gtsocial-umbx

poly.go (5720B)

---

 // 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)
 }