## `numeric::gtdata` -- weights and abscissae of Gauss-Tschebyscheff quadrature

#### Introduction

`numeric::gtdata``(n)` returns the weights and the abscissae of the Gauss-Tschebyscheff quadrature rule with n nodes.

#### Call(s)

`numeric::gtdata(n)`

#### Parameters

`n` | - | the number of nodes: a positive integer

#### Returns

A list `[b,c]` is returned. The lists `b=[b[1],..,b[n]]` and `c=[c[1],..,c[n]]` are the exact weights and abscissae of the Gauss-Tschebyscheff quadrature rule, respectively.

#### Side Effects

The function uses `option remember`. It is not sensitive to the environment variable `DIGITS`.

#### Related Functions

`numeric::gldata`, `numeric::ncdata`, `numeric::quadrature`

#### Details

- The Gauss-Tschebyscheff quadrature rule sum(b[i]*f(c[i]),i=1..n) produces the exact integral int(f(x),x=0..1) for all integrands of the form f(x)=p(x)/sqrt(x*(1-x)) with polynomials p(x) through degree 2n-1.
- The exact weights b=[b[1],..,b[n]] and abscissae c=[c[1],..,c[n]] are given by

      b[i]= PI/2/n*sin((2*i-1)*PI/2/n), c[i]=(1+cos((2*i-1)*PI/2/n))/2.

#### Example 1

The following call produces exact data for the quadrature rule with two nodes:

    >> numeric::gtdata(2)

            -- --     1/2      1/2 --  --  1/2               1/2 -- --
            |  |  PI 2     PI 2     |  |  2                 2     |  |
            |  |  -------, -------  |, |  ---- + 1/2, 1/2 - ----  |  |
            -- --    8        8    --  --  4                 4   -- --

#### Background

- The numerical integrator `numeric::quadrature` calls `numeric::gtdata` to provide the data for Gauss-Tschebyscheff quadrature.

#### Changes

- no changes

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