Gauss-Radau quadrature formula is a quadrature formula having one of the nodes in the left boundary of integration interval. At that, other N-1 nodes are located as N-point formula is accurate for the polynomials of order 2N-2 and lower (order decreased by 1 beside Gauss quadrature formula).

It is used the same algorithm as algorithm of generating Gauss quadrature formula to find nodes and weights, but instead of generating matrix J_n  it is used modified matrix J ^R_n :

Structure of matrix J_n-1  is described in algorithm of Gauss quadratre formula generating. It should be noted that position of one nodes is fixed, degree of freedom decremented by one, and number of coefficients a passed into the algorithm decremented by 1. This algorithm can find quadrature formula with fixed node both in the left and right boundaries.

Note #1
Although fixed node usually situates on the left boundary of integration interval, it is not obligatory. If we pass any number at the left of integration interval into the algorithm, it will work.

Implementation of algorithm

In fact, the algorithm given here is only the wrapper for the algorithm of finding eigenvalues and eigenvectors of a symmetric tridiagonal matrix. On the basis of passed arrays Alpha and Beta (they correspond to a_i  and b_i  coefficients) algorithm forms a tridiagonal matrix, finds its eigenvalues and eigenvectors and then calculates the result - nodes and weights of quadrature formula. At that, there is a special operating mode in the subroutine which finds eigenvectors. The algorithm can calculate only the first row of the eigenvectors matrix. It will take only O(N ²) instead of O(N ³).

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C#

C# 1.0 source.
gqgenradau.csharp.zip - Computation of Gauss-Radau quadrature rule nodes and weights

C++

C++ source.
gqgenradau.cpp.zip - Computation of Gauss-Radau quadrature rule nodes and weights
ablas.zip - optimized basic linear algebra subroutines with SSE2 support (for C++ sources only)

C++, multiple precision arithmetic

C++ source. MPFR/GMP is used.
GMP source is available from gmplib.org. MPFR source is available from www.mpfr.org.
gqgenradau.mpfr.zip - Computation of Gauss-Radau quadrature rule nodes and weights
mpfr.zip - precompiled Win32 MPFR/GMP binaries

Delphi

Delphi source.
Can be compiled under FPC (in Delphi compatibility mode).
gqgenradau.delphi.zip - Computation of Gauss-Radau quadrature rule nodes and weights

Visual Basic 6

Visual Basic 6 source.
gqgenradau.vb6.zip - Computation of Gauss-Radau quadrature rule nodes and weights

Zonnon ^beta

Zonnon source.
Zonnon is an experimental language developed at ETH Zurich.
See www.zonnon.ethz.ch for more information.
gqgenradau.zonnon.zip - Computation of Gauss-Radau quadrature rule nodes and weights