◆ Dka()

Roots of a polynomial (complex coefficients) (3rd order DKA method)

Purpose: This routine computes all roots of a polynomial p(z) with complex coefficients by 3rd order DKA (Durand-Kerner-Aberth) method.
p(z) = a0*z^n + a1*z^(n-1) + ... + an

Parameters

[in]	N	Degree of polynomial. (N >= 1)
[in]	A()	Array A(LA - 1) (LA >= N + 1) Complex coefficient vector of p(z) (a0 to an).
[out]	Z()	Array Z(LZ - 1) (LZ >= N) The obtained zeros.
[out]	Info	= 0: Successful exit. = -1: The argument N had an illegal value. (N <= 0) = -2: The argument A() is invalid. = -3: The argument Z() is invalid. = 1: Maximum number of iterations exceeded.
[out]	Iter	(Optional) Number of iterations required to converge.
[in]	MaxIter	(Optional) Maximum number of iterations. (default = 10 * N) If MaxIter <= 0, the default value will be used.

Reference: (Japanese book) Masatake Mori "FORTRAN77 Numerical Calculation Programming (augmented edition)" Iwanami Shoten (1987)

Example Program: Solve the following algebraic equation.
x^3 + (-19-14i)*x^2 + (67+191i)*x + 116-612i = 0

The exact solutions are 8 + 4i, 4 + 9i and 7 + i.
Sub Ex_Dka()

Const N As Long = 3

Dim A(N) As Complex, Z(N - 1) As Complex

Dim Iter As Long, Info As Long, I As Long

A(0) = Cmplx(1): A(1) = Cmplx(-19, -14): A(2) = Cmplx(67, 191)

A(3) = Cmplx(116, -612)

Call Dka(N, A(), Z(), Info, Iter)

For I = 0 To N - 1

Debug.Print Creal(Z(I)), Cimag(Z(I))

Next

Debug.Print "Iter =", Iter, "Info =", Info

End Sub

Cmplx
Function Cmplx(R As Double, Optional I As Double=0) As Complex
Building complex number

Cimag
Function Cimag(A As Complex) As Double
Imaginary part of complex number

Creal
Function Creal(A As Complex) As Double
Real part of complex number

Dka
Sub Dka(N As Long, A() As Complex, Z() As Complex, Info As Long, Optional Iter As Long, Optional MaxIter As Long=0)
Roots of a polynomial (complex coefficients) (3rd order DKA method)