Polynomials (Maxima 5.42.455.gc5fa8d39c Manual)

14 Polynomials

14.1 Introduction to Polynomials

Polynomials are stored in Maxima either in General Form or as Canonical Rational Expressions (CRE) form. The latter is a standard form, and is used internally by operations such as factor, ratsimp, and so on.

Canonical Rational Expressions constitute a kind of representation which is especially suitable for expanded polynomials and rational functions (as well as for partially factored polynomials and rational functions when ratfac is set to true). In this CRE form an ordering of variables (from most to least main) is assumed for each expression.

Polynomials are represented recursively by a list consisting of the main variable followed by a series of pairs of expressions, one for each term of the polynomial. The first member of each pair is the exponent of the main variable in that term and the second member is the coefficient of that term which could be a number or a polynomial in another variable again represented in this form. Thus the principal part of the CRE form of 3*x^2-1 is (X 2 3 0 -1) and that of 2*x*y+x-3 is (Y 1 (X 1 2) 0 (X 1 1 0 -3)) assuming y is the main variable, and is (X 1 (Y 1 2 0 1) 0 -3) assuming x is the main variable. "Main"-ness is usually determined by reverse alphabetical order.

The "variables" of a CRE expression needn’t be atomic. In fact any subexpression whose main operator is not +, -, *, / or ^ with integer power will be considered a "variable" of the expression (in CRE form) in which it occurs. For example the CRE variables of the expression x+sin(x+1)+2*sqrt(x)+1 are x, sqrt(X), and sin(x+1). If the user does not specify an ordering of variables by using the ratvars function Maxima will choose an alphabetic one.

In general, CRE’s represent rational expressions, that is, ratios of polynomials, where the numerator and denominator have no common factors, and the denominator is positive. The internal form is essentially a pair of polynomials (the numerator and denominator) preceded by the variable ordering list. If an expression to be displayed is in CRE form or if it contains any subexpressions in CRE form, the symbol /R/ will follow the line label.

See the rat function for converting an expression to CRE form.

An extended CRE form is used for the representation of Taylor series. The notion of a rational expression is extended so that the exponents of the variables can be positive or negative rational numbers rather than just positive integers and the coefficients can themselves be rational expressions as described above rather than just polynomials. These are represented internally by a recursive polynomial form which is similar to and is a generalization of CRE form, but carries additional information such as the degree of truncation. As with CRE form, the symbol /T/ follows the line label of such expressions.

Categories: Polynomials Rational expressions

14.2 Functions and Variables for Polynomials

Option variable: algebraic

Default value: false

algebraic must be set to true in order for the simplification of algebraic integers to take effect.

Categories: Simplification flags and variables

Option variable: berlefact

Default value: true

When berlefact is false then the Kronecker factoring algorithm will be used otherwise the Berlekamp algorithm, which is the default, will be used.

Categories: Polynomials

Function: bezout (p1, p2, x)

an alternative to the resultant command. It returns a matrix. determinant of this matrix is the desired resultant.

Examples:

(%i1) bezout(a*x+b, c*x^2+d, x);
                         [ b c  - a d ]
(%o1)                    [            ]
                         [  a     b   ]
(%i2) determinant(%);
                            2      2
(%o2)                      a  d + b  c
(%i3) resultant(a*x+b, c*x^2+d, x);
                            2      2
(%o3)                      a  d + b  c

Categories: Polynomials

Function: bothcoef (expr, x)

Returns a list whose first member is the coefficient of x in expr (as found by ratcoef if expr is in CRE form otherwise by coeff) and whose second member is the remaining part of expr. That is, [A, B] where expr = A*x + B.

Example:

(%i1) islinear (expr, x) := block ([c],
        c: bothcoef (rat (expr, x), x),
        is (freeof (x, c) and c[1] # 0))$
(%i2) islinear ((r^2 - (x - r)^2)/x, x);
(%o2)                         true

Categories: Polynomials

Function: coeff
coeff (expr, x, n)
coeff (expr, x)

Returns the coefficient of x^n in expr, where expr is a polynomial or a monomial term in x. Other than ratcoef coeff is a strictly syntactical operation and will only find literal instances of x^n in the internal representation of expr.

coeff(expr, x^n) is equivalent to coeff(expr, x, n). coeff(expr, x, 0) returns the remainder of expr which is free of x. If omitted, n is assumed to be 1.

x may be a simple variable or a subscripted variable, or a subexpression of expr which comprises an operator and all of its arguments.

It may be possible to compute coefficients of expressions which are equivalent to expr by applying expand or factor. coeff itself does not apply expand or factor or any other function.

coeff distributes over lists, matrices, and equations.

See also: factor_max_degree

Categories: Polynomials

Option variable: factorflag

Default value: false

When factorflag is false, suppresses the factoring of integer factors of rational expressions.

Categories: Polynomials

Function: factorout (expr, x_1, x_2, …)

Rearranges the sum expr into a sum of terms of the form f (x_1, x_2, …)*g where g is a product of expressions not containing any x_i and f is factored.

Note that the option variable keepfloat is ignored by factorout.

Example:

(%i1) expand (a*(x+1)*(x-1)*(u+1)^2);
             2  2          2      2      2
(%o1)     a u  x  + 2 a u x  + a x  - a u  - 2 a u - a

(%i2) factorout(%,x);
         2
(%o2) a u  (x - 1) (x + 1) + 2 a u (x - 1) (x + 1)
                                              + a (x - 1) (x + 1)

Categories: Expressions

Function: factorsum (expr)

Tries to group terms in factors of expr which are sums into groups of terms such that their sum is factorable. factorsum can recover the result of expand ((x + y)^2 + (z + w)^2) but it can’t recover expand ((x + 1)^2 + (x + y)^2) because the terms have variables in common.

Example:

(%i1) expand ((x + 1)*((u + v)^2 + a*(w + z)^2));
           2      2                            2      2
(%o1) a x z  + a z  + 2 a w x z + 2 a w z + a w  x + v  x

                                     2        2    2            2
                        + 2 u v x + u  x + a w  + v  + 2 u v + u
(%i2) factorsum (%);
                                   2          2
(%o2)            (x + 1) (a (z + w)  + (v + u) )

Categories: Expressions

Function: fasttimes (p_1, p_2)

Returns the product of the polynomials p_1 and p_2 by using a special algorithm for multiplication of polynomials. p_1 and p_2 should be multivariate, dense, and nearly the same size. Classical multiplication is of order n_1 n_2 where n_1 is the degree of p_1 and n_2 is the degree of p_2. fasttimes is of order max (n_1, n_2)^1.585.

Categories: Polynomials

Function: fullratsimp (expr)

fullratsimp repeatedly applies ratsimp followed by non-rational simplification to an expression until no further change occurs, and returns the result.

When non-rational expressions are involved, one call to ratsimp followed as is usual by non-rational ("general") simplification may not be sufficient to return a simplified result. Sometimes, more than one such call may be necessary. fullratsimp makes this process convenient.

fullratsimp (expr, x_1, ..., x_n) takes one or more arguments similar to ratsimp and rat.

Example:

(%i1) expr: (x^(a/2) + 1)^2*(x^(a/2) - 1)^2/(x^a - 1);
                       a/2     2   a/2     2
                     (x    - 1)  (x    + 1)
(%o1)                -----------------------
                              a
                             x  - 1
(%i2) ratsimp (expr);
                          2 a      a
                         x    - 2 x  + 1
(%o2)                    ---------------
                              a
                             x  - 1
(%i3) fullratsimp (expr);
                              a
(%o3)                        x  - 1
(%i4) rat (expr);
                       a/2 4       a/2 2
                     (x   )  - 2 (x   )  + 1
(%o4)/R/             -----------------------
                              a
                             x  - 1

Categories: Simplification functions Rational expressions

Function: fullratsubst (a, b, c)

is the same as ratsubst except that it calls itself recursively on its result until that result stops changing. This function is useful when the replacement expression and the replaced expression have one or more variables in common.

fullratsubst will also accept its arguments in the format of lratsubst. That is, the first argument may be a single substitution equation or a list of such equations, while the second argument is the expression being processed.

load ("lrats") loads fullratsubst and lratsubst.

Examples:

(%i1) load ("lrats")$

subst can carry out multiple substitutions. lratsubst is analogous to subst.

(%i2) subst ([a = b, c = d], a + c);
(%o2)                         d + b
(%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
(%o3)                (d + a c) e + a d + b c

If only one substitution is desired, then a single equation may be given as first argument.

(%i4) lratsubst (a^2 = b, a^3);
(%o4)                          a b

fullratsubst is equivalent to ratsubst except that it recurses until its result stops changing.

(%i5) ratsubst (b*a, a^2, a^3);
                               2
(%o5)                         a  b
(%i6) fullratsubst (b*a, a^2, a^3);
                                 2
(%o6)                         a b

fullratsubst also accepts a list of equations or a single equation as first argument.

(%i7) fullratsubst ([a^2 = b, b^2 = c, c^2 = a], a^3*b*c);
(%o7)                           b
(%i8) fullratsubst (a^2 = b*a, a^3);
                                 2
(%o8)                         a b

fullratsubst may cause an indefinite recursion.

(%i9) errcatch (fullratsubst (b*a^2, a^2, a^3));

*** - Lisp stack overflow. RESET

Categories: Rational expressions

Function: gcd (p_1, p_2, x_1, …)

Returns the greatest common divisor of p_1 and p_2. The flag gcd determines which algorithm is employed. Setting gcd to ez, subres, red, or spmod selects the ezgcd, subresultant prs, reduced, or modular algorithm, respectively. If gcd false then gcd (p_1, p_2, x) always returns 1 for all x. Many functions (e.g. ratsimp, factor, etc.) cause gcd’s to be taken implicitly. For homogeneous polynomials it is recommended that gcd equal to subres be used. To take the gcd when an algebraic is present, e.g., gcd (x^2 - 2*sqrt(2)* x + 2, x - sqrt(2)), the option variable algebraic must be true and gcd must not be ez.

The gcd flag, default: spmod, if false will also prevent the greatest common divisor from being taken when expressions are converted to canonical rational expression (CRE) form. This will sometimes speed the calculation if gcds are not required.

See also ezgcd, gcdex, gcdivide, and poly_gcd.

Example:

(%i1) p1:6*x^3+19*x^2+19*x+6; 
                        3       2
(%o1)                6 x  + 19 x  + 19 x + 6
(%i2) p2:6*x^5+13*x^4+12*x^3+13*x^2+6*x;
                  5       4       3       2
(%o2)          6 x  + 13 x  + 12 x  + 13 x  + 6 x
(%i3) gcd(p1, p2);
                            2
(%o3)                    6 x  + 13 x + 6
(%i4) p1/gcd(p1, p2), ratsimp;
(%o4)                         x + 1
(%i5) p2/gcd(p1, p2), ratsimp;
                              3
(%o5)                        x  + x

ezgcd returns a list whose first element is the greatest common divisor of the polynomials p_1 and p_2, and whose remaining elements are the polynomials divided by the greatest common divisor.

(%i6) ezgcd(p1, p2);
                    2                     3
(%o6)           [6 x  + 13 x + 6, x + 1, x  + x]

Categories: Polynomials Rational expressions

Function: gcdex
gcdex (f, g)
gcdex (f, g, x)

Returns a list [a, b, u] where u is the greatest common divisor (gcd) of f and g, and u is equal to a f + b g. The arguments f and g should be univariate polynomials, or else polynomials in x a supplied main variable since we need to be in a principal ideal domain for this to work. The gcd means the gcd regarding f and g as univariate polynomials with coefficients being rational functions in the other variables.

gcdex implements the Euclidean algorithm, where we have a sequence of L[i]: [a[i], b[i], r[i]] which are all perpendicular to [f, g, -1] and the next one is built as if q = quotient(r[i]/r[i+1]) then L[i+2]: L[i] - q L[i+1], and it terminates at L[i+1] when the remainder r[i+2] is zero.

The arguments f and g can be integers. For this case the function igcdex is called by gcdex.

See also ezgcd, gcd, gcdivide, and poly_gcd.

Examples:

(%i1) gcdex (x^2 + 1, x^3 + 4);
                       2
                      x  + 4 x - 1  x + 4
(%o1)/R/           [- ------------, -----, 1]
                           17        17

(%i2) % . [x^2 + 1, x^3 + 4, -1];
(%o2)/R/                        0

Note that the gcd in the following is 1 since we work in k(y)[x], not the y+1 we would expect in k[y, x].

(%i1) gcdex (x*(y + 1), y^2 - 1, x);
                               1
(%o1)/R/                 [0, ------, 1]
                              2
                             y  - 1

Categories: Polynomials Rational expressions

Function: gcfactor (n)

Factors the Gaussian integer n over the Gaussian integers, i.e., numbers of the form a + b %i where a and b are rational integers (i.e., ordinary integers). Factors are normalized by making a and b non-negative.

Categories: Integers

Function: gfactor (expr)

Factors the polynomial expr over the Gaussian integers (that is, the integers with the imaginary unit %i adjoined). This is like factor (expr, a^2+1) where a is %i.

Example:

(%i1) gfactor (x^4 - 1);
(%o1)           (x - 1) (x + 1) (x - %i) (x + %i)

Categories: Polynomials

Function: gfactorsum (expr)

is similar to factorsum but applies gfactor instead of factor.

Categories: Expressions

Function: hipow (expr, x)

Returns the highest explicit exponent of x in expr. x may be a variable or a general expression. If x does not appear in expr, hipow returns 0.

hipow does not consider expressions equivalent to expr. In particular, hipow does not expand expr, so hipow (expr, x) and hipow (expand (expr, x)) may yield different results.

Examples:

(%i1) hipow (y^3 * x^2 + x * y^4, x);
(%o1)                           2
(%i2) hipow ((x + y)^5, x);
(%o2)                           1
(%i3) hipow (expand ((x + y)^5), x);
(%o3)                           5
(%i4) hipow ((x + y)^5, x + y);
(%o4)                           5
(%i5) hipow (expand ((x + y)^5), x + y);
(%o5)                           0

Categories: Expressions

Option variable: intfaclim

Default value: true

If true, maxima will give up factorization of integers if no factor is found after trial divisions and Pollard’s rho method and factorization will not be complete.

When intfaclim is false (this is the case when the user calls factor explicitly), complete factorization will be attempted. intfaclim is set to false when factors are computed in divisors, divsum and totient.

Internal calls to factor respect the user-specified value of intfaclim. Setting intfaclim to true may reduce the time spent factoring large integers.

Categories: Integers

Option variable: keepfloat

Default value: false

When keepfloat is true, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

Note that the function solve and those functions calling it (eigenvalues, for example) currently ignore this flag, converting floating point numbers anyway.

Examples:

(%i1) rat(x/2.0);

rat: replaced 0.5 by 1/2 = 0.5
                                x
(%o1)/R/                        -
                                2

(%i2) rat(x/2.0), keepfloat;
(%o2)/R/                      0.5 x

solve ignores keepfloat:

(%i1) solve(1.0-x,x), keepfloat;

rat: replaced 1.0 by 1/1 = 1.0
(%o1)                        [x = 1]

Categories: Numerical evaluation

Function: lopow (expr, x)

Returns the lowest exponent of x which explicitly appears in expr. Thus

(%i1) lopow ((x+y)^2 + (x+y)^a, x+y);
(%o1)                       min(a, 2)

Categories: Expressions

Function: lratsubst (L, expr)

is analogous to subst (L, expr) except that it uses ratsubst instead of subst.

The first argument of lratsubst is an equation or a list of equations identical in format to that accepted by subst. The substitutions are made in the order given by the list of equations, that is, from left to right.

load ("lrats") loads fullratsubst and lratsubst.

Examples:

(%i1) load ("lrats")$

subst can carry out multiple substitutions. lratsubst is analogous to subst.

(%i2) subst ([a = b, c = d], a + c);
(%o2)                         d + b
(%i3) lratsubst ([a^2 = b, c^2 = d], (a + e)*c*(a + c));
(%o3)                (d + a c) e + a d + b c

If only one substitution is desired, then a single equation may be given as first argument.

(%i4) lratsubst (a^2 = b, a^3);
(%o4)                          a b

Categories: Polynomials Rational expressions

Option variable: modulus

Default value: false

When modulus is a positive number p, operations on canonical rational expressions (CREs, as returned by rat and related functions) are carried out modulo p, using the so-called "balanced" modulus system in which n modulo p is defined as an integer k in [-(p-1)/2, ..., 0, ..., (p-1)/2] when p is odd, or [-(p/2 - 1), ..., 0, ...., p/2] when p is even, such that a p + k equals n for some integer a.

If expr is already in canonical rational expression (CRE) form when modulus is reset, then you may need to re-rat expr, e.g., expr: rat (ratdisrep (expr)), in order to get correct results.

Typically modulus is set to a prime number. If modulus is set to a positive non-prime integer, this setting is accepted, but a warning message is displayed. Maxima signals an error, when zero or a negative integer is assigned to modulus.

Examples:

(%i1) modulus:7;
(%o1)                           7
(%i2) polymod([0,1,2,3,4,5,6,7]);
(%o2)            [0, 1, 2, 3, - 3, - 2, - 1, 0]
(%i3) modulus:false;
(%o3)                         false
(%i4) poly:x^6+x^2+1;
                            6    2
(%o4)                      x  + x  + 1
(%i5) factor(poly);
                            6    2
(%o5)                      x  + x  + 1
(%i6) modulus:13;
(%o6)                          13
(%i7) factor(poly);
                      2        4      2
(%o7)               (x  + 6) (x  - 6 x  - 2)
(%i8) polymod(%);
                            6    2
(%o8)                      x  + x  + 1

Categories: Integers

Function: num (expr)

Returns the numerator of expr if it is a ratio. If expr is not a ratio, expr is returned.

num evaluates its argument.

See also totaldisrep.

Categories: Rational expressions

Function: ratexpand (expr)

Option variable: ratexpand

Expands expr by multiplying out products of sums and exponentiated sums, combining fractions over a common denominator, cancelling the greatest common divisor of the numerator and denominator, then splitting the numerator (if a sum) into its respective terms divided by the denominator.

The return value of ratexpand is a general expression, even if expr is a canonical rational expression (CRE).

The switch ratexpand if true will cause CRE expressions to be fully expanded when they are converted back to general form or displayed, while if it is false then they will be put into a recursive form. See also ratsimp.

When keepfloat is true, prevents floating point numbers from being rationalized when expressions which contain them are converted to canonical rational expression (CRE) form.

Examples:

(%i1) ratexpand ((2*x - 3*y)^3);
                     3         2       2        3
(%o1)          - 27 y  + 54 x y  - 36 x  y + 8 x
(%i2) expr: (x - 1)/(x + 1)^2 + 1/(x - 1);
                         x - 1       1
(%o2)                   -------- + -----
                               2   x - 1
                        (x + 1)
(%i3) expand (expr);

                    x              1           1
(%o3)          ------------ - ------------ + -----
                2              2             x - 1
               x  + 2 x + 1   x  + 2 x + 1

(%i4) ratexpand (expr);
                        2
                     2 x                 2
(%o4)           --------------- + ---------------
                 3    2            3    2
                x  + x  - x - 1   x  + x  - x - 1

Categories: Rational expressions

Option variable: ratfac

Default value: false

When ratfac is true, canonical rational expressions (CRE) are manipulated in a partially factored form.

During rational operations the expression is maintained as fully factored as possible without calling factor. This should always save space and may save time in some computations. The numerator and denominator are made relatively prime, for example factor ((x^2 - 1)^4/(x + 1)^2) yields (x - 1)^4 (x + 1)^2, but the factors within each part may not be relatively prime.

In the ctensr (Component Tensor Manipulation) package, Ricci, Einstein, Riemann, and Weyl tensors and the scalar curvature are factored automatically when ratfac is true. ratfac should only be set for cases where the tensorial components are known to consist of few terms.

The ratfac and ratweight schemes are incompatible and may not both be used at the same time.

Categories: Rational expressions

Function: ratnumer (expr)

Returns the numerator of expr, after coercing expr to a canonical rational expression (CRE). The return value is a CRE.

expr is coerced to a CRE by rat if it is not already a CRE. This conversion may change the form of expr by putting all terms over a common denominator.

num is similar, but returns an ordinary expression instead of a CRE. Also, num does not attempt to place all terms over a common denominator, and thus some expressions which are considered ratios by ratnumer are not considered ratios by num.

Categories: Rational expressions

Function: ratp (expr)

Returns true if expr is a canonical rational expression (CRE) or extended CRE, otherwise false.

CRE are created by rat and related functions. Extended CRE are created by taylor and related functions.

Categories: Predicate functions Rational expressions

Option variable: ratprint

Default value: true

When ratprint is true, a message informing the user of the conversion of floating point numbers to rational numbers is displayed.

Categories: Rational expressions Numerical evaluation Console interaction

Function: ratsimp (expr)

Function: ratsimp (expr, x_1, …, x_n)

Simplifies the expression expr and all of its subexpressions, including the arguments to non-rational functions. The result is returned as the quotient of two polynomials in a recursive form, that is, the coefficients of the main variable are polynomials in the other variables. Variables may include non-rational functions (e.g., sin (x^2 + 1)) and the arguments to any such functions are also rationally simplified.

ratsimp (expr, x_1, ..., x_n) enables rational simplification with the specification of variable ordering as in ratvars.

When ratsimpexpons is true, ratsimp is applied to the exponents of expressions during simplification.

See also ratexpand. Note that ratsimp is affected by some of the flags which affect ratexpand.

Examples:

(%i1) sin (x/(x^2 + x)) = exp ((log(x) + 1)^2 - log(x)^2);

                                         2      2
                   x         (log(x) + 1)  - log (x)
(%o1)        sin(------) = %e
                  2
                 x  + x

(%i2) ratsimp (%);
                             1          2
(%o2)                  sin(-----) = %e x
                           x + 1
(%i3) ((x - 1)^(3/2) - (x + 1)*sqrt(x - 1))/sqrt((x - 1)*(x + 1));

                       3/2
                (x - 1)    - sqrt(x - 1) (x + 1)
(%o3)           --------------------------------
                     sqrt((x - 1) (x + 1))

(%i4) ratsimp (%);
                           2 sqrt(x - 1)
(%o4)                    - -------------
                                 2
                           sqrt(x  - 1)
(%i5) x^(a + 1/a), ratsimpexpons: true;
                               2
                              a  + 1
                              ------
                                a
(%o5)                        x

Categories: Simplification functions Rational expressions

Option variable: ratsimpexpons

Default value: false

When ratsimpexpons is true, ratsimp is applied to the exponents of expressions during simplification.

Categories: Simplification flags and variables Rational expressions

Option variable: radsubstflag

Default value: false

radsubstflag, if true, permits ratsubst to make substitutions such as u for sqrt (x) in x.

Categories: Simplification flags and variables

Function: ratsubst (a, b, c)

Substitutes a for b in c and returns the resulting expression. b may be a sum, product, power, etc.

ratsubst knows something of the meaning of expressions whereas subst does a purely syntactic substitution. Thus subst (a, x + y, x + y + z) returns x + y + z whereas ratsubst returns z + a.

When radsubstflag is true, ratsubst makes substitutions for radicals in expressions which don’t explicitly contain them.

ratsubst ignores the value true of the option variable keepfloat.

Examples:

(%i1) ratsubst (a, x*y^2, x^4*y^3 + x^4*y^8);
                              3      4
(%o1)                      a x  y + a

(%i2) cos(x)^4 + cos(x)^3 + cos(x)^2 + cos(x) + 1;
               4         3         2
(%o2)       cos (x) + cos (x) + cos (x) + cos(x) + 1

(%i3) ratsubst (1 - sin(x)^2, cos(x)^2, %);
            4           2                     2
(%o3)    sin (x) - 3 sin (x) + cos(x) (2 - sin (x)) + 3

(%i4) ratsubst (1 - cos(x)^2, sin(x)^2, sin(x)^4);
                        4           2
(%o4)                cos (x) - 2 cos (x) + 1

(%i5) radsubstflag: false$

(%i6) ratsubst (u, sqrt(x), x);
(%o6)                           x

(%i7) radsubstflag: true$

(%i8) ratsubst (u, sqrt(x), x);
                                2
(%o8)                          u

Categories: Rational expressions

Function: ratvars (x_1, …, x_n)

Function: ratvars ()

System variable: ratvars

Declares main variables x_1, …, x_n for rational expressions. x_n, if present in a rational expression, is considered the main variable. Otherwise, x_[n-1] is considered the main variable if present, and so on through the preceding variables to x_1, which is considered the main variable only if none of the succeeding variables are present.

If a variable in a rational expression is not present in the ratvars list, it is given a lower priority than x_1.

The arguments to ratvars can be either variables or non-rational functions such as sin(x).

The variable ratvars is a list of the arguments of the function ratvars when it was called most recently. Each call to the function ratvars resets the list. ratvars () clears the list.

Categories: Rational expressions

Option variable: ratvarswitch

Default value: true

Maxima keeps an internal list in the Lisp variable VARLIST of the main variables for rational expressions. If ratvarswitch is true, every evaluation starts with a fresh list VARLIST. This is the default behavior. Otherwise, the main variables from previous evaluations are not removed from the internal list VARLIST.

The main variables, which are declared with the function ratvars are not affected by the option variable ratvarswitch.

Examples:

If ratvarswitch is true, every evaluation starts with a fresh list VARLIST.

(%i1) ratvarswitch:true$

(%i2) rat(2*x+y^2);
                             2
(%o2)/R/                    y  + 2 x
(%i3) :lisp varlist
($X $Y)

(%i3) rat(2*a+b^2);
                             2
(%o3)/R/                    b  + 2 a

(%i4) :lisp varlist
($A $B)

If ratvarswitch is false, the main variables from the last evaluation are still present.

(%i4) ratvarswitch:false$

(%i5) rat(2*x+y^2);
                             2
(%o5)/R/                    y  + 2 x
(%i6) :lisp varlist
($X $Y)

(%i6) rat(2*a+b^2);
                             2
(%o6)/R/                    b  + 2 a

(%i7) :lisp varlist
($A $B $X $Y)

Categories: Rational expressions Global flags

Function: ratweight
ratweight (x_1, w_1, …, x_n, w_n)
ratweight ()

Assigns a weight w_i to the variable x_i. This causes a term to be replaced by 0 if its weight exceeds the value of the variable ratwtlvl (default yields no truncation). The weight of a term is the sum of the products of the weight of a variable in the term times its power. For example, the weight of 3 x_1^2 x_2 is 2 w_1 + w_2. Truncation according to ratwtlvl is carried out only when multiplying or exponentiating canonical rational expressions (CRE).

ratweight () returns the cumulative list of weight assignments.

Note: The ratfac and ratweight schemes are incompatible and may not both be used at the same time.

Examples:

(%i1) ratweight (a, 1, b, 1);
(%o1)                     [a, 1, b, 1]
(%i2) expr1: rat(a + b + 1)$
(%i3) expr1^2;
                  2                  2
(%o3)/R/         b  + (2 a + 2) b + a  + 2 a + 1
(%i4) ratwtlvl: 1$
(%i5) expr1^2;
(%o5)/R/                  2 b + 2 a + 1

Categories: Rational expressions

System variable: ratweights

Default value: []

ratweights is the list of weights assigned by ratweight. The list is cumulative: each call to ratweight places additional items in the list.

kill (ratweights) and save (ratweights) both work as expected.

Categories: Rational expressions

Option variable: ratwtlvl

Default value: false

ratwtlvl is used in combination with the ratweight function to control the truncation of canonical rational expressions (CRE). For the default value of false, no truncation occurs.

Categories: Rational expressions

Function: remainder
remainder (p_1, p_2)
remainder (p_1, p_2, x_1, …, x_n)

Returns the remainder of the polynomial p_1 divided by the polynomial p_2. The arguments x_1, …, x_n are interpreted as in ratvars.

remainder returns the second element of the two-element list returned by divide.

Categories: Polynomials

Function: resultant (p_1, p_2, x)

The function resultant computes the resultant of the two polynomials p_1 and p_2, eliminating the variable x. The resultant is a determinant of the coefficients of x in p_1 and p_2, which equals zero if and only if p_1 and p_2 have a non-constant factor in common.

If p_1 or p_2 can be factored, it may be desirable to call factor before calling resultant.

The option variable resultant controls which algorithm will be used to compute the resultant. See the option variable resultant.

The function bezout takes the same arguments as resultant and returns a matrix. The determinant of the return value is the desired resultant.

Examples:

(%i1) resultant(2*x^2+3*x+1, 2*x^2+x+1, x);
(%o1)                           8
(%i2) resultant(x+1, x+1, x);
(%o2)                           0
(%i3) resultant((x+1)*x, (x+1), x);
(%o3)                           0
(%i4) resultant(a*x^2+b*x+1, c*x + 2, x);
                         2
(%o4)                   c  - 2 b c + 4 a

(%i5) bezout(a*x^2+b*x+1, c*x+2, x);

                        [ 2 a  2 b - c ]
(%o5)                   [              ]
                        [  c      2    ]

(%i6) determinant(%);
(%o6)                   4 a - (2 b - c) c

Categories: Polynomials

Option variable: resultant

Default value: subres

The option variable resultant controls which algorithm will be used to compute the resultant with the function resultant. The possible values are:

subres: for the subresultant polynomial remainder sequence (PRS) algorithm,
mod: (not enabled) for the modular resultant algorithm, and
red: for the reduced polynomial remainder sequence (PRS) algorithm.

On most problems the default value subres should be best.

Categories: Polynomials

Option variable: savefactors

Default value: false

When savefactors is true, causes the factors of an expression which is a product of factors to be saved by certain functions in order to speed up later factorizations of expressions containing some of the same factors.

Categories: Polynomials

Function: showratvars (expr)

Returns a list of the canonical rational expression (CRE) variables in expression expr.

• Introduction to Polynomials:
• Functions and Variables for Polynomials: