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 );

$\left(-c^2+1\right)s$

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

$1+d+c$

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

$2bc-dc$

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

$2db$

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

${\ⅇ}^{2-a}$

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 $2c-b$ or $a+c$? What should algsubs(a+b=c, a+Pi*b) return? Should algsubs(x*y^2=c, x*y^4) return $c{y}^2$ or $\frac{c^2}{x}$? 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 $0<\mathrm{degree}\left(u,x\right)\le \mathrm{degree}\left(v,x\right)$ or $\mathrm{degree}\left(v,x\right)\le \mathrm{degree}\left(u,x\right)<0$.

For example, the monomial $\frac{x^2}{y^2}$ divides the monomial $\frac{x^3}{y^2}$ but not $\frac{x}{y^2}$ or $\frac{x^2}{y}$.

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;

$f≔\frac{a}{u^4}+\frac{b}{u^2}+c+d{u}^2+e{u}^4$

algsubs(u^2=v,f);

$e{v}^2+dv+c+\frac{b}{u^2}+\frac{a}{u^4}$

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

$e{u}^4+d{u}^2+\frac{c{v}^2+bv+a}{v^2}$

Hence, to substitute for both positive and negative powers.

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

$\frac{e{v}^4+d{v}^3+c{v}^2+bv+a}{v^2}$

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 $a_i$, and the value f it is replacing is the sum of terms $f_i$, then the replacement is made if and only if each monomial $a_i$ divides $f_i$ and $\mathrm{sum}\left(a_i\right)=c\left(\mathrm{sum}\left(f_i\right)\right)$ that is, the coefficients must all be the same. For example, algsubs( x^2+2*y=z, 3*x^2+6*y ) succeeds with $c=3$ but algsubs(x^2+2*y=z, 3*x^2+3*y) fails.

The algsubs command goes recursively through the expression $f$. 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 ${\left(x+1\right)}^3$.

$\mathrm{algsubs}\left(a+b=c\&comma;\exp \left(a+b+d\right)\right)$

${\&ExponentialE;}^{c+d}$

$\mathrm{algsubs}\left(\frac{PV}{T}=R\&comma;\frac{P^{2}V}{T^2}-PR\right)$

$-PR+\frac{RP}{T}$

$\mathrm{algsubs}\left(s+\frac{1}{t}=y\&comma;{\left(s+\frac{1}{t}+1\right)}^3-s-\frac{2}{t}\right)$

${\left(y+1\right)}^3-\frac{1}{t}-y$

$f≔{\sin \left(x\right)}^3-\cos \left(x\right){\sin \left(x\right)}^2+{\cos \left(x\right)}^2\sin \left(x\right)+{\cos \left(x\right)}^3$

$f≔{\sin \left(x\right)}^3-\cos \left(x\right){\sin \left(x\right)}^2+{\cos \left(x\right)}^2\sin \left(x\right)+{\cos \left(x\right)}^3$

$\mathrm{algsubs}\left({\sin \left(x\right)}^2=1-{\cos \left(x\right)}^2\&comma;f\right)$

$2{\cos \left(x\right)}^3+\sin \left(x\right)-\cos \left(x\right)$

$f≔\mathrm{expand}\left({\left(u+w-\frac{1}{w}\right)}^3\right)$

$f≔u^3+3u^{2}w-\frac{3u^2}{w}+3u{w}^2-6u+\frac{3u}{w^2}+w^3-3w+\frac{3}{w}-\frac{1}{w^3}$

$\mathrm{algsubs}\left(u^2=0\&comma;f\right)$

$\frac{w^6-3w^4+3w^2-1}{w^3}+\frac{3\left(w^4-2w^2+1\right)u}{w^2}$

$\mathrm{algsubs}\left(u^2=0\&comma;f\&comma;\left[u\&comma;w\right]\right)$

$3u{w}^2-6u+\frac{3u}{w^2}+w^3-3w+\frac{3}{w}-\frac{1}{w^3}$

$\mathrm{algsubs}\left(\frac{1}{w^2}=0\&comma;f\&comma;\left[u\&comma;w\right]\right)$

$u^3+3u^{2}w-\frac{3u^2}{w}+3u{w}^2-6u+w^3-3w+\frac{3}{w}$

$\mathrm{algsubs}\left(uw=0\&comma;f\right)$

$u^3-\frac{3u^2}{w}-6u+\frac{3u}{w^2}+w^3-3w+\frac{3}{w}-\frac{1}{w^3}$

$\mathrm{algsubs}\left(\frac{u}{w}=0\&comma;f\right)$

$u^3+3u^{2}w+3u{w}^2-6u+w^3-3w+\frac{3}{w}-\frac{1}{w^3}$

$f≔2a+b$

$f≔2a+b$

$\mathrm{algsubs}\left(a+b=c\&comma;f\&comma;\left[a\&comma;b\right]\right)$

$2c-b$

$\mathrm{algsubs}\left(a+b=c\&comma;f\&comma;\left[b\&comma;a\right]\right)$

$a+c$

$\mathrm{algsubs}\left(a+b=c\&comma;f\&comma;\mathrm{exact}\right)$

$2a+b$

$f≔\frac{\left(\mathrm{a1}+\mathrm{a2}-1\right)x}{\mathrm{a1}+\mathrm{a2}}+\frac{\mathrm{a1}y}{\mathrm{a1}+\mathrm{a2}}-\frac{\mathrm{a2}\left(\mathrm{a1}+\mathrm{a2}\right)}{z}$

$f≔\frac{\left(\mathrm{a1}+\mathrm{a2}-1\right)x}{\mathrm{a1}+\mathrm{a2}}+\frac{\mathrm{a1}y}{\mathrm{a1}+\mathrm{a2}}-\frac{\mathrm{a2}\left(\mathrm{a1}+\mathrm{a2}\right)}{z}$

$\mathrm{algsubs}\left(\mathrm{a1}+\mathrm{a2}=p\&comma;f\right)$

$\frac{x\left(p-1\right)}{p}+\frac{y\left(-\mathrm{a2}+p\right)}{p}-\frac{\mathrm{a2}p}{z}$

$\mathrm{algsubs}\left(\mathrm{a1}+\mathrm{a2}=p\&comma;f\&comma;\mathrm{exact}\right)$

$\frac{x\left(p-1\right)}{p}+\frac{y\mathrm{a1}}{p}-\frac{\mathrm{a2}p}{z}$