On extensions of the Newton-Raphson iterative scheme to arbitrary - - PowerPoint PPT Presentation

On extensions of the Newton-Raphson iterative scheme to arbitrary orders

Gilbert Labelle, LaCIM-UQAM, Montréal (Québec) Canada FPSAC’10, San Francisco, August 2010

1 / 15

Definition

Let tn → a. The convergence is said to be of order p if tn+1 − a = O ((tn − a)p) , as n → ∞.

Theorem (Classical Newton-Raphson)

Let U ⊆ R be open and f : U → R be twice differentiable. If a ∈ U is a simple root of f (t) = 0, then the iterative scheme, tn+1 = N (tn) , n = 0, 1, 2, ..., with N (t) = t − f (t) f ′ (t), produces a quadratically convergent (p = 2) sequence of approximations tn → a, as n → ∞, whenever the first approximation, t0, is sufficiently near to a.

2 / 15

Higher order convergence can also be achieved:

Theorem (Householder, p = 3)

N (t) = t − f (t) f ′ (t)

• 1 + f (t) f ′′ (t)

2f ′ (t)2

• .

Theorem (Halley, p = 3)

N (t) = t − 2f (t) f ′ (t) 2f ′ (t)2 − f (t) f ′′ (t) .

Theorem (Householder, p = k + 1)

N (t) = t + k (1/f )(k−1) (t) (1/f )(k) (t) .

3 / 15

Theorem (Extension of Newton-Raphson to order p = k + 1)

Let f be of class C k+1 around the simple root a and let N(t) =

k

• ν=0

(−1)ν f (t)ν ν!

• 1

f ′(t)D ν t. Then for every t0 sufficiently near to a, the sequence (tn)n≥0, defined by tn+1 = N(tn), converges to a to the order k + 1 : tn+1 − a ∼ C · (tn − a)k+1, n → ∞, C = (−1)k+1

• f ′(t)k+1

(k + 1)!

• 1

f ′(t)D k+1 t

• t:=a

.

Proof (Sketch).

a = f <−1>(0) = f <−1>(f (t) − f (t)) = f <−1>(f (t) + u)|u:=−f (t).

4 / 15

The last iteration step can also be rewritten as, N(t) =

k

• ν=0

(−1)ν f (t)

f ′(t)D

ν

• t,

where z

ν

• = z(z−1)(z−2)···(z−ν+1)

ν!

.

Corollary

Let f be analytic around the simple root a. Then, for every g, analytic around a and t sufficiently near to a: g(a) =

∞

• ν=0

(−1)ν f (t)ν ν!

• 1

f ′(t)D ν g(t) =

∞

• ν=0

(−1)ν f (t)

f ′(t)D

ν

• g(t).

5 / 15

Typical illustrations (order = k + 1) :

◮ Root extraction, f (t) = tn − c = 0, a = c1/n, g(t) = tm/n:

N(t) =

k

• ν=0

(−1)ν 1

n

ν

• t
• 1 − c

tn ν , cm/n =

∞

• ν=0

(−1)ν m

n

ν

• tm

1 − c tn ν .

◮ Computing logarithms, f (t) = et − c, a = ln(c), g analytic:

N(t) = t −

k

• ν=1

(1 − ce−t)ν ν , g(ln(c)) =

∞

• ν=0

(ce−t − 1)ν D ν

• g(t).

6 / 15

Another illustration (order = 2p + 1):

◮ Approximating π, f (t) = sin(t) = 0, a = π, g analytic:

3 4π < t0 < 5 4π, tn+1 = N(tn) → π where N(t) = t−tan(t)+tan(t)3 3 −tan(t)5 5 +· · ·+(−1)2p−1 tan(t)2p−1 2p − 1 . Moreover, g(π) =

∞

• ν=0

(−1)ν tan(t)D ν

• g(t),

for t near π.

7 / 15

COMBINATORIAL APPROACH Given a combinatorial species, R, the species, A = A(X), of R-enriched rooted trees is recursively defined by A = XR(A).

Figure: An R-enriched rooted tree (X = )

Hence, A is the solution of F(T) = 0 where F(T) = T − XR(T).

8 / 15

Let D = d/dT denote the combinatorial differentiation operator with respect to singletons of sort T. Note that −F(T) = XR(T) − T and DF(T) = F ′(T) = 1 − XR′(T). This suggests that for some actions of the symmetric groups Sν:

Theorem

Let m ≥ 0. If α coincides with the species A of R-enriched rooted trees on sets up to cardinality m, then N(α) =

k

• ν=0

1 Sν (XR(α) − α)ν

• 1

1 − XR′(T)D ν T

• T:=α

, coincides with A on sets up to cardinality (k + 1)(m + 1).

9 / 15

In other words, α = A|≤m ⇒ N(α)|≤(k+1)(m+1) = A|≤(k+1)(m+1).

Proof (m = 6 fixed).

◮ α-structures are called light R -enriched rooted trees . ◮ (XR(α) − α)-structures are called m-broccolis : α α α α heavy =

Figure: A m-broccoli for m = 6

10 / 15

◮ D = 1 1−XR′(T)D

is called an eclosion operator (T = ): K − → D K

Figure: The eclosion operator D applied to a species K(X, T)

11 / 15

Now let τ be an A-structure on a set of size ≤ (k + 1)(m + 1). Let ν be the number of broccolis contained in τ. Then 0 ≤ ν ≤ k. Number arbitrarily these broccolis from 1 to ν as in Figure (a), then detach these broccolis as in Figure (b), (here m = 6, ν = 3):

b1 b2 b3

(a) Numbering broccolis

b1 b2 b3 := α 1 3 2

(b) Detached broccolis

Figure: Visualizing (XR(α) − α)ν

1 1−XR′(T)D

ν T

• T:=α

We conclude using the fact that Sν acts on these structures.

12 / 15

Corollary

Let A = XR(A) and G be an arbitrary species. Then, according to the number ν of leaves, the following expansions hold: A =

∞

• ν=0

1 Sν X ν

• R(0)

1 − XR′(T)D ν T

• T:=0

, G(A) =

∞

• ν=0

1 Sν X ν

• R(0)

1 − XR′(T)D ν G(T)

• T:=0

.

Proof.

Take m = 0 and α = 0. The 0-broccolis become XR(0)-structures (that is, enriched singletons, or leaves). Finally let k → ∞.

13 / 15

Corollary

Let R(x) = ∞

n=0 rnxn/n! and G(x) = ∞ n=0 gnxn/n!. Let γn,ν be

the number of G-assemblies of R-enriched rooted trees on [n] having exactly ν leaves. Then, for ν ≥ 1,

∞

• n=0

γn,νxn/n! = rν

0 xν

ν!(1 − r1x)2ν−1 pν(x), where pν(x) = ων(x, 0) are polynomials defined by ω1(x, t) = G ′(t), ων(x, t) =

• (1 − xR′(t)) ∂

∂t + (2ν − 3)xR′′(t)

• ων−1(x, t).

Proof.

Use induction on ν in underlying series of the above corollary.

14 / 15

Examples (Some applications to generating series)

◮ Ordinary rooted trees having ν leaves (R = E, G = X) : xν(1 − x)−2ν+1pν(x)/ν!, p1(x) = 1, pν(x) = x

• (1 − x)p′

ν−1(x) + (2ν − 3)pν−1(x)

• .

◮ Mobiles having ν leaves (R = 1 + C, G = X) : xν(1 − x)−2ν+1qν(x)/ν!, qν(x) = Qν(x, 0), Q1(x, t) = 1, Qν(x, t) =

• (1 − t)(1 − t − x) ∂

∂t + x + (2ν − 4)(1 − t)

• Qν−1(x, t).

◮ Endofunctions having ν leaves (R = E, G = S) : xν(1 − x)−2ν+1ǫν(x)/ν!, ǫν(x) = Kν(x, 0), K1(x, t) = 1, Kν(x, t) =

• (1 − x)(1 − t)
• x ∂

∂x + ∂ ∂t

• + ν + (ν − 3)x − (2ν − 3)xt
• Kν−1(x, t).

15 / 15