| Roots | p. 1 1 0 | Polynomial |
p. c « (m;r) p.p.c « cIf e is a vector whose elements are all non-negative integers, then p.<c,.e gives the coefficients of the equivalent polynomial: (p. <c,.e)&p. « (<c,.e)&p. |
There are three cases -- coefficients; multiplier and roots;
multinomial (boxed matrix of coefficients and exponents):
c p. x « +/c*x^i.#c
(m;r) p. x « m * */x-r
(<r)&p. « (1;r)&p.
(<c,.e)p.<y « c+/ .*e*/ .(^~)y
where m is a scalar; c and r are
scalars or vectors; and e is a vector or matrix such
that ($e)-:(#c),(#y) . A scalar y is extended normally.
|
p. 1 0 0 1
+-+-----------------------------+
|1|_1 0.5j0.866025 0.5j_0.866025|
+-+-----------------------------+
]mr=: p. c=: 0 16 _12 2 Multiplier/Roots from Coefficients
+-+-----+
|2|4 2 0|
+-+-----+
x=: 0 1 2 3 4 5
(c p. x), ((<c,.i.4)p. x), (mr p. x),: 2*(x-4)*(x-2)*(x-0)
0 6 0 _6 0 30
0 6 0 _6 0 30
0 6 0 _6 0 30
0 6 0 _6 0 30
c=: 1 3 3 1
c p. x
1 8 27 64 125 216
(x+1)^3
1 8 27 64 125 216
bc=: !~/~i.5 Binomial coefficients
bc;(bc p./ x);((i.5) ^~/ x+1)
+---------+--------------------+--------------------+
|1 0 0 0 0|1 1 1 1 1 1|1 1 1 1 1 1|
|1 1 0 0 0|1 2 3 4 5 6|1 2 3 4 5 6|
|1 2 1 0 0|1 4 9 16 25 36|1 4 9 16 25 36|
|1 3 3 1 0|1 8 27 64 125 216|1 8 27 64 125 216|
|1 4 6 4 1|1 16 81 256 625 1296|1 16 81 256 625 1296|
+---------+--------------------+--------------------+
c&p. d. 1 x First derivative of polynomial
3 12 27 48 75 108
(<1 _1 ,. 5 0) p. 3 Coefficients / Exponents
242
_1 0 0 0 0 1 p. 3
242
p. <1 _1 ,. 5 0 Coefficients / Exponents to Coefficients
_1 0 0 0 0 1
c=: _1 1 2 3 [ e=: 4 2$2 1 1 1 1 2 0 2
c,.e Coefficients / Exponents
_1 2 1
1 1 1
2 1 2
3 0 2
(<c,.e) p. <y=:2.5 _1 Multinomial
11.75
c +/ .* e */ .(^~) y
11.75
Note that (<c,.e)p.<y is a "proper" multinomial
only if the elements of e are all non-negative integers.
In general the powers are not so limited, as in the weighted sum
of square root and 4-th root:] t=: <2 3,.1r2 1r4 +-----+ |2 1r2| |3 1r4| +-----+ (t p. 16), +/ 2 3 * 16 ^ 1r2 1r4 14 14The variant p.!.s is a stope polynomial; it differs from p. in that its definition is based upon the stope ^!.s instead of on ^ (power).