Multiplication of polynomials
M. Gastineau
IMCCE - Observatoire de Paris - CNRS UMR8028
77, avenue Denfert Rochereau
75014 PARIS
FRANCE
[email protected]
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Different products
Full
• All terms are computed
Truncated in the partial or total degree of the variables
Special truncation to select terms satisfying a rule
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Available methods
Naive method
• efficient for low degree or for sparse polynomials
Karatsuba’s algorithm
• efficient for intermediate degree and dense polynomials
• reduce the number of multiplications
FFT method
• efficient only for large degree
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Naive multiplication
A(x) =
da
max
!
ai xi and B(x) =
i=damin
db
max
!
bi xi
i=dbmin
Perform the multiplication of all terms
C(x) =
damax
+dbmax
!
ck x with ck =
k
k=damin +dbmin
!
ai bj
i+j=k
if A, B and C, have r, s and t terms
• rs multiplications and rs-t additions
• complexity : O(rs)
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for univariate polynomials stored as vector
Algorithm 1: Compute the full product of univariate polynomials A and
B represented with a dense vector
Input: A: polynomial {damin , damax , array of coefficients a }
Input: B: polynomial {dbmin , dbmax , array of coefficients b }
Output: C: polynomial {dcmin , dcmax , array of coefficients c }
dcmin ← damin + dbmin
dcmax ← damax + dbmax
C ← create a polynomial with minimal degree dcmin and maximal degree
dcmax
for k ← dcmin to dcmax do
c[k] ← a[damin ] × b[k − damin ]
for j ← damin + 1 to damax do
c[k] ← c[k] + a[j] × b[k − j]
end
end
return C
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for recursive dense vector 1/2
Function mulfull(A,B ) Compute the full product of multivariate polynomials A and B represented with a recursive dense vector
Input: A: multivariate polynomial {damin , damax , array of coefficients a}
Input: B: multivariate polynomial {dbmin , dbmax , array of coefficients b}
Output: C: multivariate polynomial {dcmin , dcmax , array of coefficients
c}
dcmin ← damin + dbmin
dcmax ← damax + dbmax
C ← create a polynomial with minimal dcmin and maximal dcmax degree
for k ← dcmin to dcmax do
c[k] ← mulfull (a[damin ], b[k − damin ])
for j ← damin + 1 to damax do
fmafull (a[j], b[k − j], c[k])
end
end
return C
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for recursive dense vector 2/2
Procedure fmafull(A,B,C ) Compute the full fused multiplicationaddition C = C + A × B with A, B and C multivariate polynomials
represented as recursive dense vector
Input: A: multivariate polynomial {damin , damax , array of coefficients a}
Input: B: multivariate polynomial {dbmin , dbmax , array of coefficients b}
Input: C: multivariate polynomial {dcmin , dcmax , array of coefficients c }
Output: C: multivariate polynomial
newdcmin ← damin + dbmin
newdcmax ← damax + dbmax
if newdcmin < dcmin or dcmax < newdcmax then
dcmin ←min(newdcmin , dcmin )
dcmax ←max(newdcmax , dcmax )
resize C
end
for k ← dcmin to dcmax do
c[k] ← mulfull (a[damin ], b[k − damin ])
for j ← damin + 1 to damax do
fmafull (a[j], b[k − j], c[k])
end
end
if c contains 0 at its beginning or at its end then
adjust dcmin
adjust dcmax
resize C
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for univariate polynomials stored as list
Function mulfull(A,B ) Compute the full product of univariate polynomials A and B represented with a list
Input: A: polynomial { list of ( coefficients a , degree δa ) }
Input: B: polynomial { list of ( coefficients b , degree δb ) }
Output: C: polynomial { list of ( coefficients c , degree δc )}
C ← create a empty polynomial
foreach element in A do
D ← create a empty polynomial
foreach element in B do
add to the tail of D an element (a × b, δa + δb )
end
C ←C +D
end
return C
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for recursive list
Procedure fmafull(A,B,C ) Compute the full fused multiplicationaddition C = C + A × B with A, B and C multivariate polynomials
represented as recursive list
Input: A: polynomial { list of ( coefficients a , degree δa ) }
Input: B: polynomial { list of ( coefficients b , degree δb ) }
Input: C: polynomial { list of ( coefficients c , degree δc ) }
Output: C: polynomial { list of ( coefficients c , degree δc ) }
iter ← head of C
foreach element in A do
// avoid to scan to C when the loop on B is finished
iterb ← iter
foreach element in B do
// find after iterb in C if the degree δa + δb is present
while current degree δc referenced by iterb < δa + δb do
iterb ← next element after iterb
end
if δc = δa + δb then
fmafull (a, b, c)
if c = 0 then remove the element referenced by iterb
else
insert an element (mulfull (a, b), δa + δb ) just before iterb
end
if current element is the first element of B then
iter ← iterb
end
end
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for flat vector 1/2
variables
X1
Xn
exponents
1 2
coefficients
m
1st term
2nd term
65th term
66th term
64*(m-1)+1 term
64*(m-1)+2 term
64th term
128th term
64*m term
1 2
m
How to sort terms ?
• search and shift operations too slow
• need an intermediate and adjustable storage : BURST TRIE
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Multiplication for flat vector 2/2
Burst tries
• trie node = dense container
• leaf node = sparse container
3 + 5z + 7z 3 + 11y + 9zy + 13zyx + 8z 2 x2 + 9x4
Burst trie
Coe!cients
0
4
8
1
0
0
1
2
3
3
1
4
1
5
1
0
1
6
2
7
3
5
7
11
9
13
8
9
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block
A=
da
max
!
BHδ (a)
,
B=
×
d!1 d!2
d!n
bj X1 X2 ...Xn
=
C =A×B =
with
BHδ (b)
δ=dbmin
δ=damin
ai X1d1 X2d2 ...Xndn
db
max
!
d1 +d!1 d2 +d!2
dn +d!n
ai bj X1
X2
...Xn
dc
max
!
BHδ (c)
δ=dcmin
dcmin = damin + dbmin
dcmax = damax + dbmax
!
BHδ (c) =
BHi (a) × BHj (b)
i+j=δ
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block
Function fmafull(BHδ (a), BHδ! (b), BHδ+δ! (c))
Compute the full fused multiplication-addition
BHδ+δ! (c) = BHδ+δ! (c) + BHδ (a) × BHδ! (b)
Input: BHδ (a) : homogeneous blocks { degree δ, a : array of r coeff. }
Input: BHδ! (b) : homogeneous blocks { degree δ ! , b : array of s coeff. }
Input: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff. }
Output: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff.}
for i ← 1 to r do
for j ← 1 to s do
l ← get location of the term in BHδ+δ! (c)
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block using functions
Function fmafull(BHδ (a), BHδ! (b), BHδ+δ! (c))
Compute the full fused multiplication-addition
BHδ+δ! (c) = BHδ+δ! (c) + BHδ (a) × BHδ! (b) using functions to compute
location
Input: BHδ (a) : homogeneous blocks { degree δ, a : array of r coeff. }
Input: BHδ! (b) : homogeneous blocks { degree δ ! , b : array of s coeff.s }
Input: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff. }
Output: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff.}
for i ← 1 to r do
expoa ← get array of exponents from the location i in BHδ
for j ← 1 to s do
expob ← get array of exponents from the location j in BHδ!
expoc ← expoa + expob
l ← get location of the term with exponents expoc in BHδ+δ!
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block using addressing tables
Construction of the addressing table for the product of blocks in 3
variables with exponent tables of degree 1 and 2.
+
T exp1
0
0
1
0
1
0
0
0
0
1
1
2
1
0
0
1
2
5
=
T exp2
0
1
2
0
1
0
2
1
0
1
0
0
T addr1,2
2 3 5 6
3 4 6 7
6 7 8 9
T exp3
0 0 3
0 1 2
0 2 1
0 3 0
1 0 2
1 1 1
1 2 0
2 0 1
2 1 0
3 0 0
8
9
10
T addr2,1 =t T addr1,2
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Execution time to build the tables of exponents
build the tables of exponents for homogeneous blocks in 10 variables up to the degree 20
0.4
20
0.35
0.3
19
time (s)
0.25
0.2
18
0.15
17
0.1
16
0.05
13
12
14
15
0
0
2
4
6
8
10
12
number of terms (in Millions)
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Execution time to build the addressing tables
product of two homogeneous blocks in 5 variables up to the total degree 40
initialization
after initialization
10
20+20
8
time (s)
20+19
6
20+18
20+17
4
20+16
20+15
2
20+14
20+13
20+12
20+11
20+10
0
0
20
40
60
80
100
120
muliplication operations (in Millions)
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Overhead to load the addressing tables from disk
30
execution time overhead (%)
25
20
15
10
5
0
20
22
24
26
28
30
32
34
36
38
40
total degree of the addressing table
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous blocks
BH(X,Y,Z)
BH0
BH1
BH2
BH3
BH4
3
5
11
0
0
9
0
0
0
0
0
0
0
0
0
13
0
0
0
0
0
0
0
0
0
0
0
0
0
0
8
0
0
0
0
9
P (x, y, z) = 3 + 5z + 7z 3 + 11y + 9yz + 13xyz + 8x2 z 2 + 9x4
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Compacted homogeneous blocks
BHC(X,Y,Z)
BHC0 BHC1 BHC2 BHC3 BHC4
3
5 1
11 2
9 2
13 6
8 10
9 15
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block using addressing tables
Function fmafull(BHδ (a), BHδ! (b), T addrδ,δ! , BHδ+δ! (c))
Compute the full fused multiplication-addition
BHδ+δ! (c) = BHδ+δ! (c) + BHδ (a) × BHδ! (b) using the addressing table
Input: BHδ (a) : homogeneous blocks { degree δ, a : array of r coeff. }
Input: BHδ! (b) : homogeneous blocks { degree δ ! , b : array of s coeff. }
Input: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff. }
Input: T addrδ,δ! : addressing table of degree δ, δ !
Output: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t
coefficients }
for i ← 1 to r do
for j ← 1 to s do
l ← T addrδ,δ! [i, j]
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block using addressing tables
Function fmafull(BHδ (a), BHδ! (b), T addrδ,δ! , BHδ+δ! (c))
Compute the full fused multiplication-addition
BHδ+δ! (c) = BHδ+δ! (c) + BHδ (a) × BHδ! (b) using the addressing table
Input: BHδ (a) : homogeneous blocks { degree δ, a : array of r coeff. }
Input: BHδ! (b) : homogeneous blocks { degree δ ! , b : array of s coeff. }
Input: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff. }
Input: T addrδ,δ! : addressing table of degree δ, δ !
Output: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t
coefficients }
for i ← 1 to r do
for j ← 1 to s do
! [i, j] δ (a).index[i], BHCδ ! (b).index[j]]
! [BHC
←TTaddr
addrδ,δ
ll←
δ,δ
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Homogeneous block using addressing tables
Function fmafull(BHδ (a), BHδ! (b), T addrδ,δ! , BHδ+δ! (c))
Compute the full fused multiplication-addition
BHδ+δ! (c) = BHδ+δ! (c) + BHδ (a) × BHδ! (b) using the addressing table
Input: BHδ (a) : homogeneous blocks { degree δ, a : array of r coeff. }
Input: BHδ! (b) : homogeneous blocks { degree δ ! , b : array of s coeff. }
Input: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff. }
Input: T addrδ,δ! : addressing table of degree δ, δ !
Output: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t
coefficients }
for i ← 1 to r do
for j ← 1 to s do
l ← T addrδ,δ! [i, j]
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Cache blocking technique
BH!(a)
BH!(b)
BH!(b)
BH!(a)
normal flow
flow with cache blocking
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Cache blocking technique
Input: BHδ (a) : homogeneous blocks { degree δ, a : array of r coeff. }
Input: BHδ! (b) : homogeneous blocks { degree δ ! , b : array of s coeff. }
Input: T addrδ,δ! : addressing table of degree δ, δ !
Input: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff. }
Input: chunksize : chunk size { arrays of two integers}
Output: BHδ+δ! (c) : homog. blocks { degree δ + δ ! , c : array of t coeff.}
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
iteri ← r/chunksize[1] /* number of chunks for the loop i
iterj ← s/chunksize[2] /* number of chunks for the loop j
for ci ← 0 to iteri − 1 do
for cj ← 0 to iterj − 1 do
for bi ← 1 to chunksize[1] do
for bj ← 1 to chunksize[2] do
i ← ci × iteri + bi
j ← cj × iterj + bj
l ← T addrδ,δ! [i, j]
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
end
/* if s not divisible by chunksize[2]
for bi ← 1 to chunksize[1] do
for j ← iterj × chunksize[2] to s do
i ← ci × iteri + bi
l ← T addrδ,δ! [i, j]
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] + BHδ (a)[i] × BHδ! (b)[j]
end
end
end
*/
*/
22
23
24
25
26
27
/* if r not divisible by chunksize[1]
*/
for i ← iteri × chunksize[1] to r do
for j ← 1 to s do
l ← T addrδ,δ! [i, j];
BHδ+δ! (c)[l] ← BHδ+δ! (c)[l] +
BHδ (a)[i] × BHδ! (b)[j]
end
end
*/
Function fmafull(BHδ (a), BHδ! (b), T addrδ,δ! , chunksize, BHδ+δ! (c))
Compute the full fused multiplication-addition using the addressing table
and cache blocking technique
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Benchmark of the cache blocking technique
Factor of the reduction of the execution time for the product of two
homogeneous blocks in 8 variables of degree 7
Core2 duo processor
G5 processor
1000
1.1
900
1.05
1000
1.1
900
1.05
800
800
1
1
700
700
0.95
600
500
0.9
400
0.85
300
0.95
600
500
0.9
400
0.85
300
0.8
200
0.8
200
0.75
100
0
0.7
0
100 200 300 400 500 600 700 800 900 1000
0.75
100
0
0.7
0
100 200 300 400 500 600 700 800 900 1000
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Benchmark of the cache blocking technique
Factor of the reduction of the execution time of the product of two
homogeneous blocks in 8 variables of degree 9
double precision
1000
quadruple precision
1.1
900
1.05
800
1000
1.1
900
1.05
800
1
1
700
700
0.95
600
500
0.9
400
0.85
0.95
600
500
0.9
400
0.85
300
300
0.8
0.8
200
200
0.75
100
0
0.7
0
100 200 300 400 500 600 700 800 900 1000
0.75
100
0
0.7
0
100 200 300 400 500 600 700 800 900 1000
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Benchmarks
s × (s + 1) with s = (1 + x + y + z + t + u)14
CAS
Ginac 1.3.2
Maple 10
Singular 3.0.2
Maxima 5.9.2
Mathematica 5.2
TRIP 0.99
TRIP 0.99
TRIP 0.99
TRIP 0.99
representation
tree
DAG
list
recursive list
tree
recursive vector
recursive list
flat vector
homogeneous blocks
(with initialization)
(after initialization)
time (s)
4137
2899.70
144.27
443.95
766.65
13.50
12.85
28.10
5.44
0.57
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Effect of the sparsity of the polynomials
s × s with s containing some terms of (1 + x1 + x2 + x3 + x4 + x5 )10
0.8
recursive list
recursive vector
flat vector
homogeneous block
compacted homogeneous block
0.7
0.6
time (s)
0.5
0.4
0.3
0.2
0.1
0
0
500
1000
1500
2000
2500
3000
number of terms
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
full product VxV with different representations
The serie V has 3052 terms and the result has 227453 terms
V (λ, λ , X, X, Y, Y , X , X ! , Y , Y ! ) =
!
!
!
!
d2
X d1 X Y d3 Y
d4
d6
d8
!
X !d5 X ! Y !d7 Y ! eı(k1 λ+k2 λ )
d1 + d2 + d3 + d4 + d5 + d6 + d7 + d8 = 7
CAS
Maple 10
Mathematica 5.2
TRIP 0.99
TRIP 0.99
TRIP 0.99
TRIP 0.99
representation
DAG
tree
recursive vector
recursive list
flat vector
homogeneous blocks
TRIP 0.99
compacted homogeneous blocks
TRIP 0.99
d’alembert blocks
TRIP 0.99
compacted d’alembert blocks
(with initialization)
(after initialization)
(with initialization)
(after initialization)
(with initialization)
(after initialization)
(with initialization)
(after initialization)
time (s)
345.08
149.63
2.91
2.32
1.97
2468.10
2403.45
17.07
15.11
22.55
8.55
11.01
1.25
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
full product VxV for different degrees
300
recursive vector
recursive list
flat vector
BH
BHC
BHDAL
BHDALC
250
time (s)
200
150
100
50
0
4
5
6
7
8
9
10
11
12
degree
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Karatsuba's algorithm
Idea : one multiplication could be avoided for polynomial of degree 1
Let A and B polynomials
A(X) = a0 + a1 X and B(X) = b0 + b1 X
The naive multiplication C = AB is
C(X) = a0 b0 + (a0 b1 + a1 b0 )X 1 + a1 b1 X 2
But the coefficient of X 1 could be written as
a0 b1 + a1 b0 = (a0 + a1 )(b0 + b1 ) − a0 b0 − a1 b1
we need to perform 3 multiplications and 4 additions
instead of 4 multiplications and 1 addition.
could be applied recursively to polynomials of degree 2k-1
complexity O(n1.59)
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
k is equal to 0. The Karatsuba’s multiplication algorithm 11 performs the product
k . Depending of the relative speed of the addition and
of two polynomials of degree
2
Karatsuba's algorithm
the multiplication of the coefficients, the recursive call could be stopped before. A
generalization of the karatsuba algorithm to any degree is available in [33].
Karatsuba’s multiplication could be computed at most in 9nlog2 3 or O(n1.59 ) operations [12].
Algorithm 11: Compute the full product of two polynomials A and B using the
Karatsuba’s multiplication algorithm
Input: A : polynomial of degree at most n − 1 with n = 2k for k ∈ N
Input: B : polynomial of degree at most n − 1
Output: C : polynomial
if n = 1 then return C ← AB
C1 ← A(0) B (0) by a recursive call
C2 ← A(1) B (1) by a recursive call
C3 ← A(0) + A(1)
C4 ← B (0) + B (1)
C5 ← C3 C4 by a recursive call
C6 ← C5 − C1 − C2
C ← C1 + C6 X n/2 + C2 X n
return C
4.4
multiplication using Advanced
DFT
and FFT
School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Truncated product
Univariate polynomials
!
n
n
(a0 + a1 X + ... + an X + O(X )) (b0 + b1 X + ... + bn X n + O(X n ))
=
(c0 + c1 X + ... + cn X n + O(X n ))
Multivariate polynomials
• keep the term
ai X1d1 X2d2 ...Xndn
!
d!1 d!2
d!n
bj X1 X2 ...Xn
if d1 + d!1 + d2 + d!2 + ... + dn + d!n ≤ T
• if truncation is performed only on some variables,
truncated variables must be ordered
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Truncated product on polynomials stored as recursive list
Function fmatruncated(A,B,C, T ) Compute the truncated fused multiplication-addition
C = C + A × B with A, B and C multivariate polynomials represented as recursive list
Input: A: polynomial { list of ( coefficients a , degree δa ) }
Input: B: polynomial { list of ( coefficients b , degree δb ) }
Input: C: polynomial { list of ( coefficients c , degree δc )}
Input: T : degree of the truncation
Output: C: polynomial { list of ( coefficients c , degree δc )}
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
if variable of A is truncated then
iter ← head of C
foreach element in A such that δa ≤ T do
/* avoid to scan to C when the loop on B is finished
*/
iterb ← iter
foreach element in B such that δa + δb ≤ T do
/* find after iterb in C if the degree δa + δb is present
*/
while current degree δc referenced by iterb < δa + δb do
iterb ← next element after iterb
end
if δc = δa + δb then
fmatruncated (a, b, c, T − δa + δb )
if c = 0 then remove the element referenced by iterb
else
insert an element (multruncated (a, b, T − δa + δb ), δa + δb ) just before iterb
end
if current element is the first element of B then
iter ← iterb
end
end
end
else
fmafull (A,B,C)
end
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Truncated product on homogeneous blocks
Let
A=
!
BHδ (a)
,
B=
!
BHδ (b)
Truncated product on the total degree
C=A
!
B=
T
"
δ=0
BHδ (c) with BHδ (c) =
δ
"
n=0
BHn (a) × BHδ−n (b)
• use the full product of 2 homogeneous blocks
• use less addressing tables than for the full product
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Truncated product - benchmarks
600
recursive vector
recursive list
flat vector
BH
BHC
BHDAL
BHDALC
500
400
300
200
100
0
6
8
10
12
14
16
18
20
degree
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Special truncated product in degree
Perform the product of two Poisson series
but we only want to keep terms which have specific values
for k1 and k2
e.g., S1 × S2 = S , we want only terms such that k1 = 0 and k2 = 0
S=
!
!
ai X1d1 X2d2 ...Xndn expı(k1 λ+k2 λ )
very easy for the recursive representation if the series are
correctly ordered.
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Special truncated product on magnitude
ai X1d1 X2d2 ...Xndn
⊗
d!1 d!2
d!n
bj X1 X2 ...Xn
is kept if |ai bj | ≥ !0
brut-force method
Algorithm 1: Compute the truncated product of the series A and B in the amplitude of
their coefficients
!
Input: A: serie ! ai xi ordered by decreasing amplitude
Input: B: serie
bi xi ordered by decreasing amplitude
Input: !0 : thresold > 0
Output: C: series C = AB with all coefficients greater than !0
C ← create an empty polynomial
foreach coefficient ai such that |ai b0 | ≥ !0 do
foreach coefficient bj such that |bj | ≥ !0 /|ai | do
C ← C + ai bj xi+j
end
end
return C
order the series on the magnitude of the coefficient ?
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
Special truncated product on magnitude
p
Let a variable !, and a small parameter !!0 such that !!0 = !0 with p ∈ N.
Each coefficient ai of the serie A(x) could be written as
ai = a!i xi !k with k = "
log |ai |
ai
!
#
and
a
=
i
log !!0
!!0 k
! !
A (!, x) =
(
a!j xj )!k
!
k
j
A(x) = A! (!!0 , x)
The truncated product A(x)
⊗ B(x) on the amplitude of the coefficient is transformed to a
!
truncated product A! (!, x) B ! (!, x) on the variable !.
The degree of truncation is p.
Advanced School on Specific Algebraic Manipulators, 2007, © M. Gastineau, ASD/IMCCE/CNRS
58
Special truncated product
in amplitude
CHAPTER
4. MULTIPLICATION
source code
s 1 =(1+0.05∗ x ) ˆ 3 ;
s 2 =(1+0.04∗ x ) ˆ 4 ;
/∗ i n t r o d u c e t h e v a r i a b l e e p s i n s 1 and s 2 ∗/
s 1 e=s e r e p s ( s1 , eps , 0 . 1 ) ;
s 2 e=s e r e p s ( s2 , eps , 0 . 1 ) ;
/∗ d e f i n e t h e t r u n c a t u r e on a m pl i t u d e t o ( 0 . 1 ) ˆ 2 ∗/
t r =({ eps , 2 } ) ;
usetronc ( tr ) ;
/∗ perform t h e p r o d u c t ∗/
s 3 e=s 1 e ∗ s 2 e ;
/∗ remove t h e v a r i a b l e e p s from s 3 e ∗/
s 3=i n v s e r e p s ( s3e , eps , 0 . 1 ) ;
Execution of the previous source code by trip
s1(x) = 1 + 0.15*x + 0.0075*x**2 + 0.000125*x**3
s2(x) = 1 + 0.16*x + 0.0096*x**2 + 0.000256*x**3 + 2.56E-06*x**4
s1e(x,eps) = 1 + 0.15*x + 0.75*x**2*eps**2 + 0.125*x**3*eps**3
s2e(x,eps) = 1 + 0.16*x + 0.96*x**2*eps**2 + 0.256*x**3*eps**3 + 0.256*x**4*eps**5
tr = ( { eps, 2 } )
s3e(x,eps) = 1 + 0.31*x + 0.024*x**2 + 1.71*x**2*eps**2 + 0.264*x**3*eps**2
s3(x) = 1 + 0.31*x + 0.0411*x**2 + 0.00264*x**3
Advanced
Schoolof
on Specific
Manipulators,
2007, © M.of
Gastineau,
Table 4.6: Source code and execution by
TRIP
the Algebraic
truncated
product
two ASD/IMCCE/CNRS
series