/usr/share/axiom-20170501/src/algebra/POLYVEC.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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++ Description:
++ This is a low-level package which implements operations
++ on vectors treated as univariate modular polynomials. Most
++ operations takes modulus as parameter. Modulus is machine
++ sized prime which should be small enough to avoid overflow
++ in intermediate calculations.
U32VectorPolynomialOperations() : SIG == CODE where
PA ==> U32Vector
SIG ==> with
copy_first : (PA, PA, Integer) -> Void
++ copy_first(v1, v2, n) copies first n elements
++ of v2 into n first positions in v1.
copy_slice : (PA, PA, Integer, Integer) -> Void
++ copy_first(v1, v2, m, n) copies the slice of v2 starting
++ at m elements and having n elements into corresponding
++ positions in v1.
eval_at : (PA, Integer, Integer, Integer) -> Integer
++ eval_at(v, deg, pt, p) treats v as coefficients of
++ polynomial of degree deg and evaluates the
++ polynomial at point pt modulo p
++
++X a:=new(3,1)$U32VEC
++X a.1:=2
++X eval_at(a,2,3,1024)
++X eval_at(a,2,2,8)
++X eval_at(a,2,3,10)
vector_add_mul : (PA, PA, Integer, Integer, Integer, Integer) _
-> Void
++ vector_add_mul(v1, v2, m, n, c, p) sets v1(m), ...,
++ v1(n) to corresponding extries in v1 + c*v2
++ modulo p.
mul_by_binomial : (PA, Integer, Integer) -> Void
++ mul_by_binomial(v, pt, p) treats v a polynomial
++ and multiplies in place this polynomial by binomial (x + pt).
++ Highest coefficient of product is ignored.
mul_by_binomial : (PA, Integer, Integer, Integer) -> Void
++ mul_by_binomial(v, deg, pt, p) treats v as
++ coefficients of polynomial of degree deg and
++ multiplies in place this polynomial by binomial (x + pt).
++ Highest coefficient of product is ignored.
mul_by_scalar : (PA, Integer, Integer, Integer) -> Void
++ mul_by_scalar(v, deg, c, p) treats v as
++ coefficients of polynomial of degree deg and
++ multiplies in place this polynomial by scalar c
mul : (PA, PA, Integer) -> PA
++ mul(p1,p2,i) does polynomial multiplication.
truncated_multiplication : (PA, PA, Integer, Integer) -> PA
++ truncated_multiplication(x, y, d, p) computes
++ x*y truncated after degree d
truncated_mul_add : (PA, PA, PA, Integer, Integer) -> Void
++ truncated_mul_add(x, y, z, d, p) adds to z
++ the produce x*y truncated after degree d
pow : (PA, PositiveInteger, NonNegativeInteger, Integer) -> PA
++ pow(u, n, d, p) returns u^n truncated after degree d, except if
++ n=1, in which case u itself is returned
differentiate : (PA, Integer) -> PA
++ differentiate(p,i) does polynomial differentiation.
differentiate : (PA, NonNegativeInteger, Integer) -> PA
++ differentiate(p,n,i) does polynomial differentiation.
divide! : (PA, PA, PA, Integer) -> Void
++ divide!(p1,p2,p3,i) does polynomial division.
remainder! : (PA, PA, Integer) -> Void
++ remainder!(p1,p2,i) does polynomial remainder
vector_combination : (PA, Integer, PA, Integer, _
Integer, Integer, p : Integer) -> Void
++ vector_combination(v1, c1, v2, c2, n, delta, p) replaces
++ first n + 1 entires of v1 by corresponding entries of
++ c1*v1+c2*x^delta*v2 mod p.
to_mod_pa : (SparseUnivariatePolynomial Integer, Integer) -> PA
++ to_mod_pa(s, p) reduces coefficients of polynomial
++ s modulo prime p and converts the result to vector
gcd : (PA, PA, Integer) -> PA
++ gcd(v1, v2, p) computes monic gcd of v1 and v2 modulo p.
gcd : (PrimitiveArray PA, Integer, Integer, Integer) -> PA
++ gcd(a, lo, hi, p) computes gcd of elements
++ a(lo), a(lo+1), ..., a(hi).
lcm : (PrimitiveArray PA, Integer, Integer, Integer) -> PA
++ lcm(a, lo, hi, p) computes lcm of elements
++ a(lo), a(lo+1), ..., a(hi).
degree : PA -> Integer
++ degree(v) is degree of v treated as polynomial
extended_gcd : (PA, PA, Integer) -> List(PA)
++ extended_gcd(v1, v2, p) gives [g, c1, c2] such
++ that g is \spad{gcd(v1, v2, p)}, \spad{g = c1*v1 + c2*v2}
++ and degree(c1) < max(degree(v2) - degree(g), 0) and
++ degree(c2) < max(degree(v1) - degree(g), 1)
resultant : (PA, PA, Integer) -> Integer
++ resultant(v1, v2, p) computes resultant of v1 and v2
++ modulo p.
CODE ==> add
Qmuladdmod ==> QSMULADDMOD6432$Lisp
Qmuladd ==> QSMULADD6432$Lisp
Qmul ==> QSMULMOD32$Lisp
Qdot2 ==> QSDOT2MOD6432$Lisp
Qrem ==> QSMOD6432$Lisp
modInverse ==> invmod
copy_first(np : PA, op : PA, n : Integer) : Void ==
ns := n pretend SingleInteger
for j in 0..(ns - 1) repeat
np(j) := op(j)
copy_slice(np : PA, op : PA, m : Integer, _
n : Integer) : Void ==
ms := m pretend SingleInteger
ns := n pretend SingleInteger
for j in ms..(ms + ns - 1) repeat
np(j) := op(j)
eval_at(v : PA, deg : Integer, pt : Integer, _
p : Integer) : Integer ==
i : SingleInteger := deg::SingleInteger
res : Integer := 0
while not(i < 0) repeat
res := Qmuladdmod(pt, res, v(i), p)
i := i - 1
res
to_mod_pa(s : SparseUnivariatePolynomial Integer, p : Integer) : PA ==
zero?(s) => new(1, 0)$PA
n0 := degree(s) pretend SingleInteger
ncoeffs := new((n0+1) pretend NonNegativeInteger, 0)$PA
while not(zero?(s)) repeat
n := degree(s)
ncoeffs(n) := positiveRemainder(leadingCoefficient(s), p)
s := reductum(s)
ncoeffs
vector_add_mul(v1 : PA, v2 : PA, m : Integer, n : Integer, _
c : Integer, p : Integer) : Void ==
ms := m pretend SingleInteger
ns := n pretend SingleInteger
for i in ms..ns repeat
v1(i) := Qmuladdmod(c, v2(i), v1(i), p)
mul_by_binomial(v : PA, n : Integer, pt : Integer, _
p : Integer) : Void ==
prev_coeff : Integer := 0
ns := n pretend SingleInteger
for i in 0..(ns - 1) repeat
pp := v(i)
v(i) := Qmuladdmod(pt, pp, prev_coeff, p)
prev_coeff := pp
mul_by_binomial(v : PA, pt : Integer, _
p : Integer) : Void ==
mul_by_binomial(v, #v, pt, p)
mul_by_scalar(v : PA, n : Integer, c : Integer, _
p : Integer) : Void ==
ns := n pretend SingleInteger
for i in 0..ns repeat
v(i) := Qmul(c, v(i), p)
degree(v : PA) : Integer ==
n := #v
for i in (n - 1)..0 by -1 repeat
not(v(i) = 0) => return i
-1
vector_combination(v1 : PA, c1 : Integer, _
v2 : PA, c2 : Integer, _
n : Integer, delta : Integer, _
p : Integer) : Void ==
ns := n pretend SingleInteger
ds := delta pretend SingleInteger
if not(c1 = 1) then
ns + 1 < ds =>
for i in 0..ns repeat
v1(i) := Qmul(v1(i), c1, p)
for i in 0..(ds - 1) repeat
v1(i) := Qmul(v1(i), c1, p)
for i in ds..ns repeat
v1(i) := Qdot2(v1(i), c1, v2(i - ds), c2, p)
else
for i in ds..ns repeat
v1(i) := Qmuladdmod(c2, v2(i - ds), v1(i), p)
divide!(r0 : PA, r1 : PA, res : PA, p: Integer) : Void ==
dr0 := degree(r0) pretend SingleInteger
dr1 := degree(r1) pretend SingleInteger
c0 := r1(dr1)
c0 := modInverse(c0, p)
while not(dr0 < dr1) repeat
delta := dr0 - dr1
c1 := Qmul(c0, r0(dr0), p)
res(delta) := c1
c1 := p - c1
r0(dr0) := 0
dr0 := dr0 - 1
if dr0 < 0 then break
vector_combination(r0, 1, r1, c1, dr0, delta, p)
while r0(dr0) = 0 repeat
dr0 := dr0 - 1
if dr0 < 0 then break
remainder!(r0 : PA, r1 : PA, p: Integer) : Void ==
dr0 := degree(r0) pretend SingleInteger
dr1 := degree(r1) pretend SingleInteger
c0 := r1(dr1)
c0 := modInverse(c0, p)
while not(dr0 < dr1) repeat
delta := dr0 - dr1
c1 := Qmul(c0, r0(dr0), p)
c1 := p - c1
r0(dr0) := 0
dr0 := dr0 - 1
if dr0 < 0 then break
vector_combination(r0, 1, r1, c1, dr0, delta, p)
while r0(dr0) = 0 repeat
dr0 := dr0 - 1
if dr0 < 0 then break
gcd(x : PA, y : PA, p : Integer) : PA ==
dr0 := degree(y) pretend SingleInteger
dr1 : SingleInteger
if dr0 < 0 then
tmpp := x
x := y
y := tmpp
dr1 := dr0
dr0 := degree(y) pretend SingleInteger
else
dr1 := degree(x) pretend SingleInteger
dr0 < 0 => return new(1, 0)$PA
r0 := new((dr0 + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r0, y, dr0 + 1)
dr1 < 0 =>
c := r0(dr0)
c := modInverse(c, p)
mul_by_scalar(r0, dr0, c, p)
return r0
r1 := new((dr1 + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r1, x, dr1 + 1)
while 0 < dr1 repeat
while not(dr0 < dr1) repeat
delta := dr0 - dr1
c1 := sub_SI(p, r0(dr0))$Lisp
c0 := r1(dr1)
if c0 ~= 1 and delta > 30 then
c0 := modInverse(c0, p)
mul_by_scalar(r1, dr1, c0, p)
c0 := 1
r0(dr0) := 0
dr0 := dr0 - 1
vector_combination(r0, c0, r1, c1, dr0, delta, p)
while r0(dr0) = 0 repeat
dr0 := dr0 - 1
if dr0 < 0 then break
tmpp := r0
tmp := dr0
r0 := r1
dr0 := dr1
r1 := tmpp
dr1 := tmp
not(dr1 < 0) =>
r1(0) := 1
return r1
c := r0(dr0)
c := modInverse(c, p)
mul_by_scalar(r0, dr0, c, p)
r0
gcd(a : PrimitiveArray PA, lo : Integer, hi: Integer, p: Integer) _
: PA ==
res := a(lo)
for i in (lo + 1)..hi repeat
res := gcd(a(i), res, p)
res
lcm2(v1 : PA, v2 : PA, p : Integer) : PA ==
pp := gcd(v1, v2, p)
dv2 := degree(v2)
dpp := degree(pp)
dv2 = dpp =>
v1
dpp = 0 => mul(v1, v2, p)
tmp1 := new((dv2 + 1) pretend NonNegativeInteger, 0)$PA
tmp2 := new((dv2 - dpp + 1) pretend NonNegativeInteger, 0)$PA
copy_first(tmp1, v2, dv2 + 1)
divide!(tmp1, pp, tmp2, p)
mul(v1, tmp2, p)
lcm(a : PrimitiveArray PA, lo : Integer, hi: Integer, p: Integer) _
: PA ==
res := a(lo)
for i in (lo + 1)..hi repeat
res := lcm2(a(i), res, p)
res
inner_mul : (PA, PA, PA, SingleInteger, SingleInteger, _
SingleInteger, Integer) -> Void
mul(x : PA, y : PA, p : Integer) : PA ==
xdeg := degree(x) pretend SingleInteger
ydeg := degree(y) pretend SingleInteger
if xdeg > ydeg then
tmpp := x
tmp := xdeg
x := y
xdeg := ydeg
y := tmpp
ydeg := tmp
xcoeffs := x
ycoeffs := y
xdeg < 0 => x
xdeg = 0 and xcoeffs(0) = 1 => copy(y)
zdeg : SingleInteger := xdeg + ydeg
zdeg0 := ((zdeg + 1)::Integer) pretend NonNegativeInteger
zcoeffs := new(zdeg0, 0)$PA
inner_mul(xcoeffs, ycoeffs, zcoeffs, xdeg, ydeg, zdeg, p)
zcoeffs
inner_mul(x, y, z, xdeg, ydeg, zdeg, p) ==
if ydeg < xdeg then
tmpp := x
tmp := xdeg
x := y
xdeg := ydeg
y := tmpp
ydeg := tmp
xdeg :=
zdeg < xdeg => zdeg
xdeg
ydeg :=
zdeg < ydeg => zdeg
ydeg
ss : Integer
i : SingleInteger
j : SingleInteger
for i in 0..xdeg repeat
ss := z(i)
for j in 0..i repeat
ss := Qmuladd(x(i - j), y(j), ss)
z(i) := Qrem(ss, p)
for i in (xdeg+1)..ydeg repeat
ss := z(i)
for j in 0..xdeg repeat
ss := Qmuladd(x(j), y(i-j), ss)
z(i) := Qrem(ss, p)
for i in (ydeg+1)..zdeg repeat
ss := z(i)
for j in (i-xdeg)..ydeg repeat
ss := Qmuladd(x(i - j), y(j), ss)
z(i) := Qrem(ss, p)
truncated_mul_add(x, y, z, m, p) ==
xdeg := (#x - 1) pretend SingleInteger
ydeg := (#y - 1) pretend SingleInteger
inner_mul(x, y, z, xdeg, ydeg, m pretend SingleInteger, p)
truncated_multiplication(x, y, m, p) ==
xdeg := (#x - 1) pretend SingleInteger
ydeg := (#y - 1) pretend SingleInteger
z := new((m pretend SingleInteger + 1)
pretend NonNegativeInteger, 0)$PA
inner_mul(x, y, z, xdeg, ydeg, m pretend SingleInteger, p)
z
pow(x : PA, n : PositiveInteger, d: NonNegativeInteger, _
p : Integer) : PA ==
one? n => x
odd?(n)$Integer =>
truncated_multiplication(x,
pow(truncated_multiplication(x, x, d, p),
shift(n,-1) pretend PositiveInteger,
d,
p),
d,
p)
pow(truncated_multiplication(x, x, d, p),
shift(n,-1) pretend PositiveInteger,
d,
p)
differentiate(x: PA, p: Integer): PA ==
d := #x - 1
if zero? d then empty()$PA
else
r := new(d::NonNegativeInteger, 0)$PA
for i in 0..d-1 repeat
i1 := i+1
r.i := Qmul(i1, x.i1, p)
r
differentiate(x: PA, n: NonNegativeInteger, p: Integer): PA ==
zero? n => x
d := #x - 1
if d < n then empty()$PA
else
r := new((d-n+1) pretend NonNegativeInteger, 0)$PA
for i in n..d repeat
j := i-n
f := j+1
for k in j+2..i repeat f := Qmul(f, k, p)
r.(j pretend NonNegativeInteger) := Qmul(f, x.i, p)
r
extended_gcd(x : PA, y : PA, p : Integer) : List(PA) ==
dr0 := degree(x) pretend SingleInteger
dr1 : SingleInteger
swapped : Boolean := false
t0 : PA
if dr0 < 0 then
(x, y) := (y, x)
dr1 := dr0
dr0 := degree(x) pretend SingleInteger
swapped := true
else
dr1 := degree(y) pretend SingleInteger
dr1 < 0 =>
dr0 < 0 =>
return [new(1, 0)$PA, new(1, 0)$PA, new(1, 1)$PA]
r0 := new((dr0 + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r0, x, dr0 + 1)
c := r0(dr0)
c := modInverse(c, p)
mul_by_scalar(r0, dr0, c, p)
t0 := new(1, c)$PA
if swapped then
return [r0, new(1, 0)$PA, t0]
else
return [r0, t0, new(1, 0)$PA]
swapped => error "impossible"
dt := (dr0 > 0 => dr0 - 1 ; 0)
ds := (dr1 > 0 => dr1 - 1 ; 0)
-- invariant: r0 = s0*x + t0*y, r1 = s1*x + t1*y
r0 := new((dr0 + 1) pretend NonNegativeInteger, 0)$PA
t0 := new((dt + 1) pretend NonNegativeInteger, 0)$PA
s0 := new((ds + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r0, x, dr0 + 1)
s0(0) := 1
r1 := new((dr1 + 1) pretend NonNegativeInteger, 0)$PA
t1 := new((dt + 1) pretend NonNegativeInteger, 0)$PA
s1 := new((ds + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r1, y, dr1 + 1)
t1(0) := 1
while dr1 > 0 repeat
while dr0 >= dr1 repeat
delta := dr0 - dr1
c1 := sub_SI(p, r0(dr0))$Lisp
c0 := r1(dr1)
if c0 ~= 1 and delta > 30 then
c0 := modInverse(c0, p)
mul_by_scalar(r1, dr1, c0, p)
mul_by_scalar(t1, dt, c0, p)
mul_by_scalar(s1, ds, c0, p)
c0 := 1
r0(dr0) := 0
dr0 := dr0 - 1
vector_combination(r0, c0, r1, c1, dr0, delta, p)
vector_combination(t0, c0, t1, c1, dt, delta, p)
vector_combination(s0, c0, s1, c1, ds, delta, p)
while r0(dr0) = 0 repeat
dr0 := dr0 - 1
if dr0 < 0 then break
(r0, r1) := (r1, r0)
(dr0, dr1) := (dr1, dr0)
(s0, s1) := (s1, s0)
(t0, t1) := (t1, t0)
dr1 >= 0 =>
c := r1(0)
c := modInverse(c, p)
r1(0) := 1
mul_by_scalar(s1, ds, c, p)
mul_by_scalar(t1, dt, c, p)
return [r1, s1, t1]
c := r0(dr0)
c := modInverse(c, p)
mul_by_scalar(r0, dr0, c, p)
mul_by_scalar(s0, ds, c, p)
mul_by_scalar(t0, dt, c, p)
[r0, s0, t0]
resultant(x : PA, y : PA, p : Integer) : Integer ==
dr0 := degree(x) pretend SingleInteger
dr0 < 0 => 0
dr1 := degree(y) pretend SingleInteger
dr1 < 0 => 0
r0 := new((dr0 + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r0, x, dr0 + 1)
r1 := new((dr1 + 1) pretend NonNegativeInteger, 0)$PA
copy_first(r1, y, dr1 + 1)
res : SingleInteger := 1
repeat
dr0 < dr1 =>
(r0, r1) := (r1, r0)
(dr0, dr1) := (dr1, dr0)
c0 := r1(dr1)
dr1 = 0 =>
while 0 < dr0 repeat
res := Qmul(res, c0, p)
dr0 := dr0 - 1
return res
delta := dr0 - dr1
c1 := sub_SI(p, r0(dr0))$Lisp
if c0 ~= 1 then
c1 := Qmul(c1, modInverse(c0, p), p)
r0(dr0) := 0
dr0 := dr0 - 1
vector_combination(r0, 1, r1, c1, dr0, delta, p)
res := Qmul(res, c0, p)
while r0(dr0) = 0 repeat
dr0 := dr0 - 1
dr0 < 0 => return 0
res := Qmul(res, c0, p)
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