Problem 21 - A History of Mathematics: An Introduction

Question

Solve the equation \(16 x^{2}+192 x-1863.2=0\) numerically using Qin Jiushao's procedure. This equation is taken from his text.

Step-by-Step Solution

Answer

Answer: The approximate root of the quadratic equation is \(x \approx -6.166\).

Step 1: Convert the equation into a homogenous polynomial equation

In this case, we have the equation \(16x^2 + 192x - 1863.2 = 0\). We can divide both sides of the equation by 16 to get the homogenous polynomial equation \(x^2 + 12x - 116.45 = 0\).

Step 2: Rewrite the equation in the form of \(x = \frac{constant + coefficient * x^2}{coefficient}\)

We can rewrite the above equation as \(x = \frac{116.45 - x^2}{12}\).

Step 3: Iteratively plug the value back into the rewritten equation until the solution converges

We start with an initial guess for the value of x. Let's choose \(x_0 = -1\). Now we will calculate the next values of x iteratively using the equation \(x_{n+1} = \frac{116.45 - x_n^2}{12}\) until the solution converges. \(x_1 = \frac{116.45 - (-1)^2}{12} = 9.5375\) \(x_2 = \frac{116.45 - 9.5375^2}{12} = -7.2802\) \(x_3 = \frac{116.45 - (-7.2802)^2}{12} = -5.9741\) \(x_4 = \frac{116.45 - (-5.9741)^2}{12} = -6.2165\) \(x_5 = \frac{116.45 - (-6.2165)^2}{12} = -6.1668\) Continuing this process, we can see that the solution converges to \(x \approx -6.166\), which is one of the roots of the given equation. Qin Jiushao's procedure is an iterative root-finding method and does not guarantee the convergence for all equations. It only provides an approximate solution. But for this particular example, it seems to converge to a reasonable approximation.

Key Concepts

Iterative Root-Finding Methods

When confronted with a quadratic equation like \(16 x^{2}+192 x-1863.2=0\), finding the exact roots algebraically is not always straightforward, especially with more complex equations. Iterative root-finding methods come into play as practical solutions for finding approximate roots numerically. These methods start with an initial guess and refine this guess through a sequence of computations.

An iteration involves plugging the current guess into a formula and using the result as the next guess. This process continues until the changes between successive guesses are below a predefined threshold or until a specific number of iterations have been completed. Iterative methods vary in complexity from simple approaches, like the bisection method, to more sophisticated ones, like Newton-Raphson or Secant methods. Each method has its prerequisites for effective application and convergence criteria. The goal is to strike a balance between computational effort and accuracy to reach a solution that is close enough to the true root of the equation.

Qin Jiushao's Procedure

Qin Jiushao's procedure is a historical iterative method for finding roots of equations and is attributed to the 13th-century Chinese mathematician Qin Jiushao. This method involves rewriting the equation into a form that allows for iterative substitution. In the context of the quadratic equation we've been discussing, the procedure reorganizes the terms to isolate \(x\) on one side, resulting in a relationship of the form \(x = \frac{constant + coefficient \cdot x^2}{coefficient}\).

Once this form is achieved, an initial guess is made, and further values are computed by repeatedly plugging the most current estimate back into the equation. Over successive iterations, the method seeks to refine the guess and converge to an approximate solution. While historical, the procedure illustrates a fundamental strategy for numerical analysis and predates many modern algorithmic approaches. However, it is crucial to recognize that such a procedure may not ensure convergence for all equations and its efficiency depends on the nature of the equation as well as the initial guess.

Homogeneous Polynomial Equations

A homogeneous polynomial equation is an equation that has the same degree for each term when all terms are moved to one side of the equation. For instance, \(16x^2 + 192x - 1863.2 = 0\) is made homogeneous by dividing each term by 16, leading to \(x^2 + 12x - 116.45 = 0\).

Such equations are crucial in various branches of mathematics and physics, as they represent symmetry and balance. The process of homogenization simplifies the equation, making it more amenable to certain types of analysis and solution methods. Homogenous equations are especially important in projective geometry and in the study of vector spaces. The harmonization technique is also used as a starting point in solving equations through iterative methods, optimizing the equation for progressive substitution and facilitating the convergence toward a root.

Numerical Solutions of Equations

Numerical solutions to equations are indispensable when analytical solutions are intractable or impossible to find. These solutions are approximations obtained through computational means rather than exact expressions. For solving equations like quadratic ones, numerical techniques can often quickly provide a good approximation of the roots.

Access to a vast array of numerical methods means mathematicians and scientists can tackle a broad spectrum of problems. These methods span from relatively simple concepts that mimic manual calculations to advanced algorithms that are only feasible with computer assistance. The expressions like \(x_{n+1} = \frac{116.45 - x_n^2}{12}\) are evaluated repeatedly in methods such as Qin Jiushao's procedure, demonstrating the iterative nature of finding numerical solutions. Indeed, numerical analysis bridges the gap between theoretical mathematics and practical problem-solving by offering a way to deal with imperfections in real-world data and models.