這個執行緒是我 最后一個問題的延續。
我和我的朋友正在嘗試制作一個多功能計算器，可以自動計算二次、三次甚至四次方程的根。但是，我們遇到了一些問題，以前是三次方程，現在是四次方程。
與上一個執行緒不同，我們將提供影像，包括我們的程式計算的答案和WolframAlpha計算的答案，而不是在這里將它們放入文本中。
四次方程將表示為
ax^4 bx^3 cx^2 dx e
我們找到四次方程的根的方法。 （在另一個網站的另一個執行緒上找到）
a=1 b=2 c=-41 d=-42 e=360
a=1 b=-0.25 c=-0.85 d=1.45 e=-4.35
a=1 b=2 c=3 d=4 e=5
如圖所示，程式計算的判別式始終是正確的。然而，方程的根有時會導致錯誤 (NaN)，根的性質也會導致錯誤，但與根一樣經常發生錯誤。
有了上面提供的所有資訊，我們希望知道以下問題的答案：
- 究竟是什么導致了這些錯誤和問題，
- 我應該對我的代碼進行哪些修改，以便程式能夠正常運行并按預期運行，
- 我們求四次方程根的方法是否正確，或者是否有可能通過一個通用公式找到所有四次方程的根，
- 最后但并非最不重要的一點是，有沒有一種方法可以改進或壓縮我們程式中的代碼行，但仍然可以按照我們的預期運行。
如前所述，我們剛剛開始學習 C#。
為了您的方便，我們對一些代碼行做了注釋。
代碼（C#）：
using System;
using System.Numerics;
namespace ComplexNumbers
{
public class ComplexNo
{
public static void Main()
{
// prompting for coefficients and constant
Console.WriteLine("Quartic Equation Calculator :");
Console.Write("Insert first coefficient : ");
double a = Convert.ToDouble(Console.ReadLine());
Console.Write("Insert second coefficient : ");
double b = Convert.ToDouble(Console.ReadLine());
Console.Write("Insert third coefficient : ");
double c = Convert.ToDouble(Console.ReadLine());
Console.Write("Insert fourth coefficient : ");
double d = Convert.ToDouble(Console.ReadLine());
Console.Write("Insert constant : ");
double e = Convert.ToDouble(Console.ReadLine());
// discriminant
double discriminant = 256 * Math.Pow(a, 3) * Math.Pow(e, 3) - (192 * Math.Pow(a, 2) * b * d * Math.Pow(e, 2)) -
(128 * Math.Pow(a, 2) * Math.Pow(c, 2) * Math.Pow(e, 2)) (144 * Math.Pow(a, 2) * c * Math.Pow(d, 2) * e) -
(27 * Math.Pow(a, 2) * Math.Pow(d, 4)) (144 * a * Math.Pow(b, 2) * c * Math.Pow(e, 2)) -
(6 * a * Math.Pow(b, 2) * Math.Pow(d, 2) * e) - (80 * a * b * Math.Pow(c, 2) * d * e)
(18 * a * b * c * Math.Pow(d, 3)) (16 * a * Math.Pow(c, 4) * e) - (4 * a * Math.Pow(c, 3) * Math.Pow(d, 2)) -
(27 * Math.Pow(b, 4) * Math.Pow(e, 2)) (18 * Math.Pow(b, 3) * c * d * e) -
(4 * Math.Pow(b, 3) * Math.Pow(d, 3)) - (4 * Math.Pow(b, 2) * Math.Pow(c, 3) * e)
(Math.Pow(b, 2) * Math.Pow(c, 2) * Math.Pow(d, 2));
// different parts of the roots presented in p1-p6
double p1 = (2 * Math.Pow(c, 3)) - (9 * b * c * d) (27 * a * Math.Pow(d, 2)) (27 * Math.Pow(b, 2) * e) -
(72 * a * c * e);
double p2 = p1 Math.Sqrt(-4.0 * Math.Pow(Math.Pow(c, 2) - (3 * b * d) (12 * a * e), 3) Math.Pow(p1, 2));
double p3 = ((Math.Pow(c, 2) - (3 * b * d) (12 * a * e)) / (3 * a * Math.Cbrt(p2 / 2))) (Math.Cbrt(p2 / 2) / (3 * a));
double p4 = Math.Sqrt((Math.Pow(b, 2) / (4 * Math.Pow(a, 2))) - (2 * c / (3 * a)) p3);
double p5 = (Math.Pow(b, 2) / (2 * Math.Pow(a, 2))) - (4 * c / (3 * a)) - p3;
double p6 = ((-(Math.Pow(b, 3) / Math.Pow(a, 3))) (4 * b * c / Math.Pow(a, 2)) - (8 * d / a)) / (4 * p4);
// calculating roots with p1-p6
Complex x1 = (-(b / (4 * a))) - (p4 / 2) - (Math.Sqrt(p5 - p6) / 2);
Complex x2 = (-(b / (4 * a))) - (p4 / 2) (Math.Sqrt(p5 - p6) / 2);
Complex x3 = (-(b / (4 * a))) (p4 / 2) - (Math.Sqrt(p5 - p6) / 2);
Complex x4 = (-(b / (4 * a))) (p4 / 2) (Math.Sqrt(p5 - p6) / 2);
// polynomials used for testing the nature of roots
double P = (8 * a * c) - (3 * Math.Pow(b, 2));
double R = Math.Pow(b, 3) (8 * d * Math.Pow(a, 2)) - (4 * a * b * c);
double deltazero = Math.Pow(c, 2) - (3 * b * d) (12 * a * e);
double D = (64 * Math.Pow(a, 3) * e) - (16 * Math.Pow(a, 2) * Math.Pow(c, 2))
(16 * a * Math.Pow(b, 2) * c) - (16 * Math.Pow(a, 2) * b * d) - (3 * Math.Pow(b, 4));
// results
Console.WriteLine(" ");
Console.WriteLine("Solve for " a "x^4 " " " b "x^3 " " " c "x^2 " " " d "x " " " e " :");
Console.WriteLine(" ");
Console.WriteLine("Results :");
Console.WriteLine("First Root :");
string x1rp = Convert.ToString(Math.Round(x1.Real, 5)); // rounding up both real and unreal parts
string x1ip = Convert.ToString(Math.Round(x1.Imaginary, 5));
if (x1.Imaginary == 0)
{
Console.WriteLine(x1rp);
}
else if (x1.Imaginary < 0) // testing if unreal part is negative
{
Console.WriteLine(x1rp " - " Math.Round(Complex.Abs(x1.Imaginary)) "i");
}
else
{
Console.WriteLine(x1rp " " x1ip "i");
}
string x2rp = Convert.ToString(Math.Round(x2.Real, 5)); // rounding up both real and unreal parts
string x2ip = Convert.ToString(Math.Round(x2.Imaginary, 5));
if (x2.Imaginary == 0)
{
Console.WriteLine(x2rp);
}
else if (x2.Imaginary < 0) // testing if unreal part is negative
{
Console.WriteLine(x2rp " - " Math.Round(Complex.Abs(x2.Imaginary)) "i");
}
else
{
Console.WriteLine(x2rp " " x2ip "i");
}
string x3rp = Convert.ToString(Math.Round(x3.Real, 5)); // rounding up both real and unreal parts
string x3ip = Convert.ToString(Math.Round(x3.Imaginary, 5));
if (x3.Imaginary == 0)
{
Console.WriteLine(x3rp);
}
else if (x3.Imaginary < 0) // testing if unreal part is negative
{
Console.WriteLine(x3rp " - " Math.Round(Complex.Abs(x3.Imaginary)) "i");
}
else
{
Console.WriteLine(x3rp " " x3ip "i");
}
string x4rp = Convert.ToString(Math.Round(x4.Real, 5)); // rounding up both real and unreal parts
string x4ip = Convert.ToString(Math.Round(x4.Imaginary, 5));
if (x4.Imaginary == 0)
{
Console.WriteLine(x4rp);
}
else if (x4.Imaginary < 0) // testing if unreal part is negative
{
Console.WriteLine(x4rp " - " Math.Round(Complex.Abs(x4.Imaginary)) "i");
}
else
{
Console.WriteLine(x4rp " " x4ip "i");
}
// discriminant tests
if (discriminant == 0) // if delta = 0
{
if (P < 0 && D < 0 && deltazero != 0)
{
string result4 = "The equation yields a pair of double roots and two real simple roots.";
Console.WriteLine(result4);
}
else if (D > 0 || (P > 0 && (D != 0 || R != 0)))
{
string result5 = "The equation yields a pair of double roots and two complex conjugate roots.";
Console.WriteLine(result5);
}
else if (deltazero == 0 && D != 0)
{
string result6 = "The equation yields a triple root and a simple root.";
Console.WriteLine(result6);
}
else if (D == 0)
{
if (P < 0)
{
string result7 = "The equation yields two pairs of real double roots.";
Console.WriteLine(result7);
}
else if (P > 0 && R == 0)
{
string result8 = "The equation yields two pairs of double complex conjugate roots.";
Console.WriteLine(result8);
}
else if (deltazero == 0)
{
string result9pt2 = Convert.ToString(-(b / (4 * a)));
string result9 = "All of the roots of this equation are equal to -b/4a";
Console.WriteLine(result9 "(" result9pt2 ")");
}
else
{
Console.WriteLine("ERROR");
}
}
else
{
Console.WriteLine("ERROR");
}
}
else if (discriminant > 0) // if delta > 0
{
if (P < 0 && D < 0)
{
string result2 = "The equation yields four distinct real roots.";
Console.WriteLine(result2);
}
else if (P > 0 && D > 0)
{
string result3 = "The equation yields two pairs of unreal complex conjugate roots.";
Console.WriteLine(result3);
}
else
{
Console.WriteLine("ERROR");
}
}
else if (discriminant < 0) // if delta < 0
{
string result1 = "The equation yields two distinct real roots and two complex conjugate non-real roots.";
Console.WriteLine(result1);
}
// discriminant of equation
Console.WriteLine(" ");
Console.WriteLine("Discriminant :");
Console.WriteLine(discriminant);
Console.ReadLine();
}
}
}
==========================================================
Edit #1 :
I tried another method to solve the roots.
However, the imaginary part is still wrong for the last 2 roots.
If anyone is able to spot my mistake, I would be grateful.
// different parts of the roots presented in p1-p6
double p = ((8 * a * c) - (3 * Math.Pow(b, 2))) / (8 * Math.Pow(a, 2));
double q = (Math.Pow(b, 3) - (4 * a * b * c) (8 * Math.Pow(a, 2) * d)) / (8 * Math.Pow(a, 3));
double D0 = Math.Pow(c, 2) - (3 * b * d) (12 * a * e);
double D1 = (2 * Math.Pow(c, 3)) - (9 * b * c * d) (27 * Math.Pow(b, 2) * e) (27 * a * Math.Pow(d, 2)) - (72 * a * c * e);
Complex Q = Complex.Pow((D1 Complex.Sqrt(Complex.Pow(D1, 2) - (4 * Complex.Pow(D0, 3)))) / 2.0, 1.0 / 3.0);
Complex S = 1.0 / 2.0 * Complex.Sqrt((-2.0 / 3.0 * p) (1.0 / (3 * a) * (Q (D0 / Q))));
// calculating roots with p1-p6
Complex x1 = (-(b / 4 * a)) - S (1.0 / 2.0 * Complex.Sqrt((-4.0 * Complex.Pow(S, 2)) - (2 * p) (q / S)));
Complex x2 = (-(b / 4 * a)) - S - (1.0 / 2.0 * Complex.Sqrt((-4.0 * Complex.Pow(S, 2)) - (2 * p) (q / S)));
Complex x3 = (-(b / 4 * a)) S (1.0 / 2.0 * Complex.Sqrt((-4.0 * Complex.Pow(S, 2)) - (2 * p) (q / S)));
Complex x4 = (-(b / 4 * a)) S - (1.0 / 2.0 * Complex.Sqrt((-4.0 * Complex.Pow(S, 2)) - (2 * p) (q / S)));
==========================================================
Edit #2
Problem solved.
The program is now able to solve quartic equations. The cause of the problems mentioned above is actually the minor typos I made in the code lines.
uj5u.com熱心網友回復：
我從這里得到了一個有效的演算法。它似乎比你的簡單。
double D0 = c * c - 3 * b * d 12 * a * e;
double D1 = 2 * c * c * c - 9 * b * c * d 27 * b * b * e 27 * a * d * d - 72 * a * c * e;
double p = (8 * a * c - 3 * b * b) / (8 * a * a);
double q = (b * b * b - 4 * a * b * c 8 * a * a * d) / (8 * a * a * a);
Complex Q = Complex.Pow((D1 Complex.Sqrt(D1 * D1 - 4 * D0 * D0 * D0)) / 2, 1.0 / 3.0);
Complex S = Complex.Sqrt(-2 * p / 3 (Q D0 / Q) / (3 * a)) / 2;
Complex u = Complex.Sqrt(-4 * S * S - 2 * p q / S) / 2;
Complex v = Complex.Sqrt(-4 * S * S - 2 * p - q / S) / 2;
Complex x1 = -b / (4 * a) - S u;
Complex x2 = -b / (4 * a) - S - u;
Complex x3 = -b / (4 * a) S v;
Complex x4 = -b / (4 * a) S - v;
此外，根據此頁面，您可以使用它而不是很長的判別公式來獲得它的符號：
double discriminantSign = c * c - 3 * b * d 12 * a * e;
我還注意到 4 個拼寫錯誤，您在應用 Math.Round 函式時忘記提及小數點后 5，因此您的結果被四舍五入為整數。
編輯：如果您的虛部仍有問題，那是因為您沒有糾正錯別字：
Console.WriteLine(x1rp " - " Math.Round(Complex.Abs(x1.Imaginary)) "i");
進入：
Console.WriteLine(x1rp " - " Math.Round(Complex.Abs(x1.Imaginary), 5) "i");
除了上面的 4 行之外，你在任何地方都取 5 位小數。所以它對整數進行四舍五入。全部更換。
轉載請註明出處，本文鏈接：https://www.uj5u.com/qiye/424831.html
標籤：c# math graph equation algebra
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