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Summation by parts

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Theorem to simplify sums of products of sequences "Abel transformation" redirects here. For another transformation, see Abel transform.

In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826.

Statement

Suppose { f k } {\displaystyle \{f_{k}\}} and { g k } {\displaystyle \{g_{k}\}} are two sequences. Then,

∑ k = m n f k ( g k + 1 − g k ) = ( f n + 1 g n + 1 − f m g m ) − ∑ k = m n g k + 1 ( f k + 1 − f k ) . {\displaystyle \sum _{k=m}^{n}f_{k}(g_{k+1}-g_{k})=\left(f_{n+1}g_{n+1}-f_{m}g_{m}\right)-\sum _{k=m}^{n}g_{k+1}(f_{k+1}-f_{k}).}

Using the forward difference operator Δ {\displaystyle \Delta } , it can be stated more succinctly as

∑ k = m n f k Δ g k = ( f n + 1 g n + 1 − f m g m ) − ∑ k = m n g k + 1 Δ f k , {\displaystyle \sum _{k=m}^{n}f_{k}\Delta g_{k}=\left(f_{n+1}g_{n+1}-f_{m}g_{m}\right)-\sum _{k=m}^{n}g_{k+1}\Delta f_{k},}

Summation by parts is an analogue to integration by parts:

∫ f d g = f g − ∫ g d f , {\displaystyle \int f\,dg=fg-\int g\,df,}

or to Abel's summation formula:

∑ k = m + 1 n f ( k ) ( g k − g k − 1 ) = ( f ( n ) g n − f ( m ) g m ) − ∫ m n g ⌊ t ⌋ f ′ ( t ) d t . {\displaystyle \sum _{k=m+1}^{n}f(k)(g_{k}-g_{k-1})=\left(f(n)g_{n}-f(m)g_{m}\right)-\int _{m}^{n}g_{\lfloor t\rfloor }f'(t)dt.}

An alternative statement is

f n g n − f m g m = ∑ k = m n − 1 f k Δ g k + ∑ k = m n − 1 g k Δ f k + ∑ k = m n − 1 Δ f k Δ g k {\displaystyle f_{n}g_{n}-f_{m}g_{m}=\sum _{k=m}^{n-1}f_{k}\Delta g_{k}+\sum _{k=m}^{n-1}g_{k}\Delta f_{k}+\sum _{k=m}^{n-1}\Delta f_{k}\Delta g_{k}}

which is analogous to the integration by parts formula for semimartingales.

Although applications almost always deal with convergence of sequences, the statement is purely algebraic and will work in any field. It will also work when one sequence is in a vector space, and the other is in the relevant field of scalars.

Newton series

The formula is sometimes given in one of these - slightly different - forms

∑ k = 0 n f k g k = f 0 ∑ k = 0 n g k + ∑ j = 0 n − 1 ( f j + 1 − f j ) ∑ k = j + 1 n g k = f n ∑ k = 0 n g k − ∑ j = 0 n − 1 ( f j + 1 − f j ) ∑ k = 0 j g k , {\displaystyle {\begin{aligned}\sum _{k=0}^{n}f_{k}g_{k}&=f_{0}\sum _{k=0}^{n}g_{k}+\sum _{j=0}^{n-1}(f_{j+1}-f_{j})\sum _{k=j+1}^{n}g_{k}\\&=f_{n}\sum _{k=0}^{n}g_{k}-\sum _{j=0}^{n-1}\left(f_{j+1}-f_{j}\right)\sum _{k=0}^{j}g_{k},\end{aligned}}}

which represent a special case ( M = 1 {\displaystyle M=1} ) of the more general rule

∑ k = 0 n f k g k = ∑ i = 0 M − 1 f 0 ( i ) G i ( i + 1 ) + ∑ j = 0 n − M f j ( M ) G j + M ( M ) = = ∑ i = 0 M − 1 ( − 1 ) i f n − i ( i ) G ~ n − i ( i + 1 ) + ( − 1 ) M ∑ j = 0 n − M f j ( M ) G ~ j ( M ) ; {\displaystyle {\begin{aligned}\sum _{k=0}^{n}f_{k}g_{k}&=\sum _{i=0}^{M-1}f_{0}^{(i)}G_{i}^{(i+1)}+\sum _{j=0}^{n-M}f_{j}^{(M)}G_{j+M}^{(M)}=\\&=\sum _{i=0}^{M-1}\left(-1\right)^{i}f_{n-i}^{(i)}{\tilde {G}}_{n-i}^{(i+1)}+\left(-1\right)^{M}\sum _{j=0}^{n-M}f_{j}^{(M)}{\tilde {G}}_{j}^{(M)};\end{aligned}}}

both result from iterated application of the initial formula. The auxiliary quantities are Newton series:

f j ( M ) := ∑ k = 0 M ( − 1 ) M − k ( M k ) f j + k {\displaystyle f_{j}^{(M)}:=\sum _{k=0}^{M}\left(-1\right)^{M-k}{M \choose k}f_{j+k}}

and

G j ( M ) := ∑ k = j n ( k − j + M − 1 M − 1 ) g k , {\displaystyle G_{j}^{(M)}:=\sum _{k=j}^{n}{k-j+M-1 \choose M-1}g_{k},}
G ~ j ( M ) := ∑ k = 0 j ( j − k + M − 1 M − 1 ) g k . {\displaystyle {\tilde {G}}_{j}^{(M)}:=\sum _{k=0}^{j}{j-k+M-1 \choose M-1}g_{k}.}

A particular ( M = n + 1 {\displaystyle M=n+1} ) result is the identity

∑ k = 0 n f k g k = ∑ i = 0 n f 0 ( i ) G i ( i + 1 ) = ∑ i = 0 n ( − 1 ) i f n − i ( i ) G ~ n − i ( i + 1 ) . {\displaystyle \sum _{k=0}^{n}f_{k}g_{k}=\sum _{i=0}^{n}f_{0}^{(i)}G_{i}^{(i+1)}=\sum _{i=0}^{n}(-1)^{i}f_{n-i}^{(i)}{\tilde {G}}_{n-i}^{(i+1)}.}

Here, ( n k ) {\textstyle {n \choose k}} is the binomial coefficient.

Method

For two given sequences ( a n ) {\displaystyle (a_{n})} and ( b n ) {\displaystyle (b_{n})} , with n ∈ N {\displaystyle n\in \mathbb {N} } , one wants to study the sum of the following series: S N = ∑ n = 0 N a n b n {\displaystyle S_{N}=\sum _{n=0}^{N}a_{n}b_{n}}

If we define B n = ∑ k = 0 n b k , {\textstyle \displaystyle B_{n}=\sum _{k=0}^{n}b_{k},} then for every n > 0 , {\displaystyle n>0,} b n = B n − B n − 1 {\displaystyle b_{n}=B_{n}-B_{n-1}} and S N = a 0 b 0 + ∑ n = 1 N a n ( B n − B n − 1 ) , {\displaystyle S_{N}=a_{0}b_{0}+\sum _{n=1}^{N}a_{n}(B_{n}-B_{n-1}),} S N = a 0 b 0 − a 1 B 0 + a N B N + ∑ n = 1 N − 1 B n ( a n − a n + 1 ) . {\displaystyle S_{N}=a_{0}b_{0}-a_{1}B_{0}+a_{N}B_{N}+\sum _{n=1}^{N-1}B_{n}(a_{n}-a_{n+1}).}

Finally S N = a N B N − ∑ n = 0 N − 1 B n ( a n + 1 − a n ) . {\textstyle \displaystyle S_{N}=a_{N}B_{N}-\sum _{n=0}^{N-1}B_{n}(a_{n+1}-a_{n}).}

This process, called an Abel transformation, can be used to prove several criteria of convergence for S N {\displaystyle S_{N}} .

Similarity with an integration by parts

The formula for an integration by parts is ∫ a b f ( x ) g ′ ( x ) d x = [ f ( x ) g ( x ) ] a b − ∫ a b f ′ ( x ) g ( x ) d x {\textstyle \displaystyle \int _{a}^{b}f(x)g'(x)\,dx=\left_{a}^{b}-\int _{a}^{b}f'(x)g(x)\,dx} .

Beside the boundary conditions, we notice that the first integral contains two multiplied functions, one which is integrated in the final integral ( g ′ {\displaystyle g'} becomes g {\displaystyle g} ) and one which is differentiated ( f {\displaystyle f} becomes f ′ {\displaystyle f'} ).

The process of the Abel transformation is similar, since one of the two initial sequences is summed ( b n {\displaystyle b_{n}} becomes B n {\displaystyle B_{n}} ) and the other one is differenced ( a n {\displaystyle a_{n}} becomes a n + 1 − a n {\displaystyle a_{n+1}-a_{n}} ).

Applications

Proof of Abel's test. Summation by parts gives S M − S N = a M B M − a N B N − ∑ n = N M − 1 B n ( a n + 1 − a n ) = ( a M − a ) B M − ( a N − a ) B N + a ( B M − B N ) − ∑ n = N M − 1 B n ( a n + 1 − a n ) , {\displaystyle {\begin{aligned}S_{M}-S_{N}&=a_{M}B_{M}-a_{N}B_{N}-\sum _{n=N}^{M-1}B_{n}(a_{n+1}-a_{n})\\&=(a_{M}-a)B_{M}-(a_{N}-a)B_{N}+a(B_{M}-B_{N})-\sum _{n=N}^{M-1}B_{n}(a_{n+1}-a_{n}),\end{aligned}}} where a is the limit of a n {\displaystyle a_{n}} . As ∑ n b n {\textstyle \sum _{n}b_{n}} is convergent, B N {\displaystyle B_{N}} is bounded independently of N {\displaystyle N} , say by B {\displaystyle B} . As a n − a {\displaystyle a_{n}-a} go to zero, so go the first two terms. The third term goes to zero by the Cauchy criterion for ∑ n b n {\textstyle \sum _{n}b_{n}} . The remaining sum is bounded by ∑ n = N M − 1 | B n | | a n + 1 − a n | ≤ B ∑ n = N M − 1 | a n + 1 − a n | = B | a N − a M | {\displaystyle \sum _{n=N}^{M-1}|B_{n}||a_{n+1}-a_{n}|\leq B\sum _{n=N}^{M-1}|a_{n+1}-a_{n}|=B|a_{N}-a_{M}|} by the monotonicity of a n {\displaystyle a_{n}} , and also goes to zero as N → ∞ {\displaystyle N\to \infty } .

Using the same proof as above, one can show that if

  1. the partial sums B N {\displaystyle B_{N}} form a bounded sequence independently of N {\displaystyle N} ;
  2. ∑ n = 0 ∞ | a n + 1 − a n | < ∞ {\displaystyle \sum _{n=0}^{\infty }|a_{n+1}-a_{n}|<\infty } (so that the sum ∑ n = N M − 1 | a n + 1 − a n | {\displaystyle \sum _{n=N}^{M-1}|a_{n+1}-a_{n}|} goes to zero as N {\displaystyle N} goes to infinity)
  3. lim a n = 0 {\displaystyle \lim a_{n}=0}

then S N = ∑ n = 0 N a n b n {\textstyle S_{N}=\sum _{n=0}^{N}a_{n}b_{n}} converges.

In both cases, the sum of the series satisfies: | S | = | ∑ n = 0 ∞ a n b n | ≤ B ∑ n = 0 ∞ | a n + 1 − a n | . {\displaystyle |S|=\left|\sum _{n=0}^{\infty }a_{n}b_{n}\right|\leq B\sum _{n=0}^{\infty }|a_{n+1}-a_{n}|.}

Summation-by-parts operators for high order finite difference methods

A summation-by-parts (SBP) finite difference operator conventionally consists of a centered difference interior scheme and specific boundary stencils that mimics behaviors of the corresponding integration-by-parts formulation. The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique. The combination of SBP-SAT is a powerful framework for boundary treatment. The method is preferred for well-proven stability for long-time simulation, and high order of accuracy.

See also

References

  1. Chu, Wenchang (2007). "Abel's lemma on summation by parts and basic hypergeometric series". Advances in Applied Mathematics. 39 (4): 490–514. doi:10.1016/j.aam.2007.02.001.
  2. Edmonds, Sheila M. (1957). "Sums of powers of the natural numbers". The Mathematical Gazette. 41 (337): 187–188. doi:10.2307/3609189. JSTOR 3609189. MR 0096615.
  3. Strand, Bo (January 1994). "Summation by Parts for Finite Difference Approximations for d/dx". Journal of Computational Physics. 110 (1): 47–67. doi:10.1006/jcph.1994.1005.
  4. Mattsson, Ken; Nordström, Jan (September 2004). "Summation by parts operators for finite difference approximations of second derivatives". Journal of Computational Physics. 199 (2): 503–540. doi:10.1016/j.jcp.2004.03.001.
  5. Carpenter, Mark H.; Gottlieb, David; Abarbanel, Saul (April 1994). "Time-Stable Boundary Conditions for Finite-Difference Schemes Solving Hyperbolic Systems: Methodology and Application to High-Order Compact Schemes". Journal of Computational Physics. 111 (2): 220–236. CiteSeerX 10.1.1.465.603. doi:10.1006/jcph.1994.1057.

Bibliography

  • Abel, Niels Henrik (1826). "Untersuchungen über die Reihe 1 + m x + m ⋅ ( m − 1 ) 2 ⋅ 1 x 2 + m ⋅ ( m − 1 ) ⋅ ( m − 2 ) 3 ⋅ 2 ⋅ 1 x 3 + … {\displaystyle 1+{\frac {m}{x}}+{\frac {m\cdot (m-1)}{2\cdot 1}}x^{2}+{\frac {m\cdot (m-1)\cdot (m-2)}{3\cdot 2\cdot 1}}x^{3}+\ldots } u.s.w.". J. Reine Angew. Math. 1: 311–339.
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