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)abbrev package GALFACTU GaloisGroupFactorizationUtilities
++ Author: Frederic Lehobey
++ Date Created: 30 June 1994
++ Date Last Updated: 19 October 1995
++ Basic Functions:
++ Related Constructors:
++ Also See:
++ AMS Classifications:
++ Keywords:
++ References:
++ [1] Bernard Beauzamy, Products of polynomials and a priori estimates for
++ coefficients in polynomial decompositions: a sharp result,
++ J. Symbolic Computation (1992) 13, 463-472
++ [2] David W. Boyd, Bounds for the Height of a Factor of a Polynomial in
++ Terms of Bombieri's Norms: I. The Largest Factor,
++ J. Symbolic Computation (1993) 16, 115-130
++ [3] David W. Boyd, Bounds for the Height of a Factor of a Polynomial in
++ Terms of Bombieri's Norms: II. The Smallest Factor,
++ J. Symbolic Computation (1993) 16, 131-145
++ [4] Maurice Mignotte, Some Useful Bounds,
++ Computing, Suppl. 4, 259-263 (1982), Springer-Verlag
++ [5] Donald E. Knuth, The Art of Computer Programming, Vol. 2, (Seminumerical
++ Algorithms) 1st edition, 2nd printing, Addison-Wesley 1971, p. 397-398
++ [6] Bernard Beauzamy, Vilmar Trevisan and Paul S. Wang, Polynomial
++ Factorization: Sharp Bounds, Efficient Algorithms,
++ J. Symbolic Computation (1993) 15, 393-413
++ [7] Augustin-Lux Cauchy, Exercices de Math\'ematiques Quatri\`eme Ann\'ee.
++ De Bure Fr\`eres, Paris 1829 (reprinted Oeuvres, II S\'erie, Tome IX,
++ Gauthier-Villars, Paris, 1891).
++ Description:
++ \spadtype{GaloisGroupFactorizationUtilities} provides functions
++ that will be used by the factorizer.
GaloisGroupFactorizationUtilities(R,UP,F): Exports == Implementation where
R : Ring
UP : UnivariatePolynomialCategory R
F : Join(FloatingPointSystem,RetractableTo(R),Field,
TranscendentalFunctionCategory,ElementaryFunctionCategory)
N ==> NonNegativeInteger
P ==> PositiveInteger
Z ==> Integer
Exports ==> with
beauzamyBound: UP -> Z -- See [1]
++ beauzamyBound(p) returns a bound on the larger coefficient of any
++ factor of p.
bombieriNorm: UP -> F -- See [1]
++ bombieriNorm(p) returns quadratic Bombieri's norm of p.
bombieriNorm: (UP,P) -> F -- See [2] and [3]
++ bombieriNorm(p,n) returns the nth Bombieri's norm of p.
rootBound: UP -> Z -- See [4] and [5]
++ rootBound(p) returns a bound on the largest norm of the complex roots
++ of p.
singleFactorBound: (UP,N) -> Z -- See [6]
++ singleFactorBound(p,r) returns a bound on the infinite norm of
++ the factor of p with smallest Bombieri's norm. r is a lower bound
++ for the number of factors of p. p shall be of degree higher or equal
++ to 2.
singleFactorBound: UP -> Z -- See [6]
++ singleFactorBound(p,r) returns a bound on the infinite norm of
++ the factor of p with smallest Bombieri's norm. p shall be of degree
++ higher or equal to 2.
norm: (UP,P) -> F
++ norm(f,p) returns the lp norm of the polynomial f.
quadraticNorm: UP -> F
++ quadraticNorm(f) returns the l2 norm of the polynomial f.
infinityNorm: UP -> F
++ infinityNorm(f) returns the maximal absolute value of the coefficients
++ of the polynomial f.
height: UP -> F
++ height(p) returns the maximal absolute value of the coefficients of
++ the polynomial p.
length: UP -> F
++ length(p) returns the sum of the absolute values of the coefficients
++ of the polynomial p.
Implementation ==> add
import GaloisGroupUtilities(F)
height(p:UP):F == infinityNorm(p)
length(p:UP):F == norm(p,1)
norm(f:UP,p:P):F ==
n : F := 0
for c in coefficients f repeat
n := n+abs(c::F)**p
nthRoot(n,p::N)
quadraticNorm(f:UP):F == norm(f,2)
infinityNorm(f:UP):F ==
n : F := 0
for c in coefficients f repeat
n := max(n,c::F)
n
singleFactorBound(p:UP,r:N):Z == -- See [6]
n : N := degree p
r := max(2,r)
n < r => error "singleFactorBound: Bad arguments."
nf : F := n :: F
num : F := nthRoot(bombieriNorm(p),r)
if F has Gamma: F -> F then
num := num*nthRoot(Gamma(nf+1$F),2*r)
den : F := Gamma(nf/((2*r)::F)+1$F)
else
num := num*(2::F)**(5/8+n/2)*exp(1$F/(4*nf))
den : F := (pi()$F*nf)**(3/8)
safeFloor( num/den )
singleFactorBound(p:UP):Z == singleFactorBound(p,2) -- See [6]
rootBound(p:UP):Z == -- See [4] and [5]
n := degree p
zero? n => 0
lc := abs(leadingCoefficient(p)::F)
b1 : F := 0 -- Mignotte
b2 : F := 0 -- Knuth
b3 : F := 0 -- Zassenhaus in [5]
b4 : F := 0 -- Cauchy in [7]
c : F := 0
cl : F := 0
for i in 1..n repeat
c := abs(coefficient(p,(n-i)::N)::F)
b1 := max(b1,c)
cl := c/lc
b2 := max(b2,nthRoot(cl,i))
b3 := max(b3,nthRoot(cl/pascalTriangle(n,i),i))
b4 := max(b4,nthRoot(n*cl,i))
min(1+safeCeiling(b1/lc),min(safeCeiling(2*b2),min(safeCeiling(b3/
(nthRoot(2::F,n)-1)),safeCeiling(b4))))
beauzamyBound(f:UP):Z == -- See [1]
d := degree f
zero? d => safeFloor bombieriNorm f
safeFloor( (bombieriNorm(f)*(3::F)**(3/4+d/2))/
(2*sqrt(pi()$F*(d::F))) )
bombieriNorm(f:UP,p:P):F == -- See [2] and [3]
d := degree f
b := abs(coefficient(f,0)::F)
if zero? d then return b
else b := b**p
b := b+abs(leadingCoefficient(f)::F)**p
dd := (d-1) quo 2
for i in 1..dd repeat
b := b+(abs(coefficient(f,i)::F)**p+abs(coefficient(f,(d-i)::N)::F)**p)
/pascalTriangle(d,i)
if even? d then
dd := dd+1
b := b+abs(coefficient(f, dd::N)::F)**p/pascalTriangle(d,dd)
nthRoot(b,p::N)
bombieriNorm(f:UP):F == bombieriNorm(f,2) -- See [1]
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