Zero of a Polynom and of a function
11-22-2021, 02:37 PM (This post was last modified: 11-22-2021 02:43 PM by Gil.)
Post: #1
 Gil Member Posts: 202 Joined: Oct 2019
Zero of a Polynom and of a function
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 ]

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?

Regards,
Gil Campart
11-22-2021, 03:53 PM
Post: #2
 Albert Chan Senior Member Posts: 1,676 Joined: Jul 2018
RE: Zero of a Polynom and of a function
(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.
11-22-2021, 04:34 PM
Post: #3
 Gil Member Posts: 202 Joined: Oct 2019
RE: Zero of a Polynom and of a function
For prediction, ± 6000 years round year 2000, see attached snapshot of document of IMCCE.

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

Attached File(s)