Newton’s Method for Complex Roots

6.3.2 Newton’s Method for Complex Roots

Newton’s method is based on approximating f with the linear part of its Taylor expansion. Taylor’s theorem is valid for a complex function f (z) around a point of analyticity (see Sec. 3.1.2). Thus Newton’s method applies also to an equation f (z) = 0, where f (z) is a complex function, analytic in a neighborhood of a root α. An important example is when

f is a polynomial; see Sec. 6.5. The geometry of the complex Newton iteration has been studied by Yao and Ben- Israel [387]. Let z = x + iy, f (z) = u(x, y) + iv(x, y), and consider the absolute value of

f (z) : φ (x, y) = |f (x + iy)| = u(x, y) 2 + v(x, y) 2 .

This is a differentiable function as a function of (x, y), except where f (z) = 0. The gradient of φ(x, y) is

+ vv x , uu y + vv y , (6.3.18) where u x = ∂u/∂x, u y = ∂u/∂y, etc. Using the Cauchy–Riemann equations u x =v y ,

grad φ = (φ x ,φ y ) =

uu x

u y = −v x , we calculate (see Henrici [196, Sec. 6.1.4])

f (z)

u + iv

(uu x + vv x ) + i(uu y + vv y )

A comparison with (6.3.18) shows that the Newton step

z k +1 =z k − f (z k )/f ′ (z k ), k = 0, 1, . . . ,

6.3. Methods Using Derivatives 645 is in the direction of the negative gradient of |f (z k ) |, i.e., in the direction of strongest

decrease of |f (z)|.

Theorem 6.3.5.

be a point such that f (z k )f ′ (z k )

Let the function f (z) = f (x + iy) be analytic and z k =x k + iy k

be the next iterate of Newton’s method ( 6.3.19). Then z k +1 −z k is in the direction of the negative gradient of φ(x, y) = |f (x + iy)| and therefore orthogonal to the level set of |f | at (x k ,y k ). If T k is the tangent plane of φ(x, y) at (x k ,y k ) and L k is the line of intersection of T k with the (x, y)-plane, then (x k +1 ,y k +1 ) is the point on L k closest to (x k ,y k ).

k +1

Proof. See [387]. Newton’s method is very efficient if started from an initial approximation sufficiently

close to a simple zero. If this is not the case Newton’s method may converge slowly or even diverge. In general, there is no guarantee that z n +1 is closer to the root than z n , and if

f ′ (z n ) = 0 the next iterate is not even defined. It is straightforward to generalize the results in Theorem 6.3.2 to the complex case. In the case of a complex function f (x) of a complex variable the interval I 0 = int [x 0 ,x 0 +2h 0 ]

can be replaced by a disk K 0 : |z − z 1 | ≤ |h 0 |.

Theorem 6.3.6.

Let f (z) be a complex function of a complex variable. Let f (z 0 )f ′ (z 0 )

h 0 = −f (z 0 )/f ′ (z 0 ), x 1 =x 0 +h 0 . Assume that f (z) is twice continuously differentiable in the disk K 0 : |z − z 1 | ≤ |h 0 |, and that

2 |h 0 |M 2 ≤ |f ′ (z 0 ) |, M 2 = max z |f ′′ (x) |. (6.3.20)

∈K 0

Let z k

be generated by Newton’s method:

Then z k ∈K 0 and we have lim k →∞ z k = ζ , where ζ is the only zero of f (z) in K 0 . Unless ζ lies on the boundary of K 0 , ζ is a simple zero. Further, we have the relations

|ζ − z k +1 |≤ |z k −z k −1 | , k = 1, 2, . . . . (6.3.21)

2|f ′ (z k ) |

A generalization of this theorem to systems of nonlinear equations is the famous Newton–Kantorovich theorem; see Volume II, Sec. 11.1. Since the Newton step is in the direction of the negative gradient of |f (z)| at z = z k , it will necessarily give a decrease in |f (z k ) | if a short enough step in this direction is taken. A modified Newton method based on the descent property and switching to standard Newton when the condition (6.3.20) is satisfied will be described in Sec. 6.5.5.

646 Chapter 6. Solving Scalar Nonlinear Equations