Polynomial Interpolation for HP-67

[Albert Chan]

(05-11-2022 05:31 AM)Thomas Klemm Wrote:  \(f(x) = 12 + (x+5)(\frac{1}{6} - (x-1)\frac{13}{42})\)

But you can expand it to get:

\(
-\frac{13 x^2}{42} - \frac{15 x}{14} + \frac{302}{21}
\)

We can do synthetic multiplication, to transform "offset" polynomial to "normal" polynomial.
We do it inside out, first (x-1), then (x+5)

Code:

    -13/42      1/6       12
-1            13/42
----------------------------
    -13/42    10/21       12  
+5           -65/42    50/21
----------------------------
    -13/42   -15/14   302/21

Or, we can convert 1 set of polynomial offsets, to another, in 1 shot.
see Funny Factorials and Slick Sums

We don't see much benefits here, because "normal" polynomial have zero offsets.

a(n) x^n + a(n-1) x^(n-1) + ... = (x-0) * (a(n) + (x-0) * (a(n-1) + ...

From offsets (-1,5) → (0,0):

Code:

      -13/42       1/6        12      
-1 0> -13/42     10/21    302/21
+5 0> -13/42    -15/14

(-1+0)*(-13/42) + 1/6   =  10/21
(+5+0)*(+10/21) + 12    = 302/21
(+5+0)*(-13/42) + 10/21 = -15/14