SOLVESYS 3.1
SIMULTANEOUS NON-LINEAR EQUATIONS SOLVER
for the HP48G Series Calculators
Copyright 1996, Sune Bredahl, All Rights Reserved
[ Check out http://www.gbar.dtu.dk/~c947086/hp48.html for a nicely
formatted HTML version of this document ]
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Disclaimers
SOLVESYS.LIB, STKSOLVE.LIB AND THIS MANUAL ARE PRESENTED WITHOUT
WARRANTIES, EXPRESSED OR IMPLIED. THE AUTHOR MAKES NO GUARANTEE
AS TO THE FITNESS OF THIS SOFTWARE. SOLVESYS.LIB AND STKSOLVE.LIB
CAN BE COPIED FREELY PROVIDED THIS MANUAL IS DISTRIBUTED WITH
THEM. SOLVESYS.LIB AND STKSOLVE.LIB MAY NOT BE USED FOR
COMMERCIAL PURPOSES WITHOUT WRITTEN PERMISSION FROM THE AUTHOR.
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Release Notes
A bunch of changes have been done since v3.0a, most of them
related to the functionality of the The START VALUES Dialog Box.
The solver is faster than in previous versions, especially when
numeric derivatives are used. If there are less than six
variables, SOLVESYS will monitor the variable progress in
real-time.
Users of earlier versions should note that the "Linear"
derivatives model is no longer available in v3.1. Instead,
"Numeric" derivatives can be used without any loss of
performance.
I do not support the S/SX version of the STKSOLVE library no more
(don't have the time - don't have any S/SX) - HP48S/SX users must
use the library distributed with SOLVESYS v3.0a.
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What is SOLVESYS?
SOLVESYS is an easy-to-use environment for solving systems of
nonlinear equations. Unlike MES (the built-in Multiple Equations
Solver) which only works with systems that can be solved by
recursive use of a one-dimensional solver, SOLVESYS is a true
multi-dimensional solver. SOLVESYS allows on-the-fly definitions
of constants and variables and can also solve systems with more
equations than variables and vice-versa. Also linear systems are
solved very fast. A programmable (stack-mode) version of the
solver is available via the library command STKSOLVE.
The program is far from perfect, but you'll hardly find any HP48
equation system solver more advanced and user-friendly than this.
SOLVESYS uses unsupported and undocumented features of the HP48,
so use it at your own risk. SOLVESYS.LIB works exclusively on the
HP48G/GX models.
(Earlier releases of this program were called SIMN, the last
version being SIMN 5.2. The next release was called SOLVESYS
1.0.)
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Installation
To install SOLVESYS.LIB:
1. Transfer SOLVESYS.LIB to the HP48G/GX.
2. Recall the library to the stack and press n [STO], where n
is the desired RAM port (0 if you have a HP48G). After
storing the library, you can purge the original copy to save
memory.
3. Turn the HP48 off, then on. This adds SOLVESYS to your
library menu.
The checksum for SOLVESYS.LIB is # 9973h, size: 3974.5 bytes.
To remove SOLVESYS.LIB:
1. Put the library ID (n: 1550) in the stack
2. Press: [DUP][DETACH][PURGE]
If the calc. returns an "Object in use" error at this point, then
press [ON]+[C] and repeat the above steps.
To transfer SOLVESYS.LIB to another HP48G/GX:
1. Put the library ID (n: 1550) in the stack and press [RCL].
2. Store the returned library in any variable
3. Transfer this variable to the other calc. - use binary
transfer.
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Solving Principles
The solver is based on a globally convergent modification of the
multi-dimensional Newton-Raphson method. The original source for
this is Dennis and Schnabel [1] and is also well described
(including C-source) in Press et al. [2]. SOLVESYS extends this
to a least-squares model in cases where the number of equations
is different from the number of variables.
One pitfall is that the method may converge to a non-solution. It
minimizes a 'transform' function having minima where the system
has solutions but may also have minima that are not solutions to
the system, hence the problem. The method is not based on
minimization alone. It's a Newton method adjusted according to a
minimization principle when it diverges. When it does fine, it
behaves like a general Newton method.
[1] Dennis, J.E., and Schnabel, R.B. 1983, "Numerical Methods
for Unconstrained Optimization and Nonlinear Equations"
(Englewood Cliffs, NJ: Prentice-Hall).
[2] Press, W.H. et al. 1988, "Numerical Recipes in C: The Art
of Scientific Computing" (Cambridge: University Press).
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The SOLVESYS Dialog Box
This dialog box appears when starting the program. Most of the
default settings here will usually suffice:
SYSTEM { EQ1 EQ2 ... }:
This is the list of equations to be solved. You can enter
the list directly here or store it in 'EQ' prior to starting
SOLVESYS. Equations don't have to be arranged in any
particular way, eg. 'Y*X+9=X','Y*X+9-X=0' and 'Y*X+9-X' are
all equal expressions. There is no limit as to the number of
equations in this list, also the number of variables can be
different from the number of equations.
TOL:
The error tolerance. No solution is returned until all
equations have 'function' values less than this value. A
low(er) tolerance implies a high(er) accuracy of the
solution. The default value is 0.00001 and this should be
fine for most purposes.
DER:
The solver uses partial derivatives of the equations and can
use either analytic or numeric derivatives. You can change
the setting using [CHOOSE] or the [+/-] key. "Analytic"
derivatives must first be computed by symbolic
differentiation before the solver can start, whereas
"Numeric" derivatives are computed dynamically in each
iteration. This allows the solver to start immediately at
the expense of a slightly slower iteration speed, but the
total time needed is usually less than "Analytic".
Furthermore, "Numeric" handles user-defined functions and
certain HP48 functions without built-in derivatives (such as
XROOT() or SIGN()) and is also ideal for systems of linear
equations. Note however, that "Analytic" is more numerically
stable than "Numeric" and can handle certain systems which
"Numeric" fails to solve.
COMPLEX
Checking this field enables SOLVESYS to search for complex
roots. Since that's generally not what you want, just leave
it unchecked. This setting is not required for input of
complex constants (for a possible application of complex
constants, see Example).
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The START VALUES Dialog Box
After the equations have been entered in The SOLVESYS Dialog Box,
the equations are analyzed for variables and constants and the
result displayed in the START VALUES dialog box. A maximum of 12
variables/constants will be displayed at a time, so more screens
are sometimes necessary.
Three types of 'start values' are accepted:
Guess
This is what any numerical solver needs - an initial value,
guess or seed (whatever you prefer to call it), ie.
something to guide the solver to the root you want. A guess
value is entered as a real or complex number in the
corresponding variable field. Depending on the setting of
COMPLEX, a complex number defines either a complex
guess-value (if checked) or a real-valued range (see below).
SOLVESYS will use '1' as default for all variables but
clearly this should be changed according to any knowledge
about the solution. For systems of linear equations, the
default values are sufficient.
Range
A range can be used instead of a real-valued guess value. A
range is defined by a complex number, for example (-2,2)
forces the solver to search for real-valued roots in the
interval -2 to 2 for the variable in question. You should
only use ranges if you're 100% sure that it actually
encloses a root. Note that if interrupting the solver to
input other guess values, range definitions (if any) will be
lost and must be entered again if needed.
Constant
Constants are defined by placing the value in a
single-element vector, like [0.25], in the corresponding
field. Storing the value in the variable prior to starting
SOLVESYS has the same effect (both the current and the HOME
directory are searched). Constant values are always stored
in HOME. Using complex constants do not require COMPLEX to
be checked.
Pressing the [SOLVE] menukey activates the solver. The total
number of equations, variables and constants defined is displayed
briefly. If there are less than six variables, the solving
progress for each variable displayed in real-time as well as the
current error. For more variables, only the error is displayed.
If the solver is terminated (because a solution is found or due
to a user-interrupt or some other error), the user is returned to
this dialog box again. The current values of the variables can
now be inspected and the system can be resolved with other
guesses if needed (constant definitions are not possible at this
point). Pressing the [INFO] button shows the current error,
elapsed time and the number of iterations used for the session
just finished.
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Example
A well known problem is to find the radius r and center (a,b) of
a circle given three points on the circle. The three points must
then satisfy (x - a)2 + (x - b)2 = r2. The following list of
equations can be used to solve this:
{ 'SQ(RE(P1)-A)+SQ(IM(P1)-B)=SQ(R)'
'SQ(RE(P2)-A)+SQ(IM(P2)-B)=SQ(R)'
'SQ(RE(P3)-A)+SQ(IM(P3)-B)=SQ(R)' }
These equations refer to the three complex constants P1, P2 and
P3 (rather than six real-valued constants). When entering these
values later, you'll see that it's much easier to input
coordinate sets as complex numbers rather than dealing with the y
and x coordinates individually. (Obviously, you could place the
constant values directly in the equations, but then you wouldn't
be able to re-use the equations to solve for other constant
values.)
The three known points in this example are P1=(3,5), P2=(4,1) and
P3=(2,3).
To solve this, store this list of equations in the 'EQ' variable
and start SOLVESYS. Accept the default settings in The SOLVESYS
Dialog Box and press [INIT].
The START VALUES Dialog Box now appears. First, fill in the three
complex constants in single element vectors ie. P1:[(3,5)],
P2:[(4,1)] and P3:[(2,3)]. Now provide some guess values for the
variables to solve for (A, B and R). A quick sketch of the three
points on a paper, reveals that R must be somewhere in the range
1<R<3 and the center (A,B) close to (4,3). The guess values thus
obtained are A:4, B:3 and R:(1,3).
Now press [SOLVE]. The numbers of equations, variables and
constants entered are displayed briefly and after a few
iterations, a solution is found (A=4.1666, B=3.1666 and
R=2.1730). The solution is automatically copied to the stack.
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STKSOLVE - Using Input from the Stack
It is possible to access the solver in a pure stack mode
environment from the library command STKSOLVE. Setting up the
arguments required correctly is very important and should
certainly be controlled by some program. Only simple and
superficial argument checking is done.
STKSOLVE essentially offers the same facilities as SOLVESYS as
far as the solving 'engine' is concerned. Errors, user-interrupts
etc. always terminate a stack-mode session.
STKSOLVE takes the following arguments from the stack:
4: { { eqs } { vars } { startvals } }
3: tol
2: dertype
1: complexflag
Using 0 as dertype selects numeric derivatives, other values
select analytic derivatives. If complexflag is 0, the solver will
only search for real-valued results, other values enable complex
results. Constants and ranges are allowed in the { startvals }
list. The order of the list must match the { vars } list.
STKSOLVE always returns a 'termination' code. The codes (in
hex-mode) are typically #A04 (zero found) #A03 (interrupted),
#A01 (Bad Guesses) etc. As some users will know, these codes
simply refer to the corresponding message string. With this
approach, applications can choose their own strategy according to
the termination code rather than letting STKSOLVE display some
error. Typical actions could be restarting with other guesses (if
#A01 - Bad Guess(es) is returned) or simply displaying the
termination message (via DOERR). More details about error numbers
can be found in HP48 manuals.
As an example, here's a little program designed to solve linear
equation systems. A list of linear equations in level 1 is
assumed:
<< DUP # 60E008h LIBEVAL # 63231h SYSEVAL %1 SWAP # 5E370h
SYSEVAL # 05459h SYSEVAL 3 ->LIST 0.00001 0 1 STKSOLVE >>
(size: 108,5 bytes, checksum #4C8Ch)
Using a hidden library routine in solvesys.lib and some system
calls, a list of variables and a list of guess values (all 1) are
created. This is put together in a list, the error term 0.00001
is used, numeric derivatives are selected and the complex-results
flag is enabled.
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Error Messages
Some (more or less) common error messages returned by SOLVESYS
are listed below.
Bad Guess(es):
a) Iterations converged but didn't provide any satisfying
solution or b) too many iterations without progress. In
either case, you should try again with other guesses or
ranges. This error could also occur if the error tolerance
is too low.
Constant?:
At least one equation did not respond to change(s) in
variable value(s). This error will usually be due to
incorrect constant/variable definitions.
All Variables Known:
All variables were assigned a constant value. Consult the
The START VALUES Dialog Box section for proper
variable/constant definitions.
Undefined Name:
This error will occur if "Analytic" derivatives are
selected, but one or more of the current equations contain a
function without (built-in) derivative, such as XROOT(),
SIGN(), or a user-defined function. "Numeric" derivatives
can be used instead or you can define your own derivatives
(must be placed in HOME) for the function(s) in question.
Undefined Result:
Calculation such as 0/0 occurred - change guesses.
Infinite Result:
Calculation such as 1/0 occurred - change guesses. In
general, avoid rational expressions if possible. Eg.
X/(X-1)=Y+2 should be written as X=(Y+2)*(X-1). The latter
form usually has much better convergence properties.
Inconsistent Units:
Units are not supported. Note that SOLVESYS automatically
disables units for functions such as CONST().
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How To Contact The Author
Feel free to e-mail me if you have any comments, questions,
probs., suggestions, etc. Also if you find any bugs (or what you
think is a bug), please notify me.
Hope you like the program,
Sune Bredahl
c947086@student.dtu.dk
The latest version of SOLVESYS and any relevant information
regarding the program are available from:
www.gbar.dtu.dk/~c947086/hp48.html
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