syncthing/vendor/github.com/cznic/mathutil/poly.go

// 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 (
	"fmt"
)

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 (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
}