Chebyshev series expansion - Maple Help

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numapprox[chebyshev] - Chebyshev series expansion

Calling Sequence

chebyshev(f, x=a..b, eps)

chebyshev(f, x, eps)

chebyshev(f, a..b, eps)

Parameters

f

-

procedure or expression representing the function

x

-

variable name appearing in f, if f is an expression

a, b

-

numerical values specifying the interval of approximation

eps

-

(optional) numeric value

Description

• 

This function computes the Chebyshev series expansion of f, with respect to the variable x on the interval a..b, valid to accuracy eps.

• 

If the second argument is simply a name x then the equation x=1..1 is implied.

• 

If the second argument is a range then the first argument is assumed to be a Maple operator and the result will be returned as an operator. Otherwise, the first argument is assumed to be an expression and the result will be returned as an expression.

• 

If the third argument eps is present then it specifies the desired accuracy;  otherwise, the value used is eps=10Digits. It is an error to specify eps less than 10^(-Digits).

• 

The expression or operator f must evaluate to a numerical value when x takes on a numerical value.  Moreover, it must represent a function which is analytic in a region surrounding the interval a..b.

• 

The resulting series is expressed in terms of the Chebyshev polynomials Tk,x,... with floating-point series coefficients. If 'ser' is the Chebyshev series then conversion to ordinary polynomial form can be accomplished via eval(ser, T=orthopoly[T]).

• 

The series computed is the ``infinite'' Chebyshev series, truncated by dropping all terms with coefficients smaller than eps multiplied by the largest coefficient.

• 

Note:  The name T used in representing the Chebyshev polynomials is a global name, so the user must ensure that this name has no previous value.

• 

The command with(numapprox,chebyshev) allows the use of the abbreviated form of this command.

Examples

withnumapprox:

Digits:=5:

chebyshevcosx,x

0.76520T0,x0.22981T2,x+0.0049533T4,x0.000041877T6,x

(1)

chebyshevⅇx,x=0..1,0.001

1.7534T0,2x1+0.85039T1,2x1+0.10521T2,2x1+0.0087221T3,2x1

(2)

chebyshevexp,0..1,0.001

x→1.7534T0,2x1+0.85039T1,2x1+0.10521T2,2x1+0.0087222T3,2x1

(3)

chebyshevsin+cos,1..1

x→0.76520T0,x+0.88010T1,x0.22981T2,x0.039127T3,x+0.0049533T4,x+0.00049952T5,x0.000041877T6,x

(4)

See Also

numapprox, orthopoly, series, taylor


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