Does anyone know how to find all of the eigenvalues for a 3x3 matrix on a 42s? I know you can find ONE using the solver, but I'd like to be able to find all three, like on a 48g. Can this be done with "standard" matrix operations like inverse, determinant, etc.?
Eigenvalues on 42S?


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08252003, 02:11 PM
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08252003, 02:49 PM
I can't think of a direct method. Since you are only interested in a 3x3 (not nxn) you could go back to first principles. Take an arbitrary 3x3 matrix, A =[a_{ij}], and expand the expression det(AxI) as a cubic polynomial in x. The roots of this cubic are, of course, the eigenvalues. The coefficients of the cubic should be readily expressible in terms of determinants of various submatrices of A (you could probably even find this written out explicitly in a beginner's book on eigenvectors). Write a program which computes the cubic's coefficients using the submatrix and determinant capabilities of the 42s, then feed the resulting cubic into the solver. ▼
08252003, 03:12 PM
Exactly my problem. I already knew I could get one from the solver: I want to know how to get all three. I guess I'll have to manually factor out the one result from the solver and then put the remaining result into the quadratic equation. Any ideas on how to write a program to do this that will simply output three values?
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08252003, 05:03 PM
Hi Steven! If you can wait to next day, I will help you. Too late here, and the PC's keyboard is clicking, and my girlfriend not like that in the night. I haven't got 42S, I'll write a little program for 32SII, what can solve ANY REAL roots of an equation. (I'm hoping... :) ) A part of it is ready to run, the solver part I'll make tomorrow morning...!
Good night!
08252003, 07:52 PM
The cubic equation p(x) = det(AxI) = c_{3}x^{3} + c_{2}x^{2} + c_{1}x + c_{0}has the following coefficients: c_{0} = det(A)
Now, if a cubic, p(x), has 3 real roots, then they occur between successive extrema of the cubic. If the extrema are located at x=x_{1} and x=x_{2}, with x_{1} < x_{2}, then the roots of the cubic are in the three ranges: (inf, x_{1})
You can compute x_{1} and x_{2} as the roots of a quadratic: p'(x) = 3c_{3}x^{2} + 2c_{2}x + c_{1}Once you have these roots, you can feed the ranges above into the solver as guesses. Of course, you'd have to do something intelligent in place of the infinities.
08262003, 05:26 AM
Hi Steven, this is a solution for your problem. This program can solve polinomequations ANY of order (limited by mem). It's not too long, only 92 byte ;). It work fast with builtin solver, and use Hornermethod: Only one problem with it, it's solve JUST REAL ROOTS... But in the real problems, like mechanical stress calculations, the roots are reals. This is the HP32SII version:
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08262003, 08:01 AM
Tizedes, Thanks. Very helpful and useful. I'm still amazed by the power of the pioneer series. I have used these calculators since 1988 starting with a 32s my freshman year in high school. I got a 42s as a senior in high school, and I still use that calculator today. I have a spare that I have tested, and never used which sits in a hard case without batteries. I have used this calculator all through undergraduate school and graduate school while everyone else seemed to migrate to TI's, I have yet to find something that I cannot get my 42s to do that I need it to. And yes, this eigenvalue problem was to solve three dimensional stress states. Steven
08262003, 10:57 AM
If your matrix is symmetric (which is usually the case This algorithm is very straightforward, always converges without requiring any input from the user other than the NxN matrix itself, and converges very quickly, producing accurate results (great stability as well). Further, the transformation is made *inplace*, thus requiring a minimum amount of memory. I wrote such a program for the barebones HP41C, which should run unchanged, as is, in any HP42S. It was published in the PPC Journal, circa 1980 or so, a little search will find it. It is only 100+ program steps, so you can key it in effortlessly. Else (or also), have a look at this excellent paper (PDF format): Eigenvalues of a symmetric matrix: Jacobi's method which includes a full, detailed description of the method, as well as a program listing. Best regards from V. ▼
08262003, 12:06 PM
the method is also coded for the 15c in the advanced functions handbook page 148. however, i do like csaba's program as a good programmatic use of the solve feature. something ive been doing recently is the same problem the other way around. ive been finding it much better to use an eigenvalue approach to polynomial roots rather than the reverse. you also get the complex roots a lot easier this way too. ▼
08262003, 10:45 PM
"for the 15c in the advanced functions handbook page 148" Can you send it here or to me directly? VPN
08262003, 10:47 PM
"I wrote such a program for the barebones HP41C" Can you send it here or to me directly? VPN ▼
08272003, 04:29 AM
Thanks for your interest re my "Eigenvalues for an NxN symmetrical matrix: The Jacobi Method" article and program for the barebones HP41C . I initially thought it was published in "PPC Journal", V7 or V8, but a cursory search in the PDF pages available online failed to locate it, which means it was published in "PPC Technical Notes" instead, which alas, has no online presence as far as I can tell. If you've got the first 10 issues or so of "PPC Technical Notes" (circa 1980 or 1981), you'll find it there for sure. I remember it was heavily optimized, took just 100+ steps or so (thus fitting in a single card, maybe even in just one side of a card), and could handle very large matrices as it made use of the fact that the matrix was symmetrical, so it only needed to store N*(N+1)/2 elements (i.e: only 120 elements for a 15x15 matrix) instead of N*N (i.e: 225 elements for said 15x15 matrix). The N eigenvalues were found all at once, simultaneously (not one by one), and very quickly. Best regards from V.
Edited: 27 Aug 2003, 4:39 a.m. 
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