Supplementary Information 1 Beyond Newton: a new root-finding fixed-point iteration for nonlinear equations 2

Ankush Aggarwal, Sanjay Pant 3

1 Inverse of x 4

We consider a function 5 1 r(x) = H, (S1) x − which is discontinuous at x = 0. The convergence behavior using three methods are shown (Figs. S1–S3), where 6 discontinuity leads to non-smooth results.

101 | n

x 3 10− − ∗ x

| 7 10−

10 11

Error −

15 10− 0 10 Iterations

Figure S1: Convergence of Eq. (S1) using standard Newton (red), Extended Newton (gray), and Halley’s (blue) methods for H = 0.5, x = 1, and c (1,50) 0 ∈ 7

N EN H 10 10

c 0

10 1 Iterations to converge − 10 0 10 10 0 10 10 0 10 − − − x0 x0 x0

Figure S2: Iterations to converge for Eq. (S1) using (from left) standard Newton, Extended Newton, and Halley’s methods for H = 0.5 and varying x0 and c

N EN H 10 10

c 0

10 1 Iterations to converge − 10 0 10 10 0 10 10 0 10 − − − x∗ x∗ x∗

Figure S3: Iterations to converge for Eq. (S1) using (from left) standard Newton, Extended Newton, and Halley’s methods for x0 = 1 and varying x∗ and c

1 2 Nonlinear compression 8

+ We consider a function from nonlinear elasticity, which is only defined in R : 9 ( x2 1 + H if x > 0 r(x) = − x . (S2) Not defined if x 0 ≤

If the current guess xn becomes negative, the iterations are considered to be non-converged. The resulting convergence 10 behavior is shown (Fig. S4–S6) 11

105 | ) 1 n 10 x ( r | 3 10−

7 10− Function

11 10− 0 10 Iterations

Figure S4: Convergence of Eq. (S2) using standard Newton (red), Extended Newton (gray), and Halley’s (blue) methods for H = 2.9, x = 1, and c (0.01,1) 0 ∈

N EN H 1 40

30

c 20

10

0 1 Iterations to converge 0 1 0 1 0 1 x0 x0 x0

Figure S5: Iterations to converge for Eq. (S2) using (from left) standard Newton, Extended Newton, and Halley’s methods for H = 10 and varying x0 and c

N EN H 1 40

30

c 20

10

0 1 Iterations to converge 0 1 0 1 0 1 x∗ x∗ x∗

Figure S6: Iterations to converge for Eq. (S2) using (from left) standard Newton, Extended Newton, and Halley’s methods for x0 = 0.5 and varying x∗ and c

2 3 Transcendental equation 12

We consider the following transcendental equation with restricted domain 13 ( tan(x) x if x [π,3π/2] r(x) = − ∈ . (S3) Not defined if x < π or x > 3π/2

If the current guess goes outside the domain (xn < π or xn > 3π/2), the iterations are considered to be non-converged. 14 Resulting convergence behavior is shown (Fig. S7). 15

N EN H 3 40

30

c/π 20 2 10

2 1 Iterations to converge 2 3 2 3 2 3 2x0/π 2x0/π 2x0/π

Figure S7: Iterations to converge for Eq. (S3) using (from left) standard Newton, Extended Newton, and Halley’s methods for varying x0 and c

4 Complex Plane 16

3 Applying Newton’s method to a complex function r(z) = z 1 gives us a Newton fractal, and the Extended Newton 17 − and Halley’s methods modify that fractal (Fig. S8). The new schemes not only find the root in fewer iterations, even the 18 dimension of the fractal is reduced (Fig. S9) and the basins of attraction appear more connected (generally speaking) 19 than before. Choosing c in Extended Newton breaks the three-fold symmetry of this system, and the solution close to c 20 is heavily favored. Halley’s method did not have such a bias and reduced the sensitivity equally for all the three roots. 21

4.1 Dimension of Newton Fractal 22

Furthermore, for the above Newton fractal (Fig. S8), we calculate the dimension of the boundary between basins of 23 attraction (i.e. Julia set) using box counting method (Fig. S9). 24

3 N EN H N EN H 1 40 1 z3∗ 30 ) ) 0 0 z z z2∗ 0 20 0 Img( Img(

z∗ 10 1 Converged root

1 1 Iterations to converge 1 − 1 0 1 1 0 1 1 0 1 − 1 0 1 1 0 1 1 0 1 − − − − − − Re(z0) Re(z0) Re(z0) Re(z0) Re(z0) Re(z0)

10 40 10 z3∗ 30 ) ) 0 0 z z z2∗ 0 20 0 Img( Img(

z∗ 10 1 Converged root

10 1 Iterations to converge 10 − 10 0 10 10 0 10 10 0 10 − 10 0 10 10 0 10 10 0 10 − − − − − − Re(z0) Re(z0) Re(z0) Re(z0) Re(z0) Re(z0)

Figure S8: Newton fractal for r(z) = z3 1 in complex plane, (top two rows) colored by number of iterations and − (bottom two rows) colored by the final root: the standard Newton’s, Extended Newton, and Halley’s methods (from left to right). c = 0.65 0.65i is used for EN, and z = 1 + 0i, z = 0.5 √0.75i and z = 0.5 + √0.75i. − − 1∗ 2∗ − − 3∗ −

105 N EN 104 H

103 Box count 102

101 3 2 1 10− 10− 10− Box edge length

Figure S9: Dimension d of the boundary between basins of attraction for r(z) = z3 1 calculated by box counting: − d 1.45 for standard Newton’s method, d 1.22 for Extended Newton method with c = 0.65 0.65i, and d 1.20 ≈ ≈ − − ≈ for Halley’s method

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