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PolynomialTools

  

Hurwitz

  

decide whether a polynomial has all its zeros strictly in the left half plane

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

Hurwitz(p, z,'s','g')

Parameters

p

-

polynomial with complex coefficients

z

-

variable of the polynomial p

's'

-

(optional) name

'g'

-

(optional) name

Description

• 

The Hurwitz(p, z) function determines whether the the polynomial pz 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 pp&ast;p&plus;p&ast; 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 bz&plus;a, where a0orb<0, p is not Hurwitz, either. If each subsequent polynomial in the sequence returned is of the form bz&plus;a, where 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&ast; 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.

Examples

withPolynomialTools&colon;

p1z2&plus;z&plus;1

p1:=z2&plus;z&plus;1

(1)

Hurwitzp1&comma;z

true

(2)

p23z3&plus;2z2&plus;z&plus;c

p2:=3z3&plus;2z2&plus;c&plus;z

(3)

Hurwitzp2&comma;z&comma;&apos;s2&apos;&comma;&apos;g2&apos;

FAIL

(4)

s2

32z&comma;4z3c2&comma;32z&plus;zc

(5)

g2

1

(6)

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&equals;0 and c&equals;23 give nontrivial gcds between p2 and its paraconjugate. Thus, the stability criteria are satisfied only as above.

p34z4&plus;z3&plus;z2&plus;c

p3:=4z4&plus;z3&plus;z2&plus;c

(7)

Hurwitzp3&comma;z&comma;&apos;s3&apos;&comma;&apos;g3&apos;

FAIL

(8)

s3

0&comma;4z&comma;z&comma;zc&comma;z

(9)

Notice that the last term has coefficient 1. Thus, you can say unequivocally that p3 is not Hurwitz, for any value of c.

p4z5&plus;5z4&plus;4z3&plus;3z2&plus;2z&plus;c

p4:=z5&plus;5z4&plus;4z3&plus;3z2&plus;c&plus;2z

(10)

Hurwitzp4&comma;z&comma;&apos;s4&apos;&comma;&apos;g4&apos;

FAIL

(11)

s4

15z&comma;2517z&comma;2895z5c&plus;1&comma;1175c&plus;12zc2&plus;48c2&comma;c248c&plus;2z5c&plus;1c

(12)

By inspecting s4, notice that p4 is Hurwitz only if 15<c, and c2&plus;48c<2, and 0<c. This can be simplified to the conditions 0<c<24&plus;172&equals;0.04... 

p5p2&plus;Id

p5:=3z3&plus;2z2&plus;c&plus;z&plus;Id

(13)

evalc and the Hurwitz function assume that symbolic parameters have real values.

Hurwitzp5&comma;z&comma;&apos;s5&apos;&comma;&apos;g5&apos;

FAIL

(14)

s5

32z&comma;4z3c28Id3c22&comma;123c23z9c312c28d2&plus;4c&plus;Id3c229c312c28d2&plus;4c

(15)

The coefficients of s5 can be inspected according to rules, but it is a tedious process.

p6expandx1x2&plus;2xc

p6:=cx3&plus;x4&plus;cx2x32cx&plus;2x2&plus;2c2x

(16)

Hurwitzp6&comma;x&comma;&apos;s6&apos;&comma;&apos;g6&apos;

false

(17)

g6

x2&plus;2

(18)

p7x&plus;2

p7:=x&plus;2

(19)

Hurwitzp7&comma;x

true

(20)

p8x3&plus;cx2&plus;c21x&plus;1

p8:=x3&plus;cx2&plus;c21x&plus;1

(21)

Hurwitzp8&comma;x&comma;&apos;s8&apos;&comma;&apos;g8&apos;

FAIL

(22)

s8

xc&comma;c2xc3c1&comma;c2xxxc

(23)

Examination of the above for real values of c is a way to determine whether the polynomial is Hurwitz.

p9expandcz2&plus;1z&plus;1z2&plus;2z&plus;2

p9:=cz5&plus;3cz4&plus;4cz3&plus;2cz2&plus;z3&plus;3z2&plus;4z&plus;2

(24)

Hurwitzp9&comma;z&comma;&apos;s9&apos;&comma;&apos;g9&apos;

FAIL

(25)

s9

(26)

g9

cz2&plus;1

(27)

In the previous example, c might be zero. Thus, Hurwitz cannot determine whether all the zeros are in the left half plane.

References

  

Levinson, Norman, and Redheffer, Raymond M. Complex Variables. Holden-Day, 1970.

See Also

evalc

expand

fsolve

Hurwitz Zeta Function

PolynomialTools

sqrt

subs

 


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