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OrthogonalSeries

  

Evaluate

  

evaluate a finite series

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Evaluate(S, trunc_option)

Evaluate(S, x=v, trunc_option)

Evaluate(S, [x1=v1,..., xk=vk], trunc_option)

Evaluate(S, v, trunc_option)

Evaluate(S, [v1,..., vn], trunc_option)

Parameters

S

-

orthogonal series

x, x1, .., xk

-

name

v, v1, .., vk

-

values (symbolic or numeric)

trunc_option

-

(optional) equation of the form trunc=[t1,..., tn] or trunc=t1

Description

• 

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 (nk) 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.

Examples

withOrthogonalSeries:

SCreate4,2,13,ChebyshevTn,x

S:=4ChebyshevT0,x2ChebyshevT1,x+13ChebyshevT2,x

(1)

EvaluateS;EvaluateS,1;EvaluateS,x=1

23x22x+113

73

73

(2)

If y is not a variable of S, the substitution is ignored.

EvaluateS,y=1

4ChebyshevT0,x2ChebyshevT1,x+13ChebyshevT2,x

(3)

The following examples use the multivariate case.

S1Create1,2=1,4,3=3,HermiteHn,x,HermiteHm,y

S1:=HermiteH1,xHermiteH2,y+3HermiteH4,xHermiteH3,y

(4)

EvaluateS1

384x4y3576x4y1152x2y3+1728x2y+8xy2+288y34x432y

(5)

EvaluateS1,3,4

1219764

(6)

EvaluateS1,x=3,y=4

1219764

(7)

EvaluateS1,x=3,y=4,trunc=2,2

372

(8)

If a variable is given multiple substitution values, the first is used.

EvaluateS1,y=3,y=4

34HermiteH1,x+540HermiteH4,x

(9)

EvaluateS1,y=3

34HermiteH1,x+540HermiteH4,x

(10)

EvaluateS1,y=3,trunc=2

34HermiteH1,x

(11)

An infinite series can be partially evaluated if truncated.

S2Create4,2,13,1n,n=3..∞,ChebyshevTn,x

S2:=4ChebyshevT0,x2ChebyshevT1,x+13ChebyshevT2,x+n=3∞ChebyshevTn,xn

(12)

EvaluateS2

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

EvaluateS2,trunc=5

165x5+2x483x343x22x+4712

(13)

See Also

ChebyshevT

HermiteH

OrthogonalSeries

OrthogonalSeries[Create]

 


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