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**26. Polynomials: Roots from Coefficients (Newton’s Method)**

Because the polynomials` (m;r)&PIR `and` (c=:CFR (m;r))&p. `
are identical, the parameters` m;r ``and`` c `
are said to be different *representations* of the same function.
Each representation has its own useful properties. For example,
addition of polynomials is easy in the coefficient representation
but difficult in the root representation;
the identification of the zeros of the function is
difficult in the coefficient representation but trivial
in the root representation. It is therefore useful to have
functions that transform each representation to the
other.` CFR `serves for one direction;
the inverse problem is approached by methods
of successive approximation.

For a differentiable function` f` ,` `the difference` (f r)-(f a) `
for nearby points` r `and` a `is approximately equal
to the difference` r-a `multiplied by the slope of the
tangent to the graph of` f `at the point` a,f a `,` `
that is, the derivative of` f `at` a` .` `
Conversely, the difference` r-a `is approximated
by` ((f r)-(f a))%f D a`,` `and` r `is
approximated by` a+((f r)-(f a))%f D a` .

If` f `is the polynomial` c&p. `
and` r `is one of its roots, then` f r `is zero,
and if` a `is an approximation to` r` ,` `
the expression for` r `reduces to` a-(f a)%f D a` .` `
This may provide a better approximation to` r` ,` `
and is embodied in *Newton’s method*, defined as an adverb,
and illustrated as follows:

newton=: 1 : '] - x % x D'
f=: (c=: 12 _10 2)&p.
f a=: 2.4
_0.48
f newton a
1.2
f 2
0
f newton ^:0 1 2 3 4 _ a
2.4 1.2 1.75385 1.9594 1.99848 2
]a=: (^ - 4:) newton ^: 0 1 2 3 _ a=: 1
1 1.47152 1.38982 1.3863 1.38629
^ {: a
4

For the particular case of polynomials, we may define an
adverb that applies to coefficients and uses the polynomial
derivative` pD `instead of the general derivative` D` :

pD=: 1 }. ] * i.@#
NEWTON=: 1 : '] - x&p. % (pD x)&p.'
c NEWTON ^:0 1 2 3 4 _ a=: 2.4
2.4 1.2 1.75385 1.9594 1.99848 2

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