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xAct

Efficient manipulation of tensor expressions

Jose M. Martn GarcaLaboratoire Univers et Theories, Meudon

&

Institut dAstrophysique de Paris

LUTH, Meudon, 21 April 2009

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Computers in science

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Human102 flops vs. computer109 1015 flops.

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Numerics: Approximate solutions to continuous problems.

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Computers are discrete-calculus machines!

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Computer algebra(CA): Exact solutions.

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Computer algebra(CA): Exact solutions.

Our problem:EfficientTensor Computer Algebra (TCA).

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Summary

1. General purpose computer algebra (CA)

Focus: the canonicalizer.

2. Computer algebra for tensor calculus (TCA)

Focus: types of problems.

3. AMathematica/ C implementation: xAct

4. Case example: scalars of the Riemann tensor

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1. Computer algebra (CA)

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Definition:Manipulation of symbols (even the program itself)

No truncation of information

In science: manipulation of symbolic mathematical expressions

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Definition:Manipulation of symbols (even the program itself)

No truncation of information

In science: manipulation of symbolic mathematical expressions

Early history:

1950: ALGAE (Los Alamos)1953: Kahrimanian, Nolan: differentiation systems

1963: ALTRAN / ALPAK (Bell Labs)

1960s: LISP: recursion, conditionals, dynamical allocation of memory,

garbage collection, etc.

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1b. Computer algebra systems (CAS)

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General purpose systems:

System Language Develop. Release Strengths

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Reduce LISP ? 1968

Macsyma LISP ? 1970Derive LISP 1979 1988

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Reduce LISP ? 1968


Maple C 1979 1985 hashing routines

Mathematica C 1986 1988 pattern matching

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Reduce LISP ? 1968


Maple C 1979 1985 hashing routines

Mathematica C 1986 1988 pattern matchingMaxima CLISP 1982 1998Axiom Aldor 1971-91 2002 object-oriented

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1c. CA. Memory limitations

Recursive algorithmxi+1=f(xi)with initial seedx0:Numerical: reserve fixed memory per number and

xi+1= Truncate[f(xi)]

CA: exact algorithm, reallocate memory. Generic memory growth Generic computing-time growth

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Examples:

det(A + B) n! 2n terms ( 4 109 forn = 10)

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Examples:

Intermediate expression swell:Simple input Complex intermediate steps Simple output

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Examples:

Linear systems with random integers |ci| 100. Timing (s):

1 5 10 50 100 500

0.01

1

100

ExactNumerical

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1d. CA. Intrinsic limitations

Richardsons theorem (1968):

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Let beEa set of real functions such that

IfA(x), B(x) EthenA(x)B(x), A(x)B(x), A(B(x)) E.The rational numbers are contained as constant functions.

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Then for expressionsA(x)inE,

iflog 2, , ex, sin x EthenA(x) 0for allxis unsolvable;

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iflog 2, , ex, sin x EthenA(x) 0for allxis unsolvable;if also x2 EthenA(x) 0is unsolvable.

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iflog 2, , ex, sin x EthenA(x) 0for allxis unsolvable;if also x2 EthenA(x) 0is unsolvable.If furthermore there is a functionB(x) Ewithout primitive inEthenthe integration problem is unsolvable. Example: ex

2

.

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iflog 2, , ex, sin x EthenA(x) 0for allxis unsolvable;if also x2 EthenA(x) 0is unsolvable.If furthermore there is a functionB(x) Ewithout primitive inEthenthe integration problem is unsolvable. Example: ex

2

.

Is Computer Algebra doomed to failure?

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1e. CA. The canonicalizer

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There are families of expressions for which a canonical form can always bedefined. Example: polynomials.

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Definition: A=Biff their canonical forms coincide.

There are different canonical forms for a family of expressions.

The algorithm in charge of canonicalization is called the canonicalizer.

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Definition: A=Biff their canonical forms coincide.

There are different canonical forms for a family of expressions.

The algorithm in charge of canonicalization is called the canonicalizer.

Do not confuse

canonicalization: objective, given a canonical form

simplification: largely subjective. More difficult. Mathematica:

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2 Computer algebra for (GR) tensors

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2. Computer algebra for (GR) tensors

Motivation:Long but simple problems

Perturbation theory

Bel-Robinson, super energy-momentum tensors, ...

4Babcd =CaebfCcedf+

CaebfCcedf Babcd =B(abcd)

Riemann polynomials:

RabcdRaecfRbfde =RabcdRaecfRbedf 14 RabcdRabefRcdef

Lovelock (dimension-dependent) identities

Component expansions in numerics (code generation)[ Kranc: Husa, Hinder, Lechner 04 ]

Manipulation and classification of metrics[ GRDB: Ishak, Lake 02; ICD: Skea 97]

Added benefits: reproducibility, error-free, ...

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2b TCA Early history

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2b. TCA. Early history

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R. dInverno 1969ALAM

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R.A. Russell-Clark 1971CLAM

I. Frick 1974ILAM

LAM

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I. Frick 1974ILAM

LAM

I. Frick 1977SHEEP

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I. Frick 1974ILAM

LAM

I. Frick 1977SHEEP

dInverno & Frick 1982

SHEEP2

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I. Frick 1974ILAM

LAM

I. Frick 1977SHEEP

dInverno & Frick 1982

SHEEP2

J. man 1987

CLASSI

L. Hrnfeldt 1988

STENSOR

J. E. F. Skea 1988

RSHEEPxAct p.10/28

2c. Classes: types of problems

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Component computations:

Give a metric in a coordinate system / frame.

Compute other tensors from that metric.

Key issues:

Component expansions. Many terms. Memory? [ Lake 03 ]

Symmetries. Independent components? [ Klioner 04 ]

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Component computations:

Give a metric in a coordinate system / frame.

Compute other tensors from that metric.

Key issues:

Component expansions. Many terms. Memory? [ Lake 03 ]

Symmetries. Independent components? [ Klioner 04 ]

Abstract computations:

Define tensors with symmetries, connections, etc.

Compute and simplify expressions.

Key issue: canonicalization with symmetries and dummy indices.

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2c-2. Classes: sources of complexity

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p y

General expression: tensor polynomial

. . . + 3r2RabcdRaecfTbefvd + RabcdRcdefRefab+ ...

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p y



Expand and canonicalize terms independently (parallelism).

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p y




Tensor product: sort tensors and construct equivalent single tensor withinherited symmetries [ Portugal 99 ].

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Canonicalize tensor withnindices and arbitrary symmetry:

Simple algorithms: timings areexponentialinn.

Efficient algorithms: timings are effectivelypolynomialinn.

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Canonicalize tensor withnindices and arbitrary symmetry:

Simple algorithms: timings areexponentialinn.

Efficient algorithms: timings are effectivelypolynomialinn.

Arrange dummy indices in full expression.

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Monoterm symmetries(perm groups):

Rbacd = Rabcd, Rcdab = +RabcdMost packages use ad hoc exponential algorithms.

Polynomic algorithms to manipulate the Symmetric GroupSn, based onStrong Generating Set representations[ Schreier, Sims, Knuth, ... 70s, 80s ].

Applied to tensor/spinor canonicalization: [ Portugal et al. 01 ]

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Monoterm symmetries(perm groups):

Rbacd = Rabcd, Rcdab = +RabcdMost packages use ad hoc exponential algorithms.

Polynomic algorithms to manipulate the Symmetric GroupSn, based onStrong Generating Set representations[ Schreier, Sims, Knuth, ... 70s, 80s ].

Applied to tensor/spinor canonicalization: [ Portugal et al. 01 ]

Multiterm symmetries(perm algebras):

Rabcd+ Racdb+ Radbc = 0

No known efficient algorithms. Solutions?Most elegant: Young tableaux [ Fulling et al. 92; Peeters 05 ]

Particular case: dimension-dependent identities [ Edgar et al. 02 ].

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2d. Tensor packages

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MAXIMA: itensor, stensor / ctensor

REDUCE: atensor, RicciR, ExCalc / GRG, GRLIB, RedTen

MAPLE: Riegeom, Canon, MapleTensor / Riemann, Atlas,GRTensorII

MATHEMATICA: MathTensor ($$), dhPark, Tensors in Physics ($), Tensorial ($),Ricci, TTC, EinS, xTensor / GRTensorM, xCoba

Standalone: cadabra

Prototyping: Kranc, RNPL, TeLa

Many other small packages for component computations.

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2e. Example 1

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Antisymmetric tensorFba = Fab.FabFbc n... F

ha = 0 if numbernof tensors is odd.

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2e-2. Example 1

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2e-3. Example 2

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Random monomial Riemann scalars:ga1b7 . . . gdnc5 Ra1b1c1d1. . . Ranbncndn

oooooooo

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timings

inseconds

The algorithm uses theintersection algorithm, which is known to beexponential in the worst case.xAct p.17/28

2e-4. Example 2

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A million random monomial Riemann

7

scalars:

0 0.02 0.04 0.06 0.08 0.1 0.12

Time s

0

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100000

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0 0.5 1 1.5 2Time s

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A million

Riemann^7

3.2 GHz

bin .005 s

Zero: 441324

Nonzero: 558673

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2e-5. Example 2

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A million random monomial Riemann

10

scalars:

CPU 1.7 GHz

bin: 0.0001 s

Zero: 424108

Nonzero: 575892

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3. The xAct project

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3. The xAct project

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xCore

MathematicaToolsxAct p.20/28

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3. The xAct project

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xCore

MathematicaTools

xPermPermutation Group Theory

xTensorAbstract tensor algebra

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3. The xAct project

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xCore

MathematicaTools



xCobaComponent tensor algebra

[ D. Yllanes ]

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3. The xAct project

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xCore

MathematicaTools



xCobaComponent tensor algebra

[ D. Yllanes ]

xPertPerturbation Theory

HarmonicsTensor spherical hars.

[ D. Brizuela andG. Mena Marugn ]

InvarRiemann tensor

[ R. Portugal ]

Spinors[ A. Garca-Parrado ]

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3b. Features

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3b. Features

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Strengths:Fast monoterm canonicalization.

Mathematical structure, GR-oriented.

Free software (GPL).Extensive,Mathematica-style documentation.

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3b. Features

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Current weaknesses:

Multiterm canonicalization missing.Tensor-editing interface missing.

Incomplete development of calculus with charts.

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Current weaknesses:

Multiterm canonicalization missing.Tensor-editing interface missing.

Incomplete development of calculus with charts.

Other data:c20022009, GPL. Version 0.7 in March 2004; currently in 0.9.817000 lines of Mathematica code + 2500 lines of C code.

31 articles have used it:

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3d. What xTensor can do

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Define one or several manifolds, and products of them.Define (complex) vector bundles on them.

Define tensors with arbitrary monoterm symmetries.

Define connections of any type. Automatic Christoffel, Riemann, ...generation.

Define one or several metrics on each manifold. Warped metrics. Inducedmetrics.

Parametric derivatives, Lie derivatives, Poisson brackets.

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Single canonicalization routine (ToCanonical).

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Transformation rules, which detect the character of the indices, the manifold of

the indices, the symmetries of the tensors, and take into account metrics.

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Transformation rules, which detect the character of the indices, the manifold of

the indices, the symmetries of the tensors, and take into account metrics.Use all machinery ofMathematica.

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3e. xTensor notations

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Usealwaysabstract tensor notation. Penrose abstract-indices.

Walds book conventions. Ashtekars approach to gauge.

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Symbols have a type: tensor, abstract-index, manifold, ...

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Notation forTab?

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Notation forTab?T[a,-b]

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Tensor[Name["T"], Indices[Up[a], Down[b]]]

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Translator between internal / external notations

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U l b t t t t ti P b t t i di

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Translator between internal / external notationsChoice:T[a,-b]

Notation for covariant derivatives:(Penrose & Rindler) What are

2 v3and2 v3 ?

Is it e2a e3ba vb or e2aa

e3b vb

?

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U l b t t t t ti P b t t i di

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Translator between internal / external notationsChoice:T[a,-b]

Notation for covariant derivatives:(Penrose & Rindler) What are

2 v3and2 v3 ?

Is it e2a e3ba vb or e2aa

e3b vb

?

Choice:aandaare operators: Cd[-a][expr]Lie derivatives: LieD[ v[i] ][ expr ]

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4. The Riemann invariants

Scalar invRabcdRabefRcdef, or differential invRab

c

dR

acbd.

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Used in the classification of spacetimes, GR Lagrangian expansions,QG renormalization, etc.

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c

dR

acbd.

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Haskins 1902: 14 independent scalar invs in 4d.

Narlikar and Karmarkar 1948: first list of 14 scalar invs:(R, R2, W2, R3, W3, R2W, W4, R4, R2W2,

W5, R2W3, R4W, R4W2, R4W3).

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c

dR

acbd.

U d i h l ifi i f i GR L i i

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W5, R2W3, R4W, R4W2, R4W3).

Other invariants are (possibly nonpolynomial) functions of them.Completeness:polynomicbasis for all invariants. Relations?

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c

dR

acbd.

U d i th l ifi ti f ti GR L i i

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W5, R2W3, R4W, R4W2, R4W3).


Sneddon 1999: 38 s-invs (deg 11). Complete basis. (Rotor calculus.)

Carminati-Lim 2007: graph-based construction of s-inv relations.

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c

dR

acbd.

U d i th l ifi ti f ti GR L i i

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W5, R2W3, R4W, R4W2, R4W3).


Sneddon 1999: 38 s-invs (deg 11). Complete basis. (Rotor calculus.)

Carminati-Lim 2007: graph-based construction of s-inv relations.Fulling et al 1992: bases for d-invs up to 10 derivatives andR6.

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4b. Riemann invariants up to degree 7

7 Riemann

28 indices

28!

3.0

1029 s-invs

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7 Riemann

28 indices

28!

3.0

1029 s-invs

Use monoterm symmetries: 16352 invariants

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28 indices

28!

3.0

1029 s-invs


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Use cyclic property: 1639 invariants

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28!

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1029 s-invs


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y


Antisymmetrise over sets of 5 indices in 4d: 4311 integer equations

Gauss elimination: Intermediate swell problem:Original integers: |ci| 384

Final integers: |cf| 4978120

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28!

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1029 s-invs


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Largest intermediate integer:412843208881146263121056084725879635772776056596195531210903371997452212291202639383904791907983313629332658685142520797678355583630907136564045657358541395134420154091637344181884734088835920651682654

5838550350981924166243854816386358011913196352080821446913112544207777945582431400973483457511551249

4271650405780114863779964319571108875740447236120954785394441817343113274871351581501474081446209153

5133993980627211654318697002059693685910102607365896788999230680327719504392651078493689021476459822

1917466623055176060658271638645490139036389024466375930303586688550738508615214422459534528028026604

3392962118745453989341765134526682155897073108863212658336777829476719031970391332987380834358579837

4768508365933468022268161668651405869982994847652173877241170117828300225631267244981449350418876807

8308059566617955048754211100127225300485494079978006577938025856377710049543176142178387315401497328

Required 8Gb RAM and 384-bytes integers (BigNum: GMP).

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28!

3.0

1029 s-invs


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Largest intermediate integer:412843208881146263121056084725879635772776056596195531210903371997452212291202639383904791907983313629332658685142520797678355583630907136564045657358541395134420154091637344181884734088835920651682654

5838550350981924166243854816386358011913196352080821446913112544207777945582431400973483457511551249

4271650405780114863779964319571108875740447236120954785394441817343113274871351581501474081446209153

5133993980627211654318697002059693685910102607365896788999230680327719504392651078493689021476459822

1917466623055176060658271638645490139036389024466375930303586688550738508615214422459534528028026604

3392962118745453989341765134526682155897073108863212658336777829476719031970391332987380834358579837

4768508365933468022268161668651405869982994847652173877241170117828300225631267244981449350418876807

8308059566617955048754211100127225300485494079978006577938025856377710049543176142178387315401497328

Required 8Gb RAM and 384-bytes integers (BigNum: GMP).

We get the 27 invs of Sneddons basis up to degree 7 (6 dual), plus allpolynomial expression of any other invariant.Invarpackage: CPC 2007.

Database of 645 625 relations up to 12 metric derivatives. CPC 2008.xAct p.26/28

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5. Conclusions

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5. Conclusions

Brief review of the field of Computer Algebra, focusing on the importance ofthe canonicalizer.

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5. Conclusions


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There are important problems in GR and other fields which requireTensorComputer Algebra.

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5. Conclusions


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For the first time we have tensor packages with efficient canonicalizationalgorithms for monoterm symmetries,and they are free software!

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5. Conclusions


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For the first time we have tensor packages with efficient canonicalizationalgorithms for monoterm symmetries,and they are free software!

There are multiterm algorithms, but we need something more efficient. Idea:databases of solutions (example:Invarpackage).

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