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orthpoly::curtz

The Curtz polynomials

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Syntax

orthpoly::curtz(n, x)

Description

orthpoly::curtz(n,x) computes the value of the n-th degree Curtz polynomial at the point x.

These polynomials have rational coefficients.

Evaluation for real floating-point values x is numerically stable. See Example 2.

Examples

Example 1

Polynomial expressions are returned if identifiers or indexed identifiers are specified:

orthpoly::curtz(2, x)

orthpoly::curtz(3, x[1])

Using arithmetical expressions as input, the “values” of these polynomials are returned:

orthpoly::curtz(2, 3+2*I)

orthpoly::curtz(3, exp(x[1] + 2))

“Arithmetical expressions” include numbers:

orthpoly::curtz(2, sqrt(2)), orthpoly::curtz(3, 8 + I),
orthpoly::curtz(100, 0.3)

If the degree of the polynomial is a variable or expression, then orthpoly::curtz returns itself symbolically:

orthpoly::curtz(n, x)

Example 2

If a floating-point value is desired, then a direct call such as

orthpoly::curtz(50, 1.2)

is appropriate and yields a correct result. One should not evaluate the symbolic polynomial at a floating-point value, because this may be numerically unstable:

orthpoly::curtz(50, x): evalp(%, x = 1.2)

Note that only 3 digits are correct due to numerical round-off.

Parameters

n

A nonnegative integer or an arithmetical expression representing a nonnegative integer: the degree of the polynomial.

x

An indeterminate or an arithmetical expression. An indeterminate is either an identifier (of domain type DOM_IDENT) or an indexed identifier (of type "_index").

Return Values

The value of the Curtz polynomial at point x is returned as an arithmetical expression. If n is an arithmetical expression, then orthpoly::curtz returns itself symbolically.

Algorithms

The Curtz polynomials are given by the recursion formula

with C(0, x) = 1.

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