PolynomialTools[Hurwitz] - decide whether a polynomial has all its zeros strictly in the left half plane
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Calling Sequence
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Hurwitz(p, z,'s','g')
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Parameters
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p
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polynomial with complex coefficients
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z
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variable of the polynomial p
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's'
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(optional) name
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'g'
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(optional) name
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Description
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The Hurwitz(p, z) function determines whether the the polynomial has all its zeros strictly in the left half plane.
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A polynomial is a Hurwitz polynomial if all its roots are in the left half plane.
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This is useful if has symbolic coefficients. You can decide the ranges of the coefficients that make Hurwitz.
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If the Hurwitz function can use the previous rules to determine that is Hurwitz, it returns true. If it can decide that is not Hurwitz, it returns false. Otherwise, it returns FAIL.
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If the gcd is 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.
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Examples
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The elements of are all positive if and only if , by inspection. Thus, you can use the information returned even when the direct call to Hurwitz fails.
Separate calls to Hurwitz in the cases and give nontrivial gcds between and its paraconjugate. Thus, the stability criteria are satisfied only as above.
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Notice that the last term has coefficient . Thus, you can say unequivocally that is not Hurwitz, for any value of .
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By inspecting , notice that is Hurwitz only if , and , and . This can be simplified to the conditions
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evalc and the Hurwitz function assume that symbolic parameters have real values.
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The coefficients of can be inspected according to rules, but it is a tedious process.
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Examination of the above for real values of is a way to determine whether the polynomial is Hurwitz.
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In the previous example, might be zero. Thus, Hurwitz cannot determine whether all the zeros are in the left half plane.
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References
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Levinson, Norman, and Redheffer, Raymond M. Complex Variables. Holden-Day, 1970.
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