Download An Introduction to Specialised Computer Algebra Systems

A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
An Introduction to
Specialised Computer Algebra Systems
Francesco Biscani
Advanced Concepts Team
European Space Agency (ESTEC)
Course on Differential Equations and Computer Algebra
Estella, Spain – October 29-30, 2010
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Outline
1
A historical perspective in Celestial Mechanics
2
Algebraic structures
3
Polynomial representations
4
Data structures for polynomials
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Outline
1
A historical perspective in Celestial Mechanics
2
Algebraic structures
3
Polynomial representations
4
Data structures for polynomials
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Human Algebra Systems
Euler: lunar theory (incredibili stvdio atqve indefesso labore)
Delaunay: lunar theory based on Hamiltonian formalism
Le Verrier: analytical development of the planetary disturbing
function, discovery of Neptune
Hill-Brown: lunar theory
Doodson: analytical development of the tide-generating
potential
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
The Computer Age
Herget & Musen (1959)
Broucke & Garthwaite (1969)
Deprit & Rom (1971)
...
Ivanova (2001)
San-Juan & Abad (2001)
Laskar & Gastineau (2005,2010)
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Why Specialised Manipulators?
Pros
Performance (up to multiple orders of magnitude)
Capabilities (e.g., truncated products)
Algorithms
Cons
Limited to specific contexts
Steeper learning curve
Availability, documentation, etc.
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Outline
1
A historical perspective in Celestial Mechanics
2
Algebraic structures
3
Polynomial representations
4
Data structures for polynomials
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Manipulation of what?
Polynomials:
X
Ci pi
i
Fourier series:
X
i
cos
Ci
(i · t)
sin
Poisson series:
XX
i
j
cos
Ci,j p
(i · t)
sin
j
Echeloned Poisson series:
XXX
Y
cos
Ci,j,k pk
(l · d)δj,l
(i · t)
sin
i
j
k
Francesco Biscani
l
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Why polynomials?
Arguably one of the simplest algebraic structures
Widely used in many areas of mathematics, science and
engineering
Well studied (e.g., polynomial division, GCD, Gröbner basis,
etc.)
From the computer algebra POV, they easily extend to power
series, Laurent-Puiseux series, Fourier series, etc.
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Outline
1
A historical perspective in Celestial Mechanics
2
Algebraic structures
3
Polynomial representations
4
Data structures for polynomials
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Representing polynomials
Questions to answer:
which coefficients types?
which exponents types?
which operations do I need to perform?
And ultimately:
which data structure(s)?
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
The univariate case
Dense vs sparse
P [C , x] =
n
X
Ci x i
i=0
Dense representation: all monomials are explicitly stored up to
degree n:
C0
C1
...
Cn
Sparse representation: only monomials with non-zero
coefficients are stored.
(Ci1 , i1 ) −→ (Ci2 , i2 ) −→ (Ci3 , i3 ) −→ . . .
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
The multivariate case
Recursive vs distributed
P [C , (x1 , . . . , xn )]
Recursive representation: a multivariate polynomial is a univariate
polynomial with multivariate polynomials as coefficients:
P [C , (x1 , . . . , xn )] = P [P [C , (x2 , . . . , xn )] , x1 ]
Distributed representation: a multivariate polynomial is a collection
of multivariate monomials:
X
P [C , (x1 , . . . , xn )] =
Cj xj
j∈Jn
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Summary
Bi-dimensional choice space:
(dense,sparse) × (recursive,distributed)
Each possibility has its pros & cons
Specialised CAS adopt various different representations,
depending on the purposes
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Outline
1
A historical perspective in Celestial Mechanics
2
Algebraic structures
3
Polynomial representations
4
Data structures for polynomials
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Vector
set of contiguous memory locations storing coefficients
simplest data structure
lowest memory overhead
optimal memory access patterns
random-access container
very good for univariate and dense polynomials
wasteful for multivariate and sparse polynomials:
P(x) = 1 + x 1000
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Linked list
chain of (coefficient,exponent) pairs
(Ci1 , i1 ) −→ (Ci2 , i2 ) −→ (Ci3 , i3 ) −→ . . .
each node of the list contains coefficient, exponent and a
pointer to the next node
space overhead with respect to vectors in the general case
useful for sparse and multivariate polynomials
search for a specific term is O(n)
suboptimal memory access patterns
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Binary search tree
tree structure that keeps node ordered according to a
user-defined criterion
each node has at most two children, one “greater-than”, the
other “less-than” the parent
most operations have O(log n) complexity
useful for sparse and multivariate polynomials
often-used ordering criterions: total degree, lexicographic,
reverse lexicographic
same suboptimal memory access patterns as the linked list
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Hashing containers
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Hashing containers (cont.)
best theoretical performances: amortized O(1) average
complexity for search, insertion, deletion
lots of details to make them work right: hash function,
collision resolution, resizing policy, etc.
unordered containers
standard implementations have poor memory access patterns
effective for multivariate and sparse polynomials
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems
A historical perspective in Celestial Mechanics
Algebraic structures
Polynomial representations
Data structures for polynomials
Many possibilities and variations
bloom filters,
tries (including burst tries, Patricia tries and other variations),
skip lists,
heaps,
geobuckets,
cuckoo hashing,
...
http://en.wikipedia.org/wiki/List_of_data_structures
Francesco Biscani
An Introduction to Specialised Computer Algebra Systems