Editing Newton's method (section)

===Slow convergence===
The function {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}} has a root at 0.<ref name="dennis">J. E. Dennis, Jr. and Robert B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations. SIAM</ref> Since {{mvar|f}} is continuously differentiable at its root, the theory guarantees that Newton's method as initialized sufficiently close to the root will converge. However, since the derivative {{math|''f'' ′}} is zero at the root, quadratic convergence is not ensured by the theory. In this particular example, the Newton iteration is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=\frac{1}{2}x_n.</math>
It is visible from this that Newton's method could be initialized anywhere and converge to zero, but at only a linear rate. If initialized at 1, dozens of iterations would be required before ten digits of accuracy are achieved.

The function {{math|''f''(''x'') {{=}} ''x'' + ''x''<sup>4/3</sup>}} also has a root at 0, where it is continuously differentiable. Although the first derivative {{mvar|''f'' ′}} is nonzero at the root, the second derivative {{mvar|''f'' ′′}} is nonexistent there, so that quadratic convergence cannot be guaranteed. In fact the Newton iteration is given by
:<math>x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}=\frac{x_n^{4/3}}{3+4x_n^{1/3}} \approx x_n \cdot \frac{x_n^{1/3}}{3} .</math>
From this, it can be seen that the rate of convergence is superlinear but subquadratic. This can be seen in the following tables, the left of which shows Newton's method applied to the above {{math|''f''(''x'') {{=}} ''x'' + ''x''<sup>4/3</sup>}} and the right of which shows Newton's method applied to {{math|''f''(''x'') {{=}} ''x'' + ''x''<sup>2</sup>}}. The quadratic convergence in iteration shown on the right is illustrated by the orders of magnitude in the distance from the iterate to the true root (0,1,2,3,5,10,20,39,...) being approximately doubled from row to row. While the convergence on the left is superlinear, the order of magnitude is only multiplied by about 4/3 from row to row (0,1,2,4,5,7,10,13,...).

{| class=wikitable style="border: none;"
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''x'' + ''x''{{su|p=4/3|b=''n''}}}}
| rowspan=9 style="border: none;"|
! scope=col | {{math|''x''<sub>''n''</sub>}}
! scope=col | {{math|''x'' + ''x''{{su|p=2|b=''n''}}}}
|-
| 1 || 2 || 1 || 2
|-
| 1.4286 × 10<sup>−1</sup> || 2.1754 × 10<sup>−1</sup> || 3.3333 × 10<sup>−1</sup> || 4.4444 × 10<sup>−1</sup>
|-
| 1.4669 × 10<sup>−2</sup> || 1.8260 × 10<sup>−2</sup> || 6.6666 × 10<sup>−2</sup> || 7.1111 × 10<sup>−2</sup>
|-
| 9.0241 × 10<sup>−4</sup> || 9.8961 × 10<sup>−4</sup> || 3.9216 × 10<sup>−3</sup> || 3.9369 × 10<sup>−3</sup>
|-
| 2.5750 × 10<sup>−5</sup> || 2.6511 × 10<sup>−5</sup> || 1.5259 × 10<sup>−5</sup> || 1.5259 × 10<sup>−5</sup>
|-
| 2.4386 × 10<sup>−7</sup> || 2.4539 × 10<sup>−7</sup> || 2.3283 × 10<sup>−10</sup> || 2.3283 × 10<sup>−10</sup>
|-
| 5.0366 × 10<sup>−10</sup> || 5.0406 × 10<sup>−10</sup> || 5.4210 × 10<sup>−20</sup> || 5.4210 × 10<sup>−20</sup>
|-
| 1.3344 × 10<sup>−13</sup> || 1.3344 × 10<sup>−13</sup> || 2.9387 × 10<sup>−39</sup> || 2.9387 × 10<sup>−39</sup>
|}

The rate of convergence is distinguished from the number of iterations required to reach a given accuracy. For example, the function {{math|''f''(''x'') {{=}} ''x''<sup>20</sup> − 1}} has a root at 1. Since {{math|''f'' ′(1) ≠ 0}} and {{mvar|f}} is smooth, it is known that any Newton iteration convergent to 1 will converge quadratically. However, if initialized at 0.5, the first few iterates of Newton's method are approximately 26214, 24904, 23658, 22476, decreasing slowly, with only the 200th iterate being 1.0371. The following iterates are 1.0103, 1.00093, 1.0000082, and 1.00000000065, illustrating quadratic convergence. This highlights that quadratic convergence of a Newton iteration does not mean that only few iterates are required; this only applies once the sequence of iterates is sufficiently close to the root.<ref>Anthony Ralston and Philip Rabinowitz. A first course in numerical analysis, second edition</ref>