求有利于编程的一元三次与一元四次方程的求根公式

mathematica

维基百科!

无心人

http://baike.baidu.com/view/460155.htm 看看盛金公式 好吧，貌似一不小心就发现了某几个业余数学家的争论点，那个盛金公式貌似有大量非议。。。而且那个作者 范盛金 本人声称尝试去求一元五次方程的根式解去，难道是民科？？？？

云梦

我有一套解一元四次以下方程的方法：

282842712474

卡丹公式提供了虚数开立方,这个就不利于编程,不是吗? 你知道虚数如何开立方吗? matlab 发表于 2013-2-17 09:06

那你编写一个求虚数开立方的函数不行吗？如果你说有利于，我还是同意郭老大的说法，迭代最容易，毕竟你只是代数方程，性态很良好，很少出现什么死循环等坏情况

无心人

虚数开立方很容易吧

云梦

虚数的开立方是很简单的事。 例如：（a+bi)^(1/3)=c+di a+bi=(c+di)^3 展开后逐项比较哦。

云梦

Function JFC13() Dim FH1, FH2 As String '标准形式 x3+b0x2+c0x+d0=0 Xn1(0).Bz = "" Xn1(0).St = "0" Xn1(0).Zs = 0 Xn1(1) = Xn1(0) Xn1(2) = Xn1(0) Xn2(0) = Xn1(0) Xn2(1) = Xn1(0) Xn2(2) = Xn1(0) Command45.Visible = True Command46.Visible = False An = SToo(0) Cx = SToo(1) Ax = An Call Multx_ Bn = Cx Cx = SToo(2) Ax = An Call Multx_ Cn = Cx Cx = SToo(3) Ax = An Call Multx_ Dn = Cx An.Bz = "" An.St = "1" An.Zs = 0 If Left(Bn.St, 1) = "0" And Left(Cn.St, 1) = "0" Then '1.x^3+d0=0 Ax = Dn Ax.Bz = IIf(Ax.Bz = "", "-", "") FH1 = Ax.Bz '记住其标志 Ax.Bz = "" Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ Ax = Cx Call Exp Ax.Bz = FH1 '还原标志 Xn1(0) = Ax 'x1=3√-d '一对共轭虚根 Cx = Dn Ax.Bz = "" Ax.St = "125" Ax.Zs = -1 Call Mults_ 'd/8 Ax = Cx Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ Ax = Cx Call Exp 'x2,3=3√(-d/8) Xn1(1) = Ax Xn1(2) = Ax Cx.Bz = "" Cx.St = Eng(0).Tag Cx.Zs = 0 Call Mult_ Xn2(1) = Cx Xn2(2) = Cx Xn2(2).Bz = "-" Else '2.x3+b0x2+c0x+d0=0 'A=b^2-3ac Ax = Bn Cx = Bn Call Mult_ Px = Cx Ax = An Cx = Cn Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition A1n = Ax 'A= 'B=bc－9ad Ax = Bn Cx = Cn Call Mult_ Px = Cx Ax = An Cx = Dn Call Mult_ Ax.Bz = "" Ax.St = "9" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition B1n = Ax 'B= 'C=c^2－3bd Ax = Cn Cx = Cn Call Mult_ Px = Cx Ax = Bn Cx = Dn Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition C1n = Ax 'C= '判别式 B^2-4AC Ax = B1n Cx = B1n Call Mult_ Px = Cx Ax = A1n Cx = C1n Call Mult_ Ax.Bz = "" Ax.St = "4" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition ux = Ax If Ax.Bz = "-" Then '当 B2-4AC<0 时,有三个不同的实数根 't = (2Ab - 3aB)/(2 A^(3/2)) 角度 Ax = A1n Cx = Bn Call Mult_ Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Mults_ Px = Cx '2Ab Ax = An Cx = B1n Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ '3aB Ax = Px Bx = Cx hsfx = "-" Call Addition Px = Ax '(2Ab - 3aB) Ax = A1n Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "15" Ax.Zs = 0 Call Mults_ Call Exp 'A^(3/2) Cx = Ax Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Mults_ '2A^(3/2) Ax = Cx Cx = Px Call Multx_ '(2Ab - 3aB)/2A^(3/2) Ax = Cx DEG = 1 Call ArcCosx '角度t Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ tx = Cx 'ArcCos(t)/3 '-b - 2*A^(1/2) Ax = A1n Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "5" Ax.Zs = -1 Call Mults_ Call Exp 'A^(1/2) Bx = Ax hsfx = "+" Call Addition Qx = Ax '2*A^(1/2) Cx = An Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Rx = Cx '3a 'Cos(t) Ax = tx DEG = 1 Call Cosx Cx = Qx Call Mult_ '2A^(1/2)Cos(t) Ax = Bn Ax.Bz = IIf(Ax.Bz = "-", "", "-") Bx = Cx hsfx = "-" Call Addition '-b-2A^(1/2)Cos(t) Cx = Ax Ax = Rx Call Multx_ '(-b-2A^(1/2)Cos(t))/3a Xn1(0) = Cx 'x1= 'Cos(t-2Pi/3) Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Cx.Bz = "" Cx.St = Pi.Tag Cx.Zs = 0 Call Mults_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ '2Pi/3 Ox = Cx Ax = tx Bx = Cx hsfx = "-" Call Addition 't-2Pi/3 DEG = 1 Call Cosx 'Cos(t-2Pi/3) Cx = Qx Call Mult_ ' Ax = Bn Ax.Bz = IIf(Ax.Bz = "-", "", "-") Bx = Cx hsfx = "-" Call Addition '-b-cos(t-2Pi/3) Cx = Ax Ax = Rx Call Multx_ '(-b-cos(t-2Pi/3))/3a Xn1(1) = Cx Ax = tx Bx = Ox hsfx = "+" Call Addition 't+2Pi/3 DEG = 1 Call Cosx 'Cos(t+2Pi/3) Cx = Qx Call Mult_ '2ACos(t+2Pi/3) Ax = Bn Ax.Bz = IIf(Ax.Bz = "-", "", "-") Bx = Cx hsfx = "-" Call Addition '-b-2ACos(t+2Pi/3) Cx = Ax Ax = Rx Call Multx_ '(-b-2ACos(t+2Pi/3))/3a Xn1(2) = Cx ' Text12.Text = "有三个不同的实数根,x1,x2,x3" Else 'v=0 时,有三个实数根(两个相同) If Left(ux.St, 1) = "0" Then 'X1=－b/a+K；X2=X3=－K/2，其中K=B/A，(A≠0)。 Ax = A1n Cx = B1n Call Multx_ Px = Cx Ax = An Cx = Bn Call Multx_ Bx = Cx Ax = Px hsfx = "-" Call Addition Xn1(0) = Ax Cx = Px Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Multx_ Ax = Cx Ax.Bz = IIf(Ax.Bz = "", "-", "") Xn1(1) = Ax Xn1(2) = Ax 'Text12.Text = "有三个实数根(其中有两个重根x2=x3)" End If If Left(ux.St, 1) > "0" Then 'v>0 有一个实数根 Ax = ux Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "5" Ax.Zs = -1 Call Mult_ Call Exp tx = Ax Ax = B1n Ax.Bz = IIf(Ax.Bz = "", "-", "") Bx = tx hsfx = "+" Call Addition 'Y1，Y2=Ab+3a(－B±(B^2－4AC)^(1/2))/2，i^2=－1。 Cx = An Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Multxs_ Yn1 = Cx Ax = A1n Cx = Bn Call Mult_ Ax = Yn1 Bx = Cx hsfx = "+" Call Addition FH1 = Ax.Bz Ax.Bz = "" Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ Ax = Cx Call Exp Yn1 = Ax Ax = B1n Ax.Bz = IIf(Ax.Bz = "", "-", "") Bx = tx hsfx = "-" Call Addition Cx = An Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mult_ Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Multx_ Yn2 = Cx Ax = A1n Cx = Bn Call Mult_ Ax = Yn2 Bx = Cx hsfx = "+" Call Addition FH2 = Ax.Bz Ax.Bz = "" Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multx_ Ax = Cx Call Exp Yn2 = Ax Ax = Bn Ax.Bz = IIf(Ax.Bz = "", "-", "") Bx = Yn1 Bx.Bz = FH1 hsfx = "-" Call Addition Bx = Yn2 Bx.Bz = FH2 hsfx = "-" Call Addition Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multx_ Ax = Cx Xn1(0) = Ax '虚数计算 Ax = Bn '(－2b+(Y1)^(1/3)+(Y2)^(1/3))/(6a) Bx = Bn hsfx = "+" Call Addition Ax.Bz = IIf(Ax.Bz = "", "-", "") Px = Ax Bx = Yn1 Bx.Bz = FH1 hsfx = "+" Call Addition Bx = Yn2 Bx.Bz = FH2 hsfx = "+" Call Addition Cx = Ax Ax = An Call Multx_ Ax.Bz = "" Ax.St = "6" Ax.Zs = 0 Call Multx_ Ax = Cx 'x2=实数部分 Xn1(1) = Ax Xn1(2) = Ax Ax = Yn1 '3^(1/2)((Y1)^(1/3)－(Y2)^(1/3))i/(6a) Ax.Bz = FH1 Bx = Yn2 Bx.Bz = FH2 hsfx = "-" Call Addition Cx = Ax Ax = An Call Multx_ Ax.Bz = "" Ax.St = "6" Ax.Zs = 0 Call Multx_ Ax.Bz = "" Ax.St = Eng(0).Tag '3^(1/2) Ax.Zs = 0 Call Mult_ Ax = Cx 'x2=虚数部分 Xn2(1) = Ax Xn2(2) = Ax Xn2(2).Bz = "-" 'Text12.Text = "有一个实数根 x1和一对共轭虚根" End If If Left(A1n.St, 1) = "0" And Left(B1n.St, 1) = "0" Then Cx = Bn Ax = An Call Multx_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multx_ Ax = Cx Ax.Bz = IIf(Ax.Bz = "", "-", "") Xn1(0) = Ax Xn1(1) = Ax Xn1(2) = Ax ' Text12.Text = "有三个实数根( 是一个三重根x1=x2=x3)" End If End If End If End Function Function JFC130() Dim FH1, FH2 As String '标准形式 x3+b0x2+c0x+d0=0 If Left(Bn.St, 1) = "0" And Left(Cn.St, 1) = "0" Then '1.x^3+d0=0 Ax = Dn Ax.Bz = IIf(Ax.Bz = "", "-", "") FH1 = Ax.Bz '记住其标志 Ax.Bz = "" Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ Ax = Cx Call Exp Ax.Bz = FH1 '还原标志 SToo(31) = Ax 'x1=3√-d '一对共轭虚根 Cx = Dn Ax.Bz = "" Ax.St = "125" Ax.Zs = -1 Call Mults_ 'd/8 Ax = Cx Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ Ax = Cx Call Exp 'x2,3=3√(-d/8) Xn1(0) = Ax Cx.Bz = "" Cx.St = Eng(0).Tag Cx.Zs = 0 Call Mult_ Xn1(1) = Cx Xsbz(0) = 3 Ax = SToo(31) Call sc(Ax.Bz, Ax.St, Ax.Zs) Else '2.x3+b0x2+c0x+d0=0 'A=b^2-3ac Ax = Bn Cx = Bn Call Mult_ Px = Cx Ax = An Cx = Cn Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition A1n = Ax 'A= 'B=bc－9ad Ax = Bn Cx = Cn Call Mult_ Px = Cx Ax = An Cx = Dn Call Mult_ Ax.Bz = "" Ax.St = "9" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition B1n = Ax 'B= 'C=c^2－3bd Ax = Cn Cx = Cn Call Mult_ Px = Cx Ax = Bn Cx = Dn Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition C1n = Ax 'C= '判别式 B^2-4AC Ax = B1n Cx = B1n Call Mult_ Px = Cx Ax = A1n Cx = C1n Call Mult_ Ax.Bz = "" Ax.St = "4" Ax.Zs = 0 Call Mults_ Ax = Px Bx = Cx hsfx = "-" Call Addition ux = Ax If Ax.Bz = "-" Then '当 B2-4AC<0 时,有三个不同的实数根 't = (2Ab - 3aB)/(2 A^(3/2)) 角度 Ax = A1n Cx = Bn Call Mult_ Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Mults_ Px = Cx '2Ab Ax = An Cx = B1n Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ '3aB Ax = Px Bx = Cx hsfx = "-" Call Addition Px = Ax '(2Ab - 3aB) Ax = A1n Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "15" Ax.Zs = 0 Call Mults_ Call Exp 'A^(3/2) Cx = Ax Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Mult_ '2A^(3/2) Ax = Cx Cx = Px Call Multx_ '(2Ab - 3aB)/2A^(3/2) Ax = Cx DEG = 1 Call ArcCosx '角度t Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ tx = Cx 'ArcCos(t)/3 '-b - 2*A^(1/2) Ax = A1n Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "5" Ax.Zs = -1 Call Mults_ Call Exp 'A^(1/2) Bx = Ax hsfx = "+" Call Addition Qx = Ax '2*A^(1/2) Cx = An Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Rx = Cx '3a 'Cos(t) Ax = tx DEG = 1 Call Cosx Cx = Qx Call Mult_ '2A^(1/2)Cos(t) Ax = Bn Ax.Bz = IIf(Ax.Bz = "-", "", "-") Bx = Cx hsfx = "-" Call Addition '-b-2A^(1/2)Cos(t) Cx = Ax Ax = Rx Call Multx_ '(-b-2A^(1/2)Cos(t))/3a SToo(31) = Cx 'x1= 'Cos(t-2Pi/3) Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Cx.Bz = "" Cx.St = Pi.Tag Cx.Zs = 0 Call Mults_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ '2Pi/3 Ox = Cx Ax = tx Bx = Cx hsfx = "-" Call Addition 't-2Pi/3 DEG = 1 Call Cosx 'Cos(t-2Pi/3) Cx = Qx Call Mult_ ' Ax = Bn Ax.Bz = IIf(Ax.Bz = "-", "", "-") Bx = Cx hsfx = "-" Call Addition '-b-cos(t-2Pi/3) Cx = Ax Ax = Rx Call Multx_ '(-b-cos(t-2Pi/3))/3a SToo(32) = Cx Ax = tx Bx = Ox hsfx = "+" Call Addition 't+2Pi/3 DEG = 1 Call Cosx 'Cos(t+2Pi/3) Cx = Qx Call Mult_ '2ACos(t+2Pi/3) Ax = Bn Ax.Bz = IIf(Ax.Bz = "-", "", "-") Bx = Cx hsfx = "-" Call Addition '-b-2ACos(t+2Pi/3) Cx = Ax Ax = Rx Call Multx_ '(-b-2ACos(t+2Pi/3))/3a SToo(33) = Cx ' Text12.Text = "有三个不同的实数根,x1,x2,x3" Else 'v=0 时,有三个实数根(两个相同) If Left(ux.St, 1) = "0" Then 'X1=－b/a+K；X2=X3=－K/2，其中K=B/A，(A≠0)。 Ax = A1n Cx = B1n Call Multx_ Px = Cx Ax = An Cx = Bn Call Multx_ Bx = Cx Ax = Px hsfx = "-" Call Addition SToo(31) = Ax Cx = Px Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Multx_ Ax = Cx Ax.Bz = IIf(Ax.Bz = "", "-", "") SToo(32) = Ax SToo(33) = Ax 'Text12.Text = "有三个实数根(其中有两个重根x2=x3)" End If If Left(ux.St, 1) > "0" Then 'v>0 有一个实数根 Ax = ux Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "5" Ax.Zs = -1 Call Mult_ Call Exp tx = Ax Ax = B1n Ax.Bz = IIf(Ax.Bz = "", "-", "") Bx = tx hsfx = "+" Call Addition 'Y1，Y2=Ab+3a(－B±(B^2－4AC)^(1/2))/2，i^2=－1。 Cx = An Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mults_ Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Multxs_ Yn1 = Cx Ax = A1n Cx = Bn Call Mult_ Ax = Yn1 Bx = Cx hsfx = "+" Call Addition FH1 = Ax.Bz Ax.Bz = "" Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multxs_ Ax = Cx Call Exp Yn1 = Ax 'Yn1.bz = FH1 Ax = B1n Ax.Bz = IIf(Ax.Bz = "", "-", "") Bx = tx hsfx = "-" Call Addition Cx = An Call Mult_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Mult_ Ax.Bz = "" Ax.St = "2" Ax.Zs = 0 Call Multx_ Yn2 = Cx Ax = A1n Cx = Bn Call Mult_ Ax = Yn2 Bx = Cx hsfx = "+" Call Addition FH2 = Ax.Bz Ax.Bz = "" Call Lnx Cx = Ax Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multx_ Ax = Cx Call Exp Yn2 = Ax 'Yn2.bz = FH2 Ax = Bn Ax.Bz = IIf(Ax.Bz = "", "-", "") Bx = Yn1 Bx.Bz = FH1 hsfx = "-" Call Addition Bx = Yn2 Bx.Bz = FH2 hsfx = "-" Call Addition Cx = Ax Ax = An Call Multx_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multx_ Ax = Cx SToo(31) = Ax '虚数计算 Ax = Bn '(－2b+(Y1)^(1/3)+(Y2)^(1/3))/(6a) Bx = Bn hsfx = "+" Call Addition Ax.Bz = IIf(Ax.Bz = "", "-", "") Px = Ax Bx = Yn1 Bx.Bz = FH1 hsfx = "+" Call Addition Bx = Yn2 Bx.Bz = FH2 hsfx = "+" Call Addition Cx = Ax Ax = An Call Multx_ Ax.Bz = "" Ax.St = "6" Ax.Zs = 0 Call Multx_ Ax = Cx 'x2=实数部分 Xn1(0) = Ax Ax = Yn1 '3^(1/2)((Y1)^(1/3)－(Y2)^(1/3))i/(6a) Ax.Bz = FH1 Bx = Yn2 Bx.Bz = FH2 hsfx = "-" Call Addition Cx = Ax Ax = An Call Multx_ Ax.Bz = "" Ax.St = "6" Ax.Zs = 0 Call Multx_ Ax.Bz = "" Ax.St = Eng(0).Tag '3^(1/2) Ax.Zs = 0 Call Mult_ Ax = Cx 'x2=虚数部分 Xn1(1) = Ax Xsbz(0) = 1 'Text12.Text = "有一个实数根 x1和一对共轭虚根" End If If Left(A1n.St, 1) = "0" And Left(B1n.St, 1) = "0" Then Cx = Bn Ax = An Call Multx_ Ax.Bz = "" Ax.St = "3" Ax.Zs = 0 Call Multx_ Ax = Cx Ax.Bz = IIf(Ax.Bz = "", "-", "") SToo(31) = Ax SToo(32) = Ax ' Text12.Text = "有三个实数根( 是一个三重根x1=x2=x3)" End If End If End If End Function