Re: linear algebra & LAPACK - questions and thoughts

[Dr . Jürgen Sauermann]

Hi Rowan, I would not use RATIONAL_NUMBERS_WANTED=yes in the context of linear algebra even though I don't think it makes a difference. Its probably just slower. I am getting this: 68.64208074 ¯28.28427125  ¯4.478949969E¯15 ¯9.827502707E¯15  2.737380805E¯15  0           ¯3.642080737 ¯5.573380404E¯16 ¯1.508040844E¯15  3.558371050E¯16  0            0           ¯9.190591996E¯16 ¯1.708771971E¯17  3.594337453E¯16  0            0            0.000000000E0    4.132084299E¯16  2.453537944E¯16  0            0            0.000000000E0    0.000000000E0   ¯3.088077991E¯17       ⌹[⎕CT]yy  1  0 0  0 0   68.642080737002189 ¯28.284271247461849  ¯4.4789499693386752E¯15 ¯9.8275027072133962E¯15  2.7373808045404955E¯15   0.014568322948009649 0.11313708498984769                      ¯0.0023885168387308391                 0.066518309540373144                 ¯0.56858532876428267  0 ¯1 0  0 0    0                   3.6420807370024026  5.5733804042156851E¯16  1.5080408441474691E¯15 ¯3.5583710502840566E¯16   0                    0.2745683229480095                        0.16650436787857926                   0.99517588642565933                  ¯9.1326867152187798   0  0 1  0 0    0                   0                  ¯9.1905919960774138E¯16 ¯1.7087719706038831E¯17  3.5943374532205272E¯16   0                    0                         ¯1088069191219460.7                      44995745519819.392                ¯13021972329283451                    0  0 0 ¯1 0    0                   0                   0.0000000000000000E0   ¯4.1320842994244806E¯16 ¯2.4535379437658151E¯16   0                    0                                         0                     ¯2420086153952088.8                   19228054549359863                    0  0 0  0 1    0                   0                   0.0000000000000000E0    0.0000000000000000E0   ¯3.0880779907613713E¯17   0                    0                                         0                                     0                    ¯32382601831680033                   Maybe your GNU APL is a little too old, I did some changes lately. To serialize qq put it in quotes (2 ⎕TF wants a variable name, not a value):       2 ⎕TF 'yy' yy←(5 5⍴6.8642080737002189E1 ¯2.8284271247461849E1 ¯4.4789499693386752E¯15 ¯9.8275027072133962E¯15 2.7373808045404955E¯15 0 ¯3.6420807370024026 ¯5.5733804042156851E¯16 ¯1.5080408441474691E¯15 3.5583710502840566E¯16 0 0 ¯9.1905919960774138E¯16 ¯1.7087719706038831E¯17 3.5943374532205272E¯16 0 0 0 4.1320842994244806E¯16 2.4535379437658151E¯16 0 0 0 0 ¯3.0880779907613713E¯17) Dito for 10 ⎕CR:       10 ⎕CR  'yy'  yy←5 5⍴0 ⍝ prolog ≡1 (,yy)[⍳3]←68.642080737002189 ¯28.284271247461849 ¯4.4789499693386752E¯15 (,yy)[3+⍳3]←¯9.8275027072133962E¯15 2.7373808045404955E¯15 0 (,yy)[6+⍳2]←¯3.6420807370024026 ¯5.5733804042156851E¯16 (,yy)[8+⍳4]←¯1.5080408441474691E¯15 3.5583710502840566E¯16 0 0 (,yy)[12+⍳2]←¯9.1905919960774138E¯16 ¯1.7087719706038831E¯17 (,yy)[14+⍳5]←3.5943374532205272E¯16 0 0 0 4.1320842994244806E¯16 (,yy)[19+⍳6]←2.4535379437658151E¯16 0 0 0 0 ¯3.0880779907613713E¯17 Not sure what;s wrong with your qt (character ?) since I don't have it. The Internal precision of floats is 64 bit, == standard C/C++ double. Best Regards, Jürgen          n 4/19/20 8:12 PM, Rowan Cannaday wrote: Greetings again, This question is more about decimal precision and internal storage of numeric data in APL, but in the context of these linear algebra solutions. So far I've implemented Householder reflection, Hessenberg decomposition, & Wilkinson shift. I am very close (I think!) to having a general solution to the eigenvalue problem in APL, but I am running into some problems. The below tests were done with `RATIONAL_NUMBERS_WANTED=yes` & `⎕CT ← 1E¯13`, however the results were the same without this compilation flag. Let me know if I should recompile without the flag to aid debugging. I am also unable to serialize these values to share, as noted below. While decomposing arrays, I am left with numbers that seem to be outside the system limits for decimals. See below:       yy 6.8642080737002189E1 ¯2.8284271247461845E1 ¯4.4789499693386752E¯15 ¯9.8275027072133962E¯15  2.7373808045404955E¯15 0.0000000000000000E0 ¯3.6420807370024026E0 ¯5.5733804042156851E¯16 ¯1.5080408441474686E¯15  3.5583710502840566E¯16 0.0000000000000000E0  0.0000000000000000E0 ¯9.1905919960774138E¯16 ¯1.7087719706038831E¯17  3.5943374532205272E¯16 0.0000000000000000E0  0.0000000000000000E0  0.0000000000000000E0    4.1320842994244806E¯16  2.4535379437658151E¯16 0.0000000000000000E0  0.0000000000000000E0  0.0000000000000000E0    0.0000000000000000E0   ¯3.0880779907613713E¯17       ⌹[⎕CT] yy DOMAIN ERROR       ⌹[⎕CT]yy       ^     ^       2 ⎕TF yy RANK ERROR       2 ⎕TF yy       ^     ^       10 ⎕CR yy RANK ERROR       10 ⎕CR yy       ^      ^       qt 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000 0.0000000000000000       2×qt DOMAIN ERROR       2×qt       ^ ^ In another calculation, I am left with this (seemingly) bizarre result:       {(⌈/⍳0) > ⍵} qt 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I'd be happy to generate some functions, but I'll have to clean up (& properly attribute) my workspace to do so. I have not found a trivial example to demonstrate what is happening, but I thought I'd reach out in meantime to see if there is a workaround. Thank you, - Rowan On Mon, Apr 13, 2020 at 6:32 PM Rowan Cannaday <cannad...@gmail.com> wrote: Another deficiency I've noticed in my above solution is that it does not converge for complex eigenvalues. - Rowan On Mon, Apr 13, 2020 at 6:07 PM Rowan Cannaday <cannad...@gmail.com> wrote: Here is a quick solution to the eigenvalue problem. Should be refined and extended to calculate eigenvectors. {0~⍨∈⍵×∘.=⍨⍳≢⍵}({r+.×⍉qt⊣(qt r)←⌹[⎕CT]⍵ ;qt;r}⍣{(|⍺)≡|⍵}) 3 3 ⍴ ⍳9     16.11684397 ¯1.11684397 ¯3.625973215E¯16 As a quick note, my ⎕CT was 1E¯13, however it doesn't seem to zero out the ¯3.625973215E¯16. I suspect this is due to it being a negative number. Its not that big of a hurdle, it can be easily worked around with an extra function. Worlframalpha for example rounds it to 0 in their solution. Many thanks, - Rowan On Mon, Apr 13, 2020 at 5:30 PM Peter Teeson <peter.tee...@me.com> wrote: The paper you referred to was a huge epiphany for me.  Having previously worked in the business world using COBOL and FORTRAN  the beauty and elegance of APL blew me away. It still does.  At IPSA we used to model proposed changes (mostly algorithms) to the interpreter in APL first.  And then wrote the Assembly code to implement. On Apr 13, 2020, at 11:48 AM, Dr. Jürgen Sauermann <mail@xn--jrgen-sauermann-zvb.de> wrote: That was my point. If we could establish APL as a language for describing algorithms.  I was thinking along the lines of Iverson's "Notation as a tool of thought" which is also free now: https://dl.acm.org/doi/10.1145/358896.358899