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 t n → a . The convergence is said to be of order p if t n + 1 − a = O (( t n − 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, N ( t ) = t − f ( t ) t n + 1 = N ( t n ) , n = 0 , 1 , 2 , ... , with f ′ ( t ) , produces a quadratically convergent ( p = 2 ) sequence of approximations t n → a, as n → ∞ , whenever the first approximation, t 0 , is sufficiently near to a. 2 / 15

Higher order convergence can also be achieved: Theorem (Householder, p = 3) 1 + f ( t ) f ′′ ( t ) N ( t ) = t − f ( t ) � � f ′ ( t ) 2 f ′ ( t ) 2 . Theorem (Halley, p = 3) 2 f ( t ) f ′ ( t ) N ( t ) = t − 2 f ′ ( 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 k � ν ( − 1 ) ν f ( t ) ν � 1 � N ( t ) = f ′ ( t ) D t . ν ! ν = 0 Then for every t 0 sufficiently near to a, the sequence ( t n ) n ≥ 0 , defined by t n + 1 = N ( t n ) , converges to a to the order k + 1 : t n + 1 − a ∼ C · ( t n − a ) k + 1 , n → ∞ , � f ′ ( t ) k + 1 � k + 1 � � 1 C = ( − 1 ) k + 1 f ′ ( t ) D t . ( k + 1 )! 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, � f ( t ) k � f ′ ( t ) D � ( − 1 ) ν N ( t ) = t , ν ν = 0 � z = z ( z − 1 )( z − 2 ) ··· ( z − ν + 1 ) � where . ν ν ! Corollary Let f be analytic around the simple root a. Then, for every g, analytic around a and t sufficiently near to a: ∞ � ν ( − 1 ) ν f ( t ) ν � 1 � g ( a ) = f ′ ( t ) D g ( t ) ν ! ν = 0 � f ( t ) ∞ f ′ ( t ) D � � ( − 1 ) ν = g ( t ) . ν ν = 0 5 / 15

Typical illustrations (order = k + 1) : ◮ Root extraction, f ( t ) = t n − c = 0, a = c 1 / n , g ( t ) = t m / n : � 1 k � 1 − c � ν � � ( − 1 ) ν n N ( t ) = t , ν t n ν = 0 � m ∞ � 1 − c � ν c m / n = t m � � ( − 1 ) ν n . t n ν ν = 0 ◮ Computing logarithms, f ( t ) = e t − c , a = ln ( c ) , g analytic: k ( 1 − ce − t ) ν � N ( t ) = t − , ν ν = 1 ∞ � D � ( ce − t − 1 ) ν � g ( ln ( c )) = g ( t ) . ν ν = 0 6 / 15

Another illustration (order = 2 p + 1): ◮ Approximating π , f ( t ) = sin ( t ) = 0, a = π , g analytic: 3 4 π < t 0 < 5 t n + 1 = N ( t n ) → π where 4 π, N ( t ) = t − tan ( t )+ tan ( t ) 3 − tan ( t ) 5 + · · · +( − 1 ) 2 p − 1 tan ( t ) 2 p − 1 . 3 5 2 p − 1 Moreover, ∞ � tan ( t ) D � � ( − 1 ) ν g ( π ) = g ( t ) , for t near π . ν ν = 0 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 DF ( T ) = F ′ ( T ) = 1 − XR ′ ( T ) . − F ( T ) = XR ( T ) − T and This suggests that for some actions of the symmetric groups S ν : Theorem Let m ≥ 0 . If α coincides with the species A of R-enriched rooted trees on sets up to cardinality m, then k � ν 1 �� 1 � � ( XR ( α ) − α ) ν N ( α ) = 1 − XR ′ ( T ) D T , S ν T := α ν = 0 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 = heavy α α Figure: A m -broccoli for m = 6 10 / 15

◮ D = 1 1 − XR ′ ( T ) D is called an eclosion operator ( T = ): �− → K K D 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): b 1 1 3 b 3 b 1 b 2 b 3 2 b 2 α := (a) Numbering broccolis (b) Detached broccolis � ν Figure: Visualizing ( XR ( α ) − α ) ν �� � 1 1 − XR ′ ( T ) D T T := α We conclude using the fact that S ν 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: ∞ � ν 1 �� R ( 0 ) � � X ν A = 1 − XR ′ ( T ) D T , S ν T := 0 ν = 0 ∞ � ν 1 �� R ( 0 ) � � X ν G ( A ) = G ( T ) 1 − XR ′ ( T ) D . S ν T := 0 ν = 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 r n x n / n ! and G ( x ) = � ∞ n = 0 g n x n / n ! . Let γ n ,ν be the number of G-assemblies of R-enriched rooted trees on [ n ] having exactly ν leaves. Then, for ν ≥ 1 , ∞ r ν 0 x ν � γ n ,ν x n / n ! = ν !( 1 − r 1 x ) 2 ν − 1 p ν ( x ) , n = 0 where p ν ( x ) = ω ν ( x , 0 ) are polynomials defined by G ′ ( t ) , ω 1 ( x , t ) = � ( 1 − xR ′ ( t )) ∂ � ∂ t + ( 2 ν − 3 ) xR ′′ ( t ) ω ν ( x , t ) = ω ν − 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 ) : x ν ( 1 − x ) − 2 ν + 1 p ν ( x ) /ν ! , � � p 1 ( x ) = 1 , p ν ( x ) = x ( 1 − x ) p ′ ν − 1 ( x ) + ( 2 ν − 3 ) p ν − 1 ( x ) . ◮ Mobiles having ν leaves ( R = 1 + C , G = X ) : x ν ( 1 − x ) − 2 ν + 1 q ν ( x ) /ν ! , q ν ( x ) = Q ν ( x , 0 ) , Q 1 ( x , t ) = 1 , � ( 1 − t )( 1 − t − x ) ∂ � Q ν ( x , t ) = ∂ t + x + ( 2 ν − 4 )( 1 − t ) Q ν − 1 ( x , t ) . ◮ Endofunctions having ν leaves ( R = E , G = S ) : x ν ( 1 − x ) − 2 ν + 1 ǫ ν ( x ) /ν ! , ǫ ν ( x ) = K ν ( x , 0 ) , K 1 ( x , t ) = 1 , � � x ∂ ∂ x + ∂ � � K ν ( x , t ) = ( 1 − x )( 1 − t ) + ν + ( ν − 3 ) x − ( 2 ν − 3 ) xt K ν − 1 ( x , t ) . ∂ t 15 / 15