Method.
First, you write the divisor (e.g., \(1+2x\)) and determine how many times this can "fit" into the leading term of the dividend (\(1+4x\)). This will likely include a guess similar to estimating in long division. The goal is to get the terms of the dividend to be as close to the terms of the chosen divisor times the guessed quotient term.
Here's a simple way to approach polynomial long division:

• Divide the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract from the initial dividend.
• Repeat the process with the resulting new polynomial as the dividend.

It's crucial to truncate terms beyond the specified power, in this case, \(x^8\), to maintain accuracy in calculations. This ensures that our polynomial approximation, \(g(x)\), gets closer to \(\sqrt{1+4x}\) while remaining manageable.

Square Root Approximation

Approximating a square root, especially of a polynomial, involves achieving an estimate that provides a sufficiently accurate result. In our exercise, we aim to approximate the square root of the polynomial \(f(x) = 1 + 4x\) up to the polynomial degree limit of \(x^8\).

The square root approximation is not about finding an exact root but getting closer with each iteration. Newton's iteration formula helps in reaching this approximation by refining the polynomial \(g(x)\), ensuring it satisfies \(g^2(x) \approx f(x)\) modulo \(x^8\). We begin with an initial estimate, in this case, \(g_0(x) = 1\), because at \(x = 0\), \(g(x)\) should equal 1.

The iterative approach then uses the updated formula to methodically improve the guess:\( g_{k+1}(x) = \frac{1}{2} \left(g_k(x) + \frac{f(x)}{g_k(x)}\right) \). The division in the second term requires careful treatment, often employing polynomial long division. The magic in this approach lies in its convergence, as each iteration bares down the error and improves accuracy up to the desired degree.

Iterative Algorithms

Iterative algorithms are at the heart of many numerical methods, including the approximation of square roots using Newton's Method. They involve repeated application of a formula or process, each time using the result of the previous step as the starting point for the next. This looping mechanism is perfect for handling problems where an exact solution is complex or impossible to find directly, such as in polynomial square roots.

In our exercise, iterative algorithms enable us to start with a rough estimate of \(g(x)\) and progressively zero in on the more accurate version of the square root \(\sqrt{1+4x}\). An initial guess provides a baseline, and each iteration refines this with the Newton iteration formula. The iteration stops once the polynomial is appropriately approximated within the confines of the degree (like \(x^8\)).

Benefits of iterative algorithms include:

• Improved accuracy with each step.
• Potential convergence to a correct or near-correct solution.
• Practical handling of problems with limited clear-cut solutions.

Iterative approaches also provide flexibility; adjustments can be made to factors like iteration quantity and function form, tailoring the method to touch on speed or accuracy as required for different solutions.