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Definitions for Equations

System variable: %rnum_list
Default value: []

%rnum_list is the list of variables introduced in solutions by 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).

Option variable: algexact
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 doesn't 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.

Function: algsys ([expr_1, ..., expr_m], [x_1, ..., x_n])
Function: algsys ([eqn_1, ..., eqn_m], [x_1, ..., x_n])
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:

(1) First the equations are factored and split into subsystems.

(2) For each subsystem S_i, an equation E and a variable x are selected. The variable is chosen to have lowest nonzero degree. Then the resultant of E and E_j with respect to x is computed for each of the remaining equations E_j in the subsystem S_i. This yields a new subsystem S_i' in one fewer variables, as x has been eliminated. The process now returns to (1).

(3) Eventually, a subsystem consisting of a single equation is obtained. If the equation is multivariate and no approximations in the form of floating point numbers have been introduced, then 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.

(4) Finally, the solutions obtained in step (3) are substituted into previous levels and the solution process returns to (1).

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.


(%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); 
(%o3)                   a1 (- y - x  + 1)
(%i4) e4: a2*(y - (x - 1)^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

Function: allroots (expr)
Function: allroots (eqn)
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).


(%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


(%o3)                         done
(%i4) polyfactor: true$
(%i5) allroots (eqn);
(%o5) - 13.5 (x - 1.0) (x - .8296749902129361)

 (x + 1.015755543828121) (x  + .8139194463848151 x

 + 1.098699797110288)

Option variable: backsubst
Default value: true

When backsubst is false, prevents back substitution 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.

Option variable: breakup
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.


(%i1) programmode: false$
(%i2) breakup: true$
(%i3) solve (x^3 + x^2 - 1);

                        sqrt(23)    25 1/3
(%t3)                  (--------- + --)
                        6 sqrt(3)   54

                                      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);

             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]

Function: dimension (eqn)
Function: dimension (eqn_1, ..., eqn_n)
dimen is a package for dimensional analysis. load ("dimen") loads this package. demo ("dimen") displays a short demonstration.

Option variable: dispflag
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.

Function: funcsolve (eqn, g(t))
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)
(%o2)                f(n) = ---------------
                            (n + 1) (n + 2)

Warning: this is a very rudimentary implementation -- many safety checks and obvious generalizations are missing.

Option variable: globalsolve
Default value: false

When globalsolve is true, solved-for variables are assigned the solution values found by solve.


(%i1) globalsolve: true$
(%i2) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);

(%t2)                        x : --

(%t3)                        y : - -
(%o3)                     [[%t2, %t3]]
(%i3) x;
(%o3)                          --
(%i4) y;
(%o4)                          - -
(%i5) globalsolve: false$
(%i6) kill (x, y)$
(%i7) solve ([x + 3*y = 2, 2*x - y = 5], [x, y]);

(%t7)                        x = --

(%t8)                        y = - -
(%o8)                     [[%t7, %t8]]
(%i8) x;
(%o8)                           x
(%i9) y;
(%o9)                           y

Function: ieqn (ie, unk, tech, n, guess)
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.

Option variable: ieqnprint
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).

Function: lhs (eqn)
Returns the left side of the equation eqn.

If the argument is not an equation, lhs returns the argument.

See also rhs.


(%i1) e: x^2 + y^2 = z^2;
                           2    2    2
(%o1)                     y  + x  = z
(%i2) lhs (e);
                              2    2
(%o2)                        y  + x
(%i3) rhs (e);
(%o3)                          z

Function: linsolve ([expr_1, ..., expr_m], [x_1, ..., x_n])
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 then variables which are solved for will be set to the solution of the set of simultaneous 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.

(%i1) e1: x + z = y$
(%i2) e2: 2*a*x - y = 2*a^2$
(%i3) e3: y - 2*z = 2$
(%i4) linsolve ([e1, e2, e3], [x, y, z]);
(%o4)            [x = a + 1, y = 2 a, z = a - 1]

Option variable: linsolvewarn
Default value: true

When linsolvewarn is true, linsolve prints a message "Dependent equations eliminated".

Option variable: linsolve_params
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.

System variable: multiplicities
Default value: not_set_yet

multiplicities is set to a list of the multiplicities of the individual solutions returned by solve or realroots.

Function: nroots (p, low, high)
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. infinity and plus infinity.

nroots uses the method of Sturm sequences.

(%i1) p: x^10 - 2*x^4 + 1/2$
(%i2) nroots (p, -6, 9.1);
(%o2)                           4

Function: nthroot (p, n)
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.

Option variable: programmode
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.

Option variable: realonly
Default value: false

When realonly is true, algsys returns only those solutions which are free of %i.

Function: realroots (poly, bound)
Finds all of the real roots of the real univariate polynomial poly within a tolerance of bound which, if less than 1, causes all integral roots to be found exactly. The parameter bound may be arbitrarily small in order to achieve any desired accuracy. The first argument may also be an equation. realroots sets multiplicities, useful in case of multiple roots. realroots (p) is equivalent to realroots (p, rootsepsilon). rootsepsilon is a real number used to establish the confidence interval for the roots. Do example (realroots) for an example.

Function: rhs (eqn)
Returns the right side of the equation eqn.

If the argument is not an equation, rhs returns 0.

See also lhs.


(%i1) e: x^2 + y^2 = z^2;
                           2    2    2
(%o1)                     y  + x  = z
(%i2) lhs (e);
                              2    2
(%o2)                        y  + x
(%i3) rhs (e);
(%o3)                          z

Option variable: rootsconmode
Default value: true

rootsconmode governs the behavior of the rootscontract command. See rootscontract for details.

Function: rootscontract (expr)
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.


(%i1) rootsconmode: false$
(%i2) rootscontract (x^(1/2)*y^(3/2));
(%o2)                      sqrt(x y )
(%i3) rootscontract (x^(1/2)*y^(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));
(%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

Option variable: rootsepsilon
Default value: 1.0e-7

rootsepsilon is the tolerance which establishes the confidence interval for the roots found by the realroots function.

Function: solve (expr, x)
Function: solve (expr)
Function: solve ([eqn_1, ..., eqn_n], [x_1, ..., x_n])
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. For linear systems if the given equations are not compatible, the message inconsistent will be displayed (see the solve_inconsistent_error switch); if no unique solution exists, then singular will be displayed.


(%i1) solve (asin (cos (3*x))*(f(x) - 1), x);

SOLVE is using arc-trig functions to get a solution.
Some solutions will be lost.
(%o1)                  [x = ---, f(x) = 1]
(%i2) ev (solve (5^f(x) = 125, f(x)), solveradcan);
(%o2)                   [f(x) = --------]
(%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 = .0767837852378778

 - 3.608003221870287 %i], [x = - .5202594388652008 %i

 - .1331240357358706, y = 3.608003221870287 %i

 + .0767837852378778], [x = - 1.733751846381093, 

y = - .1535675710019696]]
(%i5) solve (1 + a*x + x^3, x);
              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 = 
      sqrt(4 a  + 27)   1 1/3
   3 (--------------- - -)
         6 sqrt(3)      2

 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 = 
      sqrt(4 a  + 27)   1 1/3
   3 (--------------- - -)
         6 sqrt(3)      2

 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]);
                      (sqrt(3) %i + 1)
(%o8)                 ----------------- - 1
(%i9) expand (%);
(%o9)                           0
(%i10) x^2 - 1;
(%o10)                       x  - 1
(%i11) solve (%, x);
(%o11)                  [x = - 1, x = 1]
(%i12) ev (%th(2), %[1]);
(%o12)                          0

Option variable: solvedecomposes
Default value: true

When solvedecomposes is true, solve calls polydecomp if asked to solve polynomials.

Option variable: solveexplicit
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.

Option variable: solvefactors
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.

Option variable: solvenullwarn
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 [].

Option variable: solveradcan
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.

Option variable: solvetrigwarn
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.

Option variable: solve_inconsistent_error
Default value: true

When solve_inconsistent_error is true, solve and linsolve give an error if the equations to be solved are inconsistent.

If false, solve and linsolve return an empty list [] if the equations are inconsistent.


(%i1) solve_inconsistent_error: true$
(%i2) solve ([a + b = 1, a + b = 2], [a, b]);
Inconsistent equations:  (2)
 -- an error.  Quitting.  To debug this try debugmode(true);
(%i3) solve_inconsistent_error: false$
(%i4) solve ([a + b = 1, a + b = 2], [a, b]);
(%o4)                          []

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