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20.1 Functions and Variables for Equations |
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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.
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
Categories: Algebraic equations
Default value: 10^8
algepsilon
is used by algsys
.
Categories: Algebraic equations
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.
Categories: Algebraic equations
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
Categories: Algebraic equations
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)
Categories: Polynomials · Numerical methods
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.
Categories: Polynomials · Numerical methods
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]
Categories: Algebraic equations
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]
Categories: Algebraic equations
dimen
is a package for dimensional analysis.
load ("dimen")
loads this package.
demo ("dimen")
displays a short demonstration.
Categories: Share packages
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, $, sets dispflag
to
false
.
Categories: Algebraic equations · Display flags and variables
Returns [g(t) = ...]
or []
, depending on whether
or not there exists a rational function g(t)
satisfying
eqn, which must be a first order, linear polynomial in (for this case)
g(t)
and g(t+1)
(%i1) eqn: (n + 1)*f(n) - (n + 3)*f(n + 1)/(n + 1) = (n - 1)/(n + 2); (n + 3) f(n + 1) n - 1 (%o1) (n + 1) f(n) - ---------------- = ----- n + 1 n + 2 (%i2) funcsolve (eqn, f(n)); Dependent equations eliminated: (4 3) n (%o2) f(n) = --------------- (n + 1) (n + 2)
Warning: this is a very rudimentary implementation - many safety checks and obvious generalizations are missing.
Categories: Algebraic equations
Default value: false
When globalsolve
is true
, solved-for variables are assigned the
solution values found by linsolve
, and by solve
when solving two
or more linear equations.
When globalsolve
is false
, solutions found by linsolve
and
by solve
when solving two or more linear equations are expressed as
equations, and the solved-for variables are not assigned.
When solving anything other than two or more linear equations, solve
ignores globalsolve
. Other functions which solve equations (e.g.,
algsys
) always ignore globalsolve
.
Examples:
(%i1) globalsolve: true$ (%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
Categories: Linear equations
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 above (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
.
Default values for the 2nd thru 5th parameters 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.
Categories: Integral equations
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).
Categories: Integral equations
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
Categories: Expressions
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.
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.
(%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]
Categories: Linear equations
Default value: true
When linsolvewarn
is true
, linsolve
prints a message
"Dependent equations eliminated".
Categories: Linear equations
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.
Categories: Linear equations
Default value: not_set_yet
multiplicities
is set to a list of the multiplicities of the individual
solutions returned by solve
or realroots
.
Categories: Algebraic equations · Polynomials
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
Categories: Polynomials · Numerical methods
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
.
Categories: Polynomials
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.
Categories: Polynomials · Numerical methods
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.
Categories: Algebraic equations · Polynomials
Default value: false
When realonly
is true
, algsys
returns only those solutions
which are free of %i
.
Categories: Algebraic equations
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.
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]
Categories: Polynomials · Numerical methods
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
Categories: Expressions
Default value: true
rootsconmode
governs the behavior of the rootscontract
command.
See rootscontract
for details.
Categories: Expressions · Simplification flags and variables
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
Categories: Simplification functions
Default value: 1.0e-7
rootsepsilon
is the tolerance which establishes the confidence interval
for the roots found by the realroots
function.
Categories: Polynomials · Numerical methods
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.
Categories: Algebraic equations
Default value: true
When solvedecomposes
is true
, solve
calls
polydecomp
if asked to solve polynomials.
Categories: Algebraic equations
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.
Categories: Algebraic equations
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.
Categories: Algebraic equations
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
[]
.
Categories: Algebraic equations
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.
Categories: Algebraic equations
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.
Categories: Algebraic equations
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