Improving Calculation Accuracy

#1 Pubblicato: 24/08/2022 11:25:23

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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 (78,21 KiB) scaricato 1280 volta(e).

#2 Pubblicato: 24/08/2022 19:15:42

Alvaro

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Hi Neeky. Some notes about your questions.

Improving Calculation Accuracy.sm (33,28 KiB) scaricato 1264 volta(e).
Improving Calculation Accuracy.pdf (254,89 KiB) scaricato 755 volta(e).

Best regards.
Alvaro.

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Nikolay 25/08/2022 06:57:00, sergio 25/08/2022 05:48:00

#3 Pubblicato: 25/08/2022 07:59:00

Jean Giraud

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

Improving Calculation Accuracy RootSecant.sm (28,47 KiB) scaricato 1290 volta(e).

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Nikolay 25/08/2022 08:10:00

#4 Pubblicato: 26/08/2022 10:36:02

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 (3,47 KiB) scaricato 1225 volta(e).

#5 Pubblicato: 26/08/2022 14:24:54

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 (22,64 KiB) scaricato 1266 volta(e).

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Nikolay 29/08/2022 04:47:00

#6 Pubblicato: 26/08/2022 22:25:12

Alvaro

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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 (24,15 KiB) scaricato 1279 volta(e).
Equation with Bessel functions.pdf (77,54 KiB) scaricato 767 volta(e).

Best regards.
Alvaro

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Nikolay 29/08/2022 04:47:00

#7 Pubblicato: 27/08/2022 09:20:06

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 (22,87 KiB) scaricato 1255 volta(e).

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Nikolay 29/08/2022 04:47:00

#8 Pubblicato: 29/08/2022 09:47:41

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 Pubblicato: 29/08/2022 15:23:42

Alvaro

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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 (42,62 KiB) scaricato 1243 volta(e).
Equation with Bessel functions Explain.pdf (258,62 KiB) scaricato 854 volta(e).

Best regards.
Alvaro.

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Nikolay 30/08/2022 05:07:00, churichuro 29/08/2022 19:50:00

#10 Pubblicato: 29/08/2022 16:47:10

Jean Giraud

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Wrote
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 Pubblicato: 29/08/2022 18:07:27

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 Pubblicato: 30/08/2022 05:14:28

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 Pubblicato: 30/08/2022 11:43:23

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 (21,7 KiB) scaricato 1276 volta(e).

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Nikolay 01/09/2022 10:53:00

#14 Pubblicato: 31/08/2022 19:01:45

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 (16,25 KiB) scaricato 1284 volta(e).

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Nikolay 01/09/2022 10:53:00

#15 Pubblicato: 01/09/2022 11:23:46

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 (16,25 KiB) scaricato 1284 volta(e).

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 Pubblicato: 01/09/2022 13:55:17

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 02/09/2022 04:56:00

#17 Pubblicato: 01/09/2022 17:36:35

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 (44,48 KiB) scaricato 1268 volta(e).

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Nikolay 02/09/2022 04:56:00

#18 Pubblicato: 02/09/2022 08:37:58

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 (58,95 KiB) scaricato 1271 volta(e).

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Nikolay 02/09/2022 09:30:00

#19 Pubblicato: 02/09/2022 09:35:52

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 Pubblicato: 02/09/2022 15:50:21

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 (78,33 KiB) scaricato 1251 volta(e).

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