I would like to express my thanks to:
- Richard Petti and Jeffrey P. Golden of Macsyma, Inc. (USA)
for their interest in this work,
for supplying Macsyma licenses and the modification of
the LINSOLVE function,
- The Center for Microelectronics of the University of Kaiserslautern,
in particular
Dr. Peter Conradi and Uwe Wassenmüller, for the support
of the project,
- Clemens, Frank and Michael for the first-class WG life and
particularly Michael for the
temporary hiring of its computer,
- My parents and grandparents for their constant support during
my study,
- and quite particularly my friend Dr. Ralf Sommer for outstanding
co-operation and the
common enterprises of the last three years.
Eckhard Hennig
Braunschweig, August 1994
Summary:
Technical sizing functions require frequently the solution of sets
of equations, which describe an object mathematically, after the values
of the function defining element parameters. For the analytic solution
of smaller sizing problems the application of commercial computer algebra
systems offers itself such as Macsyma [ MAC 94 ], which is able, to manipulate
and resolve after their variables symbolically extensive equations algebraically.
Despite their high efficiency these systems are however usually already
overtaxed, if linear or weakly nonlinear, parameterized sets of equations
are to solve after only one subset of their variables or be before-processed
at least symbolically. In order to be able to treat such typically with
design functions developing set of equations, in the context of this work,
a universal symbolic equation solver based on heuristic algorithms was
developed and implemented in Macsyma. Program module SOLVER extends the
functionality of the Macsyma instruction SOLVE and LINSOLVE for the symbolic
solution of algebraic equations or linear sets of equations by the ability
for the selective solution of nonlinear, parameterized systems with degrees
of freedom.
The first section of the available work describes some ranges of application
of symbolic sizing methods, the request following from them to a symbolic
equation solver as well as the used heuristic algorithms for the extraction
of linear equations and for the complexity evaluation of algebraic functions.
The second section contains an overview of the structure the Solvers and
a guidance to its use. In the appendix is the source text of the module
implemented in the internal higher programming language of Macsyma SOLVER.mac.
Table of contents
1 heuristic algorithms~1
1.1 introduction~1
1.1.1 numeric procedures in the comparison with symbolic methods~1
1.1.2 examples of areas of application of symbolic sizing methods~2
1.2 conventional equation solver~7
1.3 requirements of a symbolic equation solver~8.
1.4 extraction and solution of linear equations~10
1.4.1 intuitive methodologies for the search of linear equations~10
1.4.2 a heuristic algorithm for the search of linear equations~11
1.4.3 solution of the linear equations~13.
1.5 evaluation strategies for the solution of nonlinear equations~14
1.5.1 substitution systems for nonlinear sets of equations~14
1.5.2 heuristic procedures for the complexity evaluation of algebraic printouts~16
1.5.3 list of the solution sequence~19
2 the Solver~21
2.1 the structure of the Solvers~21
2.2 the modules of the Solvers~23
2.2.1 the Solver Preprocessor~23
2.2.2 the Immediate Assignment Solver~25
2.2.3 the linear Solver~26
2.2.4 the Valuation Solver~28
2.2.5 the Solver Postprocessor~31
2.3 application of the Solvers~32
2.3.1 command syntax~32
2.3.2 special features of the syntax of equations~33
2.3.3 example calls of the Solvers~33
2.4 the options of the Solvers~37
2.5 user specific transformation routines~40
2.6 modification of the operator evaluations~44
2.7 user specific evaluation strategies~45
i
ii~TABLE OF CONTENTS
bibliography~47
A example calculations~49
A.1 sizing of the staff two-impact~49
A.2 sizing of the transistor amplifier~57
B program listing~65
B.1 SOLVER.mac~65
Chapter 1
heuristic algorithms for the symbolic solution specifications given
by sets of equations
1,1 introduction
1,1,1 numeric procedures in the comparison with symbolic methods the
accurate sizing of technical designs for requires frequently the solution
of nonlinear sets of equations after function defining element parameters.
Usually for the determination of the looked up variables numeric optimizing
procedures are used, which do not function reliably however already for
small design problems with a comparatively small number of unknown quantities
always. Problems prepare the complexity order of many usual optimizing
algorithms (a problem, known as the curse OF dimensionality [MIC 94] is),
exponentially dependent on the variable number, in first line, the selection
of suitable initial values, which unwanted finding locally instead of global
optima and the principle-conditioned behavior with the solution of sets
of equations with degrees of freedom, with which a clarity of the solution
vector missing in the reality can be pretended.
The optimum procedure for the accomplishment of a design function would
basically be a complete analytic solution of the appropriate set of equations.
Analytic functions, which describe explicitly the looked up item values
as functions of the specification parameters, would have to be determined
only once and would be available then to the arbitrarily frequent analysis
with modified parameters, e.g. within design data bases of CAD systems.
Besides analytic sizing formulas clarify qualitatively functional connections
between the item values and the specifications and reveal in the cases
concerned the existence of degrees of freedom. Since there are however
no generally accepted procedures for the solution of any nonlinear sets
of equations and in the special cases, in which symbolic solutions can
be calculated, which complexity of the results exceeds by hand calculation
to mastering frameworks far, the analytic handling of also only small sizing
problems plays so far a subordinated role. With the help of modern, efficient
computer algebra systems, which are able to manipulate and resolve after
any variables symbolic sets of equations algebraically,
1
2 CHAPTERS 1. HEURISTIC ALGORITHMS
with the help of modern, efficient computer algebra systems, which
are able to manipulate and resolve after any variables symbolic sets of
equations algebraically, become some the problem categories, in particular
such, specified above, which require the solution of majority linear or
multivariate polynomialen systems, nevertheless the analytic handling accessible.
Examples of appropriate applications are within many fields of the engineering
sciences, among other things this concerns the design of similar electronic
circuits, regulation-technical problem definitions, the technical mechanics
and the robotics [ PFA 94 ].
1,1,2 examples of areas of application of symbolic sizing methods on
the basis the following two examples are to be demonstrated some areas
of application of symbolic methods. At the same time exemplary the concrete
request are to be worked out at them, to which with the development of
a concept for a universal solution algorithm for symbolic sets of equations
must be considered.
Example 1,1 with the first example concerns it a simple task out of
the technical mechanics. The two staff truss represented in figure 1,1
is given [ BRO 88, P. 112 ], at which under the angle opposite the horizontals
the strength F in the point C attacks. The staffs include the angles with
the horizontals and fi, the height of the triangle stretched by the staffs
are defined with c. Cross section aechen A1 and a2 of the two staffs are
squares with the side lengths h1 and h2. Modulus of elasticity of the used
material is E. Figure 1.1: Loaded two staff truss
Due to the load by the strength F deforms the truss in such a way that
itself the point of the triangle in relation to the unloaded status around
the vector (u; w) T shifts, sees figure 1.2. On the assumption that the
staff length variations are small due to the load in relation to the output
lengths, it is to be determined now the cross section dimensions h1 and
h2 of the staffs in such a way that itself for given F, fi, c and E exactly
a prescribed shift (u; w) T adjusts, i.e. a1 a2 is looked up = f (F; fi;
c; E; u; w): (1.1)
1.1. INTRODUCTION 3
figure 1.2: Flexibly deformed two staff truss
solution: From the static balance conditions at the point C follows
after figure 1.3
figure 1.3: Force equilibrium at the point C
the material equations for the staff length variations read 1.
Fxi = 0 =) follows F cos l1 = S1l1 EA1 after figure 1,3 X; (1.4)
Fxi = 0 =) follows F cos l2 = S2l2 EA2 after figure 1,3 X; (1.5)
whereby for the staff lengths l1 and l2 as well as for the cross-section
areas A1 and a2 applies to l1 = c cos (1.6) l2 = c cos fi (1.7) A1 = h
2 1; (1.8) A2 = h 2 2: (1.9)
4 CHAPTERS 1. HEURISTIC ALGORITHMS
figure 1.4:
Because of the small geometry modifications the circular arcs, on which
the staff ends can move, were replaced to shift plan in the shift plan
drawn in figure 1,4 by for original staff direction the senkrechten tangents.
To the side lengths dashed of the drawn in parallelogram applies
From it the shifts u and v result to the function exist now therein
to resolve the symbolic set of equations from the equations (1.2) to (1.13)
after the unknown quantities h1 and h2 explicitly whereby all unnecessary
variables, i.e. S1, S2, A1, l1, l2, a and b, are to be eliminated.
Example 1,2
The second example originates from electro-technology and concerns
the sizing of similar electronic circuits. For the two-stage transistor
amplifier drawn in figure 1,5 [ NÜH 89 ] symbolic sizing formulas
are to be intended, those the values of the seven resistances g 1: R7 as
function of the operating voltage VCC, which Zi and the output resistance
Zo of the circuit for numerically determined bias points of the transistors
describe small signal reinforcement A, the input impedance:
0 B @ G 1. R7 1 C A = f (VCC;a; Zi; Zo): (1.14)
1.1. INTRODUCTION 5
Figure 1.5: Two-stage transistor amplifier
Figure 1.6: Bias point alternate circuit diagram of the amplifier with
determined transistor bias points
For this purpose with the help of symbolic network analysis procedures
[ SOM 93a] are calculated as well as symbolic approximation methods [ HEN
93 ] first (highly simplified) the transfer functions A, Zi and Zo in the
pass band of the amplifier as functions of the item parameters and the
bias point values.
A = 145303681853R2/145309663773R1 (1.15)
Zi = R7 (1.16)
Zo = 1675719398828125 R2 R7 + 394048139880824192 R1 R2/136552890630303121408
R1 (1.17)
The values of the resistances R1..R7 do not only determine the small
signal characteristics, but influence besides the bias point adjustment
of the circuit. Therefore are resistances
6 CHAPTERS 1. HEURISTIC ALGORITHMS
to dimension in such a way that the small signal and bias point specifications
are fulfilled at the same time. The equations (1.15) - (1.17) for this
reason the extensive Sparse tablet set of equations is added (1.18) - (1.51),
which comes out from the bias point alternate circuit diagram of the amplifier
represented in figure 1,6.
For the determination of the looked up sizing regulations in the form
(1.14) 1.51 from the set of equations (1.15) - (1.51) all branch voltages
are and flow V?? and I?? to eliminate
1.2. CONVENTIONAL EQUATION SOLVER 7
for the determination of the looked up sizing regulations in the form
(1.14) is from the set of equations (1.15) - (1.51) all branch voltages
and flows V?? and I?? to eliminate and the remaining equations after the
resistances g 1: To resolve R7. 2,1,2 boundaries of the application of
conventional equation solver the attempt is undertaken to insert for the
solution of the sets of equations from the two examples the standard routines
of well-known commercial computer algebra systems as Macsyma [ MAC 94 ]
or Mathematica [ WOL 91 ] then in the case the use of Macsyma in the example
1,1, results of the following type result usually, like here:
(COM1) Solve(
[ F*cos(gamma) - S1*cos(alpha) - S2*cos(beta) = 0,
F*sin(gamma) - S1*sin(alpha) + S2*sin(beta) = 0,
Delta_l1 = l1*S1/(E*A1),
Delta_l2 = l2*S2/(E*A2),
l1 = c/cos(alpha),
l2 = c/cos(beta),
a = Delta_l2/sin(alpha+beta),
b = Delta_l1/sin(alpha+beta),
u = a*sin(alpha) + b*sin(beta),
w = a*cos(alpha) + b*cos(beta),
A1 = h1^2, a2 = h2^2
],
[ h1, h2 ]);
(D1) [ ]
This behavior of the computer algebra programs is to be explained with
the fact that the sets of equations which can be solved are regarded concerning
the interesting variables h1 and h2 as over-certainly, because the equation
solving no additional information about (after possibility) which can be
eliminated variables (S1, S2, A1, a2, l1, l2, a, b) cannot be transferred
or variables, i.e. the parameters of the system (F, over under any circumstances
which can be eliminated, + fi, c, E, u, w). A possible way out consists
of arranging the equation solver intending also solutions for the not interesting
variables. This is however not always feasible and usually very inefficient,
because it in some cases much computing time necessarily for the regulation
of variables, which have no uss on the looked up unknown quantities. Even
at all no solution is found, if there is no analytic solution for a only
one not interesting variable. The latter applies for example to the following
set of equations, if only the solutions for x and y are looked up: x +
y = 1 (1.52)
8 CHAPTERS 1. HEURISTIC ALGORITHMS
from the two linear equations (1.52) and (1.53) leave themselves the
solutions untouched x = 2 and y = -1 to determine directly, not in the
contradiction with the remaining nonlinear equation (1.54). Macsyma detects
this circumstance however not and gives only error messages back | in the
first attempt (COM3) because of the apparent over certainty of the system,
in the second attempt (COM4) due to the analytically not solvable third
equation:
(COM2) Eq: [ x + y = 1, 2*x - y = 5, y*z + sin(z) = 1]$
(COM3) Solve(Eq, [ x, y ]);
Inconsistent equations: (3)
(COM4) Solve(Eq, [ x, y, z ]);
ALGSYS CAN NOT solve - system too complicated.
1.3 request of a universal symbolic equation solver
For the symbolic solution of the equation systems it is necessary that
the Solve function in none of the two above cases aborts prematurely. In
the first case the solutions should be output on x and y after a consistency
check with equation (1.54). In latter case is to require that beside the
analytically calculated solutions additionally the remaining equations,
for which no such solutions could be found in implicit form are returned,
so that these can be solve if necessary with numeric procedures. An adequate
response to the command COM4 would be in this sense e.g. an output of the
form
[ x = 2; y = ]
the functions for the solution of sets of equations, provided by Macsyma,
are not modifiable without access to the system nucleus in the way that
they show the behavior required above. The target of the available work
is therefore conceiving and implementation of a universal symbolic equation
solver, which is, which is based on the Macsyma standard routines, able,
to resolve or not make at least by elimination as much as possible necessary
variables a large symbolic preprocessing of the equations sets of equations
of the type stated in the examples after any subset of all variables, so
that numeric optimizing procedures do not have to be only applied to a
small, analytically any longer solvable nonlinear core of the system. Apart
from this general objective detailed request can be derived to the program
which can be developed from some well-known facts and a series from observations,
which are to be made 1,1 and 1,2 as well as the set of equations (1.52)
by the examples { (1.54):
1. Usually only the solution for some few variables is in demand, all
other unknown quantities is to be eliminated.
2. The sets of equations which can be solved can be simple or several
times parameterized.
3. It is not by any means guaranteed that the parameters are from each
other independent, i.e. it is possible that a set of equations has a solution
only if certain arithmetic obligation conditions between some parameters
are kept.
1.3. REQUEST OF SYMBOLIC EQUATION SOLVER 9
4. the sets of equations contained often easy, direct allocations of
the form xi = const., see equations (1.6) and (1.7) or (1.42) - (1.51).
5. A substantial proportion of the equations which can be solved is
linear concerning a not directly evident subset of all variables, sees
regarding all V?? and I?? linear equations (1.18) - (1.32).
6. The systems can contain degrees of freedom.
7. No generally accepted solution procedures for nonlinear equations
and sets of equations exist.
8. Nonlinear equations can be unsolvable (contradictory), unique solutions
have or multiple solutions with finite or infinite solution varieties possess.
9. Always all items of the solution quantity with the remaining equations
are not consistent in the case of multiple solutions.
10. For many nonlinear equations no analytic solutions exist, see equation
(1.54).
From these statements the following result, the appropriate points assigned
demands:
1. The program is to solve sets of equations only so far, as it is
absolutely necessary for the determination of individual variables. Calculated
solutions are to be checked for possible contradictions with the remaining
equations.
2. Looked up variables and parameter must be able to be processed separately
from each other indicated and. Parameters may not be under any circumstances
eliminated contrary to not interesting variables.
3. If dependencies between parameters are detected, then the user the
program run under consideration and storage of the appropriate obligation
conditions must to be continued when desired be able.
4. Direct allocations should be looked up and executed directly at
the beginning of the program run, in order to reduce the range of the remaining
set of equations as far as possible at small expenditure.
5. There there for linear equations efficient, closed solution procedures
is to solve is appropriate it to search the set of equations repeated for
linear section blocks these and to use the results into the remainder of
the equations, until no equation linear equations are more available.
6. Degrees of freedom are to be expressed automatically in variables
selected by the program.
7. The solution of nonlinear equations must be controlled with the
help of heuristic evaluation strategies.
8. In the case of multiple solutions with finite varieties is each
individual solution path separately recursively to be pursued.
10 CHAPTERS 1. HEURISTIC ALGORITHMS
9. Sections of multiple solutions, inconsistent with the remaining
equations, must be detected and the appropriate solution path be rejected.
10. As was already required for the start of the paragraph, analytically
solvable equations are not to lead not to the abort of the program. Instead
the set of equations is as far as possible on triangle form to be brought
and the not solvable remaining equations as well as the partial solutions
determined up to then to be output.
1,4 extraction and solution of linear equations
is mostly nonlinear with technical tasks the equations which can be
solved, but contains the systems concerned frequently large linear section
blocks. Since linear sets of equations can be solved very efficiently with
the help of the Gauss-Elimination simultaneously, it is advisable to process
before the solution of the nonlinear equations first the linear proportion
of the system separately. Even if a complete analytic dissolution of the
entire nonlinear system cannot be achieved after all looked up variables,
it is nevertheless meaningful not to reduce by elimination of the linear
variables and equations the system to an only small, any longer analytically
solvable core whose numeric solution is substantially less complex, than
an optimization executed on the complete system. Under point 5 required
iterative solution linear subsystems entire set of equations is to that
extent no trivial function, than that neither the equations concerned nor
the subset of the variable, concerning which these equations are linear
admits from the beginning is. Therefore a search strategy must be found,
(for efficiency reasons as large ones as possible) the linear equations
and variable blocks extracted from any nonlinear set of equations.
1,4,1 intuitive methodologies for the search of linear equations
For the elucidation of the task the following nonlinear set of equations
is consulted, which is to be solved x, y and z after the variables. At
first sight only the equation (1.55) is linear, regarding all three variables.
In the case of use of a simple search algorithm, which finds excluding
such completely linear equations, max. a variable can after appropriate
dissolution of (1.55), e.g. after x in this case, from which both remaining
equations are eliminated. A more exact view of the equations reveals however
a better alternative. After the distance of equation (1.56) and shifting
the terms dependent on z on the right pages of the equations (1.55) and
(1.57) two in the variables x and y linear equations develop:
1.4. EXTRACTION AND SOLUTION of LINEAR EQUATIONS 11
their simultaneous inversion leads to the solutions parameterized in
z after their inserting into (1.56) only one nonlinear equation which can
be solved remains:
In view of the fact that with the second version in only one iteration
two unknown quantity could be determined at the same time, it is to be
preferred latter methodology of the search for completely linear equations,
applied first, despite the additional shaping expenditure. This applies
in particular if a set of equations contains at all no equations, in which
all variables involved in purely linear form occurs. The procedure used
for the extraction of linear subsystems should interconnect therefore both
demonstrated operations for the removal of nonlinear equation proportions:
1. Distance of individual nonlinear equations
2. Shifting variables occurring in nonlinear terms to the right side
of the equations by a balanced combination of the two operations can be
achieved that the resulting linear subsystems have max. size and often
at least are approximately square.
1.4.2 a heuristic algorithm for the search of linear equations
For a computer implementation such a search for equation linear equations,
executed intuitively by humans, must be systematized and formulated algorithmic.
Since the term does not define linear section set of equations, how is
evident on the basis the different solution types, the desired result in
unique way, a heuristic strategy was developed for the imitation of the
intuitive methodology. This strategy is demonstrated for the comparison
of the results at the system, already regarded, (1.55)- (1.57).
For the set of equations first a table is set up, whose lines are assigned
to the equations and their columns the variables. The entry at the position
(i; j) of the table contains for the equation i the coefficient 1 concerning
the linear member, i.e. the first power, of the variable xj. Is available
as for z in equation (1.57) no term in first power or steps these functions
not polynomial in argument on (e.g. sin x or sqrt x), then the appropriate
position with a cross () is marked. x y z
Gl. 1)
1 Macsyma places the instruction to the determination of the coefficients
of rational printouts RATCOEFF to the order.
12 CHAPTERS 1. HEURISTIC ALGORITHMS
second, equal large evaluation matrix, is assigned to this table whose
entries are equal to zero, if the corresponding is entry in the coefficient
table a constant, and equal unity, if the linear coefficient concerned
contains looked up variables or not existed (). additionally the line and
column totals become noted at the edges of the matrix, as well as under
the sigma signs on the top right and on the left of down respectively the
line total of the column totals (P S) and the column total of the line
totals (P Z). obvious correspond to
x y z P S 0 1 2 3 1) 0 0 0 0 2) 2 0 1 1 3) 1 0 0 1 P Z 3 (1.63)
A linear section set of equations and the appropriate variables are
found if with a consequence of the operations specified at the end of paragraph
1,4,1 all ones were eliminated. Transferred to manipulations thereby the
1 corresponds to the evaluation matrix. Operation deleting the line belonging
to to a certain equation. The 2. Operation is equivalent column assigned
for the distance of some variable. The linear subsystem consists afterwards
of those equations and variables, whose lines and columns were not removed
from the matrix. The reduction of the evaluation matrix (1.63) to a zero-matrix
can take place in exactly three different ways:
1. Capers of the lines 2) and 3),
2nd capers of the columns y and z,
3rd capers of the line 2) and the column z.
of the demand for max. size and as square a form of the linear equation
blocks as possible is directly portable to the characteristics of the zero-matrix
which can be obtained. In this sense the latter of the three options is
optimal, because it supplies the system (1.58)-(1.59), already detected
in the previous paragraph as optimum solution. The two other possibilities
lead against it for the under-certain equation (1.55) or to a over-certain
3 1-System in x.
The search for an optimal sequence of line and column cancellations
is a complex combinatorial problem. In order to avoid the associated expenditure,
a heuristic, local decision criterion becomes the determination of the
line or column which can be removed in the respective step, i.e. a Greedy
strategy
[FOU 92], uses: The line or column is deleted, which contains most
ones, thus that with the largest line or column total. This criterion supplies
with however still no unique predicate,
if two or more lines have the same (largest) line total,
or two or more columns the same (largest) column total has,
or if the totals of the maximumevaluated line and the maximumevaluated
column are identical.
In the first two cases any can be selected, usually is this for the sake
of simplicity first in each case the found with max. evaluation from the
concerned lines or columns.
The third case occurs in the available example. In the evaluation matrix
(1.63) possess both the line 3) and the column z the max. total 2. At the
beginning from both possibilities arbitrarily the cancellation of the line
one select, so that in the next step the following evaluation matrix results:
x y z P S 0 0 1 1 1) 0 0 0 0 3) 1 0 0 1 P Z 1 (1.64)
again is now the max. line total equal to the max. column total. The
decision could take place according to the coincidence principle, but thereby
the demand after as square a form of the linear subsystems as possible
was not sufficiently considered also here. Is recommended either
to delete lines and columns with same evaluation alternately to meet or
the selection in favor of the possibility which brings the dimension relation
n/m to the n*m evaluation matrix with n =/= m more near at unity. According
to both criteria deleting of the column z proves as more favorable than
the distance of the line 3).
x y P S 0 0 0 1) 0 0 0 3) 0 0 0 P Z 0 (1.65)
the cancellation of column z causes the removal of the last unity in
the evaluation matrix. This expresses itself in disappearing P S and P
Z, by which the end of the algorithm is marked. From the result matrix
(1.65) it is to be read off now that the equations (1.55) are linear and
(1.57) the example set of equations concerning the variables x and y. Those
finally necessary shapings of the linear equations for the creation of
the simultaneous form (1.58) - (1.59) do not represent a problem for a
computer algebra system, them by Macsyma from the instruction LINSOLVE
for the simultaneous solution of linear equations are automatically executed.
1,4,3 solution of the linear equations is unique solvable or under
the linear section sets of equations extracted using the described algorithm,
then their subsequent treatment is unproblematic. In the case of over-certain
systems obligatory inconsistencies occur which require a more detailed
handling. E.g. from a larger set of equations in the variables, in x and
y linear subsystem (1.66)-(1.68) were inferred from x, y, z and w the over-certain.
x
14 CHAPTERS 1. HEURISTIC ALGORITHMS
after the forward of elimination ensteht the following set of equations,
which is inconsistent in the sense of linear algebra and thus no solution
has:
In the regarded case are however z and w variable of the set of equations
which can be solved. The above linear subsystem has exactly solutions if
these two variables fulfill the equation (1.71). This condition is to be
only regarded therefore as a further equation of the remainder system,
from which x and y were eliminated.
As consequence it results that with the occurrence of apparent inconsistencies
following the eliminations process generally the right sides of the consistency
conditions thereupon must be checked, whether they contain looked up variables
of the entire system. If this is the case, then the conditions concerned
are again added after the solution of the linear equations the initial
equation system. If this applies not, i.e. does not occur not fulfillable
obligation conditions between numeric values or parameters, then the set
of equations has indeed no solution, and the solution process must be aborted.
1,5 evaluation strategies for the solution of nonlinear equations
Apart from few special cases do not exist closed solution procedures
for general nonlinear sets of equations. This does not exclude however
that for many nonlinear systems analytic solutions or at least partial
solutions can be calculated, only is usually their regulation not in as
efficient way as by Gauss-Elimination in the case of linear equations possible.
1,5,1 substitution systems for nonlinear sets of equations an elementary
solution procedure, which can be applied to any sets of equations, is the
well-known substitution system:
1. Select (as simple ones as possible) an equation from the system
and resolve it after a variable xj. Abort, if all equations are solve,
or no further equation analytically to solve leaves itself.
2. Insert the result into the remaining equations, in order to eliminate
xj from the system.
3. Check the system reduced by an equation and a variable for consistency
and continue with step 1.
For the demonstration of the procedure the nonlinear set of equations
(1.72) { (1.76) one regard, which is to be solved a, b, c and d after the
variables.
ab + 2c = 0 (1.72)
c 2 + d
1.5. EVALUATION STRATEGIES FOR THE SOLUTION OF NONLINEAR EQUATIONS 15
The question, which concrete characteristics distinguish " as simple
" an equation as possible, arises already directly at the beginning of
the application of the substitution system. From know-how the among other
things following evaluation criteria can be derived, which must to apply
and be able depending upon application to be differently weighted not necessarily
at the same time: Simple equations
1. contained only few the looked up variable,
2. contain a looked up variable at exactly one position, so that the
unknown quantity lets itself be isolated relatively easily,
3. indicate concerning one or several variables only small depths of
the operation hierarchy, i.e. the formula complexity is small,
4. contains no transcendental or different one, functions which can
be inverted with difficulty.
On the basis these criteria now the simplest equation of the example
system is to be determined. Regarding the first two points this could be
the equation (1.75), because it contains only the variable a, and this
occurs at exactly one position. On the other hand the evaluation does not
precipitate due to the criteria 3 and 4 very favorably. Regarding the points
2, 3 and 4 the solution of the equation (1.73) appears after the variable
d as the best selection, because all other equations contain either more
variables or more with difficulty resolvable functions.
As relevant criterion here first the combination of the points 1 and
2 may be considered, so that with the solution of the equation (1.75) after
the variable a one begins. Only if the principal value of the arctan function
is considered, then a = 2 follows: (1.77)
Begun into the remaining four equations the system results
2b+ 2c = 0; (1.78)
c 2 + d
developed there no inconsistencies, with the selection of the next
simplest equation are continued. The valuation criteria speak now unique
2 for the solution of the equation (1.79) after d:
the first equation would probably solve 2 humans simpler would consider,
but the beforehand necessary sections of the equation is strict by the
factor 2 an additionally considering expenditure.
16 CHAPTERS 1. HEURISTIC ALGORITHMS
from it 2b follows
2b + 2c = 0; (1.83)
p b + 4
concerning all criteria is now the equation (1.83) the most favorable
candidate, so that with the solution
b =-c
still two equations with the unknown quantities c remain.
Equation (1.88) is analytically not solvable, therefore independently
equation (1.87) must supply the still which is missing solution for c of
the evaluation. In this case a multiple solution results for the first
time:
[c=0,c=-1].
In it the meaning so far of the not relevant consistency check shows
up. From the two solutions only second, c = -1, fulfills the equation (1.88).
The other solution leads to the contradictory predicate
-i pi = 0 (1.90)
and must be rejected therefore. After the back substitution the consistent
solutions read
[a=2, b=1, c=-1, d=3]. (1.91)
1.5.2 heuristic procedures for the complexity evaluation of algebraic
printouts
being supposed the evaluation of algebraic equations regarding their
" simplicity " and the solution of a nonlinear set of equations which is
based on it by a computer algebra system to be thus made automatically,
then the criteria in algorithms, formulated linguistically in the paragraph
1.5.1, must be illustrated, which supply numeric complexity evaluations
for the controlling of the solution process. The height of an evaluation
number of b should be a measure for it how difficult it is to resolve an
equation i after a variable xj analytically. The larger b is, the more
complex appears 3 the function.
If the evaluation for each equation is made regarding each variable,
then a solution sequence for the equations can be generated by means of
an assortment on the basis the yardsticks. The solution sequence is representable
by an arranged list of the form [ (i1; j1; b1); (i2; j2; b2); ] whereby
for the evaluation numbers UC applies bl to k < to l. for the equations
more loeser implies the list the following statements: Attempts first,
which equation i1 after the variable xj1
3 it here the formulation appears conscious selected, because made the
evaluation on the basis heuristic criteria, which cannot guarantee optimal
decisions.
1.5. EVALUATION STRATEGIES FOR the SOLUTION of NONLINEAR EQUATIONS 17
to resolve since this appears simplest. If this does not succeed, then
tries instead the solution from equation i2 after xj2, etc.. If the solution
attempt is successful against it, then insert the solution into the remaining
equations and begin from the front with the list of a new solution sequence.
The conversion of the first criterion to an algorithm does not represent
a serious challenge, because the number of variables contained in an equation
is already a numeric value. The implementation by programming is in a computer
algebra system just as little a problem. In Macsyma the short instruction
Length(ListOfVars(Equation[i]))
is sufficient for the statement of the number of the unknown quantities
in the ith equation. This number is however only as secondary criterion
in connection with other evaluations of use, because only one order of
rank of the equations, not however by pairs of equation/variables, is supplied.
Out later evident reasons is here first the view of the criterion 2
to be reset and with the points 3 and 4 be continued. These two criteria
were separately listed, but them can be interconnected very easily only
one procedure. The heuristic evaluation algorithm used in in the following
section described the implementation symbolic equation loesers uses the
internal representation of algebraic printouts in computer algebra systems
for complexity calculation. Assembled ones algebraic functions are administered
in hierarchically organized lists by operators and operands in prefix notation,
can which be illustrated directly into a tree structure. The nodes of such
a tree contain the operators, the operands are in the pages. For example
the figure 1,7 shows the representation of the left sides of the equations
(1.75) and (1.87) as operation trees.
Figure 1.7: Tree representation of algebraic printouts
An evaluation of the formula complexity regarding the depth of the
operation hierarchy, thus the degree of the nesting of a printout, is now
readable at the tree structures. E.g. the complexity can be intended, as
the branches of tree are counted, which must be crossed by the root of
the operation tree on, in order to arrive for the instances of the regarded
variables.
18 CHAPTERS 1. HEURISTIC ALGORITHMS
regarded variables to arrive. In the case of the printout in figure
1.7a) the complexity value is b concerning the variable a equal to the
length of the fat drawn in path, i.e. b = 4. In order to achieve from the
tree root in figure 1.7b) all instances of the variable c, altogether seven
branches must be crossed, therefore are the complexity b = 7.
This methodology for the calculation of the complexity is easy so expandable
that also the criterion 4, which is considered evaluation of a printout
regarding the operators contained in it, at the same time. Instead of the
easy counting of the branches of tree only additionally a weighting must
take place, as a typical " difficulty factor " is assigned to each operator,
by which the evaluation of its operands to be multiplied is. The height
of the difficulty factor is to reflect, like complex for the computer algebra
system the formation of the inverse function, thus the dissolution of the
function after the operands, is [TRI 91].
At the beginning each page of the tree receives the weighting 1, if
it contains the variable, concerning which the evaluation to be calculated
is. Otherwise the pages are provided with the weighting 0. During the evaluation
of the operators the operator serves " + " as reference with the weighting
1, since it is to be inverted most easily. Regarded in the comparison to
it the reversal of the operators will become " * " and " tan " arbitrarily
as four or ten times so with difficulty, accordingly therefore the weightings
set. The following table shows a selection of some operators and them assigned
weightings.
Figure 1.8: Evaluation of the operators
The calculation of the complexity value takes place bottom up via repeated
addition of the branch weights at the operator node, multiplication of
the total with the operator weight and transfer of the resulting value
onto the superordinate branch, until the tree root is achieved. For the
demonstration of the procedure the operation tree in figure 1.a) one regard.
In the squares drawn in beside the nodes the operator weights are marked,
the numbers at the branches mean the total weight part of the tree subordinated
in each case. Since the complexity of the printout is to be evaluated concerning
the variable a,
1.5. EVALUATION STRATEGIES FOR THE SOLUTION OF NONLINEAR EQUATIONS 19
only the page with the symbol a the weight 1, all other pages obtains
with 0 is evaluated. At the multiplication node over-half the variables
add themselves the branch weights to the total 0 +1 = 1, those, in accordance
with which evaluation of the operator is passed on "*", with the factor
4 multiplied to right branch of operand of the "/" operator. There likewise
a multiplication with 4 takes place, so that the intermediate result is
now alike to 16. Subsequently, are added in the process of the calculation
still the factors 10 and 1 for "tan " - and the "+" operator. The final
result, b = 160, is over the root of the tree. For the operation tree in
figure 1.8b) appropriate calculations result in a complexity value of b
= 110 concerning the variables c.
Here can now the provisionally reset criterion 2 again be taken up.
In order to determine those equations of a system, one or more variables
at exactly in each case one position contained, it is sufficient, to set
in the calculation regulation for the formula complexity, described above,
all operator weights to unity and store all pairs of equation/variables,
for which under these conditions the complexity evaluation b = 1 results.
The method is to be justified thereby that the search applies exactly to
those equations, in whose tree representation determined variable symbols
in only one page occurs. That means, there is only one path of the tree
root only in each case in these cases to the symbol concerned. With a weighting
of unity for all operators the complexity evaluation even corresponds to
the number of paths to the instances of a variable.
The procedure can be verified easily on the basis of examples. If all
operator weights are set equal unity, then it results for the printouts
in figure 1.8a) and b) complexities of b = 1 for the variable a and b =
2 for the variable c. This corresponds with the fact that the variable
a is contained in exactly one page of the tree, while the symbol c at two
positions is to be found.
1.5.3 list of the solution sequence
By different combining and weights of the appraisal procedures discussed
above develops different strategies for the list of solution orders. The
strategy named MinVarPathsFirst, implemented in the program developed in
this work, represents a combination of the criterion 2 and the complexity
evaluation by means of operator weighting. The equations, which contain
looked up variables at exactly in each case one position, receive highest
priority in the solution order, whereby within this group according to
smallest weight of the operator trees one sorts. All remaining pairs of
equation/variables are likewise arranged, according to smallest tree weight,
placed to the end of the solution sequence. Therefore in the case of the
two printouts in figure 1,8 would be first tried to resolve the equation
with the tan function after the variable a although for it a higher evaluation
was calculated than for the accompanying printout concerning the variable
c.
Chapters 2 the Solver
2.1 the structure of the Solvers
The request, in particular the points 4, set up in the paragraph 1,3,
5 and 7, put in figure 2,1 schematically represented the structuring equations
solvers into five to a large extent independent main modules close [TRI
91]. Constructing on this structure and the heuristic algorithms described
in the preceding section in the context of the available work the Macsyma
program complex SOLVER was developed for the functionality extension of
the Macsyma functions SOLVE and LINSOLVE.
The function of the module Solver Preprocessor are general, preparing
work like the check of the command syntax and semantics and the structure
internal data structures, in addition, the execution of an introductory
consistency check. With their it is to be determined whether the set of
equations directly contains number = number or obligation conditions between
parameters of the form as contradictory awarding predicates.
The Immediate Assignment Solver searches the set of equations before
the call linear of the Solvers for direct assignments of the form var =
const. or const. = var and executes these immediately, so that the cost
of computation necessary for the following Programmodule is kept as small
as possible.
The linear Solver is a Pre and a post processor for the Macsyma function
LINSOLVE for the simultaneous solution of linear sets of equations. The
module extracts linear section sets of equations according to the heuristic
algorithm described in paragraph 1,4,2 and solves the equations by call
of LINSOLVE. The determined solutions are inserted before leaving linear
the Solvers into the remaining equations.
The Valuation Solver is the core module of the Solvers. Its functions
are the use of evaluation strategies for the generation of the solution
sequences and the solution of the nonlinear equations with the help of
the Macsyma internal SOLVE function. In the case of multiple solutions
the Valuation Solver checks each single solution for consistency with the
remainder set of equations. Inconsistent solutions are rejected, while
valid solutions are inserted into the remainder set of equations and the
appropriate solution paths separately through recursive of calls of the
Valuation Solvers are pursued.
All for the editing of the solutions to the output to the user necessary
step by the Solver Postprocessor are taken over. By this fall the expansion,
of the hierarchically organized solution list supplied by the Valuation
Solver, the back substitution of the
21
22 CHAPTERS 2. The SOLVER
symbolic solutions as well as picking out, analysing and outputting
the variables and assembled printouts asked by the user.
Solver Preprocessor:
Check of the command syntax
preprocessing the equations
consistency check
Immediate Assignment Solver:
Direct allocations look up and execute
consistency check
Linear Solver: Linear equations look up and solve
consistency check
Valuation Solver: Evaluation strategy
nonlinear equation solve
consistency check
multiple solutions apply recursively pursue
Solver Postprocessor: Back substitution
editing and output of the results
Figure 2.1: Structure of the Solvers
2.2. The MODULES OF THE SOLVERS 23
2,2 The Modules of the Solvers
2,2,1 the Solver Preprocessor
SOLVER PREPROCESSOR
Check of the command syntax and semantics
structure of the internal equations, variable and parameter lists
consistency check
SolverSubstPowers = TRUE?
Search system for multiples of basic powers and replace if necessary
set variables
END
figure 2.2: Flow chart of the Solver Preprocessors
Beside the examination of the command syntax and semantics as well
as the structure of the internal equations -, variable and parameter lists
executes the Solver Preprocessor a consistency check of the equations,
whose flow diagram is represented in figure 2,3. Target of the consistency
check is it to detect from the beginning whether the set of equations is
unsolvable due to direct contradictions. Such contradictions can occur
to e.g. 0 = 1 in form of pure number equations, in addition, in the form
of obligation conditions between as parameters defined symbols.
In order to uncover direct contradictions, the routine searches the
set of equations to the consistency check for equations, which consist
exclusively of values of the number or values of the number and parameters,
but contain no variables. If a contradictory number equation is found,
then the Solver Preprocessor aborts immediately. If such an equation is
against it consistently, as for example 0 = 0, it is removed from the set
of equations, since it is not the solubility and solution of the system
influenced and thus redundant.
The handling of parameter obligation conditions is somewhat more complex.
Assumed, the symbols A and B were defined as parameters of a set of equations,
which contains the equation A + B = 1 (2.1).
24 CHAPTERS 2. THE SOLVER
CONSISTENCY CHECK
all equations checked?
get equation from list
does the equation contain variables?
Does the equation contain only and no parameters of numbers?
Does a contradiction occur?
Does the parameter obligation condition hurt before made acceptance?
Remove redundant equation
question user about validity of the obligation condition
obligation condition of ok? ABORT
stores condition in SolverAssumptions
figure 2.3: Flow chart of the routine to the consistency check
from this condition follows that A and B are not independent parameters
and therefore the set of equations is not solvable for any combinations
of their values. Since conditioned inconsistencies of this type cannot
be excluded with many technical problem definitions, it is not meaningful
to abort the solution process in such cases easily it is, the parameters
equation contradicts for its part again different obligation conditions
in the set of equations. The decision, whether a parameter obligation condition
is to be regarded as admissible, is delegated therefore by the consistency
check to the user. In the case of the equation (2.1) the system asks in
the following way for the validity of the condition:
Is B + A - 1 positive, negative, or zero?
Becomes the question with p; or n; (positive or negative) answers,
then the Solver aborts. The response reads against it z; (zero), then the
consistency check stores the obligation condition in a global list named
SolverAssumptions, those after the Solver run examined
2.2. The MODULES OF THE SOLVERS 25
will can. Subsequently, the consistency check removes the redundant
equation from the system. 2,2,2 the Immediate Assignment Solver
IMMEDIATE ASSIGNMENT SOLVER
all equations searched?
Equation out of list take
variable list update
has the equation the form var=const. or const.=var?
Solutions begin
are var already in the solution list?
Consistency check
adds var=const the solution list in addition removes equation from
the list
figure 2.4: Flow chart of the Immediate Assignment Solvers
the function of the Immediate Assignment Solvers consists of it, the
set of equations after direct allocations of the form var = const. to search
and to use such equations directly for the elimination of the betrenden
variables. With by machine set up sets of equations, as for instance in
the example 1,2, necessary cost of computation can be often reduced considerably
by such preprocessing the equations the linear equations in linear the
Solver, for the extraction and solution.
The Immediate Assignment Solvers filters all direct allocations in
a loop first var = const. and const. = var from the system off and stores
it in the form var = const. in the solution list, if a solution for var
there is entered not already. Subsequently, the set of equations with the
solution list is analysed and checked again for consistency.
26 CHAPTERS 2. THE SOLVER
2.2.3 The linear Solver
The flow linear of the Solvers begins, as in paragraph 1.4.2 described,
with the list of the complete coefficient matrix of the symbolic set of
equations. On the basis this coefficient matrix the evaluation matrix is
created, which serves for the extraction linear section set of equations.
As long as the evaluation matrix contains still one elements, i.e. P S
6 = 0 and P Z 6 = 0, is applied the described heuristic evaluation strategy,
which decides on it, which nonlinear equation is removed or which in nonlinear
sense occurring variable on the right side of the equation is brought.
If the evaluation matrix was reduced to a zero-matrix, then a list
of the remaining (linear) equations and a list of the linear variables
are created, which can be transferred to the Macsyma instruction LINSOLVE
serving for the solution of linear sets of equations as function parameters.
For efficiency increase before the call of LINSOLVE however still another
special feature is considered, which was not mentioned during the description
of the extraction algorithm. Occasionally it can occur that with the deleting
of a nonlinear equation the only instance of a linear variable is removed
at the same time xj from the entire evaluation matrix, without this was
noticed directly. Likewise equations can develop, which are no longer nonlinear
concerning as linear detected variables, but this variables any longer
do not contain, i.e. the appropriate coefficients are directly zero. Therefore
again all linear variables become before the solution of the linear subsystem
thereupon it checks whether they are contained in the linear equations
still, and only the subset of the extracted equations is passed on to LINSOLVE,
which contain actually at least one of the linear variable. That would
be linear Solver also able to solve the set of equations without these
additional measures but frequently unnecessary cost of computation can
be saved by it.
In paragraph 14.3 it was demonstrated that with the solve of over-certain
linear sets of equations inconsistencies can occur, which mean not necessarily
the insolubility of the set of equations. If such contradiction equations
become, e.g. the right side of equation (1.71), discovers 1, then they
are submitted, as at the beginning of the entire set of equations, of the
consistency checking. If the contradiction equations prove as true inconsistencies,
then the total system does not have a solution, and that linear Solver
aborts with an appropriate error message. If the contradiction equations
contain against it still looked up variables, then they are removed for
the remainder set of equations from the linear set of equations and added
as new obligation conditions. Thereupon LINSOLVE with the linear independent
part of the equations is again called (inconsistencies to be able with
the second run any longer not to occur).
If the linear equations were successfully solved, then the solutions
are inserted into the remainder set of equations and the list of the looked
up variables updates. Latter meant that all variables, for which by linear
the Solver a solution was found it are removed from the list, while new
variables, which are contained in these solutions, are added the list.
1 That access to the contradiction equations is from the outside not
possible on use of the standard design LINSOLVE commands. Therefore
one was necessary on Lisp level particularly modified version of the instruction,
those by Jeffrey P. Golden, Macsyma, Inc. (the USA), on request kindly
was made available.
2.2. The MODULES OF THE SOLVERS 27
LINEAR SOLVER
complete coefficient matrix set up
evaluation matrix set up
sigma S = sigma Z = 0?
Heuristic apply: Line or column of the evaluation matrix
deletes list of the remaining equations
and variables sets up
line and column totals sigma S and sigma Z
updates linear equations with LINSOLVE solves
set of equations inconsistently?
Consistency check to right side of the extended coefficient matrix
apply
genuine contradiction found?
Solutions into remaining equations use
variable list update
" inconsistent one " equations remove
Abort
figure 2.5: Flow chart linear of the Solvers
28 CHAPTERS 2. THE SOLVER
2.2.4 The Valuation Solver
The Valuation Solver checks first whether equations and variables still
which can be solved are available. If this is the case, then for the remaining
equations two evaluation stencils are set up, which serve the solution
sequence as basis for the list. Both stencils have the dimensions n*m,
whereby n is the number of the equations and m the number of variables
at the moment looked up. The first matrix, the variable path matrix, enhaelt
for each equation the number of variable paths (see paragraph 1.5.2) concerning
each variable. The second matrix is the evaluation matrix. Their items
are calculated by the heuristic operator tree evaluations of each equation
concerning each variable.
To these two stencils afterwards one is applied - depending whether a internal or user-defined evaluation strategy is desired, which arranges the pairs of equation/variables in such a way that the first items are in this list as promising a candidates for a following solution attempt as possible by means of the SOLVE instruction.
As long as yet all proposals for solution in the list and no correct solution for one of the suggestions were not tried be calculated could, on the basis the determined solution order the next equation from the equation list is selected and tried to solve. This succeeds not, becomes, if available, one or more user-defined transformation functions for the shaping of the equation applied and renewed in each case solution attempts made. Even if these are without result, the next proposal for solution is tried.
At all none of the equations is solvable after any variable more, then the Valuation Solver returns the unresolved equations in implicit form additionally to all solutions found up to this point, so that this with a numeric procedure can be treated possibly later. With a successful solution attempt all single solutions (nonlinear equations to be able multiple solutions to have) are checked separately for consistency with the remaining equations. Those solutions, which lead to contradictory predicates, are rejected and with them the appropriate solution path.
If check a solution does not remain after the consistency, then the set of equations is inconsistent, and the Valuation Solver aborts. Exactly if one solution remains, then this is attached to the solution list and inserted into the remaining equations. Finally the list of the looked up variables is updated, as the variable just calculated is removed from it and in the solution contained, new variable is if necessary added. The latter variables can be, which not when parameters in addition, not when up were looked indicated. The main solution loop of the Valuation Solvers begins then again from the front.
With consistent multiple solutions all solution paths must be pursued
separately. In addition the Valuation Solver with the remaining equations
calls itself and variables for each single solution recursively and stores
the gotten back results in a hierarchically structured solution list. Their
expansion is function of the Solver Postprocessors described in the next
paragraph.
2.2. The MODULES OF THE SOLVERS 29
VALUATION SOLVER
Still equations remaining?
All variables solved?
evaluation stencils set up
evaluation strategy apply, to the solution sequence generate
still proposals for solution remaining?
Equation from list unresolved
equations at solution list get attach
attempts to solve equation
solution attempt successfully?
Shaping apply?
Consistency of all single solutions
check equation transform
inconsistent solutions
reject more than one solution?
Figure 2.6: Flow chart of the Valuation Solvers (part of 1)
30 CHAPTERS 2. THE SOLVER
all solution paths pursues?
hierarchical solution list
single solution use
variable list update
Valuation Solver update recursively call
no solution remaining?
Solution list update solution
use variable list update
Figure 2.7: Flow chart of the Valuation Solvers (part of 2)
2.2. THE MODULES OF THE SOLVERS 31
2.2.5 the Solver Postprocessor
SOLVER POSTPROCESSOR
hierarchical solution list expand
unresolved equations abfiltern
back substitution required?
Back substitution
back substitution partly execute
look-ups variables and printouts with the solutions execute analyse
unresolved equations add
Figure 2.8: Flow chart of the Solver Postprocessors
The Valuation Solver in the case of multiple solutions of nonlinear equations because of the recursive pursuit of the solution paths a hierarchically structured result list as function value supplies, must do the Solver Postprocessor at the beginning the list hierarchy outer eyes there, so that the back substitution can be executed.
If not all equations could be solved within a solution path symbolically, then the result list contains the list of the remaining, unresolved nonlinear equations additionally to the variables, for which a solution was found. This will provisionally removed from the equation list and buffered, in order to the final result to be added later.
Depending upon desire of the user takes place afterwards the back substitution of the set of equations, which is present up to this point in time still in upper triangle form. In addition the solution list is analysed so long iterative with itself, until no modification of the results is to be registered more. Even if the back substitution by the user, it is not required to 2 nevertheless executed, but only so far, as necessary is, in order to eliminate everything in the instruction line not indicated variables from the solutions. Is e.g. a set of equations
2 this requires the allocation of the option variable SolverBacksubst
with FALSE (see paragraph 2.4)
32 CHAPTERS 2. THE SOLVER
in the variables x, y, z and w only after the variables x and y to
be solve and supplies the solution process the triangle form
x = f1(y; z; w) (2.2)
y = f2(z; w) (2.3)
w = f3(z) (2.4)
z = const.; (2.5)
in such a way the complete back substitution leads to the result
x = const. (2.6)
y = const. (2.7)
With switched off complete back substitution against it the two equations
x = g(y) (2.8)
y = const. (2.9)
are returned, the internal variables z and w any longer do not contain,
but are coupled by the variable y further among themselves.
After termination of the back substitution and the analysis of the
assembled printouts with the solutions, indicated in the variable list
of the command line, at the beginning of abge filtered, again to the solution
list attached unresolved equations. The finished solution list is then
transferred as function value to the user.
2.3 Application of the Solvers
2.3.1 Command syntax
The call of the Solvers from the Macsyma echelon of command with the
syntax
Solver(equation list, variable list, parameter list)
or, if the set of equations which can be solved does not contain parameters,
also with Solver(equation list, variable list).
The equation list is a list of Macsyma objects, for which EquationP
is equal to TRUE. The set of equations (1.55)-(1.57) is thus indicated
in the following way:
(COM5) Equations:
[
x + 2*y - z = 6,
2*x + y*z - z^2 = -1,
3*x - y + 2*z^2 = 3
] $
the unknown variables the Solver likewise indicated in form of a list,
e.g. as
2.3. APPLICATION OF THE SOLVERS 33
[ x, y, z ]
for the above-mentioned set of equations. Beside purely atomic variable
symbols (SymbolP) the variable list may contain printouts also assembled
in the looked up unknown quantities. For the set of equations not explicitly
if the variables x, y and z, are looked but rather x and the value up sin(ssyz),
then the variable list reads
[ x, sin(%pi*y*z) ].
The parameter list may against it excluding atomic symbols contain,
thus only objects, for the SymbolP equal TRUE is. Assembled printouts are
here neither admissible nor meaningfully.
2.3.2 Special features of the syntax of equations
Here is referred to some differences of the command syntax in the comparison
with the admissible call forms of the Macsyma internal SOLVE function.
The latters may be transferred the equations also to in Expression form,
i.e. as printouts with EquationP = FALSE, which are understood implicitly
as equations of the form Expression = 0:
[
x + 2*y - z - 6, illegally
2*x + y*z - z^2 + 1,
3*x - y + 2*z^2 - 3
] $
This form of the equation representation one uses internally, e.g. by the Valuation Solver, but is not not permitted it with the call of the Solvers in the command line. Furthermore the SOLVE function permits an omitting of the list brackets around the arguments, if only one equation and/or only one variable are to be transferred. Also this is not admissible with the use of the Solvers.
2.3.3 Example calls of the Solvers
Example 2.1
Of in COM5 input the set of equations a correct call of the Solvers is the instruction in the command line COM7. The specification of the calculated solutions effected in form of a list from solution lists, see output line D7.
(COM6) to MsgLevel: to ' DETAIL$ / * see paragraph 2,4 * /
(COM7) Solver(Equations, [ x, y, z ]);
Outputs of the Solver Preprocessors:
The variables to solved for are [ X, Y, Z ]
Checking for inconsistencies...
34 CHAPTERS 2. THE SOLVER
... none found.
Outputs of the Immediate Assignment Solvers:
Searching for immediate assignments.
No immediate assignments found.
Outputs of the Linear Solvers:
Searching for linear equations...
... with respect to: [ X, Y, Z ]
Found 2 linear equations in 2 variables.
The variables to solved for are [ X, Y ]
The equations are [ - Z + 2 Y + X - 6, 2 Z^2 - Y + 3 X - 3 ]
Solving linear equations.
4 Z^2 - Z - 12
2 Z^2 + 3 Z + 15
The solutions are [ X = - ---------------, Y = - -------------------]
7
7
Searching for linear equations...
... with respect to: [ Z ]
No linear equations found.
Outputs of the Valuation Solvers:
Checking for remaining equations.
1 equation(s) and 1 variable (s) left.
The variables to solved for are [ Z ]
Trying to solve equation 1 for Z
Is not needed here a complexity evaluation,
because there is only one equation and a variable remaining.
Valuation: (irrelevant)
The equation is 2 Z^3 - 12 Z^2 + 17 Z + 31 = 0
Checking if equation was solved correctly.
SQRT(13) %i - 7 SQRT(13) %i +
7
The solutions are [ Z = - -------------------, Z = ---------------------,
Z = - 1 ]
2
2
Solution is correct.
Individual consistency check with multiple solutions:
The solution is NOT unique. Tracing paths separately.
Solution 1 for Z
Checking for inconsistencies...
... none found.
Solution 2 for Z
Checking for inconsistencies...
... none found.
Solution 3 for Z
2.3. APPLICATION OF THE SOLVERS 35
Checking for inconsistencies...
... none found.
SQRT(13)%i - 7 SQRT(13)%i + 7
Consistent solutions for Z: [ Z = - --------------, Z =-------------,
2
2
Z = - 1 ]
Recursive pursuit of all three solution paths:
Checking for remaining equations.
All variable solved for. No equations left.
Checking for remaining equations.
All variable solved for. No equations left.
Checking for remaining equations.
All variable solved for. No equations left.
Outputs of the Solver Postprocessors:
Post processing results.
27SQRT(13)%i-41
17SQRT(13)%i-87
(D7) [[X = ---------------, Y = - ---------------,
14
14
SQRT(13)%i-7
27SQRT(13)%i+41 17SQRT(13)%i+87
Z = - ------------], [ X = - ---------------, Y =---------------,
2
14
14
SQRT(13)%i+7
Z = ------------], [ X = 1, Y = 2, Z = - 1 ]]
2
To the better outline follows the result again in the TEX-record:
Example 2.2
As the second example is the set of equations parameterized in a and
b
ax + y^2 = 1 (2.13)
bx - y = -1 (2.14)
36 CHAPTERS 2. THE SOLVER
after the variables x and y as well as zusamm the set printout x/y
to be solve. Since the set of equations does not contain direct allocations
and a repeated search for linear equations in this case is not meaningful,
the Immediate Assignment Solver and the repetition loop linear of the Solvers
with the instruction COM8 are switched off:
(COM8) SolverImmedAssign: SolverRepeatLinear: FALSE$
(COM9) ParEq: [ 3*a*x + y^2 = 1, b*x - y = -1 ] $
(COM10) Solver(ParEq, [ x, y, x/y ], [ a, b ]);
X
The variable to solved for AR [ X, -, Y ]
Y
The parameters are [ A, B ]
Checking for inconsistencies...
... none found.
Trying to solve for [ X, Y ]
X
in order to solve for the expression -
Y
Searching for linear equations...
... with respect to: [ X, Y ]
Found 1 linear equations in 2 variables.
The variables to solved for are [ X, Y ]
The equations are [ - Y + B X + 1 ]
Solving linear equations.
The solutions are [ Y = B X + 1 ]
Checking for remaining equations.
1 equation(s) and 1 variable (s) left.
The variable to solved for are [ X ]
Trying to solve equation 1 for X Valuation: (irrelevant)
2 2
The equation is B X + (2 B + 3 A) X = 0
Checking if equation which solved correctly.
2 B + 3 A
The solutions are [ X = -, X = 0 ]
2 B
Solution is correct.
The solution is not unique.
Tracing paths separately.
Solution 1 for X
Checking for inconsistencies...
... none found.
Solution 2 for X
Checking for inconsistencies...
2.4. The OPTIONS OF THE SOLVERS 37
... none found.
2 B + 3 A
Consistent solutions for X: [ X = - ---------, X = 0 ]
2
B
Checking for remaining equations.
All variable solved for. No equations left.
Checking for remaining equations.
All variable solved for. No equations left.
Post processing results.
2B+3A B + 3A X
2B + 3A
X
(D10) [[ X = - ------, Y = - ------, - = --------], [X = 0, Y =
1, - = 0]]
2 B
Y 2
Y
B
B + 3AB
2.4 The options of the Solvers
Of the Macsyma echelon of command (Macsyma toplevel) or from program
tapes the behavior of the Solvers can be usst by the non-standard allocation
of a set of option variable in, which is listed and described in the following.
The standard mappings designated in the program represent the values or
symbols indicated behind the names of the option variables in pointed parentheses,
which are set automatically accordingly on first of the module SOLVER.
MsgLevel < SHORT > (message level) controls the range of the display messages output during the program run. The allocations OFF, SHORT and DETAIL are admissible. Becomes MsgLevel: OFF set, then all program outputs are completely suppressed. In the case of an allocation with SHORT status information is only output for the pursuit of the program run. The keyword DETAIL against it causes additionally the output of all of the Solver modules calculated intermediate results and as soon as of messages over the decisions made due to the heuristics.
SolverImmedAssign < TRUE > switches the Immediate on Assignment Solver (TRUE) or out (FALSE). If the module is switched on, then before the call linear of the Solvers the set of equations is searched for direct allocations of the form var = const. or const. = var, which can be inserted immediately into the remaining equations.
SolverRepeatImmed < TRUE > determines whether the Immediate Assignment Solver is called repeated (TRUE), until no more direct allocations are found, or whether only one call takes place (FALSE).
SolverSubstPowers < FALSE > (substitute powers) controls the handling
of variables, which occur to a basic power p0 2 IN n f 1 g in powers pk
integral multiples pk = kp0, k 2 IN. The set of equations contains e.g.
the variable x exclusively in the powers x 2, x 4, x 6,: i.e. p0 = 2, then
becomes with SolverSubstPowers: TRUE x p0 = x 2 by
38 CHAPTERS 2. THE SOLVER
new variable symbol X2 substitutes, that thus only in the powers X2,
X2 2, X2 3,: occurs. In this way the degree of the equations which can
be solved and the solution variety which can be expected reduce. Is however
if necessary a rework of the solutions necessary.
SolverInconsParams < ASK > (parameter handling inconsistent) beein usst the behavior of the routine to the consistency check. Admissible allocations are ASK, BREAK and IGNORE. During the consistency if check a dependency between parameters is discovered, then becomes with SolverInconsParams: ASK of the users in demand for the validity of the appropriate obligation condition. This is stored with positive response for later analysis in the list SolverAssumptions. If dependency is not admissible or if the option variable with BREAK is occupied, then the solution process is aborted. An allocation with IGNORE arranges the consistency check to recognize basically all obligation conditions between parameters as valid as long as these directly acceptance already made do not contradict.
SolverLinear < TRUE > switches linear the Solver on (TRUE) or out (FALSE). Switching off is recommended if the set of equations which can be solved does not contain linear equations or their number in relation to the number of the nonlinear equations is very small. In these cases much computing time can be saved through going around linear the Solvers, since the algorithm is quite complex to the search for linear equations.
SolverRepeatLinear < TRUE > arranges that repeated calls linear of
the Solvers. If the variable is set on FALSE, then that is passed through
linear Solver only once.
SolverFindAllLinearVars < TRUE > decides whether that looks linear
Solver up for max. large linear section set of equations regarding all
available variables (TRUE), or whether only subsystems in the variables
are to be extracted, which are presently/immediately looked up during the
solution process (FALSE). The allocation of the variables plays particularly
a role if under-certain sets of equations are to be solved. Here SolverFindAllLinearVars
should: FALSE to be set, because otherwise no solutions for the originally
interesting variables can possibly be found due to the too small number
of equations. With the allocation FALSE it is guaranteed that first after
these variables it is solve and the degrees of freedom in other unknown
quantities are expressed.
SolverValuationStrategy < MinVarPathsFirst > contains the name of the function, which from the variable path matrix and the evaluation matrix a solution sequence generated (see paragraph 2,7). The call of the function occurs within the Valuation Solvers with:
SolveOrder: Apply(
SolverValuationStrategy, [ VarPathMatrix, ValuationMatrix ]
)
as function value is expected a list of the form
[ [ i1, j1, b1 ], [ i2, j2, b2 ], ... ]
how it was described in paragraph 1.5.2. To consider here the option
variable is SolverMaxLenValOrder.
2.4. The OPTIONS OF THE SOLVERS 39
SolverDefaultValuation < 10 > determines the quality factor for
operators, to who explicitly such a factor with the set prop. instruction
was not assigned (see paragraph 2.6).
SolverMaxLenValOrder < 5 > (maximum length OF valuation order) determines the max. length of the solution sequence. Even if the last proposal for solution in the list does not lead to success, then the Valuation Solver aborts the solution process, even if yet all pairs of equation/variables were not tried one after the other.
SolverTransforms < [ ] > contains a list of the names of functions, which can be applied after a unsuccessful solution attempt to the betrende equation, in order to increase the solution chances with a renewed attempt. The functions are executed thereby in the order of their occurring in the list. After each function call directly the next solution attempt takes place. Even if this fails, then the next transformation in the list is applied, as long as to the equation could be solve or no further transformation is more available. Since the manipulation possibilities with the transformations are quite extensive, is a specification of their definition and application in paragraph 2.5.
SolverPostprocess < TRUE > switches the Solver on Postprocessor (TRUE) or out (FALSE). In switched off status the hierarchical result list of the Solvers without rework is returned, thus without expansion, back substitution and extraction of the variables inquired by the user. This control variable serves Debug purposes primarily.
SolverBacksubst < TRUE > (back substitution) determines whether the Solver Postprocessor is to execute a back substitution of the set of equations brought on triangle form (TRUE) or not (FALSE). If the calculated symbolic solutions are very extensive, then it is often meaningful to execute no complete back substitution but some the looked up variables as functions of other calculated unknown quantities to output.
SolverDispAllSols < FALSE > (display all solutions) points, if on TRUE set, the Solver on to output for the termination of the solution process all found solutions and not only on the variables indicated by the user.
SolverRatSimpSols < TRUE > (by form rationally simplifications on solutions) instructs the Solver Postprocessor to simplify the results before the output again with the instruction FullRatSimp.
SolverDumpToFile < FALSE > compelled when desired (TRUE) the Valuation Solver to write after each successful solution attempt all gefundenden solutions as well as the remaining equations and variables into a Macsyma batch file whose name is contained in the option variable SolverDumpFile. This option is intended for extensive problems, in which Macsyma is inclined with acute lack of main and Swap memory to system crashes. At least a part of the results can be saved by the storage of the intermediate results with a crash.
SolverDumpFile < " SOLVER.dmp " > contains the name of the file,
into which the intermediate results are to be written.
40 CHAPTERS 2. THE SOLVER
2.5 definition and integration of user specific transformation routines
Encounters not rarely the Valuation Solver during the solution process
equations, which it is not able to solve, although already some simple
shapings or summaries could help, in order the desired result to receive.
Like that the SOLVE function is e.g. not in a the position to resolve the
equation
x + sin^2 x + cos^2 x = 1 (2.15)
correctly after x since it does not notice that the square sine and
Cosine terms can be combined easy into 1.
In order to be able to execute depending upon application fitting shapings of the equations, the dynamic integration of user-defined transformation functions is intended in the Solver, which are applied if necessary to unresolved equations. The Valuation Solver transfers among other things the equation and the variable to these functions, after which the equation is to be solve. The transformation function has now the function to transform and return as its function value again to the Solver the equation, so that a renewed solution attempt can be begun.
The call of a transformation function from the Solver occurs in the following way:
Transform: CopyList(SolverTransforms).
:
SOLVER LOOP
:
Trans: Pop(Transform),
TransEq: Apply(of trans, [ Equation, variable, Solution ])
:
LOOP END
Thereby contains the additionally transferred parameter Solution the result of the failed solution attempt, on the basis whose possibly helpful conclusions can be drawn. The following instruction shows an example of the definition of a simple, user specific transformation routine, which tries, to make an equation solvable by summary of trigonometric functions:
TransformTrig(Equation, variable, Solution): =
TrigSimp(Equation) $
For the integration of the function must be entered their name into the global list SolverTransforms:
SolverTransforms: [ ' TransformTrig ] $
Also the application of a transformation routine there to be unsuccessful
can, exists the possibility of displaying to the Solver the failure of
the shaping attempt by the return of an empty list ([ ]) as function value.
In this way it is prevented that a further call of the
2.5. USER SPECIFIC TRANSFORMATION ROUTINES 41
SOLVE function with the same equation takes place. Instead the Solver
tries directly, if available, to execute the next transformation in the
list.
When alternative to the shaping of the equation and following solution
by the internal Solver is it a transformation routine also permitted, to
intend the solutions for the transferred equation and return them in the
form
[ variable = Solution_1, variable = Solution_2... ].
Possible area of application for such functions would be the application
of numeric procedures for the solution of nonlinear equations, which contain
only one variable and no parameters. Or, as the transformation routine
calls a Macsyma BREAK, the equation can be manipulated by hand.
The methodology for the definition and integration of transformation
functions with success acknowledgement is to be demonstrated in the following
by a completely calculated example. To solve the set of equations (2.16)-(2.18)
after the variables x, y and z. the function is quite simple in principle,
but it causes substantial difficulties nevertheless to the Solver.
(2.16)
(2.17)
(2.18)
First the set of equations as equation list as well as the variables are input:
(COM11) TrigEq:
[ z - sin(x) = 0,
y + z^2 + cos(x)^2 = 5,
y + x = 1
] $
(COM12) Var: [ x, y, z]$
The first solution attempt is to be executed without transformation functions:
(COM13) SolverTransforms: [ ] $
(COM14) Solver(TrigEq, Var);
The variable to solved for are [ X, Y, Z ]
Checking for inconsistencies...
... none found.
Searching for linear equations...
... with respect to: [ X, Y, Z ]
Found 2 linear equations in 2 variables.
The variables to solved for are [ Y, Z ]
The equations are [ Z - SIN(x), Y + X - 1 ]
42 CHAPTERS 2. THE SOLVER
Solving linear equations.
The solutions are [ Y = 1 - X, Z = SIN(x) ]
Checking for remaining equations.
1 equation(s) and 1 variable (s) left.
The variable to solved for are [ X ]
Trying to solve equation 1 for X Valuation: (irrelevant)
2
2
The equation is SIN (x) + COS (x) - X - 4 = 0
Checking if equation was solved correctly.
2 2
The solutions are [ X = SIN (x) + COS (x) - 4 ]
Solution is not correct.
Cannot solve equation. Giving up.
Post processing results.
Cannot determine explicit on solution for X
2 2
(D14) [[ Y = 1 - X, Z = SIN(x), [ SIN (x) + COS (x)
- X - 4]]]
The Solver is able to resolve in the above-mentioned case the equation with the square sine and Cosine terms not after the variable x and returns therefore the unresolved equation in implicit form as last item of the solution list. For the simplification of the equations now two transformation functions are to be defined and merged. The first transformation serves the summary of logarithm terms, second for the simplification of printouts, which contain trigonometric functions. Both functions check whether their application was successful and returns an empty list, if the transformation did not cause modification of the equation.
(COM15) TransformLog(Equation, variable, Solution):
= BLOCK(
[ Eq ], Eq: LogContract(Equation),
IF Eq = Equation THEN RETURN([ ])
ELSE RETURN(Eq)
) $
(COM16) TransformTrig(Equation, variable, Solution):
= BLOCK(
[ Eq ],
Eq: TrigSimp(Equation),
IF Eq = Equation THEN RETURN([ ])
ELSE RETURN(Eq)
) $
2.5. USER SPECIFIC TRANSFORMATION ROUTINES 43
the transformation function TransformLog may be applied basically as
first, therefore the integration of the routines takes place in the order:
(COM17) SolverTransforms: [ ' TransformLog, ' TransformTrig
] $
now can be begun a new Solver run:
(COM18) Solver(TrigEq, Var);
The variable to solved for are [ X, Y, Z ]
Checking for inconsistencies...
... none found.
Searching for linear equations...
... with respect to: [ X, Y, Z ]
Found 2 linear equations in 2 variable.
The variable to solved for are [ Y, Z ]
The equations are [ Z - SIN(x), Y + X - 1 ]
Solving linear equations.
The solutions are [ Y = 1 - X, Z = SIN(x) ]
Checking for remaining equations.
1 equation(s) and 1 variable (s) left.
The variable to solved for are [ X ]
Trying to solve equation 1 for X Valuation: (irrelevant)
The equation is SIN (x) + COS (x) - X - 4 = 0
Checking if equation which solved correctly.
2 2
The solutions are [ X = SIN (x) + COS (x) - 4 ]
Solution is not correct.
Here the Solver determines as in the preceded case the fact that it cannot solve the equation and applies the first transformation in the list to it.
Applying transformation TRANSFORMLOG
transformation failed.
Since the equation does not contain logarithmic functions, the shaping attempt is not successful. Therefore the second transformation routine is tried.
Applying transformation TRANSFORMTRIG
The transformation yields - X - 3 = 0
Retrying with transformed equation.
Checking if equation which solved correctly.
The solutions are [ X = - 3 ]
Solution is correct.
44 CHAPTERS 2. THE SOLVER
Solution 1 for X
Checking for inconsistencies...
... none found. Consistent solutions for X: [ X =
- 3 ]
Checking for remaining equations.
All variable solved for. No equations left. Post
processing results.
(D18) [ [ X = - 3, Y = 4, Z = - SIN(3)]]
the transformation function TransformTrig succeeds the summary to the
equation, so that the Solver is following able, to intend the correct solution
for the variable x.
2.6 Modification of the operator evaluations
Over the user of the Solvers the possibility for the non-standard influence
of the heuristic complexity evaluation of algebraic printouts to give,
is stored the evaluations of the arithmetic operators not as unchangeable
constants, but as Macsyma of properties under the keyword Valuation. The
definition of a quality factor takes place with the instruction
SetProp(operator, ' Valuation, quality factor) $.
The standard mappings of the quality factors pre-defined in the Solver became with the instruction
SetProp(' sin, ' Valuation, 10) $
SetProp(' cos, ' Valuation, 10) $
SetProp(' tan, ' Valuation, 10) $
SetProp(' asin, ' Valuation, 12) $
SetProp(' acos, ' Valuation, 12) $
SetProp(' atan, ' Valuation, 12) $
SetProp(' sinh, ' Valuation, 12) $
SetProp(' cosh, ' Valuation, 12) $
SetProp(' tanh, ' Valuation, 12) $
SetProp(' asinh, ' Valuation, 12) $
SetProp(' acosh, ' Valuation, 12) $ 10) $
2.7. USER SPECIFIC EVALUATION STRATEGIES 45
determined. E.g. if the evaluation of the tanh operator on 20 is to
be modified, then this can be achieved at any time by
SetProp(' tanh, ' Valuation, 20) $.
A quality factor with the GET command leaves itself queries:
Get(operator, ' Valuation);
Example: The evaluation " * " - operator determined through:
(COM19) Get("* ", ' Valuation);
(D19) 4
if the query as result the value FALSE it supplies then this means
that no quality factor for the operator concerned was defined.
2,7 definition and integration of user specific evaluation strategies
of means of a Umbelegung of the procedure variable SolverValuationStrategy
the Valuation Solver is arranged to use in place of the internal function
a user-defined evaluation strategy to the determination of the solution
sequence. The function become by their call with
SolveOrder: Apply(SolverValuationStrategy, [ VarPathMatrix, ValuationMatrix
])
two evaluation stencils, which transfer variable path matrix and the
complexity evaluation matrix, their line dimension equal the number of
the equations and their column dimension equal the number of variables
which can be determined is.
The entry at the position (i; j) of the variable path matrix corresponds
to the number of paths to the instances the variable xj in equation i (see
paragraph 1,5,2). The complexity evaluation matrix contains at the position
(i; j) the complexity evaluation of the equation i regarding the variables
xj. For the set of equations (1.55) - (1.57) the path matrix reads thus
2 6 4 x y z (1:55) 1 1 1 (1:56) 1 1 2 (1:57) 1 1 1 3 7 5;
and the evaluation matrix on use of the standard operator quality factors
calculated to
2 6 4 x y z (1:55) 1 4 1 (1:56) 4 4 14 (1:57) 4 1 40 3 7 5:
46 CHAPTERS 2. The SOLVER
from these stencils must generate the evaluation strategy a solution
sequence (see paragraph 1,5,2) in the form
[ [ i1, j1, b1 ], [ i2, j2, b2 ]... ].
The max. length of the list should not be larger thereby than the value
of the option variables SolverMaxLenValOrder.
The basic structure of an evaluation strategy is shown by the following
pseudocode:
MyValuationStrategy(PathMat, ValMat): = BLOCK(generates a solution
order from the evaluation strategy.
limits the length of the list to SolverMaxLenValOrder of items RETURN(the
solution sequence)) $
for the integration of the function is afterwards the procedure variable
SolverValuationStrategy with the function name to be occupied:
SolverValuationStrategy: ' MyValuationStrategy$
BIBLIOGRAPHY [ADA 79] D. Adams, The Hitchhiker's Guide to the Galaxy, London:
Pan Books, 1979
[BRO 88] E. Brommundt, G. Sachs, Technische Mechanik | Eine Einführung,
Berlin: Springer Verlag, 1988
[FOU 92] L. R. Foulds, Graph Theory Applications, Berlin, New York,
Tokyo: Springer Verlag, 1992
[HEN 93] E. Hennig, \Mathematische Grundlagen und Implementation eines
symbolischen Matrixapproximationsalgorithmus mit quantitativer Fehlervorhersage",
Diplomar- beit am Institut für Netzwerktheorie und Schaltungstechnik,
Technische Universität Braunschweig, August 1993
[MAC 94] Macsyma Inc., Macsyma Reference Manual V14, Arlington, 1994
[MIC 94] Z. Michalewicz, Genetic Algorithms + Data Structures = Evolution
Programs, Berlin, New York, Tokyo: Springer Verlag, 1994
[NÜH 89] D. Nührmann, Professionelle Schaltungstechnik, München:
Franzis Verlag, 1989
[PFA 94] J. Pfalzgraf, \Some Robotics Benchmarks for Computer Algebra",
Vortrag auf der GAMM-Jahrestagung 1994, Braunschweig, April 1994
[SOM 93a] R. Sommer, \Konzepte und Verfahren zum rechnergestützten
Entwurf analoger Schaltungen", Dissertation am Institut für Netzwerktheorie
und Schaltungstechnik, Technische Universität Braunschweig, Juli 1993
[SOM 93b] R. Sommer, E. Hennig, G. Dröge, E.-H. Horneber, \Equation-based
Symbolic Approximation by Matrix Reduction with Quantitative Error Prediction",
Alta Frequenza | Rivista di Elettronica, Vol. 5, No. 6, Dez. 1993, S. 29
{ 37 [TRI 91] H. Trispel, \Symbolische Schaltungsanalyse mit Macsyma",
Studienarbeit am In- stitut für Netzwerktheorie und Schaltungstechnik,
Technische Universität Braun- schweig, Juli 1991 [WOL 91] S. Wolfram,
Mathematica | A System for Doing Mathematics by Computer, Addi- son Wesley,
1991 47
48 BIBLIOGRAPHYAppendix A example calculations A.1 sizing of the staff two-impact solve for the demonstration of the efficiency of the Solvers finally the Beipiel stated in the introduction gave up 1,1 and 1,2. In the following are the complete display outputs of the two program runs. (COM1) Two-impact: [ F*cos(gamma) - S1*cos(alpha) - S2*cos(beta) = 0, F*sin(gamma) - S1*sin(alpha) + S2*sin(beta) = 0, Delta_l1 = l1*S1/(E*A1), Delta_l2 = l2*S2/(E*A2), l1 = c/cos(alpha), l2 = c/cos(beta), a = Delta_l2/sin(alpha+beta), b = Delta_l1/sin(alpha+beta), u = a*sin(alpha) + b*sin(beta), w = a*cos(alpha) + b*cos(beta), A1 = h1^2, a2 = h2^2 ] $ (COM2) staff dimensions: [ h1, h2]$ (COM3) two-impact parameter: [ alpha, beta, gamma, F, c, E, u, w]$ (COM4) SolverRepeatImmed: SolverRepeatLinear: FALSE$ (COM5) MsgLevel: ' DETAIL$ (COM6) Solver(two-impact, staff dimensions, two-impact parameters); The variable to solved for AR [ H1, H2 ] The of parameter of AR [ ALPHA, BETA, C, E, F, U, W, GAMMA ] 49
50 APPENDIX A. EXAMPLE CALCULATIONS
Checking for inconsistencies...
... none found.
Searching for immediate assignments.
C
Assigning L1 = ----------
COS(ALPHA)
C
Assigning L2 = ---------
COS(BETA)
Checking for inconsistencies...
... none found.
Searching for linear equations...
...with respect to: [H1, H2, A, A1, A2, B, DELTA_L1, DELTA_L2, S1, S2]
Found 7 linear equations in 7 variables.
The variables to be solved for are [A, A1, B, DELTA_L1, DELTA_L2, S1, S2]
The equations are [F COS(GAMMA) - COS(BETA) S2 - COS(ALPHA) S1,
F SIN(GAMMA) + SIN(BETA) S2 - SIN(ALPHA) S1,
A SIN(BETA + ALPHA) - DELTA_L2, B SIN(BETA + ALPHA) - DELTA_L1,
2
U - B SIN(BETA) - A SIN(ALPHA), W - B COS(BETA) + A COS(ALPHA), A1 - H1 ]
Solving linear equations.
COS(BETA) U - SIN(BETA) W
The solutions are [A = -------------------------------------------,
COS(ALPHA) SIN(BETA) + SIN(ALPHA) COS(BETA)
2 SIN(ALPHA) W + COS(ALPHA) U
A1 = H1 , B = -------------------------------------------,
COS(ALPHA) SIN(BETA) + SIN(ALPHA) COS(BETA)
DELTA_L1 = (SIN(ALPHA) SIN(BETA + ALPHA) W
+ COS(ALPHA) SIN(BETA + ALPHA) U)/(COS(ALPHA) SIN(BETA)
+ SIN(ALPHA) COS(BETA)), DELTA_L2 =
COS(BETA) SIN(BETA + ALPHA) U - SIN(BETA) SIN(BETA + ALPHA) W
-------------------------------------------------------------,
COS(ALPHA) SIN(BETA) + SIN(ALPHA) COS(BETA)
COS(BETA) F SIN(GAMMA) + SIN(BETA) F COS(GAMMA)
S1 = -----------------------------------------------,
COS(ALPHA) SIN(BETA) + SIN(ALPHA) COS(BETA)
SIN(ALPHA) F COS(GAMMA) - COS(ALPHA) F SIN(GAMMA)
S2 = -------------------------------------------------]
A.1. SIZING of the STAFF TWO-IMPACT 51
COS(ALPHA) SIN(BETA) + SIN(ALPHA) COS(BETA)
Checking for inconsistencies...
... none found.
Checking for remaining equations.
3 equation(s) and 2 variable(s) left.
The variables to be solved for are [H1, H2]
Applying valuation strategy.
Trying to solve equation 3 for H2
Valuation: 10
2
The equation is A2 - H2 = 0
Checking if equation was solved correctly.
The solutions are [H2 = - SQRT(A2), H2 = SQRT(A2)]
Solution is correct.
The solution is not unique. Tracing paths separately.
Solution 1 for H2
Checking for inconsistencies...
... none found.
Solution 2 for H2
Checking for inconsistencies...
... none found.
Consistent solutions for H2 : [H2 = - SQRT(A2), H2 = SQRT(A2)]
Checking for remaining equations.
2 equation(s) and 2 variable(s) left.
The variables to be solved for are [A2, H1]
Applying valuation strategy.
Trying to solve equation 2 for A2
Valuation: 8
The equation is COS(ALPHA) C F SIN(GAMMA) - SIN(ALPHA) C F COS(GAMMA)
- A2 COS(BETA) SIN(BETA) SIN(BETA + ALPHA) E W
2
+ A2 COS (BETA) SIN(BETA + ALPHA) E U = 0
Checking if equation was solved correctly.
The solutions are [A2 = (COS(ALPHA) C F SIN(GAMMA)
- SIN(ALPHA) C F COS(GAMMA))/(COS(BETA) SIN(BETA) SIN(BETA + ALPHA) E W
2
- COS (BETA) SIN(BETA + ALPHA) E U)]
Solution is correct.
Solution 1 for A2
Checking for inconsistencies...
... none found.
Consistent solutions for A2 : [A2 = (COS(ALPHA) C F SIN(GAMMA)
- SIN(ALPHA) C F COS(GAMMA))/(COS(BETA) SIN(BETA) SIN(BETA + ALPHA) E W
52 APPENDIX A. EXAMPLE CALCULATIONS 2 - COS (BETA) SIN(beta + ALPHA) E U) ] Checking for remaining equations. 1 equation(s) and 1 variable (s) left. The variable to solved for AR [ H1 ] Trying to solve equation 1 for H1 Valuation: (irrelevant) The equation is - COS(beta) C F SIN(gamma) - SIN(beta) C F COS(gamma) 2 + COS(alpha) SIN(alpha) SIN(beta + ALPHA) E H1 W 2 2 + COS (ALPHA) SIN(beta + ALPHA) E H1 U = 0 Checking if equation which solved correctly. The solutions AR [ H1 = - SQRT(cos(beta) C F SIN(gamma) / (COS(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)), H1 = SQRT(cos(beta) C F SIN(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) 2 E W + COS (ALPHA) SIN(beta + ALPHA) E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)) ] Solution is correct. The solution is NOT unique. Tracing paths separately. Solution 1 for H1 Checking for inconsistencies...... none found. Solution 2 for H1 Checking for inconsistencies...... none found. Consistent solutions for H1: [ H1 = - SQRT(cos(beta) C F SIN(gamma) / (COS(alpha) SIN(alpha) SIN(beta + ALPHA) E W
A.1. SIZING of the STAFF TWO-IMPACT 53 2 + COS (ALPHA) SIN(beta + ALPHA) E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)), H1 = SQRT(cos(beta) C F SIN(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) 2 E W + COS (ALPHA) SIN(beta + ALPHA) E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)) ] Checking for remaining equations. All variable solved for. NO equations left. Checking for remaining equations. All variable solved for. NO equations left. Checking for remaining equations. 2 equation(s) and 2 variable (s) left. The variable to solved for AR [ a2, H1 ] Applying valuation strategy. Trying to solve equation 2 for a2 Valuation: 8 The equation is COS(alpha) C F SIN(gamma) - SIN(alpha) C F COS(gamma) - a2 COS(beta) SIN(beta) SIN(beta + ALPHA) E W 2 + a2 COS (BETA) SIN(beta + ALPHA) E U = 0 Checking if equation which solved correctly. The solutions AR [ a2 = (COS(alpha) C F SIN(gamma) - SIN(alpha) C F COS(gamma))/(cos(beta) SIN(beta) SIN(beta + ALPHA) E W 2 - COS (BETA) SIN(beta + ALPHA) E U) ] Solution is correct. Solution 1 for a2 Checking for inconsistencies...... none found. Consistent solutions for a2: [ A2 = (COS(alpha) C F SIN(gamma) - SIN(alpha) C F COS(gamma))/(cos(beta) SIN(beta) SIN(beta + ALPHA) E W
54 APPENDIX A. EXAMPLE CALCULATIONS 2 - COS (BETA) SIN(beta + ALPHA) E U) ] Checking for remaining equations. 1 equation(s) and 1 variable (s) left. The variable to solved for AR [ H1 ] Trying to solve equation 1 for H1 Valuation: (irrelevant) The equation is - COS(beta) C F SIN(gamma) - SIN(beta) C F COS(gamma) 2 + COS(alpha) SIN(alpha) SIN(beta + ALPHA) E H1 W 2 2 + COS (ALPHA) SIN(beta + ALPHA) E H1 U = 0 Checking if equation which solved correctly. The solutions AR [ H1 = - SQRT(cos(beta) C F SIN(gamma) / (COS(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)), H1 = SQRT(cos(beta) C F SIN(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) 2 E W + COS (ALPHA) SIN(beta + ALPHA) E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)) ] Solution is correct. The solution is NOT unique. Tracing paths separately. Solution 1 for H1 Checking for inconsistencies...... none found. Solution 2 for H1 Checking for inconsistencies...... none found. Consistent solutions for H1: [ H1 = - SQRT(cos(beta) C F SIN(gamma) / (COS(alpha) SIN(alpha) SIN(beta + ALPHA) E W
A.1. SIZING of the STAFF TWO-IMPACT 55 2 + COS (ALPHA) SIN(beta + ALPHA)
E U) + SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA)
E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)), H1 = SQRT(cos(beta) C F SIN(gamma)/(cos(alpha)
SIN(alpha) SIN(beta + ALPHA) 2 E W + COS (ALPHA) SIN(beta + ALPHA) E U)
+ SIN(beta) C F COS(gamma)/(cos(alpha) SIN(alpha) SIN(beta + ALPHA) E W
2 + COS (ALPHA) SIN(beta + ALPHA) E U)) ] Checking for remaining equations.
All variable solved for. NO equations left. Checking for remaining equations.
All variable solved for. NO equations left. Post processing results. (D6)
[ [ H1 = - SQRT((cos(beta) C F SIN(gamma) + SIN(beta) C F COS(gamma)) /
(COS(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta +
ALPHA) E U)), H2 = - SQRT((sin(alpha) C F COS(gamma) - COS(alpha) C F SIN(gamma))
2 / (COS (BETA) SIN(beta + ALPHA) E U - COS(beta) SIN(beta) SIN(beta +
ALPHA) E W)) ], [ H1 = SQRT((cos(beta) C F SIN(gamma) + SIN(beta) C F COS(gamma))
/ (COS(alpha) SIN(alpha) SIN(beta + ALPHA) E W 2 + COS (ALPHA) SIN(beta
+ ALPHA) E U))
56 APPENDIX A. EXAMPLE CALCULATIONS E W)) ], [ H1 = - SQRT((cos(beta) C
F SIN(gamma) + SIN(beta) C F COS(gamma)) / (COS(alpha) SIN(alpha) SIN(beta
+ ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA) E U)), H2 = SQRT((sin(alpha)
C F COS(gamma) - COS(alpha) C F SIN(gamma)) 2 / (COS (BETA) SIN(beta +
ALPHA) E U - COS(beta) SIN(beta) SIN(beta + ALPHA) E W)) ], [ H1 = SQRT((cos(beta)
C F SIN(gamma) + SIN(beta) C F COS(gamma)) / (COS(alpha) SIN(alpha) SIN(beta
+ ALPHA) E W 2 + COS (ALPHA) SIN(beta + ALPHA
only by the combination of their signs differentiate. Due to the physical
boundary condition that the lengths
h1 and h2 cannot take negative values, it is only the last solution
meaningful:
h1 = s cos fi F sin + sin fi F cos cos sin sin (fi + Ew+ cos 2 sin
(fi + E u (A.1) h2 = sh2 = s sin F cos sin F cos
A.2. SIZING of the TRANSISTOR AMPLIFIER 57 A.2 sizing of the transistor amplifier (COM7) amplifier: [ -v_v1+v_r1+v_oc_1+v_fix2_q1 = 0, -v_v1+v_r2+v_oc_1+v_i1+v_fix2_q1 = 0, -v_v1+v_oc_1+v_i1+v_fix2_q2+v_fix2_q1-v_fix1_q2-v_fix1_q1 = 0, V_v1-v_r6+v_r3-v_oc_1-v_i1-v_fix2_q1+v_fix1_q1 = 0, -v_v1+v_r7+v_r4+v_oc_1+v_i1-v_fix1_q2 = 0, -v_v1+v_r7+v_r5+v_oc_1 = 0, V_oc_2-v_i1+v_fix1_q2 = 0, v_v_cc-v_v1+v_r6+v_oc_1+v_fix2_q1-v_fix1_q1 = 0, I_v1+i_oc_1 = 0, i_r7-i_oc_1+i_fix2_q1 = 0, i_r2+i_r1-i_fix2_q1-i_fix1_q1 = 0, I_r6+i_fix2_q2+i_fix1_q1 = 0, -i_r7+i_r5+i_r4 = 0, -i_r4+i_oc_2-i_fix2_q2-i_fix1_q2 = 0, i_r3-i_r2+i_i1+i_fix1_q2 = 0, I_v_cc-i_r6-i_r3 = 0, v_v1 = 0, i_i1 = 0, i_oc_1 = 0, v_fix1_q1 = 2,72, V_fix2_q1 = 0,607, i_r1*r1-v_r1 = 0, i_r7*r7-v_r7 = 0, i_r2*r2-v_r2 = 0, I_r6*r6-v_r6 = 0, v_fix1_q2 = 6,42, v_fix2_q2 = 0,698, i_r3*r3-v_r3 = 0, I_r4*r4-v_r4 = 0, i_r5*r5-v_r5 = 0, [ g 1, R2, R3, R4, R5, R6, R7]$ (COM9) Design parameter: [ VCC, A, ZIN, ZOUT]$ (COM10) SolverRepeatImmed: SolverRepeatLinear: TRUE$ (COM11) Solver(amplifier, resistances, Design parameters); The variable to solved for AR [ g 1, R2, R3, R4, R5, R6, R7 ] The of parameter of AR [ A, VCC, ZIN, ZOUT ] Checking for inconsistencies...... none found. Searching for immediate assignments. Assigning V_v1 = 0 Assigning I_i1 = 0 Assigning I_oc_1 = 0 68 Assigning V_fix1_q1 = -- 25,607 Assigning V_fix2_q1 = --- 1000 321 Assigning V_fix1_q2 = -- 50
58 APPENDIX A. EXAMPLE CALCULATIONS 349 Assigning V_fix2_q2 = -- 500 Assigning I_oc_2 = 0 Assigning V_v_cc = VCC 111 Assigning I_fix1_q1 = ------ 1000000 5 Assigning I_fix2_q1 = ------ 8695637 401 Assigning I_fix1_q2 = ----- 100000 25 Assigning I_fix2_q2 = ------ 1984127 Assigning R7 = ZIN Checking for inconsistencies...... none found. Searching for immediate assignments. Assigning I_v1 = 0 Checking for inconsistencies...... none found. Searching for immediate assignments. NO immediate assignments found. Searching for linear equations...... with respect to: [ g 1, R2, R3, R4, R5, R6, I_r1, I_r2, I_r3, I_r4, I_r5, I_r6, I_r7, I_v_cc, V_i1, V_oc_1, V_oc_2, V_r1, V_r2, V_r3, V_r4, V_r5, V_r6, V_r7 ] Found 18 linear equations in 18 variable. The variable to solved for AR [ I_r1, I_r2, I_r3, I_r4, I_r5, I_r6, I_r7, I_v_cc, V_i1, V_oc_1, V_oc_2, V_r1, V_r2, V_r3, V_r4, V_r5, V_r6, V_r7 ] The equations AR [ 1000 V_r1 + 1000 V_oc_1 + 607, 1000 V_r2 + 1000 V_oc_1 + 1000 V_i1 + 607, 200 V_oc_1 + 200 V_i1 - 1567, - 1000 V_r6 + 1000 V_r3 - 1000 V_oc_1 - 1000 V_i1 + 2113, 50 V_r7 + 50 V_r4 + 50 V_oc_1 + 50 V_i1 - 321, V_r7 + V_r5 + V_oc_1, 50 V_oc_2 - 50 V_i1 + 321, 1000 V_r6 +
A.2. SIZING of the TRANSISTOR AMPLIFIER 59 8695637 I_r7 + 5, 8695637000000 I_r2 + 8695637000000 I_r1 - 970215707, 1984127000000 I_r6 + 245238097, - I_r7 + I_r5 + I_r4, - 198412700000 I_r4 - 798134927, 100000 I_r3 - 100000 I_r2 + 401, I_v_cc - I_r6 - I_r3, I_r7 ZIN - V_r7 ] Solving linear equations. 8695637000000 I_r3 + 33899288663 The solutions AR [ I_r1 = -, 8695637000000 100000 I_r3 + 401 798134927 I_r2 =, I_r4 = -, 100000 198412700000 6939299538713499 245238097 5 I_r5 =, I_r6 = -, I_r7 = -, 1725324815389900000 1984127000000 8695637 1984127000000 I_r3 - 245238097 200 V_oc_1 - 1567 I_v_cc =, V_i1 = -, 1984127000000 200,200 V_oc_1 - 283 1000 V_oc_1 + 607 4221 V_oc_2 = -, V_r1 = -, V_r2 = -, 200 1000 500,200 V_oc_1 + 200 VCC - 1567 1000 ZIN - 2460865271 V_r3 = -, V_r4 =, 200 1739127400 8695637 V_oc_1 - 5 ZIN 1000 V_oc_1 + 1000 VCC - 2113 V_r5 = -, Searching for linear equations...... with respect to: [ I_r3, g 1, R2, R3, R4, R5, R6, V_oc_1 ] Found 6 linear equations in 5 variable. The variable to solved for AR [ R2, R4, R5, R6, V_oc_1 ] The equations AR [ 8695637000000 V_oc_1 - 8695637000000 I_r3 g 1 - 33899288663 g 1 + 5278251659000, 1984127000000 V_oc_1 + 1984127000000 VCC
60 APPENDIX A. EXAMPLE CALCULATIONS - 245238097 R6 - 4192460351000, 200 V_oc_1 + 200 VCC + 200 I_r3 R3 - 1567, - 992063500000 ZIN - 6940291602213499 R4 + 2441334613776708500, - 992063500000 ZIN + 1725324815389900000 V_oc_1 + 6939299538713499 R5, 145309663773 A g 1 - 145303681853 R2 ] Solving linear equations. Checking for inconsistencies...... none found. 145309663773 A g 1 The solutions AR [ R2 =, 145303681853 992063500000 ZIN - 2441334613776708500 R4 = -, 6940291602213499 R5 = - (- 9920635000000 ZIN + (17253248153899000000 I_r3 + 67260493917052201) g 1 - 10472721629416693000)/69392995387134990, R6 = (17253248153899000000 VCC + (17253248153899000000 I_r3 + 67260493917052201) g 1 - 46928834978605280000)/2132501470082789, (8695637000000 I_r3 + 33899288663) g 1 - 5278251659000 V_oc_1 = ] 8695637000000 Checking for inconsistencies...... none found. Searching for linear equations...... with respect to: [ I_r3, g 1, R3 ] NO linear equations found. Checking for remaining equations. 3 equation(s) and 3 variable (s) left. The variable to solved for AR [ I_r3, g 1, R3 ] Applying valuation strategy. Trying to solve equation 1 for I_r3 Valuation: 4 The equation is 14530966377300000 A I_r3 g 1 + 58269175172973 A g 1 + 122665368220302600 = 0 Checking if equation which solved correctly. 6474352796997 A g 1 + 13629485357811400 The solutions AR [ I_r3 = - ] 1614551819700000 A g 1
A.2. SIZING of the TRANSISTOR AMPLIFIER 61 Solution is correct. Solution 1 for I_r3 Checking for inconsistencies...... none found. Consistent solutions for I_r3: [ I_r3 = 6474352796997 A g 1 + 13629485357811400 - ] 1614551819700000 A g 1 Checking for remaining equations. 2 equation(s) and 2 variable (s) left. The variable to solved for AR [ g 1, R3 ] Applying valuation strategy. Trying to solve equation 1 for R3 Valuation: 20 The equation is - g 1 (- 16062755182397110876073408478539448677201569671148 # 116383372395290156600000000 A VCC + 64411648281412414613054367998943189195 # 578294381303946697323305113527966000 A R3 + 135601779249796410015811714375 # 830025732935651163832398508429761039502017200000 A + 135596196971410595846 # 324806740533400875556129557784494104259093864172929200000) - 1355961969714 # 10595846324806740533400875556129557784494104259093864172929200000 R3 - 179 # 2201925592952751292050414582157181324535016813917972727234536207382600 A 2 g 1 = 0 Checking if equation which solved correctly. The solutions AR [ R3 = (140395565418006489000000 A g 1 VCC 2 - 15664635352383720279 A g 1 + (- 1185219363258810780138000 A - 1185170571683430488618000) R1)/(562986217326206020890 A g 1 + 1185170571683430488618000) ] Solution is correct. Solution 1 for R3 Checking for inconsistencies...... none found. Consistent solutions for R3: [ R3 = (140395565418006489000000 A G 1 VCC 2 - 15664635352383720279 A G 1 + (- 1185219363258810780138000 A
62 APPENDIX A. EXAMPLE CALCULATIONS - 1185170571683430488618000) R1)/(562986217326206020890 A g 1 + 1185170571683430488618000) ] Checking for remaining equations. 1 equation(s) and 1 variable (s) left. The variable to solved for AR [ g 1 ] Trying to solve equation 1 for g 1 Valuation: (irrelevant) The equation is g 1 (19841637776253069394020865409024 ZOUT - 243498222421608533966015625 A ZIN) 2 - 57259002716458635629644396416 A g 1 = 0 Checking if equation which solved correctly. The solutions AR [ g 1 = 19841637776253069394020865409024 ZOUT - 243498222421608533966015625 A ZIN, 57259002716458635629644396416 A g 1 = 0 ] Solution is correct. The solution is NOT unique. Tracing paths separately. Solution 1 for g 1 Checking for inconsistencies...... none found. Solution 2 for g 1 Checking for inconsistencies...... none found. Consistent solutions for g 1: [ g 1 = 19841637776253069394020865409024 ZOUT - 243498222421608533966015625 A ZIN, 57259002716458635629644396416 A g 1 = 0 ] Checking for remaining equations. All variable solved for. NO equations left. Checking for remaining equations. All variable solved for. NO equations left. Post office processing results. (D11) [ [ G 1 = - (243498222421608533966015625 A ZIN - 19841637776253069394020865409024 ZOUT)
A.2. SIZING OF THE TRANSISTOR AMPLIFIER 63 / (57259002716458635629644396416 A), R2 = 1493854125285941926171875 A ZIN - 121727839118116990147367272448 ZOUT -, 351267764122759016747201152 R3 = (334763184724568105702258732451550460395708593115792893559436793085952 2 ZOUT + VCC 2 (106256457337115768692100530787195899963103895496024350000000000000 A ZIN - 8658388208766832396126274743883820688291739892383476917089599488000000 A ZOUT) + A (73094113258409599088098011387867214250558868171501312134070398877696000 ZOUT + ZIN (- 8216483067762383568115091483320660991714242249851965644800000000 ZOUT - 896980085605567678321894471091892803815897329518698154700000000000)) + 73091104214643137701992515955008538217110816411520639295650692857856000 2 ZOUT + A (50416680420198682402077210492446131565566605185394287109375 2 ZIN - 897017012839931319298712680905507787488523085777437562700000000000 ZIN))/(a (- 34720136717154997908466361722974120960049876968457742437529293946880 ZOUT - 210926324831111746833250971831897544037167089192987941918299029504000) 2 + 426088393921834232455323128456655558852046620939057643500000000 A 992063500000 ZIN - 2441334613776708500 ZIN), R4 = -, 6940291602213499
64 APPENDIX A. EXAMPLE CALCULATIONS R5 = (38195771383195691504052086845016246794667687936
ZOUT + A (99303995957664108704965549227744192421875 ZIN + 599657596227485533261336202180916694139772288000)
+ 8339540409811836894068029655629814665587863808000) / (3973373711375163964000922192169533030064195840
A), R6 = (- 38195771383195691504052086845016246794667687936 ZOUT + A (468741670456330507974731687410699967578125
ZIN - 2687098289520198765190831087202789799110676480000) + 987903782911837781320158487942202132025984000000
A VCC - 8339540409811836894068029655629814665587863808000) / (122104907468322449250303944919525361454884224
A), R7 = ZIN ], 992063500000 ZIN - 2441334613776708500 [ g 1 = 0, R2 =
0, R3 = 0, R4 = -, 6940291602213499 992063500000 ZIN + 1047272162941669300
R5 =, 6939299538713499 1984127000000 VCC - 5396825440000 R6 =, R7 = ZIN
] ] 245238097
more than one solution of the system are calculated also here, but
place only first a physically meaningful result