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## 27.3 Products of Polynomials

Function File: conv (a, b)

Convolve two vectors.

`y = conv (a, b)` returns a vector of length equal to `length (a) + length (b) - 1`. If a and b are polynomial coefficient vectors, `conv` returns the coefficients of the product polynomial.

Function File: c = convn (a, b, shape)

N-dimensional convolution of matrices a and b.

The size of the output is determined by the shape argument. This can be any of the following character strings:

"full"

The full convolution result is returned. The size out of the output is `size (a) + size (b)-1`. This is the default behavior.

"same"

The central part of the convolution result is returned. The size out of the output is the same as a.

"valid"

The valid part of the convolution is returned. The size of the result is `max (size (a) - size (b)+1, 0)`.

Function File: 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.

Loadable Function: y = conv2 (a, b, shape)
Loadable Function: y = conv2 (v1, v2, M, shape)

Returns 2D convolution of a and b where the size of c is given by

shape= 'full'

returns full 2-D convolution

shape= 'same'

same size as a. 'central' part of convolution

shape= 'valid'

only parts which do not include zero-padded edges

By default shape is 'full'. When the third argument is a matrix returns the convolution of the matrix M by the vector v1 in the column direction and by vector v2 in the row direction

Function File: q = polygcd (b, a, tol)

Find 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. Tolerance defaults to

 ```sqrt(eps). ```

Note that this is an unstable algorithm, so don't try it on large polynomials.

Example

 ```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 ] ```

See also: poly, polyinteg, polyderiv, polyreduce, roots, conv, deconv, residue, filter, polyval, polyvalm.

Function File: [r, p, k, e] = residue (b, a)

Compute the partial fraction expansion for the quotient of the polynomials, b and a.

 ``` 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) ```
Function File: [b, a] = residue (r, p, k)
Function File: [b, a] = residue (r, p, k, e)

Compute 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 script mpoles.m.

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.m 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 ```

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