akcs.wiser1@hpcvbbs.UUCP (steven lee wiser) (03/24/91)
Simultaneous Non-Linear Equation Solver: Nonlinear and linear
This is a program description that solves simultaneous non-linear
equations, similar to the programs like TK-Solver Plus ( sold by
Universal Technical Systems, 1220 Rock St. Rockford. Il. 1-800-435-
7887). Of course this is not as powerful as TK, but I think you will be
impressed by what the HP-48Sx can do. I have solved up to 10 non-linear
equations will this routine. Patience is necessary with your HP-48sx
above 10 equations or if the equations are of complicated form.
This routine uses the solution developed in the book "Design of
Thermal Systems", third edition, by W. R. Stoecker , published by McGraw
Hill. The Newton-Raphson Method with multiple equations and unknowns
is the mathematical method used.
Approximately 1k of memory is used by the basic routines.
Additional memory is used for the equations and guess terms.
Additionally, linear equations can be easily solved with this
system. Usually only one iteration is necessary for the linear system.
With this system, the equations can be written in any form or order as
is convenient.
SIMUL.EQ Name of the directory on my HP-28S or 48SX
The following are words in the SIMUL.EQ directory:
CONT control program for solving the equations
u test for convergence number
NEW.G sequence that defines the new guesses of values
DIF.F program sequence that takes the partial derivatives
EVA.F program sequence to evaluate the primary functions at each
iteration guess value.
F list of equation names ex. { f1 f2 f3 ...}
L list of variable names ex. { x y z ....}
CONT << 0 'ITER' STO F
SIZE 'A' STO {A 1 }
1 CON { A 1 } 0 CON
TMP1' STO 'TMP2'
STO
DO NEW.G 'ITER' 1
STO+
UNTIL TMP1 TMP2 -
ARRY-> DROP A 1 - 1
START + -1
STEP ABS u <
END 440 .5 BEEP
>>
u .1 This value should vary based upon the magnitude of
the guess values. Ex. variable guess values of 1000 should indicate
that a relative large value for u should be used; for small variable
guess values a small value of u should be used. If a large guess and a
small guess are both in the problem experience should dictate which is
the most important value to govern the control value.
NEW.G << 1 A
FOR J 'L(J)' EVAL
EVAL
NEXT { A 1 } ->ARRY
'TMP1' STO 1 A
HHHH$$$$JR(J)' EVAL
EVAL
NEXT { A 1 } ->ARRY
DIF.F - ARRY-> DROP A 1
FOR K 'L(K)' EVAL
STO -1
STEP 1 A
FOR J 'L(J)' EVAL
EVAL
NEXT { A 1 } ->ARRY
'TMP2' STO
>>
DIF.F << EVA.F 1 A
FOR J 1 A
FOR K 'F(J)'
EVAL 'L(K)' EVAL d (see d explained below)
NEXT
NEXT { A A } ->ARRY
/
>>
WHERE ' d ' is the differential symbol in HP-28s
and now the HP-48SX.
EVA.F << 1 A
FOR J 'F(J)' EVAL
EVAL EVAL
NEXT { A 1 } ->ARRY
>>
Note: for very complicated equations it may be necessary to
experiment with additional numbers of "EVAL" where you see 2 or 3 of
them presently.
Example of a equation: f1
z = x^2 + y*SIN(x) with variables x , y , z
rewrite f1 to be : x^2 + y*sin(x) - z
This routine sets up the equations like this:
---------------MATRIX----------------------- / EVAL-----
f1 : d(f1)/d(x) d(f1)/d(y) d(f1)/d(z) .... /
f1(x,y,z,..)
f2 : d(f2)/d(x) d(f2)/d(y) d(f2)/d(z) .... /
f2(x,y,z...)
f3 : d(f3)/d(x) d(f3)/d(y) d(f3)/d(z) .... /
f3(x,y,z...)
| | | | | |
| | | | | |
f1 f2 f3 are initially evaluated at the initial guess values
x , y and z.... are solved based on matrix algebra
These new values of x, y, and z are then used to evaluate the
equations f1, f2, f3 again to achieve a new set of x, y, and z guess
values. This continues until the sum of the differences from one f1 to
another f1 thru f3... is less than the test value given in the term "u."
To review to final answers look at either one of the "temp" matrices
oreview the actual variables within the list.
Whenever the test value is less than the sum of the differences the
routine stops and beeps. This routine can be changed in various ways.
One is to beep at each iteration. Another is to send to the screen the
values of the variables to indicate how the solution is progressing.
Have a good time with it. Steve Wiser
Richmond Va
I assume no responsiblity for any use of this program whatsoever. Slw
Uploadable program
follows:
%%HP: T(3)A(D)F(.);
DIR
F { F1 F2 F3 }
L { X Y Z }
\Gm LLLL CONTROL
\<TIME 0 'ITER'
STO F SIZE 'A' STO
{ A 1 } 1 CON { A 1
} 0 CON 'TMP1' STO
'TMP2' STO
DO XXX 'ITER'
1 STO+
UNTIL TMP1
TMP2 - OBJ\-> DROP A
1 - 1
START + -1
STEP ABS \Gm
\<=
END 440 .5
BEEP TIME SWAP -
10000 *
\>>
XXX
\<< 1 A
FOR j 'L(j)'
EVAL EVAL
NEXT { A 1 }
\->ARRY 'TMP1' STO 1
A
FOR j 'L(j)'
EVAL EVAL
NEXT { A 1 }
\->ARRY ZZZ - OBJ\->
DROP A 1
FOR k 'L(k)'
EVAL STO -1
STEP 1 A
FOR j 'L(j)'
EVAL EVAL
NEXT { A 1 }
\->ARRY 'TMP2' STO
\>>
ZZZ
\<< YYY 1 A
FOR j 1 A
FOR k 'F(j)
' EVAL 'L(k)' EVAL
\.d
NEXT
NEXT { A A }
\->ARRY /
\>>
YYY
\<< 1 A
FOR j 'F(j)'
EVAL EVAL EVAL
NEXT { A 1 }
\->ARRY
\>>
END
An example of a set of equations if as follows:
let F1 be .... DP-5*w^2-500 from DP = 5*w + 500
let F2 be .... 1000-30*w1-4.75*w1^2-DP from DP = 1000-30 * w1 - 4.75 *
w1^2
let F3 be .... 900-45*w2-30*w2^2-DP from DP = 900 - 45 * w2 - 30 *
w2^2
let f4 be ..... w-w1-w2 from w = w1 + wlways
return a solution to just any set of equations.
Note: This was originally written when I had my HP-28c believe it or
not. And with it I was able to solve up to six or seven simple
equations.
Note that the downloadable program has a change that lets you see how
much time it takes to solve an equation set. Other changes can easily be
made. One that has recently come to mind is not using the stack as I do
now and using simply an matrix instead.
slw finally figure out how to upload. Use GeoWorks Ensemble...