An introduction to the modern theory of equations, by Florian Cajori.

ROOTS OF NUMERICAL EQUATIONS

67

quantity h. Then we have x = a + h. By Taylor's Theorem
f(x) = f (a + h) = f(a) + h/f,(a) + 2 f r(a) +..
Since h is small, we get, by neglecting higher powers of h, ah
approximate value of h from the equation f(a) + hf'(a) = 0,
namely, h = - f( We have approximately x = a - (a)
f'(a) f'().
Letting this new approximation to the value of x be represented by b, we may repeat the above process and secure a
still closer approximation, and so on.
Ex. 1. Solve x - sin x = 2.
The angle x, measured in radians, must lie between 2 and 3. Take
a = 2,5, f (a) =.5 - sin 2.5 =.5 - sin 143 14' = -.097.
f (a) = 1 - cos 2.5 = 1.801.
Hence h =.0539, b a + 7h = 2.5539.
A second approximation gives us
f(b)=-.00054, f'(b) =1.8322, h=.0002947.
Hence x = b + h = 2.554195.
58. Complex Roots of Numerical Equations. Recently methods
for approximating to the complex as well as the real roots of
numerical equations have been perfected.* The exposition of
these methods is too long for a work like this.
* See Emory McClintock, "A Method for Calculating Simultaneously All the
Roots of an Equation," in the American Journal of Mathematics, Vol. XVII.,
pp. 89-110; M. E. Carvallo, Methode pratique pour la Resolution nzmnerique
complete des Equations algebriques ou transcendantes, Paris, 1896.