◆ Cpqr79()

Roots of a polynomial (complex coefficients) (Companion matrix method)

Purpose: This routine computes all roots of a polynomial p(z) with complex coefficients by computing the eigenvalues of the companion matrix.
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: Failed to converge within maximum number (30) of QR iterations on some eigenvalue of the companion matrix.

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_Cpqr79()

Const N As Long = 3

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

Dim 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 Cpqr79(N, A(), Z(), Info)

For I = 0 To N - 1

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

Next

Debug.Print "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

Cpqr79
Sub Cpqr79(N As Long, A() As Complex, Z() As Complex, Info As Long)
Roots of a polynomial (complex coefficients) (Companion matrix method)

Example Results: 4 9.00000000000001

7.99999999999999 3.99999999999998

7.00000000000001 1.00000000000001

Info = 0