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SandBox[Polynomial Sequences: Matrix]

Edit detail for SandBox[Polynomial Sequences: Matrix] revision 3 of 11

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Editor: rrogers Time: 2018/08/15 22:11:00 GMT+0
Note:

added:
--
--

fricas --

-- Functions to facilitate Transforming Polynomial Generating Functions into -- Coefficient arrays. -- These are coefficient arrays (A) and can be rendered into a polynomial list -- by A*[1,x,x^2,...]^T -- EGF Validation can always done for f(x,t) -- By doing a Taylor series expansion indexed by t -- OGF is similiar except the factorial n! in t^n/n! -- has to be applied to the polynomial. -- -- --)Clear all FPFI ==> Fraction(Polynomial(Fraction(Integer)))

Type: Void

fricas

--
--
-- Lower strict Jordan
-- When used on the left of variables x it divides by x
-- with 1/x =0
-- When used on the right of the coefficient matrix it shifts
-- the columns to the left by one.
-- Really a horizontal shift of the coefficient array and 
-- Should be applied A*SJ_lower*[x...]^T
--
SJ_lower(dim) ==
        J : Matrix(Polynomial(Fraction(Integer))) := new(dim,dim,0)
        for i in 1..(dim-1) repeat J(i+1,i):=1
        J

Type: Void

fricas

--
--
-- Upper strict Jordan
-- When used on the left of variables x it multiplies by x
-- With last entry 0
-- When used on the right of coefficient matrix it shifts
-- the columns to the right by one.
--
SJ_upper(dim) ==
        J : Matrix(FPFI) := new(dim,dim,0)
        for i in 1..(dim-1) repeat J(i,i+1):=1
        J

Type: Void

fricas

--
--
SJ_upper(4)

fricas

Compiling function SJ_upper with type PositiveInteger -> Matrix(
      Fraction(Polynomial(Fraction(Integer))))

$\label{eq1}\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \ 0 & 0 & 0 & 0$

(1)

Type: Matrix(Fraction(Polynomial(Fraction(Integer))))