# Documentation

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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.