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Started by Salvatore A. Colomba
Wed, 18 Nov 1992 03:56
v07i036: smhl.sac - Symbolic Matrix Handling Library v1.1, Part01/02
Author: Salvatore A. Colomba
Date: Wed, 18 Nov 1992 03:56
Date: Wed, 18 Nov 1992 03:56
367 lines
19359 bytes
19359 bytes
Checksum: 2231066675 (verify with brik -cv)
Submitted-by: Salvatore A. Colomba <flynn@sabrina.dei.unipd.it>
Posting-number: Volume 7, Issue 36
Archive-name: smhl.sac/part01
BEGIN_DOC smhl1_1.doc
Symbolic Matrix Handling Library
Version 1.1 16 Nov 1992
BETA TEST RELEASE
by
Salvatore A. Colomba
0 - Warnings
This is a BETA TEST RELEASE. This means that I've not tried it in
real operative situations; I've just tested the routines. If some bug
is still present, any report will be very appreciated and immediately
followed by a fixed release.
1 - Introduction
This is version 1.1 of SMHL, an update to version 1.0. I've fixed
some bugs, speeded up most of the routines (with a look to symmat_cf;
thanks to Carlos Ferraro for a couple of good ideas), added some;
moreover, most of the routines aren't anymore hidden, in order to make
it easier to develop more optimized SMHL-based programs.
SMHL is a collection of routines to handle symbolic matrices almost
exactly as they were numerical ones. Almost every standard command that
usually operates on either algebraics or matrices is extended to deal
with the new matrix type, and a TRUE linear system solver is provided.
SMHL has been developed in User RPL, so it's not very fast, but it
still should be useful for many purposes...
SMHL is a Cardware: if you like it, I would be very happy if you send
me a picture card of your country... Thanks!!! Non-commercial
distribution of SMHL with this document (unchanged) is encouraged. My
address is:
Salvatore A. Colomba
C.so del Popolo 14
35100 Padova Italy
Comments, suggestions, bug reports, NeXT computers welcome... My e-
mail address is the following:
Flynn@sabrina.dei.unipd.it
(Flynn@147.162.2.106)
NOTE: I take no responsibility for any damage caused by any possible
bug in the SMHL library. Although in SMHL1.0 people all around the
world has discovered 'just' two bugs, version 1.1 is mostly re-written,
so I can't be sure that it's totally bug-free... And, as someone once
said: "There is always one more bug".
2 - Installation
Well I'm sure that all you out there know the way to install a
library, however here are the required steps:
1) UUdecode or ASC-> SMHL and download it in your HP48 (or
vice-versa).
2) Recall it and store in your favorite port; if you don't
have enough memory, disable LAST_STACK and LAST_ARG,
recall it on the stack, PURGE the original variable and
then press <port number> STO; by doing this way, you
avoid to get more than one copy of SMHL in memory at
the same time.
3) Turn off and then on your calculator.
4) Now SMHL is installed; enter in the SMHL directory
under the LIBRARY menu and press BlueShift-NXT (PREV);
choose KEYS to remap the new functions in your user
keyboard (DKEY gets rid of them), and enter in USER
mode.
3 - Conventions
Symbolic matrices are delimited by the {} characters, using the same
convention as the numerical matrices, i.e.:
[ 1 2 3 ] <--> { a b c } - Row vectors
[[ 1 2 3 ] {{ a b c }
[ 4 5 6 ] <--> { d E f } - Matrices
[ 7 8 9 ]] { g h I }
Note that symbols 'e' and 'i' represent two constants, and without
apostrophes they are 'built-in commands', and HP48 DOESN'T care to
convert them in computation, so keep in mind:
- You can't use them like any other symbol.
- If you meant to use them for what they
are, you MUST enclose them in
apostrophes to allow HP48 to handle them
correctly.
Symbolic and numerical matrices or arrays are fully interchangeable:
where a matrix can stay, there a list can do. So, you haven't to
perform boring conversions before, e.g., summing two matrices, one
numerical and the other symbolic; result, in general, is symbolic if
one of the arguments was symbolic. However, you cannot form matrices
like:
{[ 1 2 3 ]
{ 4 5 6 }
{ 7 8 9 }}
The first row is not always accepted, so don't use it; no checks are
performed, so it's due to you to avoid such situations.
4 - Notes
From now on, when I'll say 'matrix', I'll mean both numerical and
symbolic matrix or vector.
I'm sorry, but my routines don't have almost any error managing and
there isn't a true arguments checking; I think it would have slow down
calculations too much. Maybe in a system-rpl version (or just a
compiled one)... Anyway, the linear system solver will tell you if a
set of linear equation has no solution and Det will refuse to work on a
non-square matrix. Note that if arguments aren't correct, you won't
always incur into an error: sometimes you'll simply get wrong or bad-
formed results.
Flag -28 is used to keep track of an internal sort of 'sort by type'
routine; if you ABSOLUTELY need that flag for any other program, unpack
the SMHL library, STOre in 'SWAPPED' the flag number you wish to use
and pack again the library, or e-mail me the flag number you would
prefer and I'll send you a modified version.
As far as regards the keys, a user keymap is provided to extend the
arithmetic commands to the new matrix type, so keep the keyboard in
USER mode; note that when you are in algebraic mode, you don't need to
exit USER mode; everything that can be put in an algebraic expression
will be as the usual.
5 - Basic commands
Name [Key] What's more
---------------------------------------------------------------------
Eval [EVAL] E-valuates each matrix element.
->q [->Q] Applies ->Q to all the elements; if the input
matrix is a numerical one, it's obviously turned
into a symbolic one.
->Num [->NUM] Transforms a symbolic matrix (or vector) into a
numerical one.
Sq [x^2] If argument is an array, it returns the square of
its modulus, that is, it performs a Dot with the
array itself.
\^| [y^x] Returns the n-th power of a matrix. This is a
highly optimized routine. In the range of
exponents from 1 to 100, the maximum number of
multiplications is 10, corresponding to the
exponent value 87... Nice, isn't it? The
exponent, must be a _relative_ integer other than
zero.
PInv [INV] |\
Inv INV |->PInv is a Partial Inverse: the inverse is given in
terms of a matrix in level 2, and a coefficient in
level one; to get the inverse matrix, you'll
simply have to divide the matrix by that
coefficient. Useful you you have to operate
further on the inverse. Inv, on the other hand,
gives directly the inverse matrix. Note that if
the input matrix was a numerical one, the outputs
of PInv and Inv will be the same, i.e. you won't
get the determinant in level one.
Div [/] Performs linear system solving; see below for a
more detailed description.
Mul [*] If arguments are two vectors, it computes a
symbolic Dot.
Neg [+/-] |
Sub [-] |
Add [+] |
Con CON |
Trn TRN |
Rdm RDM |----> Simply extended to handle symbolic matrices
Det DET |
Abs ABS |
Arry-> ARRY-> |
->Arry ->ARRY |
Size SIZE |
VIEW ... Shows the matrix in level 1 using the small font;
it'll leave you in scroll mode, so that you can
view matrices larger than display. I've written a
version of this routine that shows the elements in
the Equation Writer format; nice, but it eats a
lot of memory... If you are one of those lucky
people with a memory expansion, feel free to e-
mail me to obtain a modified SMHL.
EXCT ... EXpands and CollecTs each single element.
MAT-> ... Given a matrix in level 1, it will split it in
rows and leave the number of rows in level 1.
->MAT ... It's quite the opposite than MAT->; given, say, n
arrays and the number n in level 1, it returns the
formed matrix. Note that:
1: Arrays MUST all be of the same kind, you
can't mix symbolic and numerical ones.
2: They MUST have the same number of
elements.
If you don't follow these simple rules, you'll
surely incur in some error, or get a bad formed
matrix.
Type ... Divides its input in five types:
0: Numbers, complex
1: Symbolic, expressions
2: Numerical arrays (eventually complex)
3: Symbolic arrays
4: Numerical matrices (eventually complex)
5: Symbolic matrices
Apply ... Applies program in level 1 to the SYMBOLIC matrix
or vector in level 2.
M->L ... Turn a numerical matrix into a symbolic one.
Note: most of the above commands can be put in algebraic form
(p.e.: 'Inv(X)'); this to give you maximum compatibility with
the standard functions.
6 - Symbolic linear system solving
System solving needs the same arguments than the built-in system
solver; but it works with any possible matrix; when the solution is not
unique, it will give the GENERAL solution, in terms of independent
variables xk (p.e.: x4,x5,x6,...). Let's see some example:
_________________________
| {HOME} |
|-----------------------| The rank of this matrix is 2; it
|2: [ 0 0 0 ]| means that the solution will be
|1: [[ 1 2 3 ] | a 1-dimension subspace. In more
| [ 4 5 6 ] | mathematical terms, the solution
| [ 7 8 9 ] | will be Ker(A), where A is the
|### ### ### ### ### ###| matrix in level 1.
`-----------------------'
Pressing [/], after about 20 seconds, in level 1 you'll get:
{ { x3 } { '2*-x3' } { x3 } }
Trying to multiply the original matrix for this solution, you'll
always obtain a three-elements null vector. The same principle can be
applied to solve liner systems with vector unknown quantities; for
example:
_________________________
| {HOME} |
|-----------------------|
|2: [[ 1 2 3 ] [ 4 5 6..|
|1: [[ 1 2 3 ] |
| [ 4 5 6 ] |
| [ 7 8 9 ] |
|### ### ### ### ### ###|
`-----------------------'
Pressing [/], after about 20 seconds (SMHL1.0 took 30!), you'll get
the solution of the following linear system:
x1+2*x2+3*x3 = ( 1,2,3 )
4*x1+5*x2+6*x3 = ( 4,5,6 )
7*x1+8*x2+9*x3 = ( 7,8,9 )
that is:
x1= { 'x31+1' 'x32' 'x33-1' }
x2= { '2*-x31' '2*x32+1' '2*-x33+2 }
x3= { 'x31' 'x32' 'x33' }
Call VIEW to see it better; note that the identity matrix is only one
of the oo^3 possible matrices and corresponds to x31=0, x32=0, x33=1.
Nice, eh? Symbolic matrices are _obviously_ supported, but note that in
this case the solutions found won't _always_ be REALLY general: my
routines assume that the coefficient matrix rank doesn't depend upon
symbolic elements values; some other times, you'll get a 'Incompatible
system' error, although system has some solution if some condition is
verified...
Let's see some examples:
/ a 2 3 | 0 \
( 4 5 6 | 0 )
\ 7 8 9 | 0 /
When you try to solve this system with my package, you'll get the
null solution, although the number of solution varies as a changes; if
a!=1 the solution is actually the null one, but if a=1 the rank of the
coefficient matrix becomes 2, and you'll have oo^1 solutions. The
problem resides in the PIVOT routine itself: it's not so smart to avoid
taking symbolics as special elements; I'm sorry, but I'll hardly
improve PIVOT in this direction, because it would have to solve a sort
of 'assignment problem', and I think it's not a practicable way in User
RPL. Another example:
/ 1 2 3 | a \ PIVOT / 1 0 -1 | 0 \
( 4 5 6 | b ) ---------> ( 0 1 2 | 0 )
\ 7 8 9 | c / \ 0 0 0 | c-7*a-2*(b-4*a) /
As you can see, this system has no solution for ALL the possible
values of a,b,c, and solving it by using Div will lead to an error,
although the system with:
c-7*a-2*(b-4*a)=0
has the solutions:
{ x1 '-2*x1' x3 }
7 - Program development oriented routines
The following routines constitute under certain points of view, the
very heart of SMHL; they are very stupid routines, in general they
DON'T make ANY check on their arguments, but intensive mail exchanges
with some people has convinced me to make them visible.
det Computes a raw determinant on SYMBOLIC matrices only.
Dot Performs a dot between two vectors.
MINOR Given a matrix in level 2 and a list containig row and
column, it returns the minor of the matrix.
->qu It's equivalent to ->Q, but it's extended to unit objects.
V\:-K Divides a vector by a constant; result is ALWAYS a list.
M\:-K Same as above for matrices.
SYS Solves a linear system; same syntax as the built-in solver.
K\*V Multiplies a vector by a constant; result is ALWAYS a list.
K\*M Same as above for matrices.
M\*M Multiplies two SYMBOLIC matrices.
V\+-V Sums two vectors
M\+-M Sums two SYMBOLIC matrices.
L->M Turns a symbolic matrix into a numerical one; note that if
argument is a symbolic vector, it turns it in a numerical
column vector.
PIVOT Pivotizes, as much as possible, a matrix.
STICK Given a known terms matrix or vector in level 2 and
coefficent matrix in level one, returns the complete matrix,
that is, it puts them side to side.
SPLIT Is the inverse operation. Note that if a positive integer n
is given, a matrix of n columns will be extracted.
TSRT Sorts by Type two arguments and returns a two digit number
whose first digit is the first argument's Type and second
digit is the second one.
SWAP? Turns TSRTed objects in the right order.
SWAP\? Turns TSRTed objects in the inverse order.
MULX Equivalent to the HP's MULTI
XMAKE Given two positive integers m and n, creates a mxn matrix in
the form:
/ x11 x12 ... x1n \
| x21 x22 ... x2n |
| ... ... ... ... |
\ xm1 xm2 ... xmn /
Else, if m is 1, it will return a vectror in the form:
( x1 x2 .. xn )
NULL Returns a null vector whose size is given by the integer in
level 1; if integer in level 2 is other than zero, a symbolic
vector will be returned.
VER Call it and follow the instructions, or a lighting will melt
down your HP!!! !8^) !8^) !8^)
KEYS Maps the new functions on the keyboard.
DKEY Restores the standard key definitions.
Well, I think that's all folks. Hope that your teacher allows you to
bring your HP48 at the exam...
END_DOC
v07i037: smhl.sac - Symbolic Matrix Handling Library v1.1, Part02/02
Author: Salvatore A. Colomba
Date: Wed, 18 Nov 1992 03:58
Date: Wed, 18 Nov 1992 03:58
629 lines
40601 bytes
40601 bytes
Checksum: 608322349 (verify with brik -cv)
Submitted-by: Salvatore A. Colomba <flynn@sabrina.dei.unipd.it>
Posting-number: Volume 7, Issue 37
Archive-name: smhl.sac/part02
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END_UU
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