Order-Degree Curves for Hypergeometric Creative … Telescoping INPUT: something like f(n;k) := ( 1)...

Order-Degree Curves for

Hypergeometric Creative

Telescoping

Shaoshi ChenNCSU, Raleigh, USA

Manuel KauersRISC, Linz, Austria

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Creative Telescoping

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INPUT: something like f(n, k) := (−1)k 1k+1

(nk

)(2n+k

k

)

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(nk

)(2n+k

k

)OUTPUT: something like

(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)

where g(n, k) = k(k+1)(2−2k+9n−kn+10n2)2n(1−k+n)(2n+1) f(n, k).

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(nk

)(2n+k

k


(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)


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(nk

)(2n+k

k


(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)


polynomials in n only

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(nk

)(2n+k

k


(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)



shift(s) in n only

2



(nk

)(2n+k

k


(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)



shift(s) in n only︷︸︸︷= ∆kg(n, k)

2



(nk

)(2n+k

k


(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)



shift(s) in n only︷︸︸︷= Σ−1

k g(n, k)

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(nk

)(2n+k

k


(4n+ 2)f(n, k) + (n+ 2)f(n+ 1, k) = g(n, k + 1)− g(n, k)



shift(s) in n only︷︸︸︷= Σ−1

k g(n, k)

rational function in n and k

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INPUT: a hypergeometric term f(n, k)

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i.e., f(n+1,k)f(n,k) ∈ K(n, k) and f(n,k+1)

f(n,k) ∈ K(n, k)

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OUTPUT: T ∈ K[n, Sn] \ {0} and Q ∈ K(n, k) such that

T · f(n, k) = (Sk − 1) ·Qf(n, k)

∣∣∣ ∑k

2




T · f(n, k) = (Sk − 1) ·Qf(n, k)

∣∣∣ ∑k

“telescoper”

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T · f(n, k) = (Sk − 1) ·Qf(n, k)

∣∣∣ ∑k

“telescoper” “certificate”

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T · f(n, k) = (Sk − 1) ·Qf(n, k)

∣∣∣ ∑k

2




T · f(n, k) = (Sk − 1) ·Qf(n, k)∣∣∣ ∑k

2



OUTPUT: T ∈ K[n, Sn] \ {0} and Q ∈ K(n, k) such that∑k

T · f(n, k) =∑k

(Sk − 1) ·Qf(n, k)

2




T ·∑k

f(n, k) =∑k

(Sk − 1) ·Qf(n, k)

2




T ·∑k

f(n, k) = 0 (usually)

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Focus on the telescoper

T =

degree d︷︸︸︷

(a0,0 + a0,1n+ a0,2n

2 + · · ·+ a0,dnd)

+(a1,0 + a1,1n+ a1,2n

2 + · · ·+ a1,dnd)Sn

+(a2,0 + a2,1n+ a2,2n

2 + · · ·+ a2,dnd)S2n

+ . . .

+(ar,0 + ar,1n+ ar,2n

2 + · · ·+ ar,dnd)Srn

order r

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T =

degree d︷︸︸︷

(a0,0 + a0,1n+ a0,2n

2 + · · ·+ a0,dnd)

+(a1,0 + a1,1n+ a1,2n

2 + · · ·+ a1,dnd)Sn

+(a2,0 + a2,1n+ a2,2n

2 + · · ·+ a2,dnd)S2n

+ . . .

+(ar,0 + ar,1n+ ar,2n

2 + · · ·+ ar,dnd)Srn

order r

3


T =

degree d︷︸︸︷(a0,0 + a0,1n+ a0,2n

2 + · · ·+ a0,dnd)

+(a1,0 + a1,1n+ a1,2n

2 + · · ·+ a1,dnd)Sn

+(a2,0 + a2,1n+ a2,2n

2 + · · ·+ a2,dnd)S2n

+ . . .

+(ar,0 + ar,1n+ ar,2n

2 + · · ·+ ar,dnd)Srn

order r

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Question: For a given hypergeometric term f(n, k), what are theorder r and the degree d of the corresponding telescoper?

Answer: This is not a good question. “The” telescoper is notuniquely determined by f(n, k)!

Instead, the set of all telescopers for a fixed term f(n, k) forms aleft ideal in the operator algebra K[n, Sn].

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The Order-Degree-Curve

order

degree

5


T

order

degree

5


T

︸︷︷︸r

order

degree

5


T

︸︷︷︸r

d

order

degree

5


order

degree

5


order

degree

5


order

degree

5


order

degree

5


order

degree

5


order

degree

5


Theorem.

I Consider a proper hypergeometric term

f(n, k) = pol(n, k)xnykM∏

m=1

Γ(amn+a′mk+a′′m)Γ(bmn−b′mk+b′′m)Γ(umn+u′

mk+u′′m)Γ(vmn−v′mk+v′′m) .

I There exists a telescoper of order r and degree d whenever

d >Ar +B

r + C

whereI A = ϑν − 1, B = 2 deg pol + |µ|+ 3− (1 + |µ|)ν, C = 1− ν.I µ =

∑Mm=1(am + bm − um − vm)

I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Theorem.



m=1




d >Ar +B

r + C



I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Theorem.



m=1




d >Ar +B

r + C



I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Theorem.



m=1




d >Ar +B

r + C

where

I A = ϑν − 1, B = 2 deg pol + |µ|+ 3− (1 + |µ|)ν, C = 1− ν.I µ =


I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Theorem.



m=1




d >Ar +B

r + C

whereI A = ϑν − 1, B = 2 deg pol + |µ|+ 3− (1 + |µ|)ν, C = 1− ν.

I µ =∑M

m=1(am + bm − um − vm)

I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Theorem.



m=1




d >Ar +B

r + C



I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

6


Theorem.



m=1




d >Ar +B

r + C



I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Theorem.



m=1




d >Ar +B

r + C



I ν = max{∑M

m=1(a′m + v′m),∑M

m=1(u′m + b′m)}

I ϑ = max{∑M

m=1(am + bm),∑M

m=1(um + vm)}

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Example 1: (n2+k2+1)Γ(2n+3k)Γ(2n−k)

d >7r + 5

r − 3

Example 2: Γ(2n+k)Γ(n−k+2)Γ(2n−k)Γ(n+2k)

d >8r − 1

r − 2

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d >7r + 5

r − 3


d >8r − 1

r − 2

7



d >7r + 5

r − 3

5 10 15

10

20

30


d >8r − 1

r − 2

5 10 15

10

20

30

7



d >7r + 5

r − 3

5 10 15

10

20

30


d >8r − 1

r − 2

5 10 15

10

20

30

7



d >7r + 5

r − 3

5 10 15

10

20

30

sometimes tight


d >8r − 1

r − 2

5 10 15

10

20

30

sometimes not

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The curve overshoots much when f(n, k) is a rational function.

Therefore we give a separate formula for this case in the paper.

Also the formula for this special case has the form d > Ar+Br−C .

The coefficients A,B,C now depend on quantities appearing in acertain representation of f(n, k) as computed in Le’s algorithm.

The refined formula may or may not be tight, but at least it seemsto overshoot far less than the general formula.

See the paper for details and examples.

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Consequences

Even if it may not be accurate, we can use the curve to estimatethe shapes of some interesting telescopers, before computing them.

order

degree

9

Consequences


order

degree

9

Consequences


order

degree

minimal order

9

Consequences


order

degree

minimal order

minimal degree

9

Consequences


order

degree

minimal order

minimal degree

minimal telescoper size

9

Consequences


order

degree

minimal order

minimal degree


minimal total size

9

Consequences


order

degree

minimal order

minimal degree

minimal cost


minimal total size

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Consequences

I For currently feasible input sizes, the minimal cost telescoperagrees with minimal order telescoper.

I We expect that the separation becomes measurable within thecoming few years.

I For asymptotically large input size, the difference is significant.

For τ ≥ max{ϑ, ν} and any fixed constant α > 1 we have:I O∼(τ9). . . cost for telescoper of expected minimal order rmin

I O∼(τ8). . . cost for telescoper of order α rmin.

I Under appropriate assumptions, the optimal choice of α turnsout to be 1.2.

I Similar effects happen in other summation/integrationcontexts (cf. e.g. Alin’s talk on Wednesday 12:05–12:30).

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Consequences








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Consequences








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Consequences



I For asymptotically large input size, the difference is significant.For τ ≥ max{ϑ, ν} and any fixed constant α > 1 we have:

I O∼(τ9). . . cost for telescoper of expected minimal order rmin




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Consequences








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Consequences








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Consequences








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Open Questions

I What is the smallest problem size for which it pays off tocompute a non-minimal telescoper?

I What is the “true curve” which (generically) does notovershoot? Is it also a hyperbola?

I What is the deeper reason behind all these order/degreephenomena discovered recently?

I What is the right question to be asked in the case of severalvariables?

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Open Questions





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Open Questions





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Open Questions





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Open Questions





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Date post:	21-Mar-2018
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Order-Degree Curves for Hypergeometric Creative … Telescoping INPUT: something like f(n;k) := ( 1)...

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