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residue | See Also |
Convert between partial fraction expansion and polynomial coefficients
[r,p,k] = residue(b,a) [b,a] = residue(r,p,k)The
residue
function converts a quotient of polynomials to pole-residue representation, and back again.
[r,p,k] = residue(b,a)
finds the residues, poles, and direct term of a partial fraction expansion of the ratio of two polynomials, b(s) and a(s), of the form:[b,a] = residue(r,p,k)
converts the partial fraction expansion back to the polynomials with coefficients in b
and a
.
If there are no multiple roots, then:n
is
n = length(a)-1 = length(r) = length(p)The direct term coefficient vector is empty if
length(b)
<
length(a)
; otherwise
length(k) = length(b)-length(a)+1If
p(j)
=
...
=
p(j+m-1)
is a pole of multiplicity m
, then the expansion includes terms of the formb,a |
Vectors that specify the coefficients of the polynomials in descending powers of s |
r |
Column vector of residues |
p |
Column vector of poles |
k |
Row vector of direct terms |
residue
function is an M-file. It first obtains the poles with roots
. Next, if the fraction is nonproper, the direct term k
is found using deconv
, which performs polynomial long division. Finally, the residues are determined by evaluating the polynomial with individual roots removed. For repeated roots, the M-file resi2
computes the residues at the repeated root locations.
Numerically, the partial fraction expansion of a ratio of polynomials represents an ill-posed problem. If the denominator polynomial, a(s), is near a polynomial with multiple roots, then small changes in the data, including roundoff errors, can make arbitrarily large changes in the resulting poles and residues. Problem formulations making use of state-space or zero-pole representations are preferable.
deconv
Deconvolution and polynomial division
poly
Polynomial with specified roots
roots
Polynomial roots