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The packages to_poly and to_poly_solve are experimental;
the specifications of the functions in these packages might change or
the some of the functions in these packages might be merged into other
Maxima functions.
Barton Willis (Professor of Mathematics, University of Nebraska at
Kearney) wrote the to_poly and to_poly_solve packages and the
English language user documentation for these packages.
The operator %and is a simplifying nonshort-circuited logical
conjunction. Maxima simplifies an %and expression to either true,
false, or a logically equivalent, but simplified, expression. The
operator %and is associative, commutative, and idempotent. Thus
when %and returns a noun form, the arguments of %and form
a non-redundant sorted list; for example
(%i1) a %and (a %and b); (%o1) a %and b
If one argument to a conjunction is the explicit the negation of another
argument, %and returns false:
(%i2) a %and (not a); (%o2) false
If any member of the conjunction is false, the conjunction simplifies to false even if other members are manifestly non-boolean; for example
(%i3) 42 %and false; (%o3) false
Any argument of an %and expression that is an inequation (that
is, an inequality or equation), is simplified using the Fourier
elimination package. The Fourier elimination simplifier has a
pre-processor that converts some, but not all, nonlinear inequations
into linear inequations; for example the Fourier elimination code
simplifies abs(x) + 1 > 0 to true, so
(%i4) (x < 1) %and (abs(x) + 1 > 0); (%o4) x < 1
Notes
prederror does not alter the
simplification %and expressions.
%and, %or, and not should be
fully parenthesized.
and and or are both
short-circuited. Thus and isn’t associative or commutative.
Limitations The conjunction %and simplifies inequations
locally, not globally. This means that conjunctions such as
(%i5) (x < 1) %and (x > 1); (%o5) (x > 1) %and (x < 1)
do not simplify to false. Also, the Fourier elimination code ignores the fact database;
(%i6) assume(x > 5); (%o6) [x > 5] (%i7) (x > 1) %and (x > 2); (%o7) (x > 1) %and (x > 2)
Finally, nonlinear inequations that aren’t easily converted into an equivalent linear inequation aren’t simplified.
There is no support for distributing %and over %or;
neither is there support for distributing a logical negation over
%and.
To use load("to_poly_solve")
Related functions %or, %if, and, or, not
Status The operator %and is experimental; the
specifications of this function might change and its functionality
might be merged into other Maxima functions.
The operator %if is a simplifying conditional. The
conditional bool should be boolean-valued. When the
conditional is true, return the second argument; when the conditional is
false, return the third; in all other cases, return a noun form.
Maxima inequations (either an inequality or an equality) are not
boolean-valued; for example, Maxima does not simplify 5 < 6
to true, and it does not simplify 5 = 6 to false; however, in
the context of a conditional to an %if statement, Maxima
automatically attempts to determine the truth value of an
inequation. Examples:
(%i1) f : %if(x # 1, 2, 8); (%o1) %if(x - 1 # 0, 2, 8) (%i2) [subst(x = -1,f), subst(x=1,f)]; (%o2) [2, 8]
If the conditional involves an inequation, Maxima simplifies it using the Fourier elimination package.
Notes
(%i3) %if(42,1,2); (%o3) %if(42, 1, 2)
if is nary, the operator %if isn’t
nary.
Limitations The Fourier elimination code only simplifies nonlinear inequations that are readily convertible to an equivalent linear inequation.
To use: load("to_poly_solve")
Status: The operator %if is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The operator %or is a simplifying nonshort-circuited logical
disjunction. Maxima simplifies an %or expression to either
true, false, or a logically equivalent, but simplified,
expression. The operator %or is associative, commutative, and
idempotent. Thus when %or returns a noun form, the arguments
of %or form a non-redundant sorted list; for example
(%i1) a %or (a %or b); (%o1) a %or b
If one member of the disjunction is the explicit the negation of another
member, %or returns true:
(%i2) a %or (not a); (%o2) true
If any member of the disjunction is true, the disjunction simplifies to true even if other members of the disjunction are manifestly non-boolean; for example
(%i3) 42 %or true; (%o3) true
Any argument of an %or expression that is an inequation (that
is, an inequality or equation), is simplified using the Fourier
elimination package. The Fourier elimination code simplifies
abs(x) + 1 > 0 to true, so we have
(%i4) (x < 1) %or (abs(x) + 1 > 0); (%o4) true
Notes
prederror does not alter the
simplification of %or expressions.
%and, %or, and not; the binding powers of these
operators might not match your expectations.
and and or are both short-circuited.
Thus or isn’t associative or commutative.
Limitations The conjunction %or simplifies inequations
locally, not globally. This means that conjunctions such as
(%i1) (x < 1) %or (x >= 1); (%o1) (x > 1) %or (x >= 1)
do not simplify to true. Further, the Fourier elimination code ignores the fact database;
(%i2) assume(x > 5); (%o2) [x > 5] (%i3) (x > 1) %and (x > 2); (%o3) (x > 1) %and (x > 2)
Finally, nonlinear inequations that aren’t easily converted into an equivalent linear inequation aren’t simplified.
The algorithm that looks for terms that cannot both be false is weak;
also there is no support for distributing %or over %and;
neither is there support for distributing a logical negation over
%or.
To use load("to_poly_solve")
Related functions %or, %if, and, or, not
Status The operator %or is experimental; the
specifications of this function might change and its functionality
might be merged into other Maxima functions.
The predicate complex_number_p returns true if its argument is
either a + %i * b, a, %i b, or %i,
where a and b are either rational or floating point
numbers (including big floating point); for all other inputs,
complex_number_p returns false; for example
(%i1) map('complex_number_p,[2/3, 2 + 1.5 * %i, %i]);
(%o1) [true, true, true]
(%i2) complex_number_p((2+%i)/(5-%i));
(%o2) false
(%i3) complex_number_p(cos(5 - 2 * %i));
(%o3) false
Related functions isreal_p
To use load("to_poly_solve")
Status The operator complex_number_p is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function call compose_functions(l) returns a lambda form that is
the composition of the functions in the list l. The functions are
applied from right to left; for example
(%i1) compose_functions([cos, exp]);
%g151
(%o1) lambda([%g151], cos(%e ))
(%i2) %(x);
x
(%o2) cos(%e )
When the function list is empty, return the identity function:
(%i3) compose_functions([]); (%o3) lambda([%g152], %g152) (%i4) %(x); (%o4) x
Notes
funmake (not compose_functions)
signals an error:
(%i5) compose_functions([a < b]); funmake: first argument must be a symbol, subscripted symbol, string, or lambda expression; found: a < b #0: compose_functions(l=[a < b])(to_poly_solve.mac line 40) -- an error. To debug this try: debugmode(true);
new_variable.
(%i6) compose_functions([%g0]); (%o6) lambda([%g154], %g0(%g154)) (%i7) compose_functions([%g0]); (%o7) lambda([%g155], %g0(%g155))
Although the independent variables are different, Maxima is able to to deduce that these lambda forms are semantically equal:
(%i8) is(equal(%o6,%o7)); (%o8) true
To use load("to_poly_solve")
Status The function compose_functions is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function dfloat is a similar to float, but the function
dfloat applies rectform when float fails to evaluate
to an IEEE double floating point number; thus
(%i1) float(4.5^(1 + %i));
%i + 1
(%o1) 4.5
(%i2) dfloat(4.5^(1 + %i));
(%o2) 4.48998802962884 %i + .3000124893895671
Notes
float is both an option variable (default
value false) and a function name.
Related functions float, bfloat
To use load("to_poly_solve")
Status The function dfloat is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function elim eliminates the variables in the set or list
x from the equations in the set or list l. Each member
of x must be a symbol; the members of l can either be
equations, or expressions that are assumed to equal zero.
The function elim returns a list of two lists; the first is
the list of expressions with the variables eliminated; the second
is the list of pivots; thus, the second list is a list of
expressions that elim used to eliminate the variables.
Here is an example of eliminating between linear equations:
(%i1) elim(set(x + y + z = 1, x - y - z = 8, x - z = 1),
set(x,y));
(%o1) [[2 z - 7], [y + 7, z - x + 1]]
Eliminating x and y yields the single equation 2 z - 7 = 0;
the equations y + 7 = 0 and z - z + 1 = 1 were used as pivots.
Eliminating all three variables from these equations, triangularizes the linear
system:
(%i2) elim(set(x + y + z = 1, x - y - z = 8, x - z = 1),
set(x,y,z));
(%o2) [[], [2 z - 7, y + 7, z - x + 1]]
Of course, the equations needn’t be linear:
(%i3) elim(set(x^2 - 2 * y^3 = 1, x - y = 5), [x,y]);
3 2
(%o3) [[], [2 y - y - 10 y - 24, y - x + 5]]
The user doesn’t control the order the variables are eliminated. Instead, the algorithm uses a heuristic to attempt to choose the best pivot and the best elimination order.
Notes
eliminate, the function
elim does not invoke solve when the number of equations
equals the number of variables.
elim works by applying resultants; the option
variable resultant determines which algorithm Maxima
uses. Using sqfr, Maxima factors each resultant and suppresses
multiple zeros.
elim will triangularize a nonlinear set of polynomial
equations; the solution set of the triangularized set can be larger
than that solution set of the untriangularized set. Thus, the triangularized
equations can have spurious solutions.
Related functions elim_allbut, eliminate_using, eliminate
Option variables resultant
To use load("to_poly")
Status The function elim is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
This function is similar to elim, except that it eliminates all the
variables in the list of equations l except for those variables that
in in the list x
(%i1) elim_allbut([x+y = 1, x - 5*y = 1],[]); (%o1) [[], [y, y + x - 1]] (%i2) elim_allbut([x+y = 1, x - 5*y = 1],[x]); (%o2) [[x - 1], [y + x - 1]]
To use load("to_poly")
Option variables resultant
Related functions elim, eliminate_using, eliminate
Status The function elim_allbut is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Using e as the pivot, eliminate the symbol x from the
list or set of equations in l. The function eliminate_using
returns a set.
(%i1) eq : [x^2 - y^2 - z^3 , x*y - z^2 - 5, x - y + z];
3 2 2 2
(%o1) [- z - y + x , - z + x y - 5, z - y + x]
(%i2) eliminate_using(eq,first(eq),z);
3 2 2 3 2
(%o2) {y + (1 - 3 x) y + 3 x y - x - x ,
4 3 3 2 2 4
y - x y + 13 x y - 75 x y + x + 125}
(%i3) eliminate_using(eq,second(eq),z);
2 2 4 3 3 2 2 4
(%o3) {y - 3 x y + x + 5, y - x y + 13 x y - 75 x y + x
+ 125}
(%i4) eliminate_using(eq, third(eq),z);
2 2 3 2 2 3 2
(%o4) {y - 3 x y + x + 5, y + (1 - 3 x) y + 3 x y - x - x }
Option variables resultant
Related functions elim, eliminate, elim_allbut
To use load("to_poly")
Status The function eliminate_using is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Fourier elimination is the analog of Gauss elimination for linear inequations
(equations or inequalities). The function call fourier_elim([eq1, eq2,
...], [var1, var2, ...]) does Fourier elimination on a list of linear
inequations [eq1, eq2, ...] with respect to the variables
[var1, var2, ...]; for example
(%i1) fourier_elim([y-x < 5, x - y < 7, 10 < y],[x,y]); (%o1) [y - 5 < x, x < y + 7, 10 < y] (%i2) fourier_elim([y-x < 5, x - y < 7, 10 < y],[y,x]); (%o2) [max(10, x - 7) < y, y < x + 5, 5 < x]
Eliminating first with respect to x and second with respect to y yields lower and upper bounds for x that depend on y, and lower and upper bounds for y that are numbers. Eliminating in the other order gives x dependent lower and upper bounds for y, and numerical lower and upper bounds for x.
When necessary, fourier_elim returns a disjunction of lists of
inequations:
(%i3) fourier_elim([x # 6],[x]); (%o3) [x < 6] or [6 < x]
When the solution set is empty, fourier_elim returns emptyset,
and when the solution set is all reals, fourier_elim returns universalset;
for example
(%i4) fourier_elim([x < 1, x > 1],[x]); (%o4) emptyset (%i5) fourier_elim([minf < x, x < inf],[x]); (%o5) universalset
For nonlinear inequations, fourier_elim returns a (somewhat)
simplified list of inequations:
(%i6) fourier_elim([x^3 - 1 > 0],[x]);
2 2 (%o6) [1 < x, x + x + 1 > 0] or [x < 1, - (x + x + 1) > 0]
(%i7) fourier_elim([cos(x) < 1/2],[x]); (%o7) [1 - 2 cos(x) > 0]
Instead of a list of inequations, the first argument to fourier_elim
may be a logical disjunction or conjunction:
(%i8) fourier_elim((x + y < 5) and (x - y >8),[x,y]);
3
(%o8) [y + 8 < x, x < 5 - y, y < - -]
2
(%i9) fourier_elim(((x + y < 5) and x < 1) or (x - y >8),[x,y]);
(%o9) [y + 8 < x] or [x < min(1, 5 - y)]
The function fourier_elim supports the inequation operators
<, <=, >, >=, #, and =.
The Fourier elimination code has a preprocessor that converts some nonlinear inequations that involve the absolute value, minimum, and maximum functions into linear in equations. Additionally, the preprocessor handles some expressions that are the product or quotient of linear terms:
(%i10) fourier_elim([max(x,y) > 6, x # 8, abs(y-1) > 12],[x,y]); (%o10) [6 < x, x < 8, y < - 11] or [8 < x, y < - 11] or [x < 8, 13 < y] or [x = y, 13 < y] or [8 < x, x < y, 13 < y] or [y < x, 13 < y] (%i11) fourier_elim([(x+6)/(x-9) <= 6],[x]); (%o11) [x = 12] or [12 < x] or [x < 9] (%i12) fourier_elim([x^2 - 1 # 0],[x]); (%o12) [- 1 < x, x < 1] or [1 < x] or [x < - 1]
To use load("fourier_elim")
The predicate isreal_p returns true when Maxima is able to
determine that e is real-valued on the entire real line; it
returns false when Maxima is able to determine that e isn’t
real-valued on some nonempty subset of the real line; and it returns a
noun form for all other cases.
(%i1) map('isreal_p, [-1, 0, %i, %pi]);
(%o1) [true, true, false, true]
Maxima variables are assumed to be real; thus
(%i2) isreal_p(x); (%o2) true
The function isreal_p examines the fact database:
(%i3) declare(z,complex)$ (%i4) isreal_p(z); (%o4) isreal_p(z)
Limitations
Too often, isreal_p returns a noun form when it should be able
to return false; a simple example: the logarithm function isn’t
real-valued on the entire real line, so isreal_p(log(x)) should
return false; however
(%i5) isreal_p(log(x)); (%o5) isreal_p(log(x))
To use load("to_poly_solve")
Related functions complex_number_p
Status The function isreal_p is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Return a unique symbol of the form %[z,n,r,c,g]k, where
k is an integer. The allowed values for type are
integer, natural_number, real, complex, and general.
(By natural number, we mean the nonnegative integers; thus zero is
a natural number. Some, but not all, definitions of natural number
exclude zero.)
When type isn’t one of the allowed values, type defaults to general. For integers, natural numbers, and complex numbers, Maxima automatically appends this information to the fact database.
(%i1) map('new_variable,
['integer, 'natural_number, 'real, 'complex, 'general]);
(%o1) [%z144, %n145, %r146, %c147, %g148]
(%i2) nicedummies(%);
(%o2) [%z0, %n0, %r0, %c0, %g0]
(%i3) featurep(%z0, 'integer);
(%o3) true
(%i4) featurep(%n0, 'integer);
(%o4) true
(%i5) is(%n0 >= 0);
(%o5) true
(%i6) featurep(%c0, 'complex);
(%o6) true
Note Generally, the argument to new_variable should be quoted. The quote
will protect against errors similar to
(%i7) integer : 12$
(%i8) new_variable(integer);
(%o8) %g149
(%i9) new_variable('integer);
(%o9) %z150
Related functions nicedummies
To use load("to_poly_solve")
Status The function new_variable is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
Starting with zero, the function nicedummies re-indexes the variables
in an expression that were introduced by new_variable;
(%i1) new_variable('integer) + 52 * new_variable('integer);
(%o1) 52 %z136 + %z135
(%i2) new_variable('integer) - new_variable('integer);
(%o2) %z137 - %z138
(%i3) nicedummies(%);
(%o3) %z0 - %z1
Related functions new_variable
To use load("to_poly_solve")
Status The function nicedummies is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function parg is a simplifying version of the complex argument function
carg; thus
(%i1) map('parg,[1,1+%i,%i, -1 + %i, -1]);
%pi %pi 3 %pi
(%o1) [0, ---, ---, -----, %pi]
4 2 4
Generally, for a non-constant input, parg returns a noun form; thus
(%i2) parg(x + %i * sqrt(x)); (%o2) parg(x + %i sqrt(x))
When sign can determine that the input is a positive or negative real
number, parg will return a non-noun form for a non-constant input.
Here are two examples:
(%i3) parg(abs(x)); (%o3) 0 (%i4) parg(-x^2-1); (%o4) %pi
Note The sign function mostly ignores the variables that are declared
to be complex (declare(x,complex)); for variables that are declared
to be complex, the parg can return incorrect values; for example
(%i1) declare(x,complex)$ (%i2) parg(x^2 + 1); (%o2) 0
Related function carg, isreal_p
To use load("to_poly_solve")
Status The function parg is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function real_imagpart_to_conjugate replaces all occurrences
of realpart and imagpart to algebraically equivalent expressions
involving the conjugate.
(%i1) declare(x, complex)$
(%i2) real_imagpart_to_conjugate(realpart(x) + imagpart(x) = 3);
conjugate(x) + x %i (x - conjugate(x))
(%o2) ---------------- - --------------------- = 3
2 2
To use load("to_poly_solve")
Status The function real_imagpart_to_conjugate is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function rectform_log_if_constant converts all terms of the form
log(c) to rectform(log(c)), where c is
either a declared constant expression or explicitly declared constant
(%i1) rectform_log_if_constant(log(1-%i) - log(x - %i));
log(2) %i %pi
(%o1) - log(x - %i) + ------ - ------
2 4
(%i2) declare(a,constant, b,constant)$
(%i3) rectform_log_if_constant(log(a + %i*b));
2 2
log(b + a )
(%o3) ------------ + %i atan2(b, a)
2
To use load("to_poly_solve")
Status The function rectform_log_if_constant is
experimental; the specifications of this function might change might change and its functionality
might be merged into other Maxima functions.
The function simp_inequality applies basic simplifications to inequations,
returning either a boolean value (true or false) or the original inequation.
The simplification rules used by simp_inequality
include some facts about the ranges of the absolute value, power,
and exponential functions along with some elementary algebra facts.
For conjunctions or disjunctions of inequations,
simp_inequality is applied to each individual inequation,
but no effort is made to simplify the entire logical expression.
Effectively, simp_inequality creates a new empty context, so database facts are not used to simplify inequations.
load("to_poly_solve") loads this function.
Examples:
(%i2) simp_inequality(1 # 0); (%o2) true
(%i3) simp_inequality(1 < 0); (%o3) false
(%i4) simp_inequality(a=a); (%o4) true
(%i5) simp_inequality(a # a); (%o5) false
(%i6) simp_inequality(a + 1 # a); (%o6) true
(%i7) simp_inequality(a < a+1); (%o7) true
(%i8) simp_inequality(abs(x) >= 0); (%o8) true
(%i9) simp_inequality(exp(x) > 0); (%o9) true
(%i10) simp_inequality(x^2 >= 0); (%o10) true
(%i11) simp_inequality(2^x # 0); (%o11) true
(%i12) simp_inequality(2^(x+1) > 2^x); (%o12) true
The fact database is not consulted. For example:
(%i13) assume(xx > 0)$ (%i14) simp_inequality(xx > 0); (%o14) xx>0
And finally, for conjunctions or disjunctions of inequations, each inequation is simplified, but no effort is made to simplify the entire logical expression; for example:
(%i15) simp_inequality((1 > 0) and (x < 0) and (x > 0)); (%o15) x<0 and x>0
This function applies the identities cot(x) = atan(1/x),
acsc(x) = asin(1/x), and similarly for asec, acoth, acsch
and asech to an expression. See Abramowitz and Stegun,
Eqs. 4.4.6 through 4.4.8 and 4.6.4 through 4.6.6.
To use load("to_poly_solve")
Status The function standardize_inverse_trig is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
When l is a single equation or a list of equations, substitute
the right hand side of each equation for the left hand side. The
substitutions are made in parallel; for example
(%i1) load("to_poly_solve")$
(%i2) subst_parallel([x=y,y=x], [x,y]);
(%o2) [y, x]
Compare this to substitutions made serially:
(%i3) subst([x=y,y=x],[x,y]); (%o3) [x, x]
The function subst_parallel is similar to sublis except that
subst_parallel allows for substitution of nonatoms; for example
(%i4) subst_parallel([x^2 = a, y = b], x^2 * y);
(%o4) a b
(%i5) sublis([x^2 = a, y = b], x^2 * y);
2
sublis: left-hand side of equation must be a symbol; found: x
-- an error. To debug this try: debugmode(true);
The substitutions made by subst_parallel are literal, not semantic; thus
subst_parallel does not recognize that x * y is a subexpression
of x^2 * y
(%i6) subst_parallel([x * y = a], x^2 * y);
2
(%o6) x y
The function subst_parallel completes all substitutions
before simplifications. This allows for substitutions into
conditional expressions where errors might occur if the
simplifications were made earlier:
(%i7) subst_parallel([x = 0], %if(x < 1, 5, log(x))); (%o7) 5 (%i8) subst([x = 0], %if(x < 1, 5, log(x))); log: encountered log(0). -- an error. To debug this try: debugmode(true);
Related functions subst, sublis, ratsubst
To use load("to_poly_solve_extra.lisp")
Status The function subst_parallel is experimental; the
specifications of this function might change might change and its
functionality might be merged into other Maxima functions.
The function to_poly attempts to convert the equation e
into a polynomial system along with inequality constraints; the
solutions to the polynomial system that satisfy the constraints are
solutions to the equation e. Informally, to_poly
attempts to polynomialize the equation e; an example might
clarify:
(%i1) load("to_poly_solve")$
(%i2) to_poly(sqrt(x) = 3, [x]);
2
(%o2) [[%g130 - 3, x = %g130 ],
%pi %pi
[- --- < parg(%g130), parg(%g130) <= ---], []]
2 2
The conditions -%pi/2<parg(%g130),parg(%g130)<=%pi/2 tell us that
%g130 is in the range of the square root function. When this is
true, the solution set to sqrt(x) = 3 is the same as the
solution set to %g130-3,x=%g130^2.
To polynomialize trigonometric expressions, it is necessary to
introduce a non algebraic substitution; these non algebraic substitutions
are returned in the third list returned by to_poly; for example
(%i3) to_poly(cos(x),[x]);
2 %i x
(%o3) [[%g131 + 1], [2 %g131 # 0], [%g131 = %e ]]
Constant terms aren’t polynomializied unless the number one is a member of the variable list; for example
(%i4) to_poly(x = sqrt(5),[x]);
(%o4) [[x - sqrt(5)], [], []]
(%i5) to_poly(x = sqrt(5),[1,x]);
2
(%o5) [[x - %g132, 5 = %g132 ],
%pi %pi
[- --- < parg(%g132), parg(%g132) <= ---], []]
2 2
To generate a polynomial with sqrt(5) + sqrt(7) as one of its roots, use the commands
(%i6) first(elim_allbut(first(to_poly(x = sqrt(5) + sqrt(7),
[1,x])), [x]));
4 2
(%o6) [x - 24 x + 4]
Related functions to_poly_solve
To use load("to_poly")
Status: The function to_poly is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
The function to_poly_solve tries to solve the equations e
for the variables l. The equation(s) e can either be a
single expression or a set or list of expressions; similarly, l
can either be a single symbol or a list of set of symbols. When
a member of e isn’t explicitly an equation, for example x^2 -1,
the solver assumes that the expression vanishes.
The basic strategy of to_poly_solve is to convert the input into a polynomial form and to
call algsys on the polynomial system. Internally to_poly_solve defaults algexact
to true. To change the default for algexact, append ’algexact=false to the to_poly_solve
argument list.
When to_poly_solve is able to determine the solution set, each
member of the solution set is a list in a %union object:
(%i1) load("to_poly_solve")$
(%i2) to_poly_solve(x*(x-1) = 0, x);
(%o2) %union([x = 0], [x = 1])
When to_poly_solve is unable to determine the solution set, a
%solve nounform is returned (in this case, a warning is printed)
(%i3) to_poly_solve(x^k + 2* x + 1 = 0, x);
Nonalgebraic argument given to 'to_poly'
unable to solve
k
(%o3) %solve([x + 2 x + 1 = 0], [x])
Substitution into a %solve nounform can sometimes result in the solution
(%i4) subst(k = 2, %); (%o4) %union([x = - 1])
Especially for trigonometric equations, the solver sometimes needs
to introduce an arbitrary integer. These arbitrary integers have the
form %zXXX, where XXX is an integer; for example
(%i5) to_poly_solve(sin(x) = 0, x); (%o5) %union([x = 2 %pi %z33 + %pi], [x = 2 %pi %z35])
To re-index these variables to zero, use nicedummies:
(%i6) nicedummies(%); (%o6) %union([x = 2 %pi %z0 + %pi], [x = 2 %pi %z1])
Occasionally, the solver introduces an arbitrary complex number of the
form %cXXX or an arbitrary real number of the form %rXXX.
The function nicedummies will re-index these identifiers to zero.
The solution set sometimes involves simplifying versions of various
of logical operators including %and, %or, or %if
for conjunction, disjunction, and implication, respectively; for example
(%i7) sol : to_poly_solve(abs(x) = a, x);
(%o7) %union(%if(isnonnegative_p(a), [x = - a], %union()),
%if(isnonnegative_p(a), [x = a], %union()))
(%i8) subst(a = 42, sol);
(%o8) %union([x = - 42], [x = 42])
(%i9) subst(a = -42, sol);
(%o9) %union()
The empty set is represented by %union().
The function to_poly_solve is able to solve some, but not all,
equations involving rational powers, some nonrational powers, absolute
values, trigonometric functions, and minimum and maximum. Also, some it
can solve some equations that are solvable in in terms of the Lambert W
function; some examples:
(%i1) load("to_poly_solve")$
(%i2) to_poly_solve(set(max(x,y) = 5, x+y = 2), set(x,y));
(%o2) %union([x = - 3, y = 5], [x = 5, y = - 3])
(%i3) to_poly_solve(abs(1-abs(1-x)) = 10,x);
(%o3) %union([x = - 10], [x = 12])
(%i4) to_poly_solve(set(sqrt(x) + sqrt(y) = 5, x + y = 10),
set(x,y));
3/2 3/2
5 %i - 10 5 %i + 10
(%o4) %union([x = - ------------, y = ------------],
2 2
3/2 3/2
5 %i + 10 5 %i - 10
[x = ------------, y = - ------------])
2 2
(%i5) to_poly_solve(cos(x) * sin(x) = 1/2,x,
'simpfuncs = ['expand, 'nicedummies]);
%pi
(%o5) %union([x = %pi %z0 + ---])
4
(%i6) to_poly_solve(x^(2*a) + x^a + 1,x);
2 %i %pi %z81
-------------
1/a a
(sqrt(3) %i - 1) %e
(%o6) %union([x = -----------------------------------],
1/a
2
2 %i %pi %z83
-------------
1/a a
(- sqrt(3) %i - 1) %e
[x = -------------------------------------])
1/a
2
(%i7) to_poly_solve(x * exp(x) = a, x); (%o7) %union([x = lambert_w(a)])
For linear inequalities, to_poly_solve automatically does Fourier
elimination:
(%i8) to_poly_solve([x + y < 1, x - y >= 8], [x,y]);
7
(%o8) %union([x = y + 8, y < - -],
2
7
[y + 8 < x, x < 1 - y, y < - -])
2
Each optional argument to to_poly_solve must be an equation;
generally, the order of these options does not matter.
simpfuncs = l, where l is a list of functions.
Apply the composition of the members of l to each solution.
(%i1) to_poly_solve(x^2=%i,x);
1/4 1/4
(%o1) %union([x = - (- 1) ], [x = (- 1) ])
(%i2) to_poly_solve(x^2= %i,x, 'simpfuncs = ['rectform]);
%i 1 %i 1
(%o2) %union([x = - ------- - -------], [x = ------- + -------])
sqrt(2) sqrt(2) sqrt(2) sqrt(2)
Sometimes additional simplification can revert a simplification; for example
(%i3) to_poly_solve(x^2=1,x);
(%o3) %union([x = - 1], [x = 1])
(%i4) to_poly_solve(x^2= 1,x, 'simpfuncs = [polarform]);
%i %pi
(%o4) %union([x = 1], [x = %e ]
Maxima doesn’t try to check that each member of the function list l is
purely a simplification; thus
(%i5) to_poly_solve(x^2 = %i,x, 'simpfuncs = [lambda([s],s^2)]); (%o5) %union([x = %i])
To convert each solution to a double float, use simpfunc = ['dfloat]:
(%i6) to_poly_solve(x^3 +x + 1 = 0,x,
'simpfuncs = ['dfloat]), algexact : true;
(%o6) %union([x = - .6823278038280178],
[x = .3411639019140089 - 1.161541399997251 %i],
[x = 1.161541399997251 %i + .3411639019140089])
use_grobner = true With this option, the function
poly_reduced_grobner is applied to the equations before
attempting their solution. Primarily, this option provides a workaround
for weakness in the function algsys. Here is an example of
such a workaround:
(%i7) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y],
'use_grobner = true);
sqrt(7) - 1 sqrt(7) + 1
(%o7) %union([x = - -----------, y = -----------],
2 2
sqrt(7) + 1 sqrt(7) - 1
[x = -----------, y = - -----------])
2 2
(%i8) to_poly_solve([x^2+y^2=2^2,(x-1)^2+(y-1)^2=2^2],[x,y]);
(%o8) %union()
maxdepth = k, where k is a positive integer. This
function controls the maximum recursion depth for the solver. The
default value for maxdepth is five. When the recursions depth is
exceeded, the solver signals an error:
(%i9) to_poly_solve(cos(x) = x,x, 'maxdepth = 2); Unable to solve Unable to solve (%o9) %solve([cos(x) = x], [x], maxdepth = 2)
parameters = l, where l is a list of symbols. The solver
attempts to return a solution that is valid for all members of the list
l; for example:
(%i10) to_poly_solve(a * x = x, x);
(%o10) %union([x = 0])
(%i11) to_poly_solve(a * x = x, x, 'parameters = [a]);
(%o11) %union(%if(a - 1 = 0, [x = %c111], %union()),
%if(a - 1 # 0, [x = 0], %union()))
In (%o2), the solver introduced a dummy variable; to re-index the
these dummy variables, use the function nicedummies:
(%i12) nicedummies(%);
(%o12) %union(%if(a - 1 = 0, [x = %c0], %union()),
%if(a - 1 # 0, [x = 0], %union()))
The to_poly_solve uses data stored in the hashed array
one_to_one_reduce to solve equations of the form f(a) =
f(b). The assignment one_to_one_reduce['f,'f] : lambda([a,b],
a=b) tells to_poly_solve that the solution set of f(a)
= f(b) equals the solution set of a=b; for example
(%i13) one_to_one_reduce['f,'f] : lambda([a,b], a=b)$ (%i14) to_poly_solve(f(x^2-1) = f(0),x); (%o14) %union([x = - 1], [x = 1])
More generally, the assignment one_to_one_reduce['f,'g] : lambda([a,b],
w(a, b) = 0 tells to_poly_solve that the solution set of f(a)
= f(b) equals the solution set of w(a,b) = 0; for example
(%i15) one_to_one_reduce['f,'g] : lambda([a,b], a = 1 + b/2)$ (%i16) to_poly_solve(f(x) - g(x),x); (%o16) %union([x = 2])
Additionally, the function to_poly_solve uses data stored in the hashed array
function_inverse to solve equations of the form f(a) = b.
The assignment function_inverse['f] : lambda([s], g(s))
informs to_poly_solve that the solution set to f(x) = b equals
the solution set to x = g(b); two examples:
(%i17) function_inverse['Q] : lambda([s], P(s))$ (%i18) to_poly_solve(Q(x-1) = 2009,x); (%o18) %union([x = P(2009) + 1]) (%i19) function_inverse['G] : lambda([s], s+new_variable(integer)); (%o19) lambda([s], s + new_variable(integer)) (%i20) to_poly_solve(G(x - a) = b,x); (%o20) %union([x = b + a + %z125])
Notes
fullratsubst is
able to appropriately make substitutions, the solve variables can be nonsymbols:
(%i1) to_poly_solve([x^2 + y^2 + x * y = 5, x * y = 8],
[x^2 + y^2, x * y]);
2 2
(%o1) %union([x y = 8, y + x = - 3])
(%i1) declare(x,complex)$
(%i2) to_poly_solve(x + (5 + %i) * conjugate(x) = 1, x);
%i + 21
(%o2) %union([x = - -----------])
25 %i - 125
(%i3) declare(y,complex)$
(%i4) to_poly_solve(set(conjugate(x) - y = 42 + %i,
x + conjugate(y) = 0), set(x,y));
%i - 42 %i + 42
(%o4) %union([x = - -------, y = - -------])
2 2
to_poly_solve consults the fact database to decide if the
argument to the absolute value is complex valued. When
(%i1) to_poly_solve(abs(x) = 6, x);
(%o1) %union([x = - 6], [x = 6])
(%i2) declare(z,complex)$
(%i3) to_poly_solve(abs(z) = 6, z);
(%o3) %union(%if((%c11 # 0) %and (%c11 conjugate(%c11) - 36 =
0), [z = %c11], %union()))
This is the only situation that the solver consults the fact database. If
a solve variable is declared to be an integer, for example, to_poly_solve
ignores this declaration.
Relevant option variables algexact, resultant, algebraic
Related functions to_poly
To use load("to_poly_solve")
Status: The function to_poly_solve is experimental; its
specifications might change and its functionality might be merged into
other Maxima functions.
%union(soln_1, soln_2, soln_3, ...) represents the union of its arguments,
each of which represents a solution set,
as determined by to_poly_solve.
%union() represents the empty set.
In many cases, a solution is a list of equations [x = ..., y = ..., z = ...]
where x, y, and z are one or more unknowns.
In such cases, to_poly_solve returns a %union expression
containing one or more such lists.
The solution set sometimes involves simplifying versions of various
of logical operators including %and, %or, or %if
for conjunction, disjunction, and implication, respectively.
Examples:
%union(...) represents the union of its arguments,
each of which represents a solution set,
as determined by to_poly_solve.
In many cases, a solution is a list of equations.
(%i1) load ("to_poly_solve") $
(%i2) to_poly_solve ([sqrt(x^2 - y^2), x + y], [x, y]);
(%o2) %union([x = 0, y = 0], [x = %c13, y = - %c13])
%union() represents the empty set.
(%i1) load ("to_poly_solve") $
(%i2) to_poly_solve (abs(x) = -1, x);
(%o2) %union()
The solution set sometimes involves simplifying versions of various of logical operators.
(%i1) load ("to_poly_solve") $
(%i2) sol : to_poly_solve (abs(x) = a, x);
(%o2) %union(%if(isnonnegative_p(a), [x = - a], %union()),
%if(isnonnegative_p(a), [x = a], %union()))
(%i3) subst (a = 42, sol);
(%o3) %union([x = - 42], [x = 42])
(%i4) subst (a = -42, sol);
(%o4) %union()
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