Improving Calculation Accuracy

#1 Posted: 8/24/2022 11:25:23 AM

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Dear forum users! I turn to you for advice on the SMath program. I am attaching a file in .sm format with suggestions for improving the accuracy of calculations in the SMath program and questions for the presented calculation.

Found an error in the last question. I am attaching the updated file.

Improving Calculation Accuracy.sm (78 KiB) downloaded 36 time(s).

#2 Posted: 8/24/2022 7:15:42 PM

Alvaro Diaz Falconi

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Hi Neeky. Some notes about your questions.

Improving Calculation Accuracy.sm (33 KiB) downloaded 41 time(s).
Improving Calculation Accuracy.pdf (254 KiB) downloaded 58 time(s).

Best regards.
Alvaro.

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Nikolay 8/25/2022 6:57:00 AM, sergio 8/25/2022 5:48:00 AM

#3 Posted: 8/25/2022 7:59:00 AM

Jean Giraud

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... this version for your convenience.
Previous deleted.

Improving Calculation Accuracy RootSecant.sm (28 KiB) downloaded 41 time(s).

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Nikolay 8/25/2022 8:10:00 AM

#4 Posted: 8/26/2022 10:36:02 AM

Nikolay

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Thank you very much for the tip to use the bisection method!

Please tell me how it is possible to implement in the SMath program an algorithm for calculating the roots of an equation that (an equation) contains Bessel functions.Equation with Bessel functions.sm (3 KiB) downloaded 31 time(s).

#5 Posted: 8/26/2022 2:24:54 PM

Jean Giraud

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Please tell me how it is possible to implement in the SMath program an algorithm for calculating the roots of an equation that (an equation) contains Bessel functions

Immediately, answer is no from Smath built-in Bessel... Why ?
Smath built-in Bessel has no derivative for roots solver.
Visit ► Samples ► Solve Grand TREASURY ► sheet 0
You can have Bessel from Computer Approximation ... attached.
Cheers ... Jean.

Bessel J0,Y0.sm (22 KiB) downloaded 27 time(s).

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Nikolay 8/29/2022 4:47:00 AM

#6 Posted: 8/26/2022 10:25:12 PM

Alvaro Diaz Falconi

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Wrote
...
Please tell me how it is possible to implement in the SMath program an algorithm for calculating the roots of an equation that (an equation) contains Bessel functions.

Hi. Use bisection method.

Equation with Bessel functions.sm (24 KiB) downloaded 45 time(s).
Equation with Bessel functions.pdf (77 KiB) downloaded 38 time(s).

Best regards.
Alvaro

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Nikolay 8/29/2022 4:47:00 AM

#7 Posted: 8/27/2022 9:20:06 AM

Jean Giraud

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This document solve Smath built-in mwbesselj(0,x)
1. rootDichotomy for the roots.
2. rootSecant for solving intersection with linear f(x) ... typical.
Jean.

Bessel SOLVE.sm (22 KiB) downloaded 32 time(s).

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Nikolay 8/29/2022 4:47:00 AM

#8 Posted: 8/29/2022 9:47:41 AM

Nikolay

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Thanks for the help.
Now I can determine the roots of such equations.
My calculations that contain these equations are calculated correctly.
The only thing I don't understand is the algorithm presented in Razonar's post..
Could you please tell me how to deal with this algorithm?

#9 Posted: 8/29/2022 3:23:42 PM

Alvaro Diaz Falconi

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Wrote
Thanks for the help.
Now I can determine the roots of such equations.
My calculations that contain these equations are calculated correctly.
The only thing I don't understand is the algorithm presented in Razonar's post..
Could you please tell me how to deal with this algorithm?

Hi. Some comments added.

Equation with Bessel functions Explain.sm (42 KiB) downloaded 34 time(s).
Equation with Bessel functions Explain.pdf (258 KiB) downloaded 36 time(s).

Best regards.
Alvaro.

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Nikolay 8/30/2022 5:07:00 AM, churichuro 8/29/2022 7:50:00 PM

#10 Posted: 8/29/2022 4:47:10 PM

Jean Giraud

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Now I can determine the roots of such equations.

If you provide a sample, that will help mutually.
As offered, you can solve for roots & intersections.

#11 Posted: 8/29/2022 6:07:27 PM

overlord

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Hi. Some comments added.

Best regards.
Alvaro.

Mentioned procedure for finding when solve(), diff(), etc would work is useful.

Thanks

#12 Posted: 8/30/2022 5:14:28 AM

Nikolay

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Hello!

Thank you very much for your clarification and for taking the time to answer my question.

Due to the fact that I am not a mathematician or a programmer, it is still very difficult for me to understand these algorithms.

Best regards.
Nikolai.

#13 Posted: 8/30/2022 11:43:23 AM

Jean Giraud

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Due to the fact that I am not a mathematician or a programmer, it is still very difficult for me to understand these algorithms.
Best regards.
Nikolai.

You have to be the Smth Mathematician in some of these built-in functions.

BesselJ(x,0)... is strctly 0 from x=0 ... ∞
Thus, you can multiply by anything.
As a desirable effect, it constrains
x*BesselJ(x,1)to start @ 0
for profitable inspection suite.
Psi(x)+BesselJ(x,0)renders superbly Psi(x).
What complains here is for smarter than me
Cheers ... Jean.

Neeky.sm (21 KiB) downloaded 24 time(s).

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Nikolay 9/1/2022 10:53:00 AM

#14 Posted: 8/31/2022 7:01:45 PM

Jean Giraud

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Hello Nikolai,
Your Bessel PDF is interesting.
It has a close relationship with Weibull.
As demonstrated, not optimized.
Added solve roots/datum, accurate algo style.
Cheers ... Jean.

Stat Treasury_11 PDF Weibull_Bessel.sm (16 KiB) downloaded 32 time(s).

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Nikolay 9/1/2022 10:53:00 AM

#15 Posted: 9/1/2022 11:23:46 AM

Nikolay

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Hello Nikolai,
Your Bessel PDF is interesting.
It has a close relationship with Weibull.
As demonstrated, not optimized.
Added solve roots/datum, accurate algo style.
Cheers ... Jean.

Stat Treasury_11 PDF Weibull_Bessel.sm (16 KiB) downloaded 32 time(s).

Hello Jean Giraud,

Thank you for your attention to my questions and for tips on how to improve the accuracy of calculations.

The equation whose roots I need to find is correctly written in SMath:
F(x) = x * Bessel(1;x) - 1,01 * Bessel(0;x).

Tell me, please, is it possible to apply the algorithm you proposed to determine the roots of this equation?

Best regards,
Nikolai

#16 Posted: 9/1/2022 1:55:17 PM

Jean Giraud

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... is it possible to apply the algorithm you proposed to determine the roots of this equation?

Immediate answer is NO from brute force.
I try to manage my Waterloo ... !

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Nikolay 9/2/2022 4:56:00 AM

#17 Posted: 9/1/2022 5:36:35 PM

Jean Giraud

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Tell me, please, is it possible to apply the algorithm you proposed to determine the roots of this equation?

Now, answer is YES from individual Dichotomy.
Accuracy is in the range of ~ 15 D.
Enjoy ... Jean.

Nikolai roots.sm (44 KiB) downloaded 30 time(s).

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Nikolay 9/2/2022 4:56:00 AM

#18 Posted: 9/2/2022 8:37:58 AM

Jean Giraud

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The equation whose roots I need to find is correctly written in SMath:
F(x) = x * Bessel(1;x) - 1,01 * Bessel(0;x).
Tell me, please, is it possible to apply the algorithm you proposed to determine the roots of this equation?

... here is your project Nikolai !
As far as I understand your roots demand.
f(x) is not scalar wrt 'x' for the solver/scanner solve(■,■,■,■)
but compatible with roots-Dichotomy command line.
Please, don't hesitate for more ... Jean.

Nikolai roots.sm (58 KiB) downloaded 33 time(s).

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Nikolay 9/2/2022 9:30:00 AM

#19 Posted: 9/2/2022 9:35:52 AM

Nikolay

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Jean and Alvaro thank you very much for your efforts.

It was extremely important to get feedback on this issue.

Now my knowledge of mathematics and programming has been improved.

Best regards,
Nikolai

#20 Posted: 9/2/2022 3:50:21 PM

Jean Giraud

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Jean and Alvaro thank you very much for your efforts.

This augmented version includes min/max Golden ratio.
In case you would want to fit upper/lower envelope,
you will need those support points and more.
Cheers ... Jean.

Nikolai roots.sm (78 KiB) downloaded 21 time(s).

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Improving Calculation Accuracy - Determining the zeros of an equation using "solve" function and using Newton's method - Messages