decide whether a polynomial has all its zeros strictly in the left half plane
polynomial with complex coefficients
variable of the polynomial p
The Hurwitz(p, z) function determines whether the polynomial p⁡z has all its zeros strictly in the left half plane.
A polynomial is a Hurwitz polynomial if all its roots are in the left half plane.
The parameter p is a polynomial with complex coefficients. The polynomial may have symbolic parameters, which evalc and Hurwitz assume to be real. The paraconjugate p* of p is defined as the polynomial whose roots are the roots of p reflected across the imaginary axis.
The parameter 's', if specified, is a name to which the sequence of partial fractions of the Stieltjes continued fraction of p−p*p+p* will be assigned. The first element of the sequence returned in 's' is special. If it is of higher degree than 1 in z, p is not Hurwitz. If it is of the form b⁢z+a, where ℜ⁡a≠0orb<0, p is not Hurwitz, either. If each subsequent polynomial in the sequence returned is of the form b⁢z+a, where ℜ⁡a=0and0<b, then p is a Hurwitz polynomial.
This is useful if p has symbolic coefficients. You can decide the ranges of the coefficients that make p Hurwitz.
If the Hurwitz function can use the previous rules to determine that p is Hurwitz, it returns true. If it can decide that p is not Hurwitz, it returns false. Otherwise, it returns FAIL.
The parameter 'g', if specified, is a name to which the gcd of p and its paraconjugate p* will be assigned. The zeros of this gcd are precisely the zeros of p which are symmetrical under reflection across the imaginary axis.
If the gcd is 1 while the sequence of partial fractions is empty, the conditions for being a Hurwitz polynomial are trivially satisfied. A manual check is recommended, though a warning is returned only if infolevel[Hurwitz] >= 1.
p1 ≔ z2+z+1
p2 ≔ 3⁢z3+2⁢z2+z+c
The elements of s2 are all positive if and only if 0<c<23, by inspection. Thus, you can use the information returned even when the direct call to Hurwitz fails.
Separate calls to Hurwitz in the cases c=0 and c=23 give nontrivial gcds between p2 and its paraconjugate. Thus, the stability criteria are satisfied only as above.
p3 ≔ 4⁢z4+z3+z2+c
Notice that the last term has coefficient −1. Thus, you can say unequivocally that p3 is not Hurwitz, for any value of c.
p4 ≔ z5+5⁢z4+4⁢z3+3⁢z2+2⁢z+c
By inspecting s4, notice that p4 is Hurwitz only if −15<c, and c2+48⁢c<2, and 0<c. This can be simplified to the conditions 0<c<−24+17⁢2=0.04...
p5 ≔ p2+I⁢d
evalc and the Hurwitz function assume that symbolic parameters have real values.
The coefficients of s5 can be inspected according to rules, but it is a tedious process.
p6 ≔ expand⁡x−1⁢x2+2⁢x−c
p7 ≔ x+2
p8 ≔ x3+c⁢x2+c2−1⁢x+1
Examination of the above for real values of c is a way to determine whether the polynomial is Hurwitz.
p9 ≔ expand⁡c⁢z2+1⁢z+1⁢z2+2⁢z+2
In the previous example, c might be zero. Thus, Hurwitz cannot determine whether all the zeros are in the left half plane.
Levinson, Norman, and Redheffer, Raymond M. Complex Variables. Holden-Day, 1970.
Hurwitz Zeta Function
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