Algebraic equations

Type of physical science: Mathematical methods

Field of study: Algebra

An algebraic equation is an equation f(x) = 0 where f(x) is a polynomial and used as an elegant tool to model interrelationships of variables of physical systems. Whether solved exactly or by approximation, their solution has a central role in the development of algebra, in the solution of more complex equations, and as standard methodology in all sciences.

Overview

Algebraic equations are mathematical expressions of the form: y = f(x) = a_nxⁿ+ a_n-1x^n-1 + … + a₂x² + a₁x + a₀ = 0, where the exponents are integer constants and the coefficients are algebraic numbers. The function f describes the relationship between the variables commonly termed x and y. These variables often represent physical quantities such as time and distance. Since they are the simplest form of equation, they serve as tools for expressing fundamental relationships in many systems. Consequently, algebraic equations occur in all sciences.

The solution of an algebraic equation requires determining the value of x (the independent variable) for which y (the dependent variable) becomes 0. This can be expanded to any constant value k for y, since y = k can be expressed as f(x) = k, which can be written as f(x)-k = 0, which is also an algebraic equation.

The exact solution of an algebraic equation of the first degree such as y = ax + b = 0 is simply expressed as x = -b/a. For second-degree equations (called quadratics) such as ax to the power 2 + bx + c = 0, two solutions (roots) occur, which are expressed by the familiar formula: x = [^-b±√(b²-4ac)]/2a. Third-degree equations (cubics) and fourth-degree equations (quartics) can be solved by fixed formulas also, but the formulas are far more elaborate and involve several cases.

Solving the general fifth- and higher-degree algebraic equations exactly by formulas involving +, -, *, /, and nth roots was proved impossible in the noted Abel-Ruffini theorem.

Where a general problem is not solvable, one standard practice is to solve large and important subclasses of that problem. In algebraic equations, one such class of equations is xⁿ +/- k = 0, which gives rise to the complex roots of a real number. Another important class of equations has only even powers of x; by using a substitution such as u = x², the degree of those equations can be halved. Similar reduction transformations exist for other classes of algebraic equations. While these algebraic "tricks" are mathematically elegant, scientists benefit most from an alternate approach, the approximation methods, which are not so dependent on the particular structure of an equation.

For equations of the third degree and higher, exact methods have been relegated the role of museum pieces and have not been pursued as a major research area, even by mathematicians. For most scientists, approximation methods are the accepted standard practice.

Approximation methods can solve general equations that are too large, complex, or time-consuming to solve exactly. Approximation methods provide solutions that can be computed to ten, twenty, or an arbitrary number of decimal places for equations of any degree. In the real world, applications rarely require accuracy beyond twenty digits. With the advent of high-speed computers and powerful numerical algorithms, approximation methods are the preferred technique for solving most equations.

Approximation methods generally require an initial guess, a number close to the root.

Some methods are more tolerant than others, permitting a rougher initial estimate to start the process. In some physical systems, the real-world constraints or knowledge particular to the system considered may help in formulating the initial guess. Alternatively, there are a few methods that can compute initial guesses. That initial approximate number is then repeatedly refined by using formulas to reach a number closer to the root.

Among the classical approximation methods for algebraic equations, the most popular are Horner's method, Newton's method, the secant method, and the bisection method.

Like most approximation methods, Horner's method starts off with an initial rough guess to the root. Just as long division with numbers is repeated subtraction, Horner's method repeatedly "subtracts" from polynomial f(x), though in a more complicated way. As each numerical portion (that is, digit of accuracy) of the answer is computed, the polynomial is modified to assume a root smaller by that portion. The portions get smaller as one approaches the root to the desired accuracy, the last coefficient (the constant term) becoming very close to zero as the root is approached. No more portions for the desired root are required when a user-predetermined cutoff point in accuracy is reached.

Newton's method starts with a point on a curve f(x) and uses the slope at that point to make a straight line. Where that line hits the x-axis is a good estimate of the root. To make a better estimate, that estimate is used as the new point on the curve f(x), and the process is repeated until the desired accuracy is reached. As an alternate method, the secant method works similarly but uses two points on the curve to compute that slope.

The bisection method is popular as an introductory programming exercise. If f(x) is negative for one value of x and f(x) is positive for another value of x, then f(x) must be zero somewhere in between. Consider that interval from one x value to the other. Take that interval and slice it in half, discarding the half that does not have the change in sign for f(x). Repeating this halving process eventually gives a small enough interval that a "single number" results, which is the solution.

This method corresponds to the process of elimination in deductive reasoning. A variant of bisection called regula falsi works faster by interpolating where to slice the interval, eliminating larger fruitless intervals using proportions.

Some modern methods of approximation expand on the principles of classical methods.

Muller's method is based on Newton's method but uses a parabola instead of a straight line to approximate where the root should be. Since the curvature of a parabola more closely matches that of the function, each iteration of Muller's method will get closer to the root than the line in Newton's method could. Another improved algorithm--the Bairstow method for finding complex roots--uses an enhanced version of synthetic division, the core procedure of an efficient implementation of Horner's method.

Some modern approximation methods can start with very rough initial guesses. The quotient-difference algorithm takes an innovative approach by repeatedly computing quotients and differences of a row of values to arrive at all roots simultaneously. Even more exotic is Graeffe's method. In it, each iteration squares the roots of the equation from the previous iteration until the final equation has easily computed roots.

While proximity of initial guesses to actual roots is important, other issues are also critical. Choosing between approximation methods often depends upon two criteria, speed and effectiveness. Newton's method may be faster than bisection generally, but bisection guarantees an answer. On some curves, Newton's method may flounder because of the shape of the curve for f(x), wasting much computing time and possibly never converging to a single number. Prior to electronic computers, the bisection method was rarely used because of the number of computations of f(x) on fractional values. Therefore, technology also influences the choice of approximation methods. Horner's method is easy for calculation on paper but other methods are computationally faster. Computers caused a shift from the methods easier for hand-computation to other methods that do not depend upon the simplicity of each particular arithmetic step. Newer computers exploiting parallel architecture have opened the way for new, faster algorithms to approximate solutions.

Besides computing solutions, algebraic equations can be studied with respect to the qualitative nature of the roots and factorization based on those roots. The beauty of knowing a root r of polynomial f(x) of degree n is that the polynomial can then be factored as f(x) = (x-r) * p(x), where p(x) is a polynomial of degree n-1. Repeating this can eventually produce the factorization f(x) = (x - r₁) (x - r₂) (x - r₃) … (x - r_n-1) (x - r_n). In this way, a polynomial of degree n has exactly n roots, but they need not all be distinct. For example, the equation f(x) = x² - 6x + 9 has two roots, both 3, giving the factorization f(x) = (x - 3) (x - 3).

An algebraic equation has exactly n roots. In the most general case, the coefficients will be algebraic complex numbers and the solutions will be algebraic complex numbers. Even if the coefficients are restricted to integers, complex algebraic numbers could result as roots. This near-circular relationship between algebraic equations and algebraic numbers is known as a closure property. Intuitively, one cannot leave the class of algebraic numbers when using them in algebraic equations in any way; the computations will remain algebraic.

There are numbers that are not algebraic, called transcendental numbers (such as π and e). These numbers cannot be written as finite expressions of algebraic numbers. The number π can be written only as an infinite expression (called an infinite series). These numbers can be roots only of equations called transcendental equations. For example, π is a root of sine (x) = 0. Functions such as sine, other trigonometric functions, logarithm, and exponential functions are likewise called transcendental. Their importance in science is irrefutable; however, when dealing with anything transcendental, only a finite approximation is ever used. The value 3.1415 is often used for pi. Logarithms are approximated to several decimal places. Frequently, transcendental functions are approximated by finite polynomials for computational purposes to as many decimal places as desired. Since paper and computer memory are finite, all numerical computation is inherently finite and algebraic in realty.

Many theorems associate properties of the coefficients of algebraic equations with properties of the roots. When the coefficients of f(x) are all real numbers, complex roots can still occur but only as pairs called conjugate pairs. An algebraic equation's roots will always be real or complex algebraic numbers, even if the coefficients are complex numbers.

Descartes' rule associates the signs of the roots with the signs of the coefficients. Furthermore, the coefficients can be explicitly represented by sums of products of the roots; unfortunately, this does not effectively compute the roots from the coefficients in the general case.

Prior to any discussion of applications, a qualitative perspective on the roots is required. While an nth-degree equation may have n roots, in reality, some of those roots may be irrelevant in a specific application. Sometimes, some roots have no physical meaning. At other times, several roots have real-world interpretations, but only one root is the correct one in the specific state being studied for a system, a common occurrence in systems like dampening oscillations.

Applications

Polynomials occur in sciences as a modeling tool used to represent relationships between variables. Algebraic equations occur in most scientific discourse as mathematical expressions of theoretical hypothesis formulation, derivations, and deductions. It would be difficult to imagine any scientific treatise without algebraic equations. Popular examples of quadratic problems occur in systems with acceleration such as motion influenced by gravity.

Cubic equations commonly arise in classical volume problems, often involving volumes of partially submerged objects.

Closer to the laboratory, polynomials are useful for curve-fitting, polynomial interpolation, and nonlinear regressions. For example, suppose a scientist sets input variables for a system to several values and observes several corresponding output values. The data of the several experiments can then be plotted on paper. The next step in hypothesis formation (or confirmation, if a theory is being tested) would be to determine a formula that fits the data.

Formulation as a polynomial is often chosen because the theory of algebraic equations is relatively well understood, numerous computer algorithms exist for them, and polynomials are "well behaved." Polynomials have no discontinuities, no unpleasant breaks in their curves, and no infinite values for f(x) when x is finite. Given these nice properties, modeling by polynomial is usually adopted and one of the several methods cited earlier is used to construct a polynomial that intercepts all data points.

Determining where f(x) = 0 is useful in finding maximal or minimal values of a function. Computing optimal values is useful in theoretical and applied aspects of sciences.

Finding zeros of functions is frequently associated with equilibrium points. While familiar in chemistry and economics, equilibriums are also prevalent in systems involving forces and motion in physics. Determining the forces needed to put an object into an at-rest state on an inclined plane is one example; another example is balancing forces in flight. In chemistry, calculations based on equilibrium constants frequently generate algebraic equations of degree three and above.

Sometimes, algebraic equations serve an intermediate-step role in the solution of harder equations. They are useful in solving some linear differential equations. Differential equations are frequently employed in describing motion, rates of growth, and circuit analysis. They are often difficult to solve. One standard technique is to assume the answer is an expression. When that expression is plugged into the linear differential equation, the problem may reduce to an algebraic equation, which is then solved. This tactic is also used for difference equations, a modeling tool for investigating discrete sequences. One famous application of algebraic equations is the conversion of the Fibonacci series (an important series used in the analysis of growth patterns and developmental forms) from a difference equation to an exponential function.

Solving algebraic equations is required for the computation of eigenvalues, the roots of special polynomials called characteristic polynomials. Characteristic polynomials are essential tools in linear algebraic and matrix applications in sciences. An electric circuit may be described by a set of differential equations. These equations, under the proper substitutions, can be rewritten as matrices. To determine the frequencies of vibrations (in general), eigenvalues for those matrices are computed by solving their algebraic equations. In studying springs with masses, molecular systems, circuits, or other structures, algebraic equations are used in the determination of the modes of vibration and oscillation.

Polynomial approximations appear in scientific visualization and graphics. Where line and parabola segments are not sufficient, another powerful tool is often employed, called cubic splines. Cubic splines are formed by segments of third-degree polynomial curves. These segments can fit so well (smoothly) with each other because each segment's third-degree polynomials give more control over curvature. Besides graphics, they serve an important role in curve fitting. Rather than use an nth degree polynomial to fit n data points, cubic splines can be used. Where n is large, the formulas are more manageable and are easier to process as a set of cubics than as one giant polynomial of high degree.

Algebraic equations are also a development platform for work in other equations. In the area of approximation, methods originally used for algebraic equations were later applied to other equations. Though first developed for cubic polynomials, Newton's method is used in solving transcendental (or nonalgebraic) equations and even applied in apparently unrelated areas such as nonlinear optimization. Sometimes, theorems in the well-understood and simpler theory (of algebraic equations) can be (with some modification) "lifted" into the theory of transcendental equations.

Algebraic equations contributed to the evolution and axiomatization of number systems. Complex numbers were first defined by them. Irrational numbers and even the rationals and negative numbers are usually defined and formulated using algebraic equations. If A is a positive integer, x + A = 0 can be solved by negative numbers. Likewise, for integers A and B, one can solve Ax + B = 0 for x by letting x be a fraction or rational number. To solve x² - 2 = 0, a larger set of numbers called irrational numbers is constructed. To solve equations such as x to the power of 2 + 1 = 0, complex numbers are required.

Context

The earliest references to algebraic equations date back to Babylonian antiquity.

Cuneiform tablets show the general solution quadratics and several instances of success with higher-degree equations. Ancient Greek texts demonstrate the application of geometric algebra to the solution of special cubics. Documents referred primarily to the computation of areas, volumes, arc lengths, structural engineering, and marketplace inventory. During the resurgence of reason associated with the Renaissance, the general cubic equation was finally solved.

Geronimo Cardano's publication of that solution and Lodovico Ferrari's solution to the general quartic in 1545 ushered in the modern period of mathematics. Whereas zero, negative numbers, and irrational numbers were begrudgingly accepted, formulas for the cubic made complex numbers necessary.

Concomitant with the progress in algebraic equations was the evolution of algebraic notation. The common language word problems and primitive numerical notation were being displaced by modern equational language (using +, -, *, =, other operations, variables, functions) and the decimal system. With the establishment of this specialized language of equations, mathematical and scientific progress accelerated. Unfettered by traditional language, the mental capacity for scientific reasoning and discourse was magnified. Equations could refine and isolate mathematical relationships from the problem description, clarifying theories and making them easier to modify.

Subsequent efforts at exactly solving higher-degree equations enriched mathematics but proved futile. After the cumulative work of Joseph-Louis Lagrange, Niels Henrik Abel, Paolo Rufini, and Evariste Galois, a comprehensive theory and proof of the unsolvability of quintic and higher-degree equations by radicals was completed in the first half of the nineteenth century.

Modern group theory branched off from this point. Afterward, Charles Hermite solved the general quintic exactly but required elliptic functions in the answer. Further work on the general nth-degree case continued in automorphic function and group theory.

Approximation methods have a long history. Ancient applications of approximation methods occur in early Egyptian and Babylonian texts. Presently, some methods still retain Latin names, such as regula falsi. Horner's method was routinely used in thirteenth century China. In the following centuries, growing interest in infinite series and operations approximation became a central discipline of mathematics. Sir Isaac Newton's work with derivatives led to his ubiquitous approximation method. The rise of analysis generated many other classical methods, corresponding to the growing demand by science and engineering.

The proliferation of computers led to an explosive interest in approximation methods.

A new discipline (complexity of computation) was formulated to assess the computational effectiveness of approximation techniques. New algorithms for numerically solving larger and more complex equations more efficiently resulted. High-resolution computer graphics and scientific visualization require the solution of large numbers of equations instantaneously.

More powerful methods exploiting advances in computer architecture are continually being developed. High-speed parallel processors, supercomputers, multiple CPU (central processing unit) computers, and vector processors running advanced algorithms are the equation solvers of the future.

Principal terms

ALGEBRAIC NUMBER: a number that can be written as a finite expression using +, -, *, /, and nth root operations over integers: for example, 2.3, 1/2, and (3 + √2)/4

DEGREE: the largest power in a polynomial

POLYNOMIAL: an expression of the form f(x) = a_nxⁿ+ a_n-1x^n-1 + … + a₂x² + a₁x + a₀, where the powers are positive integers and coefficients a sub i are algebraic numbers

ROOT: the synonym for the solution of an algebraic equation

SOLUTION: a numerical value of a variable (often termed x) that satisfies an algebraic equation

ZERO OF A POLYNOMIAL: the synonym for solution of the algebraic equation using that polynomial

Bibliography

Aleksandrov, A. D., A. N. Kolmogorov, and M. A. Lavrentev. MATHEMATICS: ITS CONTENT, METHODS, AND MEANING. Cambridge, Mass.: MIT Press, 1963. Among the most accessible sourcebooks in mathematics, this multivolume text presents algebraic equations at a leisurely pace and a clear expository style in chapter 4. Unfortunately, reading pleasure is somewhat marred by a biased perspective. Chapters 8 and 14 (on approximation methods and computing) are seriously dated. The approach is descriptive.

Cajori, Florian. AN INTRODUCTION TO THE THEORY OF EQUATIONS. 1904. Reprint. New York: Dover, 1969. This classic is a lucid exposition of the area. Numerous simple exercises occur throughout. Most exercises are calculations or basic reasoning problems, not unfilled gaps in proofs, a common practice in some modern texts. Some topics may seem esoteric and dated, remnants of a more practical and concrete era of mathematics. Classical algebra is emphasized.

Childs, Lindsay. A CONCRETE INTRODUCTION TO HIGHER ALGEBRA. New York: Springer-Verlag, 1979. An excellent collection of topical chapters in modern algebra, each well written and self-contained, conducive to armchair readings. Stresses discrete, number theoretic (especially fields), and combinatorial applications, but not approximation methods.

Conkwright, Nelson Bush. INTRODUCTION TO THE THEORY OF EQUATIONS. Boston: Ginn, 1957. Filled with examples, worked-out problems, and many exercises, this text is a readable and practical introduction to the techniques and theorems involved in algebraic equations and, foremost, their solution.

Dobbs, David, and Robert Hanks. A MODERN COURSE ON THE THEORY OF EQUATIONS. Passaic, N.J.: Polygonal Publishing, 1980. One of the few books devoted to the theory of equations exclusively. While not totally easy reading, the polished style is coupled with occasional humor and flair. Some chapters are quite readable; others may prove a bit too algebraic for general tastes. With the decline in interest in exact solutions, new texts in the area are rare. On the other hand, approximation methods occur in nearly every numerical analysis text.

MacDuffee, Cyrus. THEORY OF EQUATIONS. New York: Wiley, 1954. A practical text that is easier to read than the Conkwright book but written in the same spirit and style. The proofs are occasionally rather pithy but are fine.

Uspensky, J. V. THEORY OF EQUATIONS. New York: McGraw-Hill, 1948. An old favorite similar in style to Conkwright or MacDuffee but with a more comprehensive coverage of theory. The deeper, more extensive content is balanced with more careful exposition and numerous exercises.

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