Jean,

thank you for your quaternion file. I have had my interest in these things increased because so far, I cannot find your interesting equations.

So i stared writing out standard matrix complex no. representations of quaternions.





Question, i am getting strange results for the k element of a quaternion set.


I have tried writing out the equations several ways and the k vector with a simple complex nos. does not look right.

Would appreciate a second opinion from the community.


Kind Regards,

John

quaternionsMatrix.sm (288,09 KiB) scaricato 1161 volta(e).

Hello John,
I'm not familiar with with Quaternion Matrix [Wikipedia].
Only 3 D rotation as previously posted.
You can collect all you Maple command lines for easy inspection.
Few examples of [2 x 2] applications attached.
Sunny today ... Jean

SolveEquationDotNumerics.sm (27,23 KiB) scaricato 1140 volta(e).
ODE Space Saddle_0.sm (31,23 KiB) scaricato 1140 volta(e).
Solve GivenFind Cyclic Solve Coplanar Critical READ FIRST.sm (77,38 KiB) scaricato 1274 volta(e).
Maths Matrix map.sm (237,05 KiB) scaricato 1166 volta(e).

Wrote

Hello John,
I'm not familiar with with Quaternion Matrix [Wikipedia].
Only 3 D rotation as previously posted.
You can collect all you Maple command lines for easy inspection.
Few examples of [2 x 2] applications attached.
Sunny today ... Jean

SolveEquationDotNumerics.sm (27,23 KiB) scaricato 1140 volta(e).
ODE Space Saddle_0.sm (31,23 KiB) scaricato 1140 volta(e).
Solve GivenFind Cyclic Solve Coplanar Critical READ FIRST.sm (77,38 KiB) scaricato 1274 volta(e).
Maths Matrix map.sm (237,05 KiB) scaricato 1166 volta(e).

Thank you very much for your time Jean, Quaternions were used in the first set of Maxwell equations, or electricity equations.
Later, vector analysis replaced quaternions because they were much easier to understand. Unfortunately some information of the electricity equations got lost by doing this.
However, there has been a renewed interest in quaternions because they prevent 'gimbal lock' that can happen with Euler equations. Also quaternions are a subset of Geometric Algebra that contains even more information than quaternions and is really taking off all over the world. GA is where i would like to get to through Smath eventually. I found a set of files for Maxima which contain enough of the GA that i am interested in. So thats the direction i am slowly going in.


Thank you for all these interesting files too.
kind regards, John

(The link below shows a connection between Q and GA)
https://geometricalgebra.org/quaternions.html

Wrote

Wrote

Hello John,
I'm not familiar with with Quaternion Matrix [Wikipedia].
Only 3 D rotation as previously posted.
You can collect all you Maple command lines for easy inspection.
Few examples of [2 x 2] applications attached.
Sunny today ... Jean

SolveEquationDotNumerics.sm (27,23 KiB) scaricato 1140 volta(e).
ODE Space Saddle_0.sm (31,23 KiB) scaricato 1140 volta(e).
Solve GivenFind Cyclic Solve Coplanar Critical READ FIRST.sm (77,38 KiB) scaricato 1274 volta(e).
Maths Matrix map.sm (237,05 KiB) scaricato 1166 volta(e).

Thank you very much for your time Jean, Quaternions were used in the first set of Maxwell equations, or electricity equations.
Later, vector analysis replaced quaternions because they were much easier to understand. Unfortunately some information of the electricity equations got lost by doing this.
However, there has been a renewed interest in quaternions because they prevent 'gimbal lock' that can happen with Euler equations. Also quaternions are a subset of Geometric Algebra that contains even more information than quaternions and is really taking off all over the world. GA is where i would like to get to through Smath eventually. I found a set of files for Maxima which contain enough of the GA that i am interested in. So thats the direction i am slowly going in.


Thank you for all these interesting files too.
kind regards, John

(The link below shows a connection between Q and GA)
https://geometricalgebra.org/quaternions.html

quaternions.sm (67,8 KiB) scaricato 1171 volta(e). updated...
very sunny today.

Thanks John for the Geometric Algebra lecture.
As well for the Pseudo Matrix examples.
Under the hood, Cholesky solver is a pseudo matrix.
It is the core component of the iterative Conjugate Gradient
for fitting model function(s) to data sets in Smath.
Be good ... Jean.

Wrote

Thanks John for the Geometric Algebra lecture.
As well for the Pseudo Matrix examples.
Under the hood, Cholesky solver is a pseudo matrix.
It is the core component of the iterative Conjugate Gradient
for fitting model function(s) to data sets in Smath.
Be good ... Jean.

Jean, hopefully, there is good oil under the hood.

Does the Cholesky require a square matrix,
or is there a method to take any m by n and convert it sufficiently for
Cholesky to process it. I found a Cholesky algorithm in Smath, and it requires a square matrix,
as does Maple.

(Complex no.s work brilliantly at one frequency, but tricks must be used to get correct results at harmonics as far as electricity goes. GA handles all of that in its stride).

All the best...

Wrote

Does the Cholesky require a square matrix,
or is there a method to take any m by n and convert it sufficiently for
Cholesky to process it. I found a Cholesky algorithm in Smath, and it requires a square matrix,
as does Maple.

Cholesky takes the X,Y vectors, not a matrix.
It creates its own matrix for pseudo-solve.
The pseudo-solve matrix is square [n x n]
where n is the order of the polynomial.
If you have Y only, create an artificial X.
Observe Cholesky in the Fréchet iterative fit.
Have a good visit of these two attachments.
Best to you and your next ... Jean.

PolyFit Orphan X.sm (18,25 KiB) scaricato 1152 volta(e).
Genfit Frechet DIFFICULT.sm (128,06 KiB) scaricato 1194 volta(e).

Wrote

Wrote

Does the Cholesky require a square matrix,
or is there a method to take any m by n and convert it sufficiently for
Cholesky to process it. I found a Cholesky algorithm in Smath, and it requires a square matrix,
as does Maple.

Cholesky takes the X,Y vectors, not a matrix.
It creates its own matrix for pseudo-solve.
The pseudo-solve matrix is square [n x n]
where n is the order of the polynomial.
If you have Y only, create an artificial X.
Observe Cholesky in the Fréchet iterative fit.
Have a good visit of these two attachments.
Best to you and your next ... Jean.

PolyFit Orphan X.sm (18,25 KiB) scaricato 1152 volta(e).
Genfit Frechet DIFFICULT.sm (128,06 KiB) scaricato 1194 volta(e).

Awesome! so i see PolyFit Orphan X is like left pseudoinverse matrix, but it completes the use of it .

I will have to study Frechet.

Thank you very much for your time Jean.

take care...

Wrote

I will have to study Frechet.

Let me collect more examples that illustrate
fitting techniques ... 2/3 days.
Viva Australia & John ... Jean.

Let me collect more examples that illustrate
fitting techniques ... 2/3 days.


Thank you Jean for your time, very kind.
Also, long live Canada, maple trees, and Jean!

Wrote

Thank you Jean for your time, very kind.
Also, long live Canada, maple trees, and Jean!

In the mean time John, visit these two remarkable Cholesky.
Be good ... Jean.

Genfit [Algo Ancestor Part123 Cholesky].sm (88,48 KiB) scaricato 1055 volta(e).

In the mean time John, visit these two remarkable Cholesky.
Be good ... Jean.


Thank you Jean, i will study these tonight,

Kind Regards as always,

John

Wrote

Thank you Jean, i will study these tonight,

These 4 are interesting wrt handling Cholesky directly
and handling Chelesky via CG [Conjugate Gradient]
Now, drink your beer Irish style: one for John, one for Jean.
Cheers ... Jean.

Genfit [Algo Minimize(Rational Type_J].sm (63,43 KiB) scaricato 1020 volta(e).
Genfit [Algo Ancestor Part123 Cholesky].sm (88,48 KiB) scaricato 1025 volta(e).
Genfit 0000000000 Rational Num_Denom Order.sm (88,18 KiB) scaricato 1178 volta(e).
Genfit 0000000 Rational Auto-init ITER.sm (149,38 KiB) scaricato 1160 volta(e).

Wrote

Wrote

Thank you Jean, i will study these tonight,

These 4 are interesting wrt handling Cholesky directly
and handling Chelesky via CG [Conjugate Gradient]
Now, drink your beer Irish style: one for John, one for Jean.
Cheers ... Jean.

Genfit [Algo Minimize(Rational Type_J].sm (63,43 KiB) scaricato 1020 volta(e).
Genfit [Algo Ancestor Part123 Cholesky].sm (88,48 KiB) scaricato 1025 volta(e).
Genfit 0000000000 Rational Num_Denom Order.sm (88,18 KiB) scaricato 1178 volta(e).
Genfit 0000000 Rational Auto-init ITER.sm (149,38 KiB) scaricato 1160 volta(e).

Thank you Jean for your time .

For this file: Genfit 0000000000 Rational Num_Denom Order.sm , I am getting an error with writing and reading.

I am wondering if you know what i have done wrong, kind regards, John



I have some Bunderberg ginger beer which has a nice bite, so will drink yours first then mine.

what about if you create simpler ? Maybe your long path is not recognized.
CWD:=CurrentDirectory("C:\SmathFile")
Otherwise: disable the 3 spool to file commands.
Bed time soon, Cheers Irish style ... Jean.
French from Brittany converse with Irish in Gaelic as well with Scots.

Wrote

what about if you create simpler ? Maybe your long path is not recognized.
CWD:=CurrentDirectory("C:\SmathFile")
Otherwise: disable the 3 spool to file commands.
Bed time soon, Cheers Irish style ... Jean.
French from Brittany converse with Irish in Gaelic as well with Scots.

Ah yes, the disabling worked :d thank you!


The French language is very pleasant to hear in song...

Hello John,
Here is Levenberg-Marquardt in two styles.
Not preferred vs Cholesky Conjugate Gradient.
Once Cholesky is initialized, being iterative
the final best fit is warranted,faster as well.
Have a good day ... Jean.

Genfit_10 Frechet LMSA ..SS6179.sm (112,32 KiB) scaricato 1175 volta(e).

... a good example of difficult to initialize
ended to exceptional fit. The data set could be noisy in Y's.
In Process Control Instrumentation, we collect lot of data
that we have to best render in some visual way.
Cheers John ... Jean.

GenfitMatrix Copy Master ToolBox.sm (59,37 KiB) scaricato 1145 volta(e).

Wrote

Hello John,
Here is Levenberg-Marquardt in two styles.
Not preferred vs Cholesky Conjugate Gradient.
Once Cholesky is initialized, being iterative
the final best fit is warranted,faster as well.
Have a good day ... Jean.

Genfit_10 Frechet LMSA ..SS6179.sm (112,32 KiB) scaricato 1175 volta(e).

Thank you Jean! Very helpful and thank you for teaching me

Kind Regards as always

John

Wrote

Thank you Jean! Very helpful and thank you for teaching me

You said you have Mathcad 15, there is a most useful piece of code
about rational fit to noisy data. You may still find in the PTC.
I called Genfit Matrix or GenfitMatrix. It is based on Mathsoft
genfit(vx,vy,vg,F). It was tested vs other software at the time
of developing the DAEP [Data Analysis Extension Pack].
If you can't find, I can make images of the code.
My Mathcad 11 is not connected to Internet anymore for last 6 years.
GenfitMatrix auto initializes itself, only Num/Den order.
Rational fit is most useful for data that may not have model function.
Cheers as usual ... Jean