Numerical Solution of Equations

Numerical Solution of Equations 

the process of finding approximate solutions of algebraic and transcendental equations. The numerical solution of equations reduces to performing arithmetic operations on the coefficients of equations and on the values of their constituent functions; the process makes it possible to find solutions of equations to any predefined accuracy. Many problems of mathematics and its applications reduce to the numerical solution of equations.

Although I. Newton (17th century) was the first to develop a general method of numerical solution of equations, in the early 13th century L. Fibonacci calculated a root of the equation x³ + 2x² + 10x = 20 with an error of less than ⅓ × 10^–10. In the late 16th century J. Bürgi of Switzerland calculated a root of the equation 9 – 30x² + 27x⁴ – 9x⁶ + x⁸ = 0, which defines the length of a side of a regular nonagon. At approximately the same time F. Vieta gave a method of calculating roots of algebraic equations that was similar to Newton’s method.

The numerical solution of an algebraic equation may be divided into the following stages; (1) identification of multiple roots, which reduces the problem to solving an equation with simple roots; (2) determination of boundaries between which roots of the equation may lie; (3) isolation of roots, that is, the determination of intervals such that each contains no more than one simple root (seeSTURWS THEOREM);(4) rough determination of the approximate value of a root, which is performed graphically or by some other method—for example, by studying the changes in sign of the left-hand side of the equation; (5) calculation of the root to the specified accuracy. Among the most widely used methods for the numerical solution of algebraic equations are the method of false position, Newton’s method, Lobachev-skii’s method, the method of successive approximations, and expansion in series.

The numerical solution of transcendental equations is limited to stages (4) and (5). The numerical solution of differential equations is discussed in APPROXIMATE SOLUTION OF DIFFERENTIAL EQUATIONS.

REFERENCES

Entsiklopediia elementarnoi matematiki, book 2: Algebra. Moscow-Leningrad, 1951.
Kurosh, A. G. Kurs vysshei algebry, 11 th ed. Moscow, 1975.