◆ Rpzero()

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

Purpose: This routine computes all roots of a polynomial p(z) with real 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) Real 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^5 + 2*x^3 + 2*x^2 - 15*x + 10 = 0

The exact solutions are 1(double root), -2 and ±√5i.
Sub Ex_Rpzero()

Const N As Long = 5

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

Dim IFlag As Long, Info As Long, I As Long

A(0) = 1: A(1) = 0: A(2) = 2: A(3) = 2: A(4) = -15: A(5) = 10

IFlag = 0

Call Rpzero(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: 1.0000000135981 2.14625358292393E-08 1.26597932122911E-07

1.48298437403299E-18 2.23606797749979 8.99528943854828E-15

-2 1.15041631099093E-19 7.53855142372064E-15

7.94241311875315E-17 -2.23606797749979 9.06723080094738E-15

0.999999987046461 -2.24784161738852E-08 1.26578540844987E-07

Info = 0

Note - Only half precision for double roots.