Default value: 0
%rnum is the counter for the %r variables introduced in solutions by
solve and algsys.. The next %r variable is numbered
%rnum+1.
See also %rnum_list.
Default value: []
%rnum_list is the list of variables introduced in solutions by
solve and algsys. %r variables are added to
%rnum_list in the order they are created. This is convenient for doing
substitutions into the solution later on.
See also %rnum.
It’s recommended to use this list rather than doing concat ('%r, j).
(%i1) solve ([x + y = 3], [x,y]); (%o1) [[x = 3 - %r1, y = %r1]]
(%i2) %rnum_list; (%o2) [%r1]
(%i3) sol : solve ([x + 2*y + 3*z = 4], [x,y,z]); (%o3) [[x = - 2 %r3 - 3 %r2 + 4, y = %r3, z = %r2]]
(%i4) %rnum_list; (%o4) [%r2, %r3]
(%i5) for i : 1 thru length (%rnum_list) do
sol : subst (t[i], %rnum_list[i], sol)$
(%i6) sol;
(%o6) [[x = - 2 t - 3 t + 4, y = t , z = t ]]
2 1 2 1
Default value: 10^8
algepsilon is used by algsys.
Default value: false
algexact affects the behavior of algsys as follows:
If algexact is true, algsys always calls solve and
then uses realroots on solve’s failures.
If algexact is false, solve is called only if the
eliminant was not univariate, or if it was a quadratic or biquadratic.
Thus algexact: true does not guarantee only exact solutions, just that
algsys will first try as hard as it can to give exact solutions, and
only yield approximations when all else fails.
Solves the simultaneous polynomials expr_1, …, expr_m or
polynomial equations eqn_1, …, eqn_m for the variables
x_1, …, x_n. An expression expr is equivalent to an
equation expr = 0. There may be more equations than variables or
vice versa.
algsys returns a list of solutions, with each solution given as a list
of equations stating values of the variables x_1, …, x_n
which satisfy the system of equations. If algsys cannot find a solution,
an empty list [] is returned.
The symbols %r1, %r2, …, are introduced as needed to
represent arbitrary parameters in the solution; these variables are also
appended to the list %rnum_list.
The method is as follows:
solve is called to find an exact
solution.
In some cases, solve is not be able to find a solution, or if it does
the solution may be a very large expression.
If the equation is univariate and is either linear, quadratic, or biquadratic,
then again solve is called if no approximations have been introduced.
If approximations have been introduced or the equation is not univariate and
neither linear, quadratic, or biquadratic, then if the switch
realonly is true, the function realroots is called to find
the real-valued solutions. If realonly is false, then
allroots is called which looks for real and complex-valued solutions.
If algsys produces a solution which has fewer significant digits than
required, the user can change the value of algepsilon to a higher value.
If algexact is set to true, solve will always be called.
When algsys encounters a multivariate equation which contains floating
point approximations (usually due to its failing to find exact solutions at an
earlier stage), then it does not attempt to apply exact methods to such
equations and instead prints the message:
"algsys cannot solve - system too complicated."
Interactions with radcan can produce large or complicated expressions.
In that case, it may be possible to isolate parts of the result with
pickapart or reveal.
Occasionally, radcan may introduce an imaginary unit %i into a
solution which is actually real-valued.
Examples:
(%i1) e1: 2*x*(1 - a1) - 2*(x - 1)*a2;
(%o1) 2 (1 - a1) x - 2 a2 (x - 1)
(%i2) e2: a2 - a1;
(%o2) a2 - a1
(%i3) e3: a1*(-y - x^2 + 1);
2
(%o3) a1 (- y - x + 1)
(%i4) e4: a2*(y - (x - 1)^2);
2
(%o4) a2 (y - (x - 1) )
(%i5) algsys ([e1, e2, e3, e4], [x, y, a1, a2]);
(%o5) [[x = 0, y = %r1, a1 = 0, a2 = 0],
[x = 1, y = 0, a1 = 1, a2 = 1]]
(%i6) e1: x^2 - y^2;
2 2
(%o6) x - y
(%i7) e2: -1 - y + 2*y^2 - x + x^2;
2 2
(%o7) 2 y - y + x - x - 1
(%i8) algsys ([e1, e2], [x, y]);
1 1
(%o8) [[x = - -------, y = -------],
sqrt(3) sqrt(3)
1 1 1 1
[x = -------, y = - -------], [x = - -, y = - -], [x = 1, y = 1]]
sqrt(3) sqrt(3) 3 3
Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.
The flag polyfactor when true causes allroots to factor
the polynomial over the real numbers if the polynomial is real, or over the
complex numbers, if the polynomial is complex.
allroots may give inaccurate results in case of multiple roots.
If the polynomial is real, allroots (%i*p) may yield
more accurate approximations than allroots (p), as allroots
invokes a different algorithm in that case.
allroots rejects non-polynomials. It requires that the numerator
after rat’ing should be a polynomial, and it requires that the
denominator be at most a complex number. As a result of this allroots
will always return an equivalent (but factored) expression, if
polyfactor is true.
For complex polynomials an algorithm by Jenkins and Traub is used (Algorithm 419, Comm. ACM, vol. 15, (1972), p. 97). For real polynomials the algorithm used is due to Jenkins (Algorithm 493, ACM TOMS, vol. 1, (1975), p.178).
Examples:
(%i1) eqn: (1 + 2*x)^3 = 13.5*(1 + x^5);
3 5
(%o1) (2 x + 1) = 13.5 (x + 1)
(%i2) soln: allroots (eqn);
(%o2) [x = .8296749902129361, x = - 1.015755543828121,
x = .9659625152196369 %i - .4069597231924075,
x = - .9659625152196369 %i - .4069597231924075, x = 1.0]
(%i3) for e in soln
do (e2: subst (e, eqn), disp (expand (lhs(e2) - rhs(e2))));
- 3.5527136788005E-15
- 5.32907051820075E-15
4.44089209850063E-15 %i - 4.88498130835069E-15
- 4.44089209850063E-15 %i - 4.88498130835069E-15
3.5527136788005E-15
(%o3) done
(%i4) polyfactor: true$
(%i5) allroots (eqn);
(%o5) - 13.5 (x - 1.0) (x - .8296749902129361)
2
(x + 1.015755543828121) (x + .8139194463848151 x
+ 1.098699797110288)
Computes numerical approximations of the real and complex roots of the polynomial expr or polynomial equation eqn of one variable.
In all respects, bfallroots is identical to allroots except
that bfallroots computes the roots using bigfloats. See
allroots for more information.
Default value: true
When backsubst is false, prevents back substitution in
linsolve after the equations have been triangularized. This may
be helpful in very big problems where back substitution would cause
the generation of extremely large expressions.
(%i1) eq1 : x + y + z = 6$ (%i2) eq2 : x - y + z = 2$ (%i3) eq3 : x + y - z = 0$ (%i4) backsubst : false$
(%i5) linsolve ([eq1, eq2, eq3], [x,y,z]); (%o5) [x = z - y, y = 2, z = 3]
(%i6) backsubst : true$
(%i7) linsolve ([eq1, eq2, eq3], [x,y,z]); (%o7) [x = 1, y = 2, z = 3]
Default value: true
When breakup is true, solve expresses solutions of cubic
and quartic equations in terms of common subexpressions, which are assigned to
intermediate expression labels (%t1, %t2, etc.).
Otherwise, common subexpressions are not identified.
breakup: true has an effect only when programmode is false.
Examples:
(%i1) programmode: false$
(%i2) breakup: true$
(%i3) solve (x^3 + x^2 - 1);
sqrt(23) 25 1/3
(%t3) (--------- + --)
6 sqrt(3) 54
Solution:
sqrt(3) %i 1
---------- - -
sqrt(3) %i 1 2 2 1
(%t4) x = (- ---------- - -) %t3 + -------------- - -
2 2 9 %t3 3
sqrt(3) %i 1
- ---------- - -
sqrt(3) %i 1 2 2 1
(%t5) x = (---------- - -) %t3 + ---------------- - -
2 2 9 %t3 3
1 1
(%t6) x = %t3 + ----- - -
9 %t3 3
(%o6) [%t4, %t5, %t6]
(%i6) breakup: false$
(%i7) solve (x^3 + x^2 - 1);
Solution:
sqrt(3) %i 1
---------- - -
2 2 sqrt(23) 25 1/3
(%t7) x = --------------------- + (--------- + --)
sqrt(23) 25 1/3 6 sqrt(3) 54
9 (--------- + --)
6 sqrt(3) 54
sqrt(3) %i 1 1
(- ---------- - -) - -
2 2 3
sqrt(23) 25 1/3 sqrt(3) %i 1
(%t8) x = (--------- + --) (---------- - -)
6 sqrt(3) 54 2 2
sqrt(3) %i 1
- ---------- - -
2 2 1
+ --------------------- - -
sqrt(23) 25 1/3 3
9 (--------- + --)
6 sqrt(3) 54
sqrt(23) 25 1/3 1 1
(%t9) x = (--------- + --) + --------------------- - -
6 sqrt(3) 54 sqrt(23) 25 1/3 3
9 (--------- + --)
6 sqrt(3) 54
(%o9) [%t7, %t8, %t9]
dimen is a package for dimensional analysis.
load ("dimen") loads this package.
demo ("dimen") displays a short demonstration.
Default value: true
If set to false within a block will inhibit the display of output
generated by the solve functions called from within the block.
Termination of the block with a dollar sign, $, sets dispflag to
false.
Returns [g(t) = ...] or [], depending on whether
or not there exists a rational function g(t) satisfying
eqn, which must be a first order, linear polynomial in (for this case)
g(t) and g(t+1)
(%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) =
(n - 1)/(n + 2);
(n + 3) f(n + 1) n - 1
(%o1) (n + 1) f(n) - ---------------- = -----
n + 1 n + 2
(%i2) funcsolve (eqn, f(n));
Dependent equations eliminated: (4 3)
n
(%o2) f(n) = ---------------
(n + 1) (n + 2)
Warning: this is a very rudimentary implementation – many safety checks and obvious generalizations are missing.
Default value: false
When globalsolve is true, solved-for variables are assigned the
solution values found by linsolve, and by solve when solving two
or more linear equations.
When globalsolve is false, solutions found by linsolve and
by solve when solving two or more linear equations are expressed as
equations, and the solved-for variables are not assigned.
When solving anything other than two or more linear equations, solve
ignores globalsolve. Other functions which solve equations (e.g.,
algsys) always ignore globalsolve.
Examples:
(%i1) globalsolve: true$
(%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
Solution
17
(%t2) x : --
7
1
(%t3) y : - -
7
(%o3) [[%t2, %t3]]
(%i3) x;
17
(%o3) --
7
(%i4) y;
1
(%o4) - -
7
(%i5) globalsolve: false$
(%i6) kill (x, y)$
(%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);
Solution
17
(%t7) x = --
7
1
(%t8) y = - -
7
(%o8) [[%t7, %t8]]
(%i8) x;
(%o8) x
(%i9) y;
(%o9) y
inteqn is a package for solving integral equations.
load ("inteqn") loads this package.
ie is the integral equation; unk is the unknown function;
tech is the technique to be tried from those given in the lists
below; (tech = first means: try the first technique which
finds a solution; tech = all means: try all applicable
techniques); n is the maximum number of terms to take for
taylor, neumann, firstkindseries, or
fredseries (it is also the maximum depth of recursion for the
differentiation method); guess is the initial guess for
neumann or firstkindseries.
Two types of equations are considered. A second-kind equation of the following form,
b(x)
/
[
p(x) = q(x, p(x), I w(x, u, p(x), p(u)) du)
]
/
a(x)
and a first-kind equation with the form
b(x)
/
[
f(x) = I w(x, u, p(u)) du
]
/
a(x)
The different solution techniques used require particular forms of the expressions q and w. The techniques available are the following:
Second-kind equations
flfrnk2nd: For fixed-limit, finite-rank integrands.
vlfrnk: For variable-limit, finite-rank integrands.
transform: Laplace transform for convolution types.
fredseries: Fredholm-Carleman series for linear equations.
tailor: Taylor series for quasi-linear variable-limit equations.
neumann: Neumann series for quasi-second kind equations.
collocate: Collocation using a power series form for p(x)
evaluated at equally spaced points.
First-kind equations
flfrnk1st: For fixed-limit, finite-rank integrands.
vlfrnk: For variable-limit, finite-rank integrands.
abel: For singular integrands
transform: See above
collocate: See above
firstkindseries: Iteration technique similar to neumann series.
The default values for the 2nd thru 5th parameters in the calling form are:
unk: p(x), where p is the first function
encountered in an integrand which is unknown to Maxima and x is the
variable which occurs as an argument to the first occurrence of p found
outside of an integral in the case of secondkind equations, or is the
only other variable besides the variable of integration in firstkind
equations. If the attempt to search for x fails, the user will be asked
to supply the independent variable. tech: first. n: 1.
guess: none which will cause neumann and
firstkindseries to use f(x) as an initial guess.
Examples:
(%i1) load("inteqn")$
(%i2) e: p(x) - 1 -x + cos(x) + 'integrate(cos(x-u)*p(u),u,0,x)$
(%i3) ieqn(e, p(x), 'transform);
default 4th arg, number of iterations or coll. parms.: 1
default 5th arg, initial guess: none
(%t3) [x, transform]
(%o3) [%t3]
(%i4) e: 2*'integrate(p(x*sin(u)), u, 0, %pi/2) - a*x - b$
(%i5) ieqn(e, p(x), 'firstkindseries);
default 4th arg, number of iterations or coll. parms.: 1
default 5th arg, initial guess: none
(%t5) [2 a x + %pi b, firstkindseries, 1, approximate]
(%o5) [%t5]
Default value: true
ieqnprint governs the behavior of the result returned by the
ieqn command. When ieqnprint is false, the lists returned
by the ieqn function are of the form
[solution, technique used, nterms, flag]
where flag is absent if the solution is exact.
Otherwise, it is the word approximate or incomplete corresponding
to an inexact or non-closed form solution, respectively. If a series method was
used, nterms gives the number of terms taken (which could be less than
the n given to ieqn if an error prevented generation of further terms).
Returns the left-hand side (that is, the first argument) of the expression
expr, when the operator of expr is one of the relational operators
< <= = # equal notequal >= >,
one of the assignment operators := ::= : ::, or a user-defined binary
infix operator, as declared by infix.
When expr is an atom or its operator is something other than the ones
listed above, lhs returns expr.
See also rhs.
Examples:
(%i1) e: aa + bb = cc;
(%o1) bb + aa = cc
(%i2) lhs (e);
(%o2) bb + aa
(%i3) rhs (e);
(%o3) cc
(%i4) [lhs (aa < bb), lhs (aa <= bb), lhs (aa >= bb),
lhs (aa > bb)];
(%o4) [aa, aa, aa, aa]
(%i5) [lhs (aa = bb), lhs (aa # bb), lhs (equal (aa, bb)),
lhs (notequal (aa, bb))];
(%o5) [aa, aa, aa, aa]
(%i6) e1: '(foo(x) := 2*x);
(%o6) foo(x) := 2 x
(%i7) e2: '(bar(y) ::= 3*y);
(%o7) bar(y) ::= 3 y
(%i8) e3: '(x : y);
(%o8) x : y
(%i9) e4: '(x :: y);
(%o9) x :: y
(%i10) [lhs (e1), lhs (e2), lhs (e3), lhs (e4)];
(%o10) [foo(x), bar(y), x, x]
(%i11) infix ("][");
(%o11) ][
(%i12) lhs (aa ][ bb);
(%o12) aa
Solves the list of simultaneous linear equations for the list of variables. The expressions must each be polynomials in the variables and may be equations. If the length of the list of variables doesn’t match the number of linearly-independent equations to solve the result will be an empty list.
When globalsolve is true, each solved-for variable is bound to
its value in the solution of the equations.
When backsubst is false, linsolve does not carry out back
substitution after the equations have been triangularized. This may be
necessary in very big problems where back substitution would cause the
generation of extremely large expressions.
When linsolve_params is true, linsolve also generates the
%r symbols used to represent arbitrary parameters described in the manual
under algsys. Otherwise, linsolve solves an under-determined
system of equations with some variables expressed in terms of others.
When programmode is false, linsolve displays the solution
with intermediate expression (%t) labels, and returns the list of labels.
See also algsys, eliminate. and solve.
Examples:
(%i1) e1: x + z = y;
(%o1) z + x = y
(%i2) e2: 2*a*x - y = 2*a^2;
2
(%o2) 2 a x - y = 2 a
(%i3) e3: y - 2*z = 2;
(%o3) y - 2 z = 2
(%i4) [globalsolve: false, programmode: true];
(%o4) [false, true]
(%i5) linsolve ([e1, e2, e3], [x, y, z]);
(%o5) [x = a + 1, y = 2 a, z = a - 1]
(%i6) [globalsolve: false, programmode: false];
(%o6) [false, false]
(%i7) linsolve ([e1, e2, e3], [x, y, z]);
Solution
(%t7) z = a - 1
(%t8) y = 2 a
(%t9) x = a + 1
(%o9) [%t7, %t8, %t9]
(%i9) ''%;
(%o9) [z = a - 1, y = 2 a, x = a + 1]
(%i10) [globalsolve: true, programmode: false];
(%o10) [true, false]
(%i11) linsolve ([e1, e2, e3], [x, y, z]);
Solution
(%t11) z : a - 1
(%t12) y : 2 a
(%t13) x : a + 1
(%o13) [%t11, %t12, %t13]
(%i13) ''%;
(%o13) [z : a - 1, y : 2 a, x : a + 1]
(%i14) [x, y, z];
(%o14) [a + 1, 2 a, a - 1]
(%i15) [globalsolve: true, programmode: true];
(%o15) [true, true]
(%i16) linsolve ([e1, e2, e3], '[x, y, z]);
(%o16) [x : a + 1, y : 2 a, z : a - 1]
(%i17) [x, y, z];
(%o17) [a + 1, 2 a, a - 1]
Default value: true
When linsolvewarn is true, linsolve prints a message
"Dependent equations eliminated".
Default value: true
When linsolve_params is true, linsolve also generates
the %r symbols used to represent arbitrary parameters described in
the manual under algsys. Otherwise, linsolve solves an
under-determined system of equations with some variables expressed in terms of
others.
Default value: not_set_yet
multiplicities is set to a list of the multiplicities of the individual
solutions returned by solve or realroots.
Returns the number of real roots of the real univariate polynomial p in
the half-open interval (low, high]. The endpoints of the
interval may be minf or inf.
nroots uses the method of Sturm sequences.
(%i1) p: x^10 - 2*x^4 + 1/2$ (%i2) nroots (p, -6, 9.1); (%o2) 4
where p is a polynomial with integer coefficients and n is a
positive integer returns q, a polynomial over the integers, such that
q^n = p or prints an error message indicating that p is not a
perfect nth power. This routine is much faster than factor or even
sqfr.
Default value: false
The option variable polyfactor when true causes
allroots and bfallroots to factor the polynomial over the real
numbers if the polynomial is real, or over the complex numbers, if the
polynomial is complex.
See allroots for an example.
Default value: true
When programmode is true, solve,
realroots, allroots, and linsolve return solutions
as elements in a list.
(Except when backsubst is set to false, in which case
programmode: false is assumed.)
When programmode is false, solve, etc. create intermediate
expression labels %t1, %t2, etc., and assign the solutions to them.
Default value: false
When realonly is true, algsys returns only those solutions
which are free of %i.
Computes rational approximations of the real roots of the polynomial expr
or polynomial equation eqn of one variable, to within a tolerance of
bound. Coefficients of expr or eqn must be literal numbers;
symbol constants such as %pi are rejected.
realroots assigns the multiplicities of the roots it finds
to the global variable multiplicities.
realroots constructs a Sturm sequence to bracket each root, and then
applies bisection to refine the approximations. All coefficients are converted
to rational equivalents before searching for roots, and computations are carried
out by exact rational arithmetic. Even if some coefficients are floating-point
numbers, the results are rational (unless coerced to floats by the
float or numer flags).
When bound is less than 1, all integer roots are found exactly.
When bound is unspecified, it is assumed equal to the global variable
rootsepsilon.
When the global variable programmode is true, realroots
returns a list of the form [x = x_1, x = x_2, ...].
When programmode is false, realroots creates intermediate
expression labels %t1, %t2, …,
assigns the results to them, and returns the list of labels.
See also allroots, bfallroots, guess_exact_value,
and lhs.
Examples:
(%i1) realroots (-1 - x + x^5, 5e-6);
612003
(%o1) [x = ------]
524288
(%i2) ev (%[1], float);
(%o2) x = 1.167303085327148
(%i3) ev (-1 - x + x^5, %);
(%o3) - 7.396496210176905E-6
(%i1) realroots (expand ((1 - x)^5 * (2 - x)^3 * (3 - x)), 1e-20); (%o1) [x = 1, x = 2, x = 3] (%i2) multiplicities; (%o2) [5, 3, 1]
Returns the right-hand side (that is, the second argument) of the expression
expr, when the operator of expr is one of the relational operators
< <= = # equal notequal >= >,
one of the assignment operators := ::= : ::, or a user-defined binary
infix operator, as declared by infix.
When expr is an atom or its operator is something other than the ones
listed above, rhs returns 0.
See also lhs.
Examples:
(%i1) e: aa + bb = cc;
(%o1) bb + aa = cc
(%i2) lhs (e);
(%o2) bb + aa
(%i3) rhs (e);
(%o3) cc
(%i4) [rhs (aa < bb), rhs (aa <= bb), rhs (aa >= bb),
rhs (aa > bb)];
(%o4) [bb, bb, bb, bb]
(%i5) [rhs (aa = bb), rhs (aa # bb), rhs (equal (aa, bb)),
rhs (notequal (aa, bb))];
(%o5) [bb, bb, bb, bb]
(%i6) e1: '(foo(x) := 2*x);
(%o6) foo(x) := 2 x
(%i7) e2: '(bar(y) ::= 3*y);
(%o7) bar(y) ::= 3 y
(%i8) e3: '(x : y);
(%o8) x : y
(%i9) e4: '(x :: y);
(%o9) x :: y
(%i10) [rhs (e1), rhs (e2), rhs (e3), rhs (e4)];
(%o10) [2 x, 3 y, y, y]
(%i11) infix ("][");
(%o11) ][
(%i12) rhs (aa ][ bb);
(%o12) bb
Default value: true
rootsconmode governs the behavior of the rootscontract command.
See rootscontract for details.
Converts products of roots into roots of products. For example,
rootscontract (sqrt(x)*y^(3/2)) yields sqrt(x*y^3).
When radexpand is true and domain is real,
rootscontract converts abs into sqrt, e.g.,
rootscontract (abs(x)*sqrt(y)) yields sqrt(x^2*y).
There is an option rootsconmode affecting rootscontract as
follows:
Problem Value of Result of applying
rootsconmode rootscontract
x^(1/2)*y^(3/2) false (x*y^3)^(1/2)
x^(1/2)*y^(1/4) false x^(1/2)*y^(1/4)
x^(1/2)*y^(1/4) true (x*y^(1/2))^(1/2)
x^(1/2)*y^(1/3) true x^(1/2)*y^(1/3)
x^(1/2)*y^(1/4) all (x^2*y)^(1/4)
x^(1/2)*y^(1/3) all (x^3*y^2)^(1/6)
When rootsconmode is false, rootscontract contracts only
with respect to rational number exponents whose denominators are the same. The
key to the rootsconmode: true examples is simply that 2 divides into 4
but not into 3. rootsconmode: all involves taking the least common
multiple of the denominators of the exponents.
rootscontract uses ratsimp in a manner similar to
logcontract.
Examples:
(%i1) rootsconmode: false$
(%i2) rootscontract (x^(1/2)*y^(3/2));
3
(%o2) sqrt(x y )
(%i3) rootscontract (x^(1/2)*y^(1/4));
1/4
(%o3) sqrt(x) y
(%i4) rootsconmode: true$
(%i5) rootscontract (x^(1/2)*y^(1/4));
(%o5) sqrt(x sqrt(y))
(%i6) rootscontract (x^(1/2)*y^(1/3));
1/3
(%o6) sqrt(x) y
(%i7) rootsconmode: all$
(%i8) rootscontract (x^(1/2)*y^(1/4));
2 1/4
(%o8) (x y)
(%i9) rootscontract (x^(1/2)*y^(1/3));
3 2 1/6
(%o9) (x y )
(%i10) rootsconmode: false$
(%i11) rootscontract (sqrt(sqrt(x) + sqrt(1 + x))
*sqrt(sqrt(1 + x) - sqrt(x)));
(%o11) 1
(%i12) rootsconmode: true$
(%i13) rootscontract (sqrt(5+sqrt(5)) - 5^(1/4)*sqrt(1+sqrt(5)));
(%o13) 0
Default value: 1.0e-7
rootsepsilon is the tolerance which establishes the confidence interval
for the roots found by the realroots function.
Solves the algebraic equation expr for the variable x and returns a
list of solution equations in x. If expr is not an equation, the
equation expr = 0 is assumed in its place.
x may be a function (e.g. f(x)), or other non-atomic expression
except a sum or product. x may be omitted if expr contains only one
variable. expr may be a rational expression, and may contain
trigonometric functions, exponentials, etc.
The following method is used:
Let E be the expression and X be the variable. If E is linear
in X then it is trivially solved for X. Otherwise if E is of
the form A*X^N + B then the result is (-B/A)^1/N) times the
N’th roots of unity.
If E is not linear in X then the gcd of the exponents of X in
E (say N) is divided into the exponents and the multiplicity of the
roots is multiplied by N. Then solve is called again on the
result. If E factors then solve is called on each of the factors.
Finally solve will use the quadratic, cubic, or quartic formulas where
necessary.
In the case where E is a polynomial in some function of the variable to be
solved for, say F(X), then it is first solved for F(X) (call the
result C), then the equation F(X)=C can be solved for X
provided the inverse of the function F is known.
breakup if false will cause solve to express the solutions
of cubic or quartic equations as single expressions rather than as made
up of several common subexpressions which is the default.
multiplicities - will be set to a list of the multiplicities of the
individual solutions returned by solve, realroots, or
allroots. Try apropos (solve) for the switches which affect
solve. describe may then by used on the individual switch names
if their purpose is not clear.
solve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])
solves a system of simultaneous (linear or non-linear) polynomial equations by
calling linsolve or algsys and returns a list of the solution
lists in the variables. In the case of linsolve this list would contain
a single list of solutions. It takes two lists as arguments. The first list
represents the equations to be solved; the second list is a
list of the unknowns to be determined. If the total number of
variables in the equations is equal to the number of equations, the
second argument-list may be omitted.
When programmode is false, solve displays solutions with
intermediate expression (%t) labels, and returns the list of labels.
When globalsolve is true and the problem is to solve two or more
linear equations, each solved-for variable is bound to its value in the solution
of the equations.
Examples:
(%i1) solve (asin (cos (3*x))*(f(x) - 1), x);
solve: using arc-trig functions to get a solution.
Some solutions will be lost.
%pi
(%o1) [x = ---, f(x) = 1]
6
(%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
log(125)
(%o2) [f(x) = --------]
log(5)
(%i3) [4*x^2 - y^2 = 12, x*y - x = 2];
2 2
(%o3) [4 x - y = 12, x y - x = 2]
(%i4) solve (%, [x, y]);
(%o4) [[x = 2, y = 2], [x = .5202594388652008 %i
- .1331240357358706, y = .07678378523787788
- 3.608003221870287 %i], [x = - .5202594388652008 %i
- .1331240357358706, y = 3.608003221870287 %i
+ .07678378523787788], [x = - 1.733751846381093,
y = - .1535675710019696]]
(%i5) solve (1 + a*x + x^3, x);
3
sqrt(3) %i 1 sqrt(4 a + 27) 1 1/3
(%o5) [x = (- ---------- - -) (--------------- - -)
2 2 6 sqrt(3) 2
sqrt(3) %i 1
(---------- - -) a
2 2
- --------------------------, x =
3
sqrt(4 a + 27) 1 1/3
3 (--------------- - -)
6 sqrt(3) 2
3
sqrt(3) %i 1 sqrt(4 a + 27) 1 1/3
(---------- - -) (--------------- - -)
2 2 6 sqrt(3) 2
sqrt(3) %i 1
(- ---------- - -) a
2 2
- --------------------------, x =
3
sqrt(4 a + 27) 1 1/3
3 (--------------- - -)
6 sqrt(3) 2
3
sqrt(4 a + 27) 1 1/3 a
(--------------- - -) - --------------------------]
6 sqrt(3) 2 3
sqrt(4 a + 27) 1 1/3
3 (--------------- - -)
6 sqrt(3) 2
(%i6) solve (x^3 - 1);
sqrt(3) %i - 1 sqrt(3) %i + 1
(%o6) [x = --------------, x = - --------------, x = 1]
2 2
(%i7) solve (x^6 - 1);
sqrt(3) %i + 1 sqrt(3) %i - 1
(%o7) [x = --------------, x = --------------, x = - 1,
2 2
sqrt(3) %i + 1 sqrt(3) %i - 1
x = - --------------, x = - --------------, x = 1]
2 2
(%i8) ev (x^6 - 1, %[1]);
6
(sqrt(3) %i + 1)
(%o8) ----------------- - 1
64
(%i9) expand (%);
(%o9) 0
(%i10) x^2 - 1;
2
(%o10) x - 1
(%i11) solve (%, x);
(%o11) [x = - 1, x = 1]
(%i12) ev (%th(2), %[1]);
(%o12) 0
The symbols %r are used to denote arbitrary constants in a solution.
(%i1) solve([x+y=1,2*x+2*y=2],[x,y]); solve: dependent equations eliminated: (2) (%o1) [[x = 1 - %r1, y = %r1]]
See algsys and %rnum_list for more information.
Default value: true
When solvedecomposes is true, solve calls
polydecomp if asked to solve polynomials.
Default value: false
When solveexplicit is true, inhibits solve from returning
implicit solutions, that is, solutions of the form F(x) = 0 where
F is some function.
Default value: true
When solvefactors is false, solve does not try to factor
the expression. The false setting may be desired in some cases where
factoring is not necessary.
Default value: true
When solvenullwarn is true, solve prints a warning message
if called with either a null equation list or a null variable list. For
example, solve ([], []) would print two warning messages and return
[].
Default value: false
When solveradcan is true, solve calls radcan
which makes solve slower but will allow certain problems containing
exponentials and logarithms to be solved.
Default value: true
When solvetrigwarn is true, solve may print a message
saying that it is using inverse trigonometric functions to solve the equation,
and thereby losing solutions.