octave: Products of Polynomials
28.3 Products of Polynomials
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-- : conv (A, B)
-- : conv (A, B, SHAPE)
Convolve two vectors A and B.
The output convolution is a vector with length equal to ‘length (A)
+ length (B) - 1’. When A and B are the coefficient vectors of two
polynomials, the convolution represents the coefficient vector of
the product polynomial.
The optional SHAPE argument may be
SHAPE = "full"
Return the full convolution. (default)
SHAPE = "same"
Return the central part of the convolution with the same size
as A.
DONTPRINTYET See also: 
deconv XREFdeconv, conv2 XREFconv2, *noteDONTPRINTYET See also: deconv XREFdeconv, conv2 XREFconv2,
convn XREFconvn, fftconv XREFfftconv.
-- : C = convn (A, B)
-- : C = convn (A, B, SHAPE)
Return the n-D convolution of A and B.
The size of the result is determined by the optional SHAPE argument
which takes the following values
SHAPE = "full"
Return the full convolution. (default)
SHAPE = "same"
Return central part of the convolution with the same size as
A. The central part of the convolution begins at the indices
‘floor ([size(B)/2] + 1)’.
SHAPE = "valid"
Return only the parts which do not include zero-padded edges.
The size of the result is ‘max (size (A) - size (B) + 1, 0)’.
See also: conv2 XREFconv2, conv XREFconv.
-- : deconv (Y, A)
Deconvolve two vectors.
‘[b, r] = deconv (y, a)’ solves for B and R such that ‘y = conv (a,
b) + r’.
If Y and A are polynomial coefficient vectors, B will contain the
coefficients of the polynomial quotient and R will be a remainder
polynomial of lowest order.
See also: conv XREFconv, residue XREFresidue.
-- : conv2 (A, B)
-- : conv2 (V1, V2, M)
-- : conv2 (..., SHAPE)
Return the 2-D convolution of A and B.
The size of the result is determined by the optional SHAPE argument
which takes the following values
SHAPE = "full"
Return the full convolution. (default)
SHAPE = "same"
Return the central part of the convolution with the same size
as A. The central part of the convolution begins at the
indices ‘floor ([size(B)/2] + 1)’.
SHAPE = "valid"
Return only the parts which do not include zero-padded edges.
The size of the result is ‘max (size (A) - size (B) + 1, 0)’.
When the third argument is a matrix, return the convolution of the
matrix M by the vector V1 in the column direction and by the vector
V2 in the row direction.
See also: conv XREFconv, convn XREFconvn.
-- : Q = polygcd (B, A)
-- : Q = polygcd (B, A, TOL)
Find the greatest common divisor of two polynomials.
This is equivalent to the polynomial found by multiplying together
all the common roots. Together with deconv, you can reduce a ratio
of two polynomials.
The tolerance TOL defaults to ‘sqrt (eps)’.
*Caution:* This is a numerically unstable algorithm and should not
be used on large polynomials.
Example code:
polygcd (poly (1:8), poly (3:12)) - poly (3:8)
⇒ [ 0, 0, 0, 0, 0, 0, 0 ]
deconv (poly (1:8), polygcd (poly (1:8), poly (3:12))) - poly (1:2)
⇒ [ 0, 0, 0 ]
DONTPRINTYET See also: poly XREFpoly, roots XREFroots, *noteconv:
DONTPRINTYET See also: poly XREFpoly, roots XREFroots, conv
XREFconv, deconv XREFdeconv, residue XREFresidue.
-- : [R, P, K, E] = residue (B, A)
-- : [B, A] = residue (R, P, K)
-- : [B, A] = residue (R, P, K, E)
The first calling form computes the partial fraction expansion for
the quotient of the polynomials, B and A.
The quotient is defined as
B(s) M r(m) N
---- = SUM ------------- + SUM k(i)*s^(N-i)
A(s) m=1 (s-p(m))^e(m) i=1
where M is the number of poles (the length of the R, P, and E), the
K vector is a polynomial of order N-1 representing the direct
contribution, and the E vector specifies the multiplicity of the
m-th residue’s pole.
For example,
b = [1, 1, 1];
a = [1, -5, 8, -4];
[r, p, k, e] = residue (b, a)
⇒ r = [-2; 7; 3]
⇒ p = [2; 2; 1]
⇒ k = [](0x0)
⇒ e = [1; 2; 1]
which represents the following partial fraction expansion
s^2 + s + 1 -2 7 3
------------------- = ----- + ------- + -----
s^3 - 5s^2 + 8s - 4 (s-2) (s-2)^2 (s-1)
The second calling form performs the inverse operation and computes
the reconstituted quotient of polynomials, B(s)/A(s), from the
partial fraction expansion; represented by the residues, poles, and
a direct polynomial specified by R, P and K, and the pole
multiplicity E.
If the multiplicity, E, is not explicitly specified the
multiplicity is determined by the function ‘mpoles’.
For example:
r = [-2; 7; 3];
p = [2; 2; 1];
k = [1, 0];
[b, a] = residue (r, p, k)
⇒ b = [1, -5, 9, -3, 1]
⇒ a = [1, -5, 8, -4]
where mpoles is used to determine e = [1; 2; 1]
Alternatively the multiplicity may be defined explicitly, for
example,
r = [7; 3; -2];
p = [2; 1; 2];
k = [1, 0];
e = [2; 1; 1];
[b, a] = residue (r, p, k, e)
⇒ b = [1, -5, 9, -3, 1]
⇒ a = [1, -5, 8, -4]
which represents the following partial fraction expansion
-2 7 3 s^4 - 5s^3 + 9s^2 - 3s + 1
----- + ------- + ----- + s = --------------------------
(s-2) (s-2)^2 (s-1) s^3 - 5s^2 + 8s - 4
DONTPRINTYET See also: mpoles XREFmpoles, poly XREFpoly, *noteDONTPRINTYET See also: mpoles XREFmpoles, poly XREFpoly,
roots XREFroots, conv XREFconv, deconv XREFdeconv.
Info Catalog
octave: Finding Roots
octave: Polynomial Manipulations
octave: Derivatives / Integrals / Transforms