The function algsubs performs an algebraic substitution, replacing occurrences of a with b in the expression f.  It is a generalization of the subs command, which only handles syntactic substitution. The purpose of algsubs can be seen from these five examples where subs fails.

algsubs( s^2=1-c^2, s^3 );

algsubs( a+b=d, 1+a+b+c );

algsubs( a*b=c, 2*a*b^2-a*b*d );

algsubs( a*b/c=d, 2*a*b^2/c );

algsubs( a^2=0, exp(2-a+a^2/2-a^3/6) );

Where the expression being replaced has more than one variable, the result is ambiguous. For example, should algsubs(a+b=c, 2*a+b) return  or ? What should algsubs(a+b=c, a+Pi*b) return? Should algsubs(x*y^2=c, x*y^4) return  or ? The algsubs command provides two modes for breaking ambiguities, an exact mode and a remainder mode (the default).

Both modes depend on the ordering of the variables that appear in a. This can be set to a specific ordering by specifying the variables v in a list as a third argument. If the variables are not explicitly given, the set of indeterminates in a, b which are functions and names defines the variables and their order. Also, the result is collected in the variables given.

Both modes require that monomials in a divide monomials in f for a substitution to occur. A monomial u in a divides a monomial v in f if for each variable x in u, then either  or .

For example, the monomial  divides the monomial  but not  or .

Note: The algsubs command currently works only with integer exponents.

Note that the requirement for monomials in a to divide monomials in f means that the negative powers of u in the following example are not substituted, and must be handled separately as shown.

f := a/u^4+b/u^2+c+d*u^2+e*u^4;

algsubs(u^2=v,f);

algsubs(1/u^2=1/v,f);

Hence, to substitute for both positive and negative powers.

algsubs(u^2=v,algsubs(1/u^2=1/v,f));

If the option remainder is specified, or no option is specified, a generalized remainder is computed. If the leading monomial of a divides the leading monomial of f then the leading monomial of f is replaced with the appropriate value, and this is repeated until the leading monomial in f is less than the leading monomial in a.

If the option exact is specified, then if the value a being replaced is a sum of terms , and the value f it is replacing is the sum of terms , then the replacement is made if and only if each monomial  divides  and  that is, the coefficients must all be the same. For example, algsubs( x^2+2*y=z, 3*x^2+6*y ) succeeds with  but algsubs(x^2+2*y=z, 3*x^2+3*y) fails.

The algsubs command goes recursively through the expression . Unlike the subs command it does not substitute inside indexed names, and function calls are applied to the result of a substitution. Like subs, it does not expand products or powers before substitution. Hence algsubs( x^2=0, (x+1)^3 ) yields .