The Evaluate routine evaluates finite orthogonal series of one or more variables using a generalization of the Horner scheme.

The generalized Horner scheme accepts only finite series, that is, polynomials, but S can be a infinite series if the truncation option is used. This option has the form trunc=[t1,...,tn], where $n$ is the dimension of the series S and t1,...,tn are non-negative integers. The Evaluate(S, arguments, trunc=[t1,..., tn]) calling sequence is equivalent to Evaluate(Truncate(S, [t1,..., tn]), arguments). For $n$ equal to 1 the list format is not required. You can replace trunc=[t1] with trunc=t1.

The Evaluate(S) calling sequence returns the series S in the canonical basis.

The Evaluate(S, x=v) calling sequence evaluates the series S after substituting the value v for the variable x. More generally, the Evaluate(S, [x1=v1,..., xk=vk]) calling sequence evaluates the series S after substituting each value vi for the corresponding variable xi. If x or xi is not a variable of S, the substitution is ignored. If there exists i and j such that xi=xj in the substitution list, only the first substitution is performed. If the number of substitutions is less than the dimension of S, the result of the Evaluate function is a new orthogonal series with ($n-k$) variables. Otherwise, the result is an algebraic expression.

The Evaluate(S, v) calling sequence evaluates the univariate series S after substituting the value v for the variable. The Evaluate(S, [v1,..., vn]) calling sequence evaluates the series of dimension n S after substituting each vi for the corresponding ith variable. If n is not the dimension of S, an error is returned.

$\mathrm{with}\left(\mathrm{OrthogonalSeries}\right)\:$

$S≔\mathrm{Create}\left(\left[4\,-2\,\frac{1}{3}\right]\,\mathrm{ChebyshevT}\left(n\,x\right)\right)$

$S≔4\mathrm{ChebyshevT}\left(0\,x\right)-2\mathrm{ChebyshevT}\left(1\,x\right)+\frac{\mathrm{ChebyshevT}\left(2\,x\right)}{3}$

$\mathrm{Evaluate}\left(S\right)\;$$\mathrm{Evaluate}\left(S\,1\right)\;$$\mathrm{Evaluate}\left(S\,x=1\right)$

$\frac{2}{3}{x}^2-2x+\frac{11}{3}$

$\frac{7}{3}$

$\frac{7}{3}$

$\mathrm{Evaluate}\left(S\,y=1\right)$

$4\mathrm{ChebyshevT}\left(0\,x\right)-2\mathrm{ChebyshevT}\left(1\,x\right)+\frac{\mathrm{ChebyshevT}\left(2\,x\right)}{3}$

$\mathrm{S1}≔\mathrm{Create}\left(\left[\left(1\,2\right)=1\,\left(4\,3\right)=3\right]\,\mathrm{HermiteH}\left(n\,x\right)\,\mathrm{HermiteH}\left(m\,y\right)\right)$

$\mathrm{S1}≔\mathrm{HermiteH}\left(1\,x\right)\mathrm{HermiteH}\left(2\,y\right)+3\mathrm{HermiteH}\left(4\,x\right)\mathrm{HermiteH}\left(3\,y\right)$

$\mathrm{Evaluate}\left(\mathrm{S1}\right)$

$384x^4{y}^3-576x^{4}y-1152x^2{y}^3+1728x^{2}y+8x{y}^2+288y^3-4x-432y$

$\mathrm{Evaluate}\left(\mathrm{S1}\,\left[3\,4\right]\right)$

$1219764$

$\mathrm{Evaluate}\left(\mathrm{S1}\,\left[x=3\,y=4\right]\right)$

$1219764$

$\mathrm{Evaluate}\left(\mathrm{S1}\,\left[x=3\,y=4\right]\,\mathrm{trunc}=\left[2\,2\right]\right)$

$372$

$\mathrm{Evaluate}\left(\mathrm{S1}\,\left[y=3\,y=4\right]\right)$

$34\mathrm{HermiteH}\left(1\,x\right)+540\mathrm{HermiteH}\left(4\,x\right)$

$\mathrm{Evaluate}\left(\mathrm{S1}\,\left[y=3\right]\right)$

$34\mathrm{HermiteH}\left(1\,x\right)+540\mathrm{HermiteH}\left(4\,x\right)$

$\mathrm{Evaluate}\left(\mathrm{S1}\,y=3\,\mathrm{trunc}=2\right)$

$34\mathrm{HermiteH}\left(1\,x\right)$

$\mathrm{S2}≔\mathrm{Create}\left(\left\{\frac{1}{n}\,n=3..\mathrm{\infty }\,\left[4\,-2\,\frac{1}{3}\right]\right\}\,\mathrm{ChebyshevT}\left(n\,x\right)\right)$

$\mathrm{S2}≔4\mathrm{ChebyshevT}\left(0\,x\right)-2\mathrm{ChebyshevT}\left(1\,x\right)+\frac{\mathrm{ChebyshevT}\left(2\,x\right)}{3}+\left(\sum _{n=3}^{\mathrm{\infty }}\frac{\mathrm{ChebyshevT}\left(n\,x\right)}{n}\right)$

$\mathrm{Evaluate}\left(\mathrm{S2}\right)$

Error, (in OrthogonalSeries:-Evaluate) infinite number of terms

$\mathrm{Evaluate}\left(\mathrm{S2}\,\mathrm{trunc}=5\right)$

$\frac{16}{5}{x}^5+2x^4-\frac{8}{3}{x}^3-\frac{4}{3}{x}^2-2x+\frac{47}{12}$