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moveoutroots(theta), REAL vector theta

Keywords: time domain
moveoutroots(theta) where theta is a REAL vector defining the
polynomial P(z) = 1 - sum(theta*z^run(length(theta))), returns a new
vector theta1 defining a new polynomial P1(z) of the same form which
has the same zeros as P(z) except that any zero z0 with |z0| < 1 is
replaced by 1/z0.

Thus all the zeros of P1(z) are outside the unit circle in the complex
plane and moveoutroots(theta) are the coefficients of an invertible MA
operator and, when no zero has modulus 1, the coefficients of a
stationary (causal) AR operator.

moveoutroots assumes the sign convention used by Box and Jenkins and
implicit in movavg(), autoreg() and polyroot() (Arsign = -1, Masign =
-1).  When theta are MA coefficients using the sign convention of
Brockwell and Davis (Masign = +1), you should use moveoutroots(-theta).
See topic 'MASIGN'.

  movoutroots(2) returns 1/2 because 1 - 2*z has zero .5 < 1 and 1 -
  .5*z has zero 2 > 1.

  moveoutroots(vector(2,-1.25)) returns vector(1.6, -.8) because 1 - 2*z
  + 1.25*z^2 has zeros 8 +- .4*i and 1 - 1.6*z + .8*z^2) has roots (1 +-
  .5*i) = 1/(.8 -+ .4*i)

See polyroot(), movavg(), autoreg().

Gary Oehlert 2003-01-15