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arithmetic.go
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arithmetic.go
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package bivariate
import (
"github.com/ReneBoedker/algobra/errors"
"github.com/ReneBoedker/algobra/finitefield/ff"
)
// Plus returns the sum of the two polynomials f and g.
//
// If f and g are defined over different rings, a new polynomial is returned
// with an ArithmeticIncompat-error as error status.
//
// When f or g has a non-nil error status, its error is wrapped and the same
// polynomial is returned.
func (f *Polynomial) Plus(g *Polynomial) *Polynomial {
return f.Copy().Add(g)
}
// Add sets f to the sum of the two polynomials f and g and returns f.
//
// If f and g are defined over different rings, a new polynomial is returned
// with an ArithmeticIncompat-error as error status.
//
// When f or g has a non-nil error status, its error is wrapped and the same
// polynomial is returned.
func (f *Polynomial) Add(g *Polynomial) *Polynomial {
const op = "Adding polynomials"
if tmp := checkErrAndCompatible(op, f, g); tmp != nil {
return tmp
}
for deg, c := range g.coefs {
f.IncrementCoef(deg, c)
}
return f
}
// Neg returns the polynomial obtained by scaling f by -1 (modulo the
// characteristic).
func (f *Polynomial) Neg() *Polynomial {
g := f.baseRing.zeroWithCap(len(f.coefs))
for deg, c := range f.coefs {
g.coefs[deg] = c.Neg()
}
return g
}
// Sub sets f to the difference of the two polynomials f and g and returns f.
//
// If f and g are defined over different rings, a new polynomial is returned
// with an ArithmeticIncompat-error as error status.
//
// When f or g has a non-nil error status, its error is wrapped and the same
// polynomial is returned.
func (f *Polynomial) Sub(g *Polynomial) *Polynomial {
const op = "Subtracting polynomials"
if tmp := checkErrAndCompatible(op, f, g); tmp != nil {
return tmp
}
for deg, c := range g.coefs {
f.DecrementCoef(deg, c)
}
return f
}
// Minus returns polynomial difference f-g.
//
// If f and g are defined over different rings, a new polynomial is returned
// with an ArithmeticIncompat-error as error status.
//
// When f or g has a non-nil error status, its error is wrapped and the same
// polynomial is returned.
func (f *Polynomial) Minus(g *Polynomial) *Polynomial {
return f.Copy().Sub(g)
}
// Internal method. Multiplies the two polynomials f and g, but does not reduce
// the result according to the specified ring.
func (f *Polynomial) multNoReduce(g *Polynomial) *Polynomial {
const op = "Multiplying polynomials"
if tmp := checkErrAndCompatible(op, f, g); tmp != nil {
return tmp
}
h := f.baseRing.zeroWithCap(len(f.coefs) * len(g.coefs))
tmp := f.BaseField().One()
for degf, cf := range f.coefs {
for degg, cg := range g.coefs {
degSum, err := addDegs(degf, degg)
if err != nil {
h = f.baseRing.Zero()
h.err = errors.Wrap(op, errors.Inherit, err)
return h
}
tmp.Prod(cf, cg)
h.IncrementCoef(degSum, tmp)
}
}
return h
}
// subWithShiftAndScale sets f to the polynomial f-a*X^i*g. This is done without
// allocating a new polynomial
func (f *Polynomial) subWithShiftAndScale(g *Polynomial, i [2]uint, a ff.Element) {
switch {
case a.IsZero():
return
case a.IsOne():
for d, c := range g.coefs {
if c.IsZero() {
continue
}
f.DecrementCoef([2]uint{d[0] + i[0], d[1] + i[1]}, c)
}
default:
tmp := f.BaseField().Zero()
for d, c := range g.coefs {
if c.IsZero() {
continue
}
tmp.Prod(a, c)
f.DecrementCoef([2]uint{d[0] + i[0], d[1] + i[1]}, tmp)
}
}
}
// Times returns the product of the polynomials f and g
//
// If f and g are defined over different rings, a new polynomial is returned
// with an ArithmeticIncompat-error as error status.
//
// When f or g has a non-nil error status, its error is wrapped and the same
// polynomial is returned.
func (f *Polynomial) Times(g *Polynomial) *Polynomial {
h := f.multNoReduce(g)
if h.Err() != nil {
return h
}
h.reduce()
return h
}
// Mult sets f to the product of the polynomials f and g and returns f.
//
// If f and g are defined over different rings, a new polynomial is returned
// with an ArithmeticIncompat-error as error status.
//
// When f or g has a non-nil error status, its error is wrapped and the same
// polynomial is returned.
func (f *Polynomial) Mult(g *Polynomial) *Polynomial {
*f = *f.multNoReduce(g)
if f.Err() != nil {
return f
}
f.reduce()
return f
}
// Normalize creates a new polynomial obtained by normalizing f. That is,
// f.Normalize() multiplied by f.Lc() is f.
//
// If f is the zero polynomial, a copy of f is returned.
func (f *Polynomial) Normalize() *Polynomial {
if f.IsZero() {
return f.Copy()
}
return f.Scale(f.lcPtr().Inv())
}
// Scale scales all coefficients of f by the field element c and returns the
// result as a new polynomial. See also SetScale.
func (f *Polynomial) Scale(c ff.Element) *Polynomial {
if c.IsZero() {
return f.baseRing.Zero()
}
g := f.Copy()
for d := range g.coefs {
g.coefs[d].Mult(c)
}
return g
}
// SetScale scales all coefficients of f by the field element c and returns
// f. See also Scale.
func (f *Polynomial) SetScale(c ff.Element) *Polynomial {
if c.IsZero() {
return f.baseRing.Zero()
}
for d := range f.coefs {
f.coefs[d].Mult(c)
}
return f
}
// Pow raises f to the power of n.
//
// If the computation causes the degree of f to overflow, the returned
// polynomial has an Overflow-error as error status.
func (f *Polynomial) Pow(n uint) *Polynomial {
const op = "Computing polynomial power"
out := f.baseRing.Polynomial(map[[2]uint]ff.Element{
{0, 0}: f.BaseField().One(),
})
g := f.Copy()
for n > 0 {
if n%2 == 1 {
out.Mult(g)
if out.Err() != nil {
out = f.baseRing.Zero()
out.err = errors.Wrap(op, errors.Inherit, out.Err())
return out
}
}
n /= 2
g.Mult(g)
}
return out
}
// QuoRem returns the polynomial quotient and remainder under division by the
// given list of polynomials.
func (f *Polynomial) QuoRem(list ...*Polynomial) (q []*Polynomial, r *Polynomial, err error) {
return f.quoRemWithIgnore(-1, list...)
}
func (f *Polynomial) quoRemWithIgnore(
ignoreIndex int,
list ...*Polynomial,
) (q []*Polynomial, r *Polynomial, err error) {
const op = "Computing polynomial quotient and remainder"
if tmp := checkErrAndCompatible(op, f, list...); tmp != nil {
err = tmp.Err()
return
}
r = f.baseRing.Zero()
p := f.Copy()
q = make([]*Polynomial, len(list), len(list))
for i := range list {
q[i] = f.baseRing.Zero()
}
tmp := f.BaseField().Zero()
outer:
for p.IsNonzero() {
pLd := p.Ld()
for i, g := range list {
gLd := g.Ld()
if i == ignoreIndex {
continue
}
degDiff, ok := subtractDegs(pLd, g.Ld())
if !ok {
// Lt of g does not divide Lt of p
continue
}
if g.coefPtr(gLd).IsOne() {
tmp.Prod(p.coefPtr(pLd), g.coefPtr(gLd))
} else {
tmp.Prod(p.coefPtr(pLd), g.coefPtr(gLd).Inv())
}
q[i].IncrementCoef(degDiff, tmp)
p.subWithShiftAndScale(g, degDiff, tmp)
continue outer
}
// No generators divide
r.IncrementCoef(pLd, p.coefPtr(pLd))
p.removeCoef(pLd)
}
return q, r, nil
}
// Rem returns the polynomial remainder under division by the given list of
// polynomials.
func (f *Polynomial) Rem(list ...*Polynomial) (r *Polynomial, err error) {
const op = "Computing polynomial remainder"
if tmp := checkErrAndCompatible(op, f, list...); tmp != nil {
err = tmp.Err()
return
}
r = f.baseRing.Zero()
p := f.Copy()
tmp := f.BaseField().Zero()
outer:
for p.IsNonzero() {
pLd := p.Ld()
for _, g := range list {
gLd := g.Ld()
degDiff, ok := subtractDegs(pLd, gLd)
if !ok {
// Lt of g does not divide Lt of p
continue
}
if g.coefPtr(gLd).IsOne() {
tmp.Prod(p.coefPtr(pLd), g.coefPtr(gLd))
} else {
tmp.Prod(p.coefPtr(pLd), g.coefPtr(gLd).Inv())
}
p.subWithShiftAndScale(g, degDiff, tmp)
//degs = degs[1:]
continue outer
}
// No generators divide
r.IncrementCoef(pLd, p.coefPtr(pLd))
p.removeCoef(pLd)
}
return r, nil
}
/* Copyright 2019 René Bødker Christensen
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* 1. Redistributions of source code must retain the above copyright notice, this
* list of conditions and the following disclaimer.
*
* 2. Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
*
* 3. Neither the name of the copyright holder nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
* SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
* CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
* OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
* OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/