bo198214 Wrote:Thats something!

Can you explain your modified euler summation?

A disclaimer first: the "modified" Euler-summation, which I'll describe here, is not yet proven to show correct results for divergent summation. I'm currently only using it, when it gives some reasonable results in example problems of related rate of divergence. I hope, I can come up with a better foundation some day.

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A) basics of naive Euler-summation

The basic idea is simple and implements the usual method of Euler-summation via a matrix-method, translated to matrix-speak from the description of, for instance, K. Knopp.

Assume the pascalmatrix P and the matrix-product of V(x)~* P, which implements by its first column the function f(x)=1+x+x^2+...= 1/(1-x)

The c'th column of P provides the coefficients for the c'th derivative of f(x) (with a linear scaling), so the result is

Code:

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(1) V(x)~ * P = [f(x),x f'(x)/1!, x^2 f"(x)/2!,...] [update, the previous was corrected by inserting the powers of x]

Code:

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1/4 V(1/4)~ * P = 1/3*V(1/3)~

1/3 V(1/3)~ * P = 1/2*V(1/2)~

1/2 V(1/2)~ * P = V(1)~

The last example may serve as key-idea.

If I have to sum a series with sum s, whose terms are described as column vector A, then the sum of the series can be expressed by the vector/matrix-product

Code:

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(3) V(1)~ * A = s

Code:

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V(1)~ = 1/2* V(1/2)* P

Code:

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V(1)~ * A = s

(1/2* V(1/2)~ * P) * A = s

Now, using matrix-associativity, I change parentheses

Code:

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1/2* V(1/2)~ * (P * A) = s

If A itself contains the terms of a powerseries, then the effect can be dramatical, since

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P * V(x) = V(x+1) // by binomial-theorem

Code:

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1/2*V(1/2)~ * P * V(-1) = 1/2*V(1/2)~ * V(1+ (-1))

= 1/2 V(1/2)~ * V(0)

= 1/2 V(1/2)~ * [1,0,0,....]

= 1/2

Code:

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P^q * V(x) = V(q+x) // by binomial-theorem

Code:

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1/(1+q) * V(1/(1+q))~ * P^q = V(1)~

Code:

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let s = V(1)~ * A // the assumed sum to be determined

then s = (1/(1+q) * V(1/(1+q))~ * P^q) * A

= 1/(1+q) * V(1/(1+q))~ * (P^q * A)

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B) the approach via partial sums

The more general approach uses the concept of convergence of partial sums, so first from A the vector of partial sums B has to be computed; then second one checks, whether the partial sums, or some transforms, converge to a final value.

In matrix-notation we may use the triangular matrix DR

Code:

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DR = 1 . . .

1 1 . .

1 1 1 .

1 1 1 1 ...

...

(B.1) Hölder-means

The most simple approach is the Hölder-mean, which computes the partial means of B in two steps

Code:

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partial-sums of B: PS = DR * B

partial-means of B: HM = dZ(1)* PS // Hölder-means

where dZ(1) = diag(1/1,1/2, 1/3, 1/4,...)

and dZ(x) = diag( (1/1)^x, (1/2)^x, (1/3)^x, ...)

Code:

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rn(matrix) * V(1) = V(1)

This process of computing of partial means of partial sums in B can be iterated to higher orders, just by repeating the matrix-multiplication

Code:

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HM_1 = dZ(1)*DR * B

HM_2 = dZ(1)*DR * HM_1

HM_3 = dZ(1)*DR * HM_2

...

HM(order) = (dZ(1)*DR)^order * B

(B.2) Euler-means

However, this method is not powerful enough to sum series of geometric growth. Here, instead of the Hölder-means, again the Euler-transform using powers of P is applied.

if s = V(1)~ * A is assumed, then the Euler-means are in the result-vector

Code:

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EM(B) = rn(P) * B = rn(P) * DR * A

Denote the r'th entry in the result

Code:

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s_r = EM(B)[r] // r: rowindex

Code:

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s = (r->inf) s_r

and (r->inf) |s_(r-1)-s_r| -> 0

Such orders of this type of Euler-summation are then simply

Code:

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EM(B,order) = rn(P^order) * B = rn(P^order) * DR * A

C) the variant of Euler-summation

But again - this is only sufficient for series of geometric growth; divergent series of hypergeometric growth will exceed the power of the Euler-means-method of any order, and leave us with an eventually diverging sequence of means in EM(B,order).

In investigating variants of the matrix P I came across the observation, that log(P)= subdiag(1,[1,2,3,4,...]) and studied Pk, where log(Pk)=subdiag(1,[1^k,2^k,3^k,...]) naming them Pk-orders. (Using k=2 we get btw. a scaling of the Laguerre-matrix)

It comes out, that also

Code:

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Pk = dF(k) * P * dF(-k)

where dF(k) = diag(0!,1!, 2!)^k

Using Pk intead of P in the Euler-means-formula

Code:

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EM_k(B) = rn(P_k) * B ( = rn(P_k) * DR * A )

EM_k(B,order) = rn(P_k^order) * B ( = rn(P_k^order) * DR * A )

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However, this summation is not regular. While in the first example of naive Euler-summation it is

Code:

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1/(1+q) *V(1/(1+q)) * P^q * A = V(1)~ * A

Code:

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VE(1/(1+q))~ * Pk^q = VE(1)~

where VE(x) = [1,x/1! , x^2/2!, ...] if k=2; for higher k the factorials are to be replaced by their k'th powers

The consequence is, that the elements in A, translated via partial sums in B, are not really equally summed, but are additionally weighted, and this may introduce a distortion.

However - a distortion of this type is principally already introduced in the Hölder-means method, and thus does not always prevent a valid result.

The resulting summation-vector, call it S~, in the last row of HM if computed by

Code:

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rm(DR)*B = rn(DR)*DR *A = (rn(DR)*DR) * A = HM * A

thus

HM = dZ(1)* DR * DR

then

S_r~ = HM[r,] // [r,] indicates the r'th row

Here S_r contains decreasing weights for the elements of A, which seems forbidden for a regular summation. K. Knopp mentions this problem explicitely, but states then "in the limit case this effect loses its influence and can thus be neglected".

This Euler-sum variant with its implicite factorial scaling reminds to the Borel-summation, which also introduces exponential-series VE(x) for averaging the partial sums with implicite weighting of A, but assumes x->inf, which I cannot implement.

Surpringly this method works fine for summation of some usual hypergeometric series like 0! - 1! + 2! - 3! +- ... or the series of bernoulli-numbers and others, if k is selected appropriately.

In the current case of f°(1/2)(x) where f°1(x)=exp(x)-1 we have apparently a growth of terms in the powerseries of something like exp(j^2) (**) (where j indicates the index) after a minimum at a certain j. Thus I use Pk with k=1.4 to get the best result (which supposedly catches the hypergeometric effect of exp(j^2) (**)), and I get a continuously improving approximation of the partial sums to the last displayed value at s_96 (using 96 terms)

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Well - it is high time to analyze the theory in more depth and to determine the conditions and the range of applicability, but... it seems to be impossible to introduce a deeper discussion in the math-groups to which I'm subscribed, so this remains all home-brewn so far.

Gottfried

(**) [update] the earlier description was wrongly including a factorial which I removed here

Gottfried Helms, Kassel