gotosocial/vendor/modernc.org/mathutil/poly.go

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// Copyright (c) 2016 The mathutil Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package mathutil // import "modernc.org/mathutil"
import (
"fmt"
"math/big"
)
func abs(n int) uint64 {
if n >= 0 {
return uint64(n)
}
return uint64(-n)
}
// QuadPolyDiscriminant returns the discriminant of a quadratic polynomial in
// one variable of the form a*x^2+b*x+c with integer coefficients a, b, c, or
// an error on overflow.
//
// ds is the square of the discriminant. If |ds| is a square number, d is set
// to sqrt(|ds|), otherwise d is < 0.
func QuadPolyDiscriminant(a, b, c int) (ds, d int, _ error) {
if 2*BitLenUint64(abs(b)) > IntBits-1 ||
2+BitLenUint64(abs(a))+BitLenUint64(abs(c)) > IntBits-1 {
return 0, 0, fmt.Errorf("overflow")
}
ds = b*b - 4*a*c
s := ds
if s < 0 {
s = -s
}
d64 := SqrtUint64(uint64(s))
if d64*d64 != uint64(s) {
return ds, -1, nil
}
return ds, int(d64), nil
}
// PolyFactor describes an irreducible factor of a polynomial in one variable
// with integer coefficients P, Q of the form P*x+Q.
type PolyFactor struct {
P, Q int
}
// QuadPolyFactors returns the content and the irreducible factors of the
// primitive part of a quadratic polynomial in one variable with integer
// coefficients a, b, c of the form a*x^2+b*x+c in integers, or an error on
// overflow.
//
// If the factorization in integers does not exists, the return value is (0,
// nil, nil).
//
// See also:
// https://en.wikipedia.org/wiki/Factorization_of_polynomials#Primitive_part.E2.80.93content_factorization
func QuadPolyFactors(a, b, c int) (content int, primitivePart []PolyFactor, _ error) {
content = int(GCDUint64(abs(a), GCDUint64(abs(b), abs(c))))
switch {
case content == 0:
content = 1
case content > 0:
if a < 0 || a == 0 && b < 0 {
content = -content
}
}
a /= content
b /= content
c /= content
if a == 0 {
if b == 0 {
return content, []PolyFactor{{0, c}}, nil
}
if b < 0 && c < 0 {
b = -b
c = -c
}
if b < 0 {
b = -b
c = -c
}
return content, []PolyFactor{{b, c}}, nil
}
ds, d, err := QuadPolyDiscriminant(a, b, c)
if err != nil {
return 0, nil, err
}
if ds < 0 || d < 0 {
return 0, nil, nil
}
x1num := -b + d
x1denom := 2 * a
gcd := int(GCDUint64(abs(x1num), abs(x1denom)))
x1num /= gcd
x1denom /= gcd
x2num := -b - d
x2denom := 2 * a
gcd = int(GCDUint64(abs(x2num), abs(x2denom)))
x2num /= gcd
x2denom /= gcd
return content, []PolyFactor{{x1denom, -x1num}, {x2denom, -x2num}}, nil
}
// QuadPolyDiscriminantBig returns the discriminant of a quadratic polynomial
// in one variable of the form a*x^2+b*x+c with integer coefficients a, b, c.
//
// ds is the square of the discriminant. If |ds| is a square number, d is set
// to sqrt(|ds|), otherwise d is nil.
func QuadPolyDiscriminantBig(a, b, c *big.Int) (ds, d *big.Int) {
ds = big.NewInt(0).Set(b)
ds.Mul(ds, b)
x := big.NewInt(4)
x.Mul(x, a)
x.Mul(x, c)
ds.Sub(ds, x)
s := big.NewInt(0).Set(ds)
if s.Sign() < 0 {
s.Neg(s)
}
if s.Bit(1) != 0 { // s is not a square number
return ds, nil
}
d = SqrtBig(s)
x.Set(d)
x.Mul(x, x)
if x.Cmp(s) != 0 { // s is not a square number
d = nil
}
return ds, d
}
// PolyFactorBig describes an irreducible factor of a polynomial in one
// variable with integer coefficients P, Q of the form P*x+Q.
type PolyFactorBig struct {
P, Q *big.Int
}
// QuadPolyFactorsBig returns the content and the irreducible factors of the
// primitive part of a quadratic polynomial in one variable with integer
// coefficients a, b, c of the form a*x^2+b*x+c in integers.
//
// If the factorization in integers does not exists, the return value is (nil,
// nil).
//
// See also:
// https://en.wikipedia.org/wiki/Factorization_of_polynomials#Primitive_part.E2.80.93content_factorization
func QuadPolyFactorsBig(a, b, c *big.Int) (content *big.Int, primitivePart []PolyFactorBig) {
content = bigGCD(bigAbs(a), bigGCD(bigAbs(b), bigAbs(c)))
switch {
case content.Sign() == 0:
content.SetInt64(1)
case content.Sign() > 0:
if a.Sign() < 0 || a.Sign() == 0 && b.Sign() < 0 {
content = bigNeg(content)
}
}
a = bigDiv(a, content)
b = bigDiv(b, content)
c = bigDiv(c, content)
if a.Sign() == 0 {
if b.Sign() == 0 {
return content, []PolyFactorBig{{big.NewInt(0), c}}
}
if b.Sign() < 0 && c.Sign() < 0 {
b = bigNeg(b)
c = bigNeg(c)
}
if b.Sign() < 0 {
b = bigNeg(b)
c = bigNeg(c)
}
return content, []PolyFactorBig{{b, c}}
}
ds, d := QuadPolyDiscriminantBig(a, b, c)
if ds.Sign() < 0 || d == nil {
return nil, nil
}
x1num := bigAdd(bigNeg(b), d)
x1denom := bigMul(_2, a)
gcd := bigGCD(bigAbs(x1num), bigAbs(x1denom))
x1num = bigDiv(x1num, gcd)
x1denom = bigDiv(x1denom, gcd)
x2num := bigAdd(bigNeg(b), bigNeg(d))
x2denom := bigMul(_2, a)
gcd = bigGCD(bigAbs(x2num), bigAbs(x2denom))
x2num = bigDiv(x2num, gcd)
x2denom = bigDiv(x2denom, gcd)
return content, []PolyFactorBig{{x1denom, bigNeg(x1num)}, {x2denom, bigNeg(x2num)}}
}
func bigAbs(n *big.Int) *big.Int {
n = big.NewInt(0).Set(n)
if n.Sign() >= 0 {
return n
}
return n.Neg(n)
}
func bigDiv(a, b *big.Int) *big.Int {
a = big.NewInt(0).Set(a)
return a.Div(a, b)
}
func bigGCD(a, b *big.Int) *big.Int {
a = big.NewInt(0).Set(a)
b = big.NewInt(0).Set(b)
for b.Sign() != 0 {
c := big.NewInt(0)
c.Mod(a, b)
a, b = b, c
}
return a
}
func bigNeg(n *big.Int) *big.Int {
n = big.NewInt(0).Set(n)
return n.Neg(n)
}
func bigMul(a, b *big.Int) *big.Int {
r := big.NewInt(0).Set(a)
return r.Mul(r, b)
}
func bigAdd(a, b *big.Int) *big.Int {
r := big.NewInt(0).Set(a)
return r.Add(r, b)
}