The X-Philes (2nd Revision)/The X-Philes Number 1 (1995).iso/xphiles/hp48_1/symmat_c

Internet Message Format | 1995-03-31 | 30KB

From: Carlos Ferraro <[e-mail]>
Subject:  v05i029:  symmat_cf - Manipulate Symbolic Matrices v1.0, Part01/01
Newsgroups: comp.sources.hp48
Followup-To: comp.sys.hp48
Approved: [e-mail]

Checksum: 3490516001 (verify with brik -cv)
Submitted-by: Carlos Ferraro <[e-mail]>
Posting-number: Volume 5, Issue 29
Archive-name: symmat_cf/part01


BEGIN_DOC symmat.doc
                    Gothemburg, 1/6/1992

Hello everybody!

Disappointed by the lack of a good program for dealing with symbolic matrices,
I programmed one myself. I have studied different programming languages
like ML , Modula-2 and Turbo-Pascal, and most of all, I've spent hundreds of
 hours optimizing Forth in the old 28C (because of it's stupidly little
memory).  That's why I begun to program this set of utilities and programs
half a year ago and have been improving it constantly since then. For about
a month ago I had no part of the code left that could be improved. Seems that
every little row of the code is optimized ("upspeeded") to the possible
limit (using RPN Forth). I didn't want to use other languages because of
the always present possibility of failures in compatibility.
I've now achieved :

1) greatest possible speed "in the market"
2) most effektive, elegant and short code
3) completeness (all kind of tools)
4) free of bugs (well, well,...)
5) CEAP = (Complex Exam-Aware Program)

IT'S OBVIOUS THAT YOU MUST HAVE THE FLAGS -2 AND -3 CLEARED SO THAT
THE CALCULATOR IS ABLE TO HANDLE WITH SYMBOLS !!
BEFORE YOU DOWNLOAD IT, READ ALL THE INSTRUCTIONS AT THE BOTTOM OF
THIS TEXT (BELOW EXAMPLES) !!

For all the text down here, the following symbols will represent :

S1 = a symbolic matrix in the form of a list of lists, for ex. :
       { {a b 1} {0 a 2 } {b 1 3} } represents a matrix:     [[a b 1]
                                     [0 a 2]
                                     [b 1 3]]
Q1 = like S1 but ALLWAYS quadratic.

V1 = a symbolic vector in the form of a list (wich I will call "simple" troughout the
    instructions) or a list of one list,
              for ex. : { '2*x' '1/y' z } or { { '2*x' '1/y' z } }
         that is : [ '2*x' '1/y' z ]

sy = a symbol, f. ex. : 't' , 'k' , etc.

n = an integer

AVAILABLE PROGRAMS IN THE DIRECTORY SYMB :

( Sorry, this editor doesn't write greek symbols!! So (lambda) means
the corresponding greek symbol as you'll see it at the HP48 )


Power, (lambda)Eqv , Cross , M->L , L->M , AplyPrg , DetExCo
Inv , AXB , Mmult , Smult , ID , ADD , Transp , Jacobi , |^Stack
DetNN , D33 , D22 , Minor , ChCol , DelN , ->Mat .
You will also get a directory called OWN and a CST. Reasons for this
are given below.

INSTRUCTIONS:
                        In:        Out:

1) Power  - raises S1 to the power n        2:Q1
                        1:n        1:Q1^n


2) (lambda)Eqv  - gives the characteristical equation of any of the
               following inputs :     a real (HP48) matrix (quadratic)
                    a complex (HP48) matrix   "
                    a symb. matrix as described above.
    This program is extremely usefull and you don't even need to convert your
    real or complex matrix into a list (it is done automatically).
    The equation you get is in (lambda) variable. Automatically the program stores     the
    following matrix : (S1-(lambda)*I) in the variable (lambda)Mat, where I is
    the identity matrix, in the directory OWN, there the program ends. Why?
    Because it's ugly to get a program directory full of variables!! Anyway, I
    hope you weren't sleeping during the linear algebra lessons and understand
    what I     mean...
    In:                Out:
    1: Q1 or (see above)        1: the char. equation (in OWN)

3) Cross  -      returns the cross-product of two symb. vectors, as a new symb. vector.
            In:            Out:
            2:V1            1:V1 X V2
            1:V2

4) M->L  -  converts a real or compl. matrix in HP48-shape into one in
        "symbolic" shape (list of lists)

5) L->M  -  the opposite of M->L

6) AplyPrg  - This is a "different" prog., in the means that it takes
        a symb. matrix (or vector) and another program as its
        arguments, and it applies the given prog. to the
        matrix (or vector), element by element. This is especially
        usefull when you want to have all the elements in a matrix
        beautifully collected. or evaluated or whatever.
        In:                Out:
        2:S1                1: (see above)
        1:<< a Program here >>



7) DetExCo  -      This gives the determinant of a quadr. matrix fully collected.

        In:            Out:
        1:Q1            1: (See above)

8) Inv  -     Gives the invers of a matrix using Cramer's rule.
        If you wonder why the output is in this form, the answer is
        speed and visual optimization.
        In:        Out:
        1:Q1        2:The inverted matrix where each element has to be
                  divided by the determinant (I think
                  it's called the adjoint matrix (?) )
                1:the determinant .

9) AXB  -      Solves the system of equations using the I-don't-remeber-what-
        algorithm, but which is faster than taking the invers and
        then multiplying with the vector.
        In:            Out:
        2:V1            2:The solution-vector where each element has
                         to be divided by the determinant.
        1:Q1            1:The determinant.

10) Mmult  -    Matrix multiplication. It's your responsability to check
        the dimensions of the matrices.
        In:            Out:
        2:S1            1:S1*S2
        1:S2

11) Smult  -    Multiplies a matrix by a scalar (which can be a symbol as well).
        In:            Out:
        2:S1            1: symbol*S1
        1: (any symbol here)

12) ID  -    Makes an "identity matrix" with the given symbol as the
        elements of the diagonal.
        2: (any symbol)        1: (see above), size n*n
        1:n

13) ADD  -    Addition of matrices, check dimensions.
        2:S1            1:S1+S2
        1:S2

14) Transp -    Transponat of a matrix or vector, any dimensions!
        1:S1 or V1        1: S1*  or  V1*


15) Jacobi -    This program gives the functional matrix of an inputed
        vector, i.e. the Jacobian matrix. This is extremely usefull
        when you want to obtain the Jacobi-determinant to use it
        for ex. in a two or three dimensional integration where you must
        change coordinates!
        This prog. is designed to work only with max. 3 dim.,
        because it is what we normally use at the exams. That is
        why you MUST have 3 variables of derivation in a list at level
        1 !! If you use less variables input zeros instead!! For ex. { x y 0 } .
        It's obvious that the elements of the vector must be differen-
        tiable in terms of the variables in the list !!!!
        Observe that you can choose any variables, but make shore that
        they are empty  or derivation won't work - sorry not my fault!
        2:V1
        1: { a list of three variables of derivation }

        Out:    1: (the Jacobi-matrix)

All the other programs are functions used in these progs above.
DO NOT REMOVE ANY OF THEM !!!!
Most of them are also usefull independently, for ex. :

16) |^Stack -    splits an S1 into its elements, corresponds to OBJ-> to normal matrices
        but without giving the size of the S1.
        In: S1            Out: (the elements in order in the stack)

17) DetNN -    The determinant of an n*n matrix NOT COLLECTED.
18) D33 and D22 are the basic cases of DetNN.

19) Minor  -    The minor of a matrix. Used in Inv .
        3: n1 (row position)
        2: Q1                    Out:
        1: n2 (column position)            1: (the minor at that position)

20) ChCol -    Substitutes one column of a matrix by another given.
        Note that no check of simensions is done, because the program
        who calls this function will never provide a vector of wrong size.
        3: (a simple vector, NOT a list of one list)
        2: S1
        1: n (the column number to be replaced)

        Out: (the "new" S1)

22) DelN -    Deletes a given element of a list, i.e. deletes a row of an S1
        or a column of a (simple) V1 .
        2: S1 or V1
        1:  n                Out: (the "new" list)

23) ->(lambda)Mat  -  Substitutes an eigenvalue in the (lambda)Mat and
            evaluates it so that you can reduce this matrix
            and get the eigenvectors. It is available from
            the CST in OWN as well !!
            1: n            1: (Q1-n*I)


EXAMPLES:
1)        Calculate the 5:th power of the matrix :    { { a b } { 0 a } }
        In:                Out:
        2: { { a b } { 0 a } }
        1: 5                1: { { 'a^5' '5*a^4*b' } { 0 'a^5' } }

        The operation takes only about 4 seconds, but ExCo takes about 28 s.
        Total time aprx. 32 seconds


2)        Calculate the characteristical equation of the following matrix.
        In:    [ [1 0 1]
              [2 4 3]
              [6 .5 5] ]

        Push (lambda)Eqv and you'll get

        Out: '-4.5+10*(lambda)^2-(lambda)^3-21.5*(lambda)'

        The program itself takes less then 3 seconds.
        (The determinant takes only 0.7 seconds !!!!!! Beat if you can !!)
        Again it's ExCo who takes time (27 seconds).
        Using for ex. the grat polynom-solving program found in News you'll
        get the eigenvalues of the matrix. As you then go back to OWN
        (there is when you are when execution is ready) and use ->(lambda)Mat
        you'll get the matrices there you can do some echelon and direct find
        the eigenvectors. For ex. one eigenvalue here is 3.51064322082
        In:3.51064322082
        Press the CST-menu, press  ->(lambda)Mat
        and you'll get: { { -2.5106...  0  1 }
                  { 2  0.4893...  3  }
                  { 6 0.5  1.4893... } }

        I'm sorry I haven't got  a clearer example, but you have certainly
        lots of them in your linear algebra books.


3)        Taking the cross product of two vectors is really ease:
        2: { a 0 q }
        1: { 1 '2*a' '.5*q' }        1: { '-2*a*q' '.5*a*q+q' '2*a^2' }
        The total execution time is 5.55 seconds, icluding ExCo !!!

4)        Input [ [ 1 2 3 4 ]    Press M->L and you get  { { 1 2 3 4 }
            [ 8 0 9 6 ]                  { 8 0 9 6 }
            [ 3 5 7 1 ] ]                  { 3 5 7 1 } }

5)        Suppose you have a symbolic matrix S1 with "ugly" terms like those
        you get after multiplication. Then you do like this :
        2: S1
        1: 1 ( or 2 or << desired program  >> )

        Press  AplyPrg  and you'll get the matrix now with every term
        fully collected ( if you entered a 1 at level 1 ). That is, AplyPrg
        applies the desired program to every element of the matrix.
        Inputting a 1 or a 2 instead of a program at level 1 are only
        shortcuts to  ExCo  and  FrColct.

6)        FrColct works like this; suppose you have :
        1: 'A/B/C*INV(G/H/I*INV(A/G/B/H/C/I))'
        Pressing   ExCo  will return an expression full of "...^2" and "...^-2"
        but pressin  FrColct instead  will return
        1: 'A*A*H*I/(B*C*G*G*B*H*C*I)'       in 1min. 41 sec.
        That is : A^2/(B^2*C^2*G^2) . You may not find this example very well
        thought, but as you experiment with ExCo and FrColct, you'll
        find out they're a good complement to each other, specially when ExCo returns
        an expression full of '.../INV(...INV(.../.../...))...' and so on.

7)        Suppose you want to integrate the following function: f(x,y)=x^2+y^2
        in an interval defined by: D={ (x,y) : x^2+y^2+2*x }. Then the following
        substitution will be best to use: x= -1+r*COS(t) , y= r*SIN(t) .
        Now you'll need to calculate the functional matrix and then the determinat
        of that change of coordinates to use it in the integration. There's
        when Jacobi comes in the picture. You need only to input a list with
        the substitution expressions, and another one with the variables
        to be used in the derivations, in this case:
        In:    2: { '-1+r*COS(t)' 'r*SIN(t)' }
            1: { r t 0 }

        Why the zero as the third element in the list? As I explained
        before, the program is designed to work with three variables,i.e.
        it needs to have 3 elements in the list at level 1 !! Press Jacobi
        and  2.15 seconds later you'll have the Jacobi-matrix in the stack.
        Now you press DetExCo and get the collected determinant in 3.8 seconds,
        That is     1: 'COS(t)^2*r+SIN(t)^2*r'
        And  cos(a)^2+sin(a)^2=1  so the answer is simply   r  !!


As I'm extremely tired of writing this program documentation, I hope you'll do
without more examples. If there's anything you wonder, here's my e-adress:

            [e-mail]

and here's my adress:    Carlos Ferraro            because I won't be here
            Godhemsgatan 60 A        very much during the summer.
            414 68  GOTHEMBURG
            SWEDEN        Tel: [phone]



INSTRUCTIONS FOR DOWNLOADING:

You must have the following programs,( best if you have them at HOME), to run SYMB.
Notice that you must apply   ASC->  to all of them !!
Once you have done this, the     checksums  and         bytes   will be :

FrColct :             # 14123d          471
ExCo :                # 30286d        61
SYMB :                # 45696d        4303.5
END_DOC

BEGIN_RDME symmat.rdm
[
Make sure you load exco as ExCo.

Make sure you load frcolct as FrColct.
-chris ]
END_RDME

BEGIN_ASC ExCo.asc
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END_ASC


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