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WOLFRAM|DEMONSTRATIONS PROJECT

Newton's Polynomial Solver

​
|variables|
signs
a
11
1
-1
b
6
1
-1
x
1
1
1
-1
x
2
2
1
-1
x
3
3
1
-1
zoom
auxiliary primary rule
rule
term
x
1
x
2
x
3
auxiliary
x
1
2
3
primary
ax
11
22
33
secondary
2
bx
-6
-24
-54
tertiary
3
x
1
8
27
Σ
6
6
6
This Demonstration shows Newton's method of finding approximate roots of an equation
3
x
+b
2
x
+ax=d
by using three slide rules, called primary, secondary, and tertiary. We can read
x
directly with an auxiliary primary rule. We calculate the value of polynomial
3
x
+b
2
x
+ax
for
x
1
,
x
2
, and
x
3
. If its value for
x
i
is near
d
,
x
i
is approximately a root of the equation.
The essence of the original construction was to use slide rules to perform multiplication, while addition was left to a person. We automated addition as well by recording values of polynomial terms and the value of the polynomial in a grid.
The current construction works for coefficients and arguments with absolute value at least 1. Suppose we want to solve the equation
3
x
-6
2
x
+11x=6
. We enter
a=11,b=-6
and move sliders
x
i
​
to get polynomial value 6 for different positions of
x
. On the auxiliary rule we read
x
1
=1
,
x
2
=2
,
x
3
=3
.
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