Zero of a Polynom and of a function

[Gil]

According to astronomical theory of planets VSOP2013,
the tropical year is given at an instant t by the following expression:

f=.2421904482-
.00006116623*t-
.000000065922*t^2+
2.667909E-7*t^3+
9.47396E-10*t^4+
1.06233E-10*t^5+
3.71993E-10*t^6,
with t= (Julian Day - 2451545)/365250.

I look for a possible minimum value of f, and therefore take the derivative of f and set it equal to 0:

fp=2.231958E-9*t^5+
5.31165E-10*t^4+
3.789584E-9*t^3+
8.003727E-7*t^2-
.000000131844*t-
.00006116623.

Two ways to get a solution on the HP50G:

a) with ROOT
'2.231958E-9*t^5+
5.31165E-10*t^4+
3.789584E-9*t^3+
8.003727E-7*t^2-
.000000131844*t-
.00006116623'

't' 6 ROOT
And get the correct output 6.51025795855 for t.

b) with the Solver for Polynoms (Right Shift Arrow followed by the key 7, and choose option 3)
Let's put the array [an... a0] in the order, as requested:
[ 2.231958E-9 5.31165E-10 3.789584E-9 8.003727E-7 .000000131844 -.00006116623 ]

and ask for the "solution".

Look for real solutions/output:
—> first element of the array is real : 6.45537286594 for t in this case.

Why this huge discrepancy in the real solution?

I thank you in advance for your answer.

Regards,
Gil Campart

[Albert Chan]

(11-22-2021 02:37 PM)Gil Wrote:  Let's put the array [an... a0] in the order, as requested:
[ 2.231958E-9 5.31165E-10 3.789584E-9 8.003727E-7 .000000131844 -.00006116623 ]

Based on your posted f, you missed negative sign for df/dt linear term.

A trick to avoid typo mistake is to wrrite equation with sign "stick" to the number, like this:
At a glance, we know df/dt has 2 negative terms.

f=
+0.2421904482
−0.00006116623*t
−0.000000065922*t^2
+2.667909E-7*t^3
+9.47396E-10*t^4
+1.06233E-10*t^5
+3.71993E-10*t^6,

with t= (Julian Day - 2451545)/365250.

Also, min(t) = 6.51025795855 is likely meaningless.
I seriously doubt the equation can predict this far out , year 2000+6510 = 8510 !

It may be just an artifact to curve fit available data to polynomial.

[Gil]

For prediction, ± 6000 years round year 2000, see attached snapshot of document of IMCCE.

See also
following link

https://www.imcce.fr/news/parution-intro...ronomiques.

Thanks for having spotted the sign error : I did a program... that automated the process of returning the coefficients in form of array — and, clearly, it has a bug. I will try and fix it up for a new version.

Regards and thanks for your reactiveness.

Regards,
Gil