◆ Cpzero()

Roots of a polynomial (complex coefficients) (Netwon method)

Purpose: This routine computes all roots of a polynomial p(z) with complex coefficients by Netwon 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).
[in,out]	Z()	Array Z(LZ - 1) (LZ >= N) [in] Initial estimates for zeros. If these are unknown, set IFlag = 0 and it is not necessary to set estimates in Z(). Note - Initial estimates must be separated, that is, distinct or not repeated. [out] The obtained zeros.
[in]	IFlag	Flag to indicate if initial estimates of zeros are input. = 0: No estimates are input. <> 0: Zr() and Zi() contain estimates of zeros.
[out]	S()	Array S(LS - 1) (LS >= N) Error bound for Zr() and Zi().
[out]	Info	= 0: Successful exit. = -1: The argument N had an illegal value. (N < 1) = -2: The argument A() is invalid or had an illegal value. (A(0) = 0) = -3: The argument Z() is invalid. = -5: The argument S() is invalid. = 1: Maximum number of iterations exceeded. Best current estimates of the zeros are in Z(). Error bounds in S() are not calculated.
[out]	Iter	(Optional) Number of iterations required to converge.
[in]	MaxIter	(Optional) Maximum number of iterations. (default = 25 * N) If MaxIter <= 0, the default value will be used.

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

The exact soplutions are 8 + 4i, 4 + 9i and 7 + i.
Sub Ex_Cpzero()

Const N As Long = 3

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

Dim IFlag 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)

IFlag = 0

Call Cpzero(N, A(), Z(), IFlag, S(), Info)

For I = 0 To N - 1

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

Next

Debug.Print "Info =", Info

End Sub

Example Results: 8 4 6.12410216962375E-13

4 9 2.62419907575987E-13

7 1 2.49188840222312E-13

Info = 0