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