An extended procedure for extrapolation to the limit Work in progress, not yet finished Michela Redivo Zaglia University of Padua - Italy Claude Brezinski University of Lille - France ➜ The E –algorithm ➜ The extended procedure ➜ Some particular cases 1
Let ( S n ) be a sequence of real or complex numbers converging to a limit S . If the convergence is slow, and if one has no access to the process producing the sequence (that is, if it is a black box), ( S n ) can be transformed into a new sequence ( T n ) converging to the same limit by a sequence transformation T . Under some assumptions on ( S n ) and T , ( T n ) can converge to S faster than ( S n ) , that is T n − S lim S n − S = 0 . n →∞ 2
The idea behind a sequence transformation is extrapolation to the limit . It is assumed that ( S n ) behaves as a model sequence ( � S n ) depending on p parameters, and belonging to a given class K T of sequences. These p parameters are obtained by interpolation , requiring that S i = � S i for i = n, . . . , n + p − 1 , thus defining a unique model sequence in K T depending on the index n (the first index used in the interpolation process). Then, the limit of this model sequence is considered as an approximation of S . Since this limit depends on n , it is denoted by T n , and, therefore, the sequence ( S n ) has been transformed into the new sequence ( T n ) . 3
An example: Aitken’s ∆ 2 process K T = { � S i = S + αλ i } . S + αλ n S n = S + αλ n +1 S n +1 = S + αλ n +2 S n +2 = Solve this system for the unknowns α, λ and S . They depend on n . Thus, set T n = S ( S n ) has been transformed into ( T n ) 4
For any transformation, if the sequence ( S n ) to be accelerated belongs itself to K T , then, by construction, for all n , T n = S , the limit of the sequence ( S n ) if it converges, its antilimit otherwise. The set K T is called the kernel of the transformation T . It is the set of sequences which are transformed into a constant sequence ( S ) (usually their limit, or their antilimit). The study of the kernel of a transformation is based on the notion of linear annihilation operator for a sequence introduced by Weniger (1989). 5
There exist many approaches to sequence transformations • by defining a kernel, then constructing the corresponding transformation, and constructing a recursive algorithm for its implementation. • the construction can be based on error estimates, • they can be obtained by modifying the rules of existing algorithms, • it can make use of annihilation operators, • it can be based on the relation between extrapolation and asymptotic expansions, • by means of the theory of triangular recursive schemes, • by composing together several transformations, • by Schur complements. 6
We will make use of the following abbreviate notation for determinants � � � � � a n b n c n · · · � � � � � a n +1 b n +1 c n +1 · · · � � | a i b i c i · · · | i = n + k � � = . . . . � � i = n . . . � � . . . � � � � � � a n + k b n + k c n + k · · · The symbol ∆ will denote the usual forward difference operator whose powers are defined by ∆ 0 u n = u n , and k � k ! ∆ k u n = ∆(∆ k − 1 u n ) = ( − 1) i C i C i k u n + k − i with k = i !( k − i )! . i =0 7
T HE E – ALGORITHM The E –algorithm is the most general extrapolation algorithm known so far. It is built from the kernel S n = S + a 1 g 1 ( n ) + · · · + a k g k ( n ) , n = 0 , 1 , . . . where the ( g i ( n )) ’s are given auxiliary sequences . Writing this relation for n, . . . , n + k leads to a system of k + 1 linear equations in the k + 1 unknowns a 1 , . . . , a k , and S . Since these unknowns depend on n and k , S will be denoted by E ( n ) , and it is given as a ratio of determinants k = | S i g 1 ( i ) · · · g k ( i ) | i = n + k E ( n ) i = n . k | 1 g 1 ( i ) · · · g k ( i ) | i = n + k i = n E ( n ) = S for all n if and only if ( S n ) satisfies the preceding k relation (kernel). T HE E – ALGORITHM 8
These quantities can be recursively computed by the E –algorithm ∆( E ( n ) k − 1 /g ( n ) k − 1 ,k ) E ( n ) (main rule) = k ∆(1 /g ( n ) k − 1 ,k ) ∆( g ( n ) k − 1 ,i /g ( n ) k − 1 ,k ) g ( n ) (auxiliary rule) , = , i > k k,i ∆(1 /g ( n ) k − 1 ,k ) with E ( n ) = S n and g ( n ) 0 ,i = g i ( n ) . 0 The operator ∆ acts on the upper index n : ∆ u ( n ) = u ( n +1) − u ( n ) . T HE E – ALGORITHM 9
T HE EXTENDED PROCEDURE For k = 1 , 2 , . . . , let L k be a linear operator on a set of real or complex functions L k on R or C , such that ∀ a k ∈ L k , L k ( a k ( x n )) = 0 , n = 0 , 1 , . . . , where ( x n ) is a sequence of points in R or C . L k is an annihilation operator for L k . For k = 1 , 2 , . . . , we consider the linear operators Λ k , acting on a sequence ( u n ) , which are recursively defined by Λ k ( u n ) = L k (Λ k − 1 ( u n ) / Λ k − 1 ( g k ( n ))) , n = 0 , 1 , . . . , L k (1 / Λ k − 1 ( g k ( n ))) with Λ 0 ( u n ) = u n , for n = 0 , 1 , . . . T HE EXTENDED PROCEDURE 10
Then, the extended transformation, called the Λ k –transformation (we can identify the operator and the transformation without a risk of confusion), is defined by → (Λ ( n ) Λ k : ( S n ) �− = Λ k ( S n )) , k = 0 , 1 , . . . , k for k ≥ 0 fixed, where ( S n ) is the sequence to the accelerated (that is extrapolated). The implementation of this transformation requires the computation of the auxiliary quantities Λ k ( g i ( n )) for different values of the three indexes. Thus, any algorithm for its implementation will depend on the properties (in general, the recursive ones) of the operators L k , and it does not exist in the general case. T HE EXTENDED PROCEDURE 11
Denote Λ ( n ) by Λ k ( S n ; g k ) for indicating its dependence on k the auxiliary sequence g k . By linearity of the operators L k , and the definition of Λ 0 , it holds, by induction, Property 1 (Quasi-linearity) Λ k ( aS n + b ; αg k ) = a Λ k ( S n ; g k ) + b, ∀ a, b, and α � = 0 . According to the theory of sequence transformations, it could be written under the form f ( S n , . . . , S m ) Λ ( n ) = Df ( S n , . . . , S m ) , k for some function f depending on the operators L k , where n and m are respectively the first and the last indexes of the terms used for computing Λ ( n ) k , where Df denotes the sum of the partial derivatives of f , and where D 2 f is identically zero. T HE EXTENDED PROCEDURE 12
Property 2 The kernel of the Λ k –transformation ( k ≥ 1 ) is the set of sequences satisfying, for all n , Λ k − 1 ( S n − S ) = a k ( x n )Λ k − 1 ( g k ( n )) . Property 3 (by replacing Λ k − 1 by its definition) The kernel of the Λ k –transformation ( k ≥ 2 ) is the set of sequences satisfying, for all n , L k − 1 (Λ k − 2 ( S n − S ) / Λ k − 2 ( g k − 1 ( n ))) = a k ( x n ) L k − 1 (Λ k − 2 ( g k ( n )) / Λ k − 2 ( g k − 1 ( n ))) . T HE EXTENDED PROCEDURE 13
Property 4 (Remainder formula) Assume that, for all n , S n = � S n + r n , where ( � S n ) belongs to the kernel of the Λ k − 1 –transformation. Then, for all n, = Λ k ( S n ) = S + L k (Λ k − 1 ( r n ) / Λ k − 1 ( g k ( n )) Λ ( n ) . k L k (1 / Λ k − 1 ( g k ( n )) Property 5 For all k , the kernel of the Λ k –transformation includes the kernel of the Λ k − 1 –transformation. T HE EXTENDED PROCEDURE 14
P ARTICULAR CASES E - ALGORITHM : For the choice ∀ k, L k = ∆ we recover the E -algorithm, and we have Λ ( n ) = Λ k ( S n ) = E ( n ) k k and Λ k ( g k +1 ( n )) = g ( n ) k,k +1 . P ARTICULAR CASES 15
D RUMMOND ’ S PROCESS : In 1972, Drummond proposed the sequence transformation → (∆ ( n ) ∆ m : ( S n ) �− m ) , for m fixed, where m = ∆ m ( S n / ∆ S n ) ∆ ( n ) ∆ m (1 / ∆ S n ) , m, n = 0 , 1 , . . . . It corresponds to k = 1 , L 1 = ∆ m Property 6 The kernel of Drummond’s ∆ m –transformation is the set of sequences such that there exist S and a polynomial P m − 1 of degree at most m − 1 satisfying S n − S = P m − 1 ( n )∆ S n , n = 0 , 1 , . . . The kernel of the ∆ m +1 –transformation includes the kernel of the ∆ m –transformation. P ARTICULAR CASES 16
Solving this difference equation, gives Property 7 The kernel of Drummond’s ∆ m –transformation is the set of sequences of the form (assuming that ∀ i ≥ 0 , P m − 1 ( i ) � = − 1 , 0 ) � � n − 1 � 1 S n = S + α 1 + , n = 0 , 1 , . . . P m − 1 ( i ) i =0 Property 8 For all m, n = 0 , 1 , . . . , it holds m = | S i ∆ S i i ∆ S i · · · i m − 1 ∆ S i | i = n + m ∆ ( n ) i = n . | 1 ∆ S i i ∆ S i · · · i m − 1 ∆ S i | i = n + m i = n P ARTICULAR CASES 17
A PPLICATION TO FORMAL POWER SERIES : We consider the formal power series ∞ � c i x i , S ( x ) = i =0 and apply Drummond’s ∆ m –transformation to its partial sums n � c i x i , S n ( x ) = n = 0 , 1 , . . . i =0 ∆ ( n ) m ( x ) = N ( n ) m ( x ) /D ( n ) m ( x ) is a rational function with a numerator of degree n + m and a denominator of degree m at most, and we have N ( n ) m ( x ) − S ( x ) D ( n ) m ( x ) = O ( x n + m +1 ) , which shows that ∆ ( n ) m ( x ) is a Padé–type approximant of S . P ARTICULAR CASES 18
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