基函数与数值积分¶
基函数¶
The basis functions used here are Lagrange basis functions, wherein each function evaluates to 1 when assessed at its respective node and yields 0 at all other nodes.
The basis functions are defined on the simplices. The \(M\) dimensional simplex is defined by \(M+1\) vertices where every points within the simplex are convex combinations of those vertices.
A shape is considered convex if and only if any two points on the shape always form a segment that lies entirely within that shape. When a shape is convex, any point on that shape can be represented as the linear combination of the shape's vertices. For example, the triangle \(T\) can be represented as
\[
T(v_1, v_2, v_3)=\{ p: p = \xi_1 v_1 + \xi_2 v_2 + \xi_3 v_3; \quad \xi_1+\xi_2+\xi_3 \leq 1 \}
\]
where the \(\xi_n\) are coefficients and \(v_n\) are three vertices.
If the \(M\) dimensional simplex has the basis function with highest order \(d\), then a basis function can be formed by \((d+1)^M\) monomials, with \(d+1\) nodes in each dimension.
基函数求导¶
假设 reference cell \(S_R\) 的坐标是 \(\vec{u}=(r,s,t)\)
有映射从 reference cell 到 real cell: \(F(\vec{u}): S_R \rightarrow Q\)
\(Q\) 的坐标是 \(F(\vec{u})=\vec{R}=(x,y,z)\), 则有 Jacobian matrix \(J=\frac{\partial F}{\partial u}\)
所以积分为
\[
\int_{Q} g(\vec{R}) \, dS = \int_{S_R} g(F(\vec{u})) |\det(J)| \, du.
\]
注意，reference cell 上基函数的值与 real cell 上基函数的值相等:
\[
\hat{\phi}(\vec{u}) = \phi(F(\vec{u})) = \phi(\vec{R})
\]
Jacobian 除了出现在积分中，也出现在 gradient 中:
\[
\begin{aligned}
\nabla \hat{\phi}(\vec{u}) &= F^\prime(\vec{u})^T \nabla\phi(\vec{R}) \\
&= J^T\nabla \phi(\vec{R})
\end{aligned}
\]
因此也有:
\[
\nabla \phi(\vec{R}) = J^{-T} \nabla \hat{\phi}(\vec{u})
\]
展开写的话，就比较好理解为什么有转置了 (\(\hat{\nabla}_u = J^T \nabla_R\), \(\nabla_R=J^{-T} \hat{\nabla}_u\)) ：
\[
\begin{aligned}
& \frac{\partial \phi}{\partial r}=\frac{\partial \phi}{\partial x} \frac{\partial x}{\partial r}+\frac{\partial \phi}{\partial y} \frac{\partial y}{\partial r}+\frac{\partial \phi}{\partial z} \frac{\partial z}{\partial r} \\
& \frac{\partial \phi}{\partial s}=\frac{\partial \phi}{\partial x} \frac{\partial x}{\partial s}+\frac{\partial \phi}{\partial y} \frac{\partial y}{\partial s}+\frac{\partial \phi}{\partial y} \frac{\partial z}{\partial s} \\
& \frac{\partial \phi}{\partial t}=\frac{\partial \phi}{\partial x} \frac{\partial x}{\partial t}+\frac{\partial \phi}{\partial y} \frac{\partial y}{\partial t}+\frac{\partial \phi}{\partial z} \frac{\partial z}{\partial t}
\end{aligned} \Rightarrow\left[\begin{array}{l}
\frac{\partial \phi}{\partial r} \\
\frac{\partial \phi}{\partial s} \\
\frac{\partial \phi}{\partial t}
\end{array}\right]=\left[\begin{array}{lll}
\frac{\partial x}{\partial r} & \frac{\partial y}{\partial r} & \frac{\partial z}{\partial r} \\
\frac{\partial x}{\partial s} & \frac{\partial y}{\partial s} & \frac{\partial z}{\partial s} \\
\frac{\partial x}{\partial t} & \frac{\partial y}{\partial t} & \frac{\partial z}{\partial t}
\end{array}\right]\left[\begin{array}{l}
\frac{\partial \phi}{\partial x} \\
\frac{\partial \phi}{\partial y} \\
\frac{\partial \phi}{\partial z}
\end{array}\right]
\]
1D¶
The local basis function is \(\phi_i (x)\)
\[
\phi_i(x) = \prod_{j\neq i} \frac{x - x_j}{x_i - x_j} .
\]
For the linear basis functions on the interval \([-1,1]\), we just have two linear piecewise functions \(\phi_1(x) = \frac{1}{2} (1-x)\), \(\phi_2(x)=\frac{1}{2}(x+1)\).
In theory, the accuracy of Gaussian quadrature is solely determined by the number of quadrature points, i.e. the quadrature of shape functions whose polynomial degree is less than \(2N+1\) is always exact using \(N+1\) Gauss quadrature points. Therefore, regardless of how interpolation points of local shape functions are selected, as long as the points are non-repetitive, we will just have the equivalent FEM results. The uniform points are the simplest case. However, I use Gauss-Lobatto points here for shape functions.
The automatic derivation of the 1D basis functions of any order based on Gauss-Lobatto points has been implemented in Mathematica (basis.nb).
2D¶
triangle¶
If the basis function has monomials with the highest order \(d\), then the triangle has \(d+1\) nodes in each dimension, and the grid point number is \(1+2+3+...+(d+1)=\frac{(d+2)(d+1)}{2}\). For instance, the degree-5 gridlines has the form:
We can get expressions of basis functions on this website: https://defelement.com/elements/lagrange.html
affine mapping¶
Let the \(\mathcal{F}\) be the transformation that maps the reference element \(\vec{u}\) to the actual mesh \(\vec{x}\), namely \(\mathcal{F}: \vec{u} \rightarrow \vec{x}\), \(\mathcal{F}(\vec{u})=\vec{x}\) or \(\vec{u}=\mathcal{F}^{-1} (\vec{x})\). For the 2D case,
\[
\mathcal{F}\left( \begin{bmatrix}
u_1\\
u_2
\end{bmatrix} \right) = \begin{bmatrix}
\mathcal{F}_1 (u_1, u_2)\\
\mathcal{F}_2 (u_1, u_2)
\end{bmatrix} =
\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}
\]
Because the mapping is linear,
\[
\begin{bmatrix}
x\\
y
\end{bmatrix} = \mathcal{F}\left( \begin{bmatrix}
u_1\\
u_2
\end{bmatrix} \right) = A_T \begin{bmatrix}
u_1\\
u_2
\end{bmatrix} + \begin{bmatrix}
x_{0}\\
y_{0}
\end{bmatrix}
\]
The integral of any function \(\phi(\vec{x})\) is
\[
\int_V \phi(\vec{x}) d\vec{x} = \int_{\hat{V}} \phi\left(\mathcal{F}(\vec{u})\right) \det\left(\frac{\partial \mathcal{F}}{\partial \vec{u}}\right) d\vec{u} = \int_{\hat{V}} \hat{\phi}(\vec{u}) \mathcal{J} d\vec{u}
\]
where the \(J=\frac{\partial \mathcal{F}}{\partial \vec{u}}\) is the Jacobian matrix \(J_{ij}=\frac{\partial \mathcal{F}_i}{\partial u_j} = \frac{\partial A_{Ti}}{\partial u_j}\), \(\mathcal{J}=\det(J)=\det(A_T)\) is the Jacobian factor, and \(\hat{V}\) denotes the volume of the reference element. (The basis function satisfy \(\hat{\phi}=\phi \circ \mathcal{F}\) ? This conclusion is intuitive without finding a rigorous proof at the moment, which can also be checked in (4.44) eqn. \(\phi(x)=\hat{\phi}(\hat{x})\) in book Automated Solution of Differential Equations by the Finite Element Method, The FEniCS Book, 2011. )
1D¶
For the 1D case, we map the interval \(\hat{x} \in [-1,1]\) to \(x \in [x_1, x_2]\) where \(x_1 < x_2\). If we assume that \(x = a \hat{x} + b\), then we have
\[
x = \frac{x_2 - x_1}{2} \hat{x} + \frac{x_2 + x_1}{2}
\]
Thus the Jacobian factor \(\mathcal{J} = \det(\frac{\partial x}{\partial \hat{x}}) = \frac{x_2 - x_1}{2}\). The integral of any function \(\phi(x)\) defined in \([x_1, x_2]\) equals
\[
\int_{x_1}^{x_2} \phi(x) dx = \int_{-1}^{1} \phi \left(\mathcal{F}(\hat{x}) \right) \mathcal{J} d\hat{x} = \mathcal{J} \sum_{i} \phi \left(\mathcal{F}(\hat{x_i}) \right) w_i
\]
If the function \(\phi(x)\) is the FEM basis function, then we have
\[
\int_{x_1}^{x_2} \phi(x) dx = \mathcal{J} \int_{-1}^{1} \hat{\phi}(\hat{x}) d\hat{x} = \mathcal{J} \sum_{i} \hat{\phi}(\hat{x}_i) w_i
\]
2D¶
For the 2D case, we have the triangle area \(T\) corresponds to the reference triangle \(\hat{T}\) whose vertices are \((0,0)\), \((1,0)\) and \((0,1)\). If we assume
\[
\begin{bmatrix}
x\\
y
\end{bmatrix} = A_T \begin{bmatrix}
\hat{x} \\
\hat{y}
\end{bmatrix} + \begin{bmatrix}
c_1 \\
c_2
\end{bmatrix}
\]
and map \((0, 0)\) to \((x_1,y_1)\), \((1, 0)\) to \((x_2,y_2)\), \((0, 1)\) to \((x_3,y_3)\), then we have
\[
\begin{bmatrix}
x\\
y
\end{bmatrix} =\begin{bmatrix}
x_2 - x_1 & x_3 - x_1 \\
y_2 - y_1 & y_3 - y_1
\end{bmatrix}
\begin{bmatrix}
\hat{x} \\
\hat{y}
\end{bmatrix} + \begin{bmatrix}
x_1 \\
y_1
\end{bmatrix}
\]
and
\[
\hat{x} = \frac{(y_3 - y_1) (x - x_1) - (x_3 - x_1) (y-y_1)}{\mathcal{J}} \\
\hat{y} = \frac{-(y_2 - y_1) (x - x_1) + (x_2 - x_1) (y-y_1)}{\mathcal{J}}
\]
Thus the Jacobian factor \(\mathcal{J} = \det(A_T) = \det(J) = (x_2 - x_1) (y_3 - y_1) - (x_3 - x_1)(y_2 - y_1)\), where the \(J\) is the Jacobian matrix. The integral of any function \(\phi(x, y)\) defined on \(T\) equals (the \(\frac{1}{2}\) corresponds to https://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tri/quadrature_rules_tri.html)
\[
\int_T \phi(x, y) dS = \int_{\hat{T}} \phi \left( \mathcal{F}(\hat{x}, \hat{y}) \right) \mathcal{J} d\hat{S} = \frac{1}{2} \mathcal{J} \sum_i \phi \left( \mathcal{F}(\hat{x}_i, \hat{y}_i) \right) w_i .
\]
If the function \(\phi(x, y)\) is the FEM basis function, then we have
\[
\int_T \phi(x, y) dS = \int_{\hat{T}} \phi \left( \mathcal{F}(\hat{x}, \hat{y}) \right) \mathcal{J} d\hat{S} = \frac{1}{2} \mathcal{J} \sum_i \hat{\phi}(\hat{x}, \hat{y}) w_i
\]
Note that ,
\[
\frac{\partial}{\partial y} = \frac{\partial}{\partial \hat{y}}
\frac{\partial \hat{y}}{\partial y} + \frac{\partial}{\partial \hat{x}}
\frac{\partial \hat{x}}{\partial y}
\]
\[
\frac{\partial}{\partial x} = \frac{\partial}{\partial \hat{x}}
\frac{\partial \hat{x}}{\partial x} + \frac{\partial}{\partial \hat{y}}
\frac{\partial \hat{y}}{\partial x}
\]
\[
\frac{\partial \phi}{\partial y} = \frac{\partial \hat{\phi}}{\partial \hat{y}}
\frac{\partial \hat{y}}{\partial y} + \frac{\partial \hat{\phi}}{\partial \hat{x}}
\frac{\partial \hat{x}}{\partial y}
\]
\[
\frac{\partial \phi}{\partial x} = \frac{\partial \hat{\phi}}{\partial \hat{x}}
\frac{\partial \hat{x}}{\partial x} + \frac{\partial \hat{\phi}}{\partial \hat{y}}
\frac{\partial \hat{y}}{\partial x}
\]
\[
\frac{\partial^2 \phi}{\partial y^2} = \left( \frac{\partial}{\partial
\hat{y}} \frac{\partial \hat{y}}{\partial y} + \frac{\partial}{\partial
\hat{x}} \frac{\partial \hat{x}}{\partial y} \right) \left( \frac{\partial
\hat{\phi}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial y} + \frac{\partial
\hat{\phi}}{\partial \hat{x}} \frac{\partial \hat{x}}{\partial y} \right) =
\frac{\partial^2 \hat{\phi}}{\partial \hat{y}^2} \left( \frac{\partial
\hat{y}}{\partial y} \right)^2 + 2 \frac{\partial^2 \hat{\phi}}{\partial \hat{x}
\partial \hat{y}} \frac{\partial \hat{x}}{\partial y} \frac{\partial
\hat{y}}{\partial y} + \frac{\partial^2 \hat{\phi}}{\partial \hat{x}^2} \left(
\frac{\partial \hat{x}}{\partial y} \right)^2
\]
\[
\frac{\partial^2 \phi}{\partial x^2} = \left( \frac{\partial}{\partial
\hat{x}} \frac{\partial \hat{x}}{\partial x} + \frac{\partial}{\partial
\hat{y}} \frac{\partial \hat{y}}{\partial x} \right) \left( \frac{\partial
\hat{\phi}}{\partial \hat{x}} \frac{\partial \hat{x}}{\partial x} + \frac{\partial
\hat{\phi}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial x} \right) =
\frac{\partial^2 \hat{\phi}}{\partial \hat{x}^2} \left( \frac{\partial
\hat{x}}{\partial x} \right)^2 + 2 \frac{\partial^2 \hat{\phi}}{\partial \hat{x}
\partial \hat{y}} \frac{\partial \hat{y}}{\partial x} \frac{\partial
\hat{x}}{\partial x} + \frac{\partial^2 \hat{\phi}}{\partial \hat{y}^2} \left(
\frac{\partial \hat{y}}{\partial x} \right)^2
\]
\[
\frac{\partial^2 \phi}{\partial x \partial y} = \frac{\partial}{\partial x}
\left( \frac{\partial \phi}{\partial y} \right) = \left(
\frac{\partial}{\partial \hat{x}} \frac{\partial \hat{x}}{\partial x} +
\frac{\partial}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial x} \right)
\left( \frac{\partial \hat{\phi}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial
y} + \frac{\partial \hat{\phi}}{\partial \hat{x}} \frac{\partial \hat{x}}{\partial
y} \right)\\
= \frac{\partial^2 \hat{\phi}}{\partial \hat{x} \partial \hat{y}}
\frac{\partial \hat{y}}{\partial y} \frac{\partial \hat{x}}{\partial x} +
\frac{\partial^2 \hat{\phi}}{\partial \hat{x}^2} \frac{\partial \hat{x}}{\partial y}
\frac{\partial \hat{x}}{\partial x} + \frac{\partial^2 \hat{\phi}}{\partial
\hat{y}^2} \frac{\partial \hat{y}}{\partial y} \frac{\partial
\hat{y}}{\partial x} + \frac{\partial^2 \hat{\phi}}{\partial \hat{x} \partial
\hat{y}} \frac{\partial \hat{x}}{\partial y} \frac{\partial \hat{y}}{\partial
x}
\]
and
\[
\frac{\partial \hat{x}}{\partial x} = \frac{y_3 - y_1}{| J |}
\]
\[
\frac{\partial \hat{y}}{\partial x} = \frac{y_1 - y_2}{| J |}
\]
\[
\frac{\partial \hat{x}}{\partial y} = \frac{x_1 - x_3}{| J |}
\]
\[
\frac{\partial \hat{y}}{\partial y} = \frac{x_2 - x_1}{| J |}
\]
Thus in the Cartesian coordinate
\[
\nabla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial
\phi}{\partial y} \right) = \left( \frac{\partial \hat{\phi}}{\partial \hat{x}}
\frac{\partial \hat{x}}{\partial x} + \frac{\partial \hat{\phi}}{\partial \hat{y}}
\frac{\partial \hat{y}}{\partial x}, \frac{\partial \hat{\phi}}{\partial \hat{y}}
\frac{\partial \hat{y}}{\partial y} + \frac{\partial \hat{\phi}}{\partial \hat{x}}
\frac{\partial \hat{x}}{\partial y} \right)
\]
Gaussian quadrature¶
The Gaussian quadrature is based on several definitions and theorems.
- Definition: inner product (Boyd, p65).
Let \(f(x)\) and \(g(x)\) be arbitrary functions. Then the inner product of \(f(x)\) with \(g(x)\) with respect to the weight function \(\omega(x)\) on the interval \([a,b]\) is defined by
\[
(f,g) = \int_a^b f(x) g(x) \omega(x) dx
\]
- Definition: orthogonality (Boyd, p65)
A set of basis functions \(\phi_n(x)\) is said to be orthogonal with respect to a given inner product if
\[
(\phi_m, \phi_n) = \delta_{mn} \nu_n^2
\]
where the \(\nu_n\) is called "normalization constants" and \(\nu_n \equiv \sqrt{(\phi_n, \phi_n)}\). If \(\nu_n = 1\), the set of basis functions are orthonormal.
- Guass-Jacobi integration theorem (Boyd, p87).
If the \(N+1\) "interpolation points" or "abscissas" \({x_i}\) are chosen to be zero of \(P_{N+1}(x)\) where \(P_{N+1}(x)\) is the polynomial of degree \(N+1\) of the set of polynomials which are orthogonal on \(x\in[a,b]\) with respect to the weight function \(\rho(x)\), then the quadrature formula
\[
\int_a^b f(x) \rho(x) dx = \sum_{i=0}^{N} w_i f(x_i)
\]
is exact for all \(f(x)\) which are polynomials of at most degree \(2N+1\).
The Legendre, Chebyshev, Gegenbauer, Hermite, and Laguerre polynomials merely correspond to different weights \(w_i\) where (https://mathworld.wolfram.com/GaussianQuadrature.html)
\[
\pi(x) = \prod_{i=1}^m (x-x_i) \\
w_i = \frac{1}{\pi^\prime(x_i)} \int_a^b \frac{\pi(x) \rho(x)}{x-x_i} dx
\]
In summary, given a set of basis functions \(P_{N+1}(x)\) and its corresponding integral weight function \(\rho(x)\), the Gaussian quadrature is completely determined then (the abscissas are the roots of \(P_{N+1}(x)\), and the weights are determined by the weight function and the abscissas).
Gauss-Legendre quadrature¶
1D quadrature¶
For the Legendre polynomials \(P_n(x)\), the integral interval is \([-1,1]\), the weight function is \(\rho(x)=1\), and the quadrature is
\[
\int_{-1}^{1} f(x) dx = \sum_{i=0}^{N} w_i f(x_i) .
\]
The weights and abscissas can be computed using the following code:
void legendre_com ( int norder, double xtab[], double weight[] )
//****************************************************************************80
//
// Purpose:
//
// LEGENDRE_COM computes abscissas and weights for Gauss-Legendre quadrature.
//
// Integration interval:
//
// [ -1, 1 ]
//
// Weight function:
//
// 1.
//
// Integral to approximate:
//
// Integral ( -1 <= X <= 1 ) F(X) dX.
//
// Approximate integral:
//
// sum ( 1 <= I <= NORDER ) WEIGHT(I) * F ( XTAB(I) ).
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 31 March 2005
//
// Author:
//
// John Burkardt
//
// Parameters:
//
// Input, int NORDER, the order of the rule.
// NORDER must be greater than 0.
//
// Output, double XTAB[NORDER], the abscissas of the rule.
//
// Output, double WEIGHT[NORDER], the weights of the rule.
// The weights are positive, symmetric, and should sum to 2.
//
{
# define PI 3.141592653589793
double d1;
double d2pn;
double d3pn;
double d4pn;
double dp;
double dpn;
double e1;
double fx;
double h;
int i;
int iback;
int k;
int m;
int mp1mi;
int ncopy;
int nmove;
double p;
double pk;
double pkm1;
double pkp1;
double t;
double u;
double v;
double x0;
double xtemp;
if ( norder < 1 )
{
cerr << "\n";
cerr << "LEGENDRE_COM - Fatal error!\n";
cerr << " Illegal value of NORDER = " << norder << "\n";
exit ( 1 );
}
e1 = ( double ) ( norder * ( norder + 1 ) );
m = ( norder + 1 ) / 2;
for ( i = 1; i <= ( norder + 1 ) / 2; i++ )
{
mp1mi = m + 1 - i;
t = PI * ( double ) ( 4 * i - 1 ) / ( double ) ( 4 * norder + 2 );
x0 = cos(t) * ( 1.0 - ( 1.0 - 1.0 /
( double ) ( norder ) ) / ( double ) ( 8 * norder * norder ) );
pkm1 = 1.0;
pk = x0;
for ( k = 2; k <= norder; k++ )
{
pkp1 = 2.0 * x0 * pk - pkm1 - ( x0 * pk - pkm1 ) / ( double ) ( k );
pkm1 = pk;
pk = pkp1;
}
d1 = ( double ) ( norder ) * ( pkm1 - x0 * pk );
dpn = d1 / ( 1.0 - x0 * x0 );
d2pn = ( 2.0 * x0 * dpn - e1 * pk ) / ( 1.0 - x0 * x0 );
d3pn = ( 4.0 * x0 * d2pn + ( 2.0 - e1 ) * dpn ) / ( 1.0 - x0 * x0 );
d4pn = ( 6.0 * x0 * d3pn + ( 6.0 - e1 ) * d2pn ) / ( 1.0 - x0 * x0 );
u = pk / dpn;
v = d2pn / dpn;
//
// Initial approximation H:
//
h = - u * ( 1.0 + 0.5 * u * ( v + u * ( v * v - d3pn
/ ( 3.0 * dpn ) ) ) );
//
// Refine H using one step of Newton's method:
//
p = pk + h * ( dpn + 0.5 * h * ( d2pn + h / 3.0
* ( d3pn + 0.25 * h * d4pn ) ) );
dp = dpn + h * ( d2pn + 0.5 * h * ( d3pn + h * d4pn / 3.0 ) );
h = h - p / dp;
xtemp = x0 + h;
xtab[mp1mi-1] = xtemp;
fx = d1 - h * e1 * ( pk + 0.5 * h * ( dpn + h / 3.0
* ( d2pn + 0.25 * h * ( d3pn + 0.2 * h * d4pn ) ) ) );
weight[mp1mi-1] = 2.0 * ( 1.0 - xtemp * xtemp ) / ( fx * fx );
}
if ( ( norder % 2 ) == 1 )
{
xtab[0] = 0.0;
}
//
// Shift the data up.
//
nmove = ( norder + 1 ) / 2;
ncopy = norder - nmove;
for ( i = 1; i <= nmove; i++ )
{
iback = norder + 1 - i;
xtab[iback-1] = xtab[iback-ncopy-1];
weight[iback-1] = weight[iback-ncopy-1];
}
//
// Reflect values for the negative abscissas.
//
for ( i = 0; i < norder - nmove; i++ )
{
xtab[i] = - xtab[norder-1-i];
weight[i] = weight[norder-1-i];
}
return;
# undef PI
}
The weights and abscissas can be checked in following code:
void legendre_set ( int n, double x[], double w[] )
//****************************************************************************80
//
// Purpose:
//
// LEGENDRE_SET sets abscissas and weights for Gauss-Legendre quadrature.
//
// Discussion:
//
// The integration interval is [ -1, 1 ].
//
// The weight function w(x-1] = 1.0;
//
// The integral to approximate:
//
// Integral ( -1 <= X <= 1 ) F(X) dX
//
// Quadrature rule:
//
// Sum ( 1 <= I <= N ) W(I) * F ( X(I) )
//
// The quadrature rule will integrate exactly all polynomials up to
// X**(2*N-1).
//
// The abscissas of the rule are the zeroes of the Legendre polynomial
// P(N)(X).
//
// The integral produced by a Gauss-Legendre rule is equal to the
// integral of the unique polynomial of degree N-1 which
// agrees with the function at the N abscissas of the rule.
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 19 October 2009
//
// Author:
//
// John Burkardt
//
// Reference:
//
// Milton Abramowitz, Irene Stegun,
// Handbook of Mathematical Functions,
// National Bureau of Standards, 1964,
// ISBN: 0-486-61272-4,
// LC: QA47.A34.
//
// Vladimir Krylov,
// Approximate Calculation of Integrals,
// Dover, 2006,
// ISBN: 0486445798.
//
// Arthur Stroud, Don Secrest,
// Gaussian Quadrature Formulas,
// Prentice Hall, 1966,
// LC: QA299.4G3S7.
//
// Daniel Zwillinger, editor,
// CRC Standard Mathematical Tables and Formulae,
// 30th Edition,
// CRC Press, 1996,
// ISBN: 0-8493-2479-3.
//
// Parameters:
//
// Input, int N, the order of the rule.
// N must be between 1 and 33, 63, 64, 65, 127 or 255.
//
// Output, double X[N], the abscissas of the rule.
//
// Output, double W[N], the weights of the rule.
// The weights are positive, symmetric and should sum to 2.
//
{
if ( n == 1 )
{
x[0] = 0.0;
w[0] = 2.0;
}
else if ( n == 2 )
{
x[0] = - 0.577350269189625764509148780502;
x[1] = 0.577350269189625764509148780502;
w[0] = 1.0;
w[1] = 1.0;
}
else if ( n == 3 )
{
x[0] = - 0.774596669241483377035853079956;
x[1] = 0.0;
x[2] = 0.774596669241483377035853079956;
w[0] = 5.0 / 9.0;
w[1] = 8.0 / 9.0;
w[2] = 5.0 / 9.0;
}
else if ( n == 4 )
{
x[0] = - 0.861136311594052575223946488893;
x[1] = - 0.339981043584856264802665759103;
x[2] = 0.339981043584856264802665759103;
x[3] = 0.861136311594052575223946488893;
w[0] = 0.347854845137453857373063949222;
w[1] = 0.652145154862546142626936050778;
w[2] = 0.652145154862546142626936050778;
w[3] = 0.347854845137453857373063949222;
}
else if ( n == 5 )
{
x[0] = - 0.906179845938663992797626878299;
x[1] = - 0.538469310105683091036314420700;
x[2] = 0.0;
x[3] = 0.538469310105683091036314420700;
x[4] = 0.906179845938663992797626878299;
w[0] = 0.236926885056189087514264040720;
w[1] = 0.478628670499366468041291514836;
w[2] = 0.568888888888888888888888888889;
w[3] = 0.478628670499366468041291514836;
w[4] = 0.236926885056189087514264040720;
}
else if ( n == 6 )
{
x[0] = - 0.932469514203152027812301554494;
x[1] = - 0.661209386466264513661399595020;
x[2] = - 0.238619186083196908630501721681;
x[3] = 0.238619186083196908630501721681;
x[4] = 0.661209386466264513661399595020;
x[5] = 0.932469514203152027812301554494;
w[0] = 0.171324492379170345040296142173;
w[1] = 0.360761573048138607569833513838;
w[2] = 0.467913934572691047389870343990;
w[3] = 0.467913934572691047389870343990;
w[4] = 0.360761573048138607569833513838;
w[5] = 0.171324492379170345040296142173;
}
else if ( n == 7 )
{
x[0] = - 0.949107912342758524526189684048;
x[1] = - 0.741531185599394439863864773281;
x[2] = - 0.405845151377397166906606412077;
x[3] = 0.0;
x[4] = 0.405845151377397166906606412077;
x[5] = 0.741531185599394439863864773281;
x[6] = 0.949107912342758524526189684048;
w[0] = 0.129484966168869693270611432679;
w[1] = 0.279705391489276667901467771424;
w[2] = 0.381830050505118944950369775489;
w[3] = 0.417959183673469387755102040816;
w[4] = 0.381830050505118944950369775489;
w[5] = 0.279705391489276667901467771424;
w[6] = 0.129484966168869693270611432679;
}
else if ( n == 8 )
{
x[0] = - 0.960289856497536231683560868569;
x[1] = - 0.796666477413626739591553936476;
x[2] = - 0.525532409916328985817739049189;
x[3] = - 0.183434642495649804939476142360;
x[4] = 0.183434642495649804939476142360;
x[5] = 0.525532409916328985817739049189;
x[6] = 0.796666477413626739591553936476;
x[7] = 0.960289856497536231683560868569;
w[0] = 0.101228536290376259152531354310;
w[1] = 0.222381034453374470544355994426;
w[2] = 0.313706645877887287337962201987;
w[3] = 0.362683783378361982965150449277;
w[4] = 0.362683783378361982965150449277;
w[5] = 0.313706645877887287337962201987;
w[6] = 0.222381034453374470544355994426;
w[7] = 0.101228536290376259152531354310;
}
else if ( n == 9 )
{
x[0] = - 0.968160239507626089835576202904;
x[1] = - 0.836031107326635794299429788070;
x[2] = - 0.613371432700590397308702039341;
x[3] = - 0.324253423403808929038538014643;
x[4] = 0.0;
x[5] = 0.324253423403808929038538014643;
x[6] = 0.613371432700590397308702039341;
x[7] = 0.836031107326635794299429788070;
x[8] = 0.968160239507626089835576202904;
w[0] = 0.812743883615744119718921581105E-01;
w[1] = 0.180648160694857404058472031243;
w[2] = 0.260610696402935462318742869419;
w[3] = 0.312347077040002840068630406584;
w[4] = 0.330239355001259763164525069287;
w[5] = 0.312347077040002840068630406584;
w[6] = 0.260610696402935462318742869419;
w[7] = 0.180648160694857404058472031243;
w[8] = 0.812743883615744119718921581105E-01;
}
else if ( n == 10 )
{
x[0] = - 0.973906528517171720077964012084;
x[1] = - 0.865063366688984510732096688423;
x[2] = - 0.679409568299024406234327365115;
x[3] = - 0.433395394129247190799265943166;
x[4] = - 0.148874338981631210884826001130;
x[5] = 0.148874338981631210884826001130;
x[6] = 0.433395394129247190799265943166;
x[7] = 0.679409568299024406234327365115;
x[8] = 0.865063366688984510732096688423;
x[9] = 0.973906528517171720077964012084;
w[0] = 0.666713443086881375935688098933E-01;
w[1] = 0.149451349150580593145776339658;
w[2] = 0.219086362515982043995534934228;
w[3] = 0.269266719309996355091226921569;
w[4] = 0.295524224714752870173892994651;
w[5] = 0.295524224714752870173892994651;
w[6] = 0.269266719309996355091226921569;
w[7] = 0.219086362515982043995534934228;
w[8] = 0.149451349150580593145776339658;
w[9] = 0.666713443086881375935688098933E-01;
}
else if ( n == 11 )
{
x[0] = - 0.978228658146056992803938001123;
x[1] = - 0.887062599768095299075157769304;
x[2] = - 0.730152005574049324093416252031;
x[3] = - 0.519096129206811815925725669459;
x[4] = - 0.269543155952344972331531985401;
x[5] = 0.0;
x[6] = 0.269543155952344972331531985401;
x[7] = 0.519096129206811815925725669459;
x[8] = 0.730152005574049324093416252031;
x[9] = 0.887062599768095299075157769304;
x[10] = 0.978228658146056992803938001123;
w[0] = 0.556685671161736664827537204425E-01;
w[1] = 0.125580369464904624634694299224;
w[2] = 0.186290210927734251426097641432;
w[3] = 0.233193764591990479918523704843;
w[4] = 0.262804544510246662180688869891;
w[5] = 0.272925086777900630714483528336;
w[6] = 0.262804544510246662180688869891;
w[7] = 0.233193764591990479918523704843;
w[8] = 0.186290210927734251426097641432;
w[9] = 0.125580369464904624634694299224;
w[10] = 0.556685671161736664827537204425E-01;
}
else if ( n == 12 )
{
x[0] = - 0.981560634246719250690549090149;
x[1] = - 0.904117256370474856678465866119;
x[2] = - 0.769902674194304687036893833213;
x[3] = - 0.587317954286617447296702418941;
x[4] = - 0.367831498998180193752691536644;
x[5] = - 0.125233408511468915472441369464;
x[6] = 0.125233408511468915472441369464;
x[7] = 0.367831498998180193752691536644;
x[8] = 0.587317954286617447296702418941;
x[9] = 0.769902674194304687036893833213;
x[10] = 0.904117256370474856678465866119;
x[11] = 0.981560634246719250690549090149;
w[0] = 0.471753363865118271946159614850E-01;
w[1] = 0.106939325995318430960254718194;
w[2] = 0.160078328543346226334652529543;
w[3] = 0.203167426723065921749064455810;
w[4] = 0.233492536538354808760849898925;
w[5] = 0.249147045813402785000562436043;
w[6] = 0.249147045813402785000562436043;
w[7] = 0.233492536538354808760849898925;
w[8] = 0.203167426723065921749064455810;
w[9] = 0.160078328543346226334652529543;
w[10] = 0.106939325995318430960254718194;
w[11] = 0.471753363865118271946159614850E-01;
}
else if ( n == 13 )
{
x[0] = - 0.984183054718588149472829448807;
x[1] = - 0.917598399222977965206547836501;
x[2] = - 0.801578090733309912794206489583;
x[3] = - 0.642349339440340220643984606996;
x[4] = - 0.448492751036446852877912852128;
x[5] = - 0.230458315955134794065528121098;
x[6] = 0.0;
x[7] = 0.230458315955134794065528121098;
x[8] = 0.448492751036446852877912852128;
x[9] = 0.642349339440340220643984606996;
x[10] = 0.801578090733309912794206489583;
x[11] = 0.917598399222977965206547836501;
x[12] = 0.984183054718588149472829448807;
w[0] = 0.404840047653158795200215922010E-01;
w[1] = 0.921214998377284479144217759538E-01;
w[2] = 0.138873510219787238463601776869;
w[3] = 0.178145980761945738280046691996;
w[4] = 0.207816047536888502312523219306;
w[5] = 0.226283180262897238412090186040;
w[6] = 0.232551553230873910194589515269;
w[7] = 0.226283180262897238412090186040;
w[8] = 0.207816047536888502312523219306;
w[9] = 0.178145980761945738280046691996;
w[10] = 0.138873510219787238463601776869;
w[11] = 0.921214998377284479144217759538E-01;
w[12] = 0.404840047653158795200215922010E-01;
}
else if ( n == 14 )
{
x[0] = - 0.986283808696812338841597266704;
x[1] = - 0.928434883663573517336391139378;
x[2] = - 0.827201315069764993189794742650;
x[3] = - 0.687292904811685470148019803019;
x[4] = - 0.515248636358154091965290718551;
x[5] = - 0.319112368927889760435671824168;
x[6] = - 0.108054948707343662066244650220;
x[7] = 0.108054948707343662066244650220;
x[8] = 0.319112368927889760435671824168;
x[9] = 0.515248636358154091965290718551;
x[10] = 0.687292904811685470148019803019;
x[11] = 0.827201315069764993189794742650;
x[12] = 0.928434883663573517336391139378;
x[13] = 0.986283808696812338841597266704;
w[0] = 0.351194603317518630318328761382E-01;
w[1] = 0.801580871597602098056332770629E-01;
w[2] = 0.121518570687903184689414809072;
w[3] = 0.157203167158193534569601938624;
w[4] = 0.185538397477937813741716590125;
w[5] = 0.205198463721295603965924065661;
w[6] = 0.215263853463157790195876443316;
w[7] = 0.215263853463157790195876443316;
w[8] = 0.205198463721295603965924065661;
w[9] = 0.185538397477937813741716590125;
w[10] = 0.157203167158193534569601938624;
w[11] = 0.121518570687903184689414809072;
w[12] = 0.801580871597602098056332770629E-01;
w[13] = 0.351194603317518630318328761382E-01;
}
else if ( n == 15 )
{
x[0] = - 0.987992518020485428489565718587;
x[1] = - 0.937273392400705904307758947710;
x[2] = - 0.848206583410427216200648320774;
x[3] = - 0.724417731360170047416186054614;
x[4] = - 0.570972172608538847537226737254;
x[5] = - 0.394151347077563369897207370981;
x[6] = - 0.201194093997434522300628303395;
x[7] = 0.0;
x[8] = 0.201194093997434522300628303395;
x[9] = 0.394151347077563369897207370981;
x[10] = 0.570972172608538847537226737254;
x[11] = 0.724417731360170047416186054614;
x[12] = 0.848206583410427216200648320774;
x[13] = 0.937273392400705904307758947710;
x[14] = 0.987992518020485428489565718587;
w[0] = 0.307532419961172683546283935772E-01;
w[1] = 0.703660474881081247092674164507E-01;
w[2] = 0.107159220467171935011869546686;
w[3] = 0.139570677926154314447804794511;
w[4] = 0.166269205816993933553200860481;
w[5] = 0.186161000015562211026800561866;
w[6] = 0.198431485327111576456118326444;
w[7] = 0.202578241925561272880620199968;
w[8] = 0.198431485327111576456118326444;
w[9] = 0.186161000015562211026800561866;
w[10] = 0.166269205816993933553200860481;
w[11] = 0.139570677926154314447804794511;
w[12] = 0.107159220467171935011869546686;
w[13] = 0.703660474881081247092674164507E-01;
w[14] = 0.307532419961172683546283935772E-01;
}
else if ( n == 16 )
{
x[0] = - 0.989400934991649932596154173450;
x[1] = - 0.944575023073232576077988415535;
x[2] = - 0.865631202387831743880467897712;
x[3] = - 0.755404408355003033895101194847;
x[4] = - 0.617876244402643748446671764049;
x[5] = - 0.458016777657227386342419442984;
x[6] = - 0.281603550779258913230460501460;
x[7] = - 0.950125098376374401853193354250E-01;
x[8] = 0.950125098376374401853193354250E-01;
x[9] = 0.281603550779258913230460501460;
x[10] = 0.458016777657227386342419442984;
x[11] = 0.617876244402643748446671764049;
x[12] = 0.755404408355003033895101194847;
x[13] = 0.865631202387831743880467897712;
x[14] = 0.944575023073232576077988415535;
x[15] = 0.989400934991649932596154173450;
w[0] = 0.271524594117540948517805724560E-01;
w[1] = 0.622535239386478928628438369944E-01;
w[2] = 0.951585116824927848099251076022E-01;
w[3] = 0.124628971255533872052476282192;
w[4] = 0.149595988816576732081501730547;
w[5] = 0.169156519395002538189312079030;
w[6] = 0.182603415044923588866763667969;
w[7] = 0.189450610455068496285396723208;
w[8] = 0.189450610455068496285396723208;
w[9] = 0.182603415044923588866763667969;
w[10] = 0.169156519395002538189312079030;
w[11] = 0.149595988816576732081501730547;
w[12] = 0.124628971255533872052476282192;
w[13] = 0.951585116824927848099251076022E-01;
w[14] = 0.622535239386478928628438369944E-01;
w[15] = 0.271524594117540948517805724560E-01;
}
else if ( n == 17 )
{
x[0] = - 0.990575475314417335675434019941;
x[1] = - 0.950675521768767761222716957896;
x[2] = - 0.880239153726985902122955694488;
x[3] = - 0.781514003896801406925230055520;
x[4] = - 0.657671159216690765850302216643;
x[5] = - 0.512690537086476967886246568630;
x[6] = - 0.351231763453876315297185517095;
x[7] = - 0.178484181495847855850677493654;
x[8] = 0.0;
x[9] = 0.178484181495847855850677493654;
x[10] = 0.351231763453876315297185517095;
x[11] = 0.512690537086476967886246568630;
x[12] = 0.657671159216690765850302216643;
x[13] = 0.781514003896801406925230055520;
x[14] = 0.880239153726985902122955694488;
x[15] = 0.950675521768767761222716957896;
x[16] = 0.990575475314417335675434019941;
w[0] = 0.241483028685479319601100262876E-01;
w[1] = 0.554595293739872011294401653582E-01;
w[2] = 0.850361483171791808835353701911E-01;
w[3] = 0.111883847193403971094788385626;
w[4] = 0.135136368468525473286319981702;
w[5] = 0.154045761076810288081431594802;
w[6] = 0.168004102156450044509970663788;
w[7] = 0.176562705366992646325270990113;
w[8] = 0.179446470356206525458265644262;
w[9] = 0.176562705366992646325270990113;
w[10] = 0.168004102156450044509970663788;
w[11] = 0.154045761076810288081431594802;
w[12] = 0.135136368468525473286319981702;
w[13] = 0.111883847193403971094788385626;
w[14] = 0.850361483171791808835353701911E-01;
w[15] = 0.554595293739872011294401653582E-01;
w[16] = 0.241483028685479319601100262876E-01;
}
else if ( n == 18 )
{
x[0] = - 0.991565168420930946730016004706;
x[1] = - 0.955823949571397755181195892930;
x[2] = - 0.892602466497555739206060591127;
x[3] = - 0.803704958972523115682417455015;
x[4] = - 0.691687043060353207874891081289;
x[5] = - 0.559770831073947534607871548525;
x[6] = - 0.411751161462842646035931793833;
x[7] = - 0.251886225691505509588972854878;
x[8] = - 0.847750130417353012422618529358E-01;
x[9] = 0.847750130417353012422618529358E-01;
x[10] = 0.251886225691505509588972854878;
x[11] = 0.411751161462842646035931793833;
x[12] = 0.559770831073947534607871548525;
x[13] = 0.691687043060353207874891081289;
x[14] = 0.803704958972523115682417455015;
x[15] = 0.892602466497555739206060591127;
x[16] = 0.955823949571397755181195892930;
x[17] = 0.991565168420930946730016004706;
w[0] = 0.216160135264833103133427102665E-01;
w[1] = 0.497145488949697964533349462026E-01;
w[2] = 0.764257302548890565291296776166E-01;
w[3] = 0.100942044106287165562813984925;
w[4] = 0.122555206711478460184519126800;
w[5] = 0.140642914670650651204731303752;
w[6] = 0.154684675126265244925418003836;
w[7] = 0.164276483745832722986053776466;
w[8] = 0.169142382963143591840656470135;
w[9] = 0.169142382963143591840656470135;
w[10] = 0.164276483745832722986053776466;
w[11] = 0.154684675126265244925418003836;
w[12] = 0.140642914670650651204731303752;
w[13] = 0.122555206711478460184519126800;
w[14] = 0.100942044106287165562813984925;
w[15] = 0.764257302548890565291296776166E-01;
w[16] = 0.497145488949697964533349462026E-01;
w[17] = 0.216160135264833103133427102665E-01;
}
else if ( n == 19 )
{
x[0] = - 0.992406843843584403189017670253;
x[1] = - 0.960208152134830030852778840688;
x[2] = - 0.903155903614817901642660928532;
x[3] = - 0.822714656537142824978922486713;
x[4] = - 0.720966177335229378617095860824;
x[5] = - 0.600545304661681023469638164946;
x[6] = - 0.464570741375960945717267148104;
x[7] = - 0.316564099963629831990117328850;
x[8] = - 0.160358645640225375868096115741;
x[9] = 0.0;
x[10] = 0.160358645640225375868096115741;
x[11] = 0.316564099963629831990117328850;
x[12] = 0.464570741375960945717267148104;
x[13] = 0.600545304661681023469638164946;
x[14] = 0.720966177335229378617095860824;
x[15] = 0.822714656537142824978922486713;
x[16] = 0.903155903614817901642660928532;
x[17] = 0.960208152134830030852778840688;
x[18] = 0.992406843843584403189017670253;
w[0] = 0.194617882297264770363120414644E-01;
w[1] = 0.448142267656996003328381574020E-01;
w[2] = 0.690445427376412265807082580060E-01;
w[3] = 0.914900216224499994644620941238E-01;
w[4] = 0.111566645547333994716023901682;
w[5] = 0.128753962539336227675515784857;
w[6] = 0.142606702173606611775746109442;
w[7] = 0.152766042065859666778855400898;
w[8] = 0.158968843393954347649956439465;
w[9] = 0.161054449848783695979163625321;
w[10] = 0.158968843393954347649956439465;
w[11] = 0.152766042065859666778855400898;
w[12] = 0.142606702173606611775746109442;
w[13] = 0.128753962539336227675515784857;
w[14] = 0.111566645547333994716023901682;
w[15] = 0.914900216224499994644620941238E-01;
w[16] = 0.690445427376412265807082580060E-01;
w[17] = 0.448142267656996003328381574020E-01;
w[18] = 0.194617882297264770363120414644E-01;
}
else if ( n == 20 )
{
x[0] = - 0.993128599185094924786122388471;
x[1] = - 0.963971927277913791267666131197;
x[2] = - 0.912234428251325905867752441203;
x[3] = - 0.839116971822218823394529061702;
x[4] = - 0.746331906460150792614305070356;
x[5] = - 0.636053680726515025452836696226;
x[6] = - 0.510867001950827098004364050955;
x[7] = - 0.373706088715419560672548177025;
x[8] = - 0.227785851141645078080496195369;
x[9] = - 0.765265211334973337546404093988E-01;
x[10] = 0.765265211334973337546404093988E-01;
x[11] = 0.227785851141645078080496195369;
x[12] = 0.373706088715419560672548177025;
x[13] = 0.510867001950827098004364050955;
x[14] = 0.636053680726515025452836696226;
x[15] = 0.746331906460150792614305070356;
x[16] = 0.839116971822218823394529061702;
x[17] = 0.912234428251325905867752441203;
x[18] = 0.963971927277913791267666131197;
x[19] = 0.993128599185094924786122388471;
w[0] = 0.176140071391521183118619623519E-01;
w[1] = 0.406014298003869413310399522749E-01;
w[2] = 0.626720483341090635695065351870E-01;
w[3] = 0.832767415767047487247581432220E-01;
w[4] = 0.101930119817240435036750135480;
w[5] = 0.118194531961518417312377377711;
w[6] = 0.131688638449176626898494499748;
w[7] = 0.142096109318382051329298325067;
w[8] = 0.149172986472603746787828737002;
w[9] = 0.152753387130725850698084331955;
w[10] = 0.152753387130725850698084331955;
w[11] = 0.149172986472603746787828737002;
w[12] = 0.142096109318382051329298325067;
w[13] = 0.131688638449176626898494499748;
w[14] = 0.118194531961518417312377377711;
w[15] = 0.101930119817240435036750135480;
w[16] = 0.832767415767047487247581432220E-01;
w[17] = 0.626720483341090635695065351870E-01;
w[18] = 0.406014298003869413310399522749E-01;
w[19] = 0.176140071391521183118619623519E-01;
}
else if ( n == 21 )
{
x[ 0] = -0.9937521706203896E+00;
x[ 1] = -0.9672268385663063E+00;
x[ 2] = -0.9200993341504008E+00;
x[ 3] = -0.8533633645833173E+00;
x[ 4] = -0.7684399634756779E+00;
x[ 5] = -0.6671388041974123E+00;
x[ 6] = -0.5516188358872198E+00;
x[ 7] = -0.4243421202074388E+00;
x[ 8] = -0.2880213168024011E+00;
x[ 9] = -0.1455618541608951E+00;
x[10] = 0.0000000000000000E+00;
x[11] = 0.1455618541608951E+00;
x[12] = 0.2880213168024011E+00;
x[13] = 0.4243421202074388E+00;
x[14] = 0.5516188358872198E+00;
x[15] = 0.6671388041974123E+00;
x[16] = 0.7684399634756779E+00;
x[17] = 0.8533633645833173E+00;
x[18] = 0.9200993341504008E+00;
x[19] = 0.9672268385663063E+00;
x[20] = 0.9937521706203896E+00;
w[ 0] = 0.1601722825777420E-01;
w[ 1] = 0.3695378977085242E-01;
w[ 2] = 0.5713442542685715E-01;
w[ 3] = 0.7610011362837928E-01;
w[ 4] = 0.9344442345603393E-01;
w[ 5] = 0.1087972991671484E+00;
w[ 6] = 0.1218314160537285E+00;
w[ 7] = 0.1322689386333373E+00;
w[ 8] = 0.1398873947910731E+00;
w[ 9] = 0.1445244039899700E+00;
w[10] = 0.1460811336496904E+00;
w[11] = 0.1445244039899700E+00;
w[12] = 0.1398873947910731E+00;
w[13] = 0.1322689386333373E+00;
w[14] = 0.1218314160537285E+00;
w[15] = 0.1087972991671484E+00;
w[16] = 0.9344442345603393E-01;
w[17] = 0.7610011362837928E-01;
w[18] = 0.5713442542685715E-01;
w[19] = 0.3695378977085242E-01;
w[20] = 0.1601722825777420E-01;
}
else if ( n == 22 )
{
x[ 0] = -0.9942945854823994E+00;
x[ 1] = -0.9700604978354287E+00;
x[ 2] = -0.9269567721871740E+00;
x[ 3] = -0.8658125777203002E+00;
x[ 4] = -0.7878168059792081E+00;
x[ 5] = -0.6944872631866827E+00;
x[ 6] = -0.5876404035069116E+00;
x[ 7] = -0.4693558379867570E+00;
x[ 8] = -0.3419358208920842E+00;
x [9] = -0.2078604266882213E+00;
x[10] = -0.6973927331972223E-01;
x[11] = 0.6973927331972223E-01;
x[12] = 0.2078604266882213E+00;
x[13] = 0.3419358208920842E+00;
x[14] = 0.4693558379867570E+00;
x[15] = 0.5876404035069116E+00;
x[16] = 0.6944872631866827E+00;
x[17] = 0.7878168059792081E+00;
x[18] = 0.8658125777203002E+00;
x[19] = 0.9269567721871740E+00;
x[20] = 0.9700604978354287E+00;
x[21] = 0.9942945854823994E+00;
w[ 0] = 0.1462799529827203E-01;
w[ 1] = 0.3377490158481413E-01;
w[ 2] = 0.5229333515268327E-01;
w[ 3] = 0.6979646842452038E-01;
w[ 4] = 0.8594160621706777E-01;
w[ 5] = 0.1004141444428809E+00;
w[ 6] = 0.1129322960805392E+00;
w[ 7] = 0.1232523768105124E+00;
w[ 8] = 0.1311735047870623E+00;
w[ 9] = 0.1365414983460152E+00;
w[10] = 0.1392518728556321E+00;
w[11] = 0.1392518728556321E+00;
w[12] = 0.1365414983460152E+00;
w[13] = 0.1311735047870623E+00;
w[14] = 0.1232523768105124E+00;
w[15] = 0.1129322960805392E+00;
w[16] = 0.1004141444428809E+00;
w[17] = 0.8594160621706777E-01;
w[18] = 0.6979646842452038E-01;
w[19] = 0.5229333515268327E-01;
w[20] = 0.3377490158481413E-01;
w[21] = 0.1462799529827203E-01;
}
else if ( n == 23 )
{
x[ 0] = -0.9947693349975522E+00;
x[ 1] = -0.9725424712181152E+00;
x[ 2] = -0.9329710868260161E+00;
x[ 3] = -0.8767523582704416E+00;
x[ 4] = -0.8048884016188399E+00;
x[ 5] = -0.7186613631319502E+00;
x[ 6] = -0.6196098757636461E+00;
x[ 7] = -0.5095014778460075E+00;
x[ 8] = -0.3903010380302908E+00;
x[ 9] = -0.2641356809703449E+00;
x[10] = -0.1332568242984661E+00;
x[11] = 0.0000000000000000E+00;
x[12] = 0.1332568242984661E+00;
x[13] = 0.2641356809703449E+00;
x[14] = 0.3903010380302908E+00;
x[15] = 0.5095014778460075E+00;
x[16] = 0.6196098757636461E+00;
x[17] = 0.7186613631319502E+00;
x[18] = 0.8048884016188399E+00;
x[19] = 0.8767523582704416E+00;
x[20] = 0.9329710868260161E+00;
x[21] = 0.9725424712181152E+00;
x[22] = 0.9947693349975522E+00;
w[ 0] = 0.1341185948714167E-01;
w[ 1] = 0.3098800585697944E-01;
w[ 2] = 0.4803767173108464E-01;
w[ 3] = 0.6423242140852586E-01;
w[ 4] = 0.7928141177671895E-01;
w[ 5] = 0.9291576606003514E-01;
w[ 6] = 0.1048920914645414E+00;
w[ 7] = 0.1149966402224114E+00;
w[ 8] = 0.1230490843067295E+00;
w[ 9] = 0.1289057221880822E+00;
w[10] = 0.1324620394046967E+00;
w[11] = 0.1336545721861062E+00;
w[12] = 0.1324620394046967E+00;
w[13] = 0.1289057221880822E+00;
w[14] = 0.1230490843067295E+00;
w[15] = 0.1149966402224114E+00;
w[16] = 0.1048920914645414E+00;
w[17] = 0.9291576606003514E-01;
w[18] = 0.7928141177671895E-01;
w[19] = 0.6423242140852586E-01;
w[20] = 0.4803767173108464E-01;
w[21] = 0.3098800585697944E-01;
w[22] = 0.1341185948714167E-01;
}
else if ( n == 24 )
{
x[ 0] = -0.9951872199970213E+00;
x[ 1] = -0.9747285559713095E+00;
x[ 2] = -0.9382745520027327E+00;
x[ 3] = -0.8864155270044011E+00;
x[ 4] = -0.8200019859739029E+00;
x[ 5] = -0.7401241915785544E+00;
x[ 6] = -0.6480936519369755E+00;
x[ 7] = -0.5454214713888396E+00;
x[ 8] = -0.4337935076260451E+00;
x[ 9] = -0.3150426796961634E+00;
x[10] = -0.1911188674736163E+00;
x[11] = -0.6405689286260562E-01;
x[12] = 0.6405689286260562E-01;
x[13] = 0.1911188674736163E+00;
x[14] = 0.3150426796961634E+00;
x[15] = 0.4337935076260451E+00;
x[16] = 0.5454214713888396E+00;
x[17] = 0.6480936519369755E+00;
x[18] = 0.7401241915785544E+00;
x[19] = 0.8200019859739029E+00;
x[20] = 0.8864155270044011E+00;
x[21] = 0.9382745520027327E+00;
x[22] = 0.9747285559713095E+00;
x[23] = 0.9951872199970213E+00;
w[ 0] = 0.1234122979998730E-01;
w[ 1] = 0.2853138862893375E-01;
w[ 2] = 0.4427743881741982E-01;
w[ 3] = 0.5929858491543672E-01;
w[ 4] = 0.7334648141108031E-01;
w[ 5] = 0.8619016153195320E-01;
w[ 6] = 0.9761865210411380E-01;
w[ 7] = 0.1074442701159656E+00;
w[ 8] = 0.1155056680537256E+00;
w[ 9] = 0.1216704729278035E+00;
w[10] = 0.1258374563468283E+00;
w[11] = 0.1279381953467521E+00;
w[12] = 0.1279381953467521E+00;
w[13] = 0.1258374563468283E+00;
w[14] = 0.1216704729278035E+00;
w[15] = 0.1155056680537256E+00;
w[16] = 0.1074442701159656E+00;
w[17] = 0.9761865210411380E-01;
w[18] = 0.8619016153195320E-01;
w[19] = 0.7334648141108031E-01;
w[20] = 0.5929858491543672E-01;
w[21] = 0.4427743881741982E-01;
w[22] = 0.2853138862893375E-01;
w[23] = 0.1234122979998730E-01;
}
else if ( n == 25 )
{
x[ 0] = -0.9955569697904981E+00;
x[ 1] = -0.9766639214595175E+00;
x[ 2] = -0.9429745712289743E+00;
x[ 3] = -0.8949919978782754E+00;
x[ 4] = -0.8334426287608340E+00;
x[ 5] = -0.7592592630373577E+00;
x[ 6] = -0.6735663684734684E+00;
x[ 7] = -0.5776629302412229E+00;
x[ 8] = -0.4730027314457150E+00;
x[ 9] = -0.3611723058093879E+00;
x[10] = -0.2438668837209884E+00;
x[11] = -0.1228646926107104E+00;
x[12] = 0.0000000000000000E+00;
x[13] = 0.1228646926107104E+00;
x[14] = 0.2438668837209884E+00;
x[15] = 0.3611723058093879E+00;
x[16] = 0.4730027314457150E+00;
x[17] = 0.5776629302412229E+00;
x[18] = 0.6735663684734684E+00;
x[19] = 0.7592592630373577E+00;
x[20] = 0.8334426287608340E+00;
x[21] = 0.8949919978782754E+00;
x[22] = 0.9429745712289743E+00;
x[23] = 0.9766639214595175E+00;
x[24] = 0.9955569697904981E+00;
w[ 0] = 0.1139379850102617E-01;
w[ 1] = 0.2635498661503214E-01;
w[ 2] = 0.4093915670130639E-01;
w[ 3] = 0.5490469597583517E-01;
w[ 4] = 0.6803833381235694E-01;
w[ 5] = 0.8014070033500101E-01;
w[ 6] = 0.9102826198296370E-01;
w[ 7] = 0.1005359490670506E+00;
w[ 8] = 0.1085196244742637E+00;
w[ 9] = 0.1148582591457116E+00;
w[10] = 0.1194557635357847E+00;
w[11] = 0.1222424429903101E+00;
w[12] = 0.1231760537267154E+00;
w[13] = 0.1222424429903101E+00;
w[14] = 0.1194557635357847E+00;
w[15] = 0.1148582591457116E+00;
w[16] = 0.1085196244742637E+00;
w[17] = 0.1005359490670506E+00;
w[18] = 0.9102826198296370E-01;
w[19] = 0.8014070033500101E-01;
w[20] = 0.6803833381235694E-01;
w[21] = 0.5490469597583517E-01;
w[22] = 0.4093915670130639E-01;
w[23] = 0.2635498661503214E-01;
w[24] = 0.1139379850102617E-01;
}
else if ( n == 26 )
{
x[ 0] = -0.9958857011456169E+00;
x[ 1] = -0.9783854459564710E+00;
x[ 2] = -0.9471590666617142E+00;
x[ 3] = -0.9026378619843071E+00;
x[ 4] = -0.8454459427884981E+00;
x[ 5] = -0.7763859488206789E+00;
x[ 6] = -0.6964272604199573E+00;
x[ 7] = -0.6066922930176181E+00;
x[ 8] = -0.5084407148245057E+00;
x[ 9] = -0.4030517551234863E+00;
x[10] = -0.2920048394859569E+00;
x[11] = -0.1768588203568902E+00;
x[12] = -0.5923009342931320E-01;
x[13] = 0.5923009342931320E-01;
x[14] = 0.1768588203568902E+00;
x[15] = 0.2920048394859569E+00;
x[16] = 0.4030517551234863E+00;
x[17] = 0.5084407148245057E+00;
x[18] = 0.6066922930176181E+00;
x[19] = 0.6964272604199573E+00;
x[20] = 0.7763859488206789E+00;
x[21] = 0.8454459427884981E+00;
x[22] = 0.9026378619843071E+00;
x[23] = 0.9471590666617142E+00;
x[24] = 0.9783854459564710E+00;
x[25] = 0.9958857011456169E+00;
w[ 0] = 0.1055137261734304E-01;
w[ 1] = 0.2441785109263173E-01;
w[ 2] = 0.3796238329436282E-01;
w[ 3] = 0.5097582529714782E-01;
w[ 4] = 0.6327404632957484E-01;
w[ 5] = 0.7468414976565967E-01;
w[ 6] = 0.8504589431348521E-01;
w[ 7] = 0.9421380035591416E-01;
w[ 8] = 0.1020591610944255E+00;
w[ 9] = 0.1084718405285765E+00;
w[10] = 0.1133618165463197E+00;
w[11] = 0.1166604434852967E+00;
w[12] = 0.1183214152792622E+00;
w[13] = 0.1183214152792622E+00;
w[14] = 0.1166604434852967E+00;
w[15] = 0.1133618165463197E+00;
w[16] = 0.1084718405285765E+00;
w[17] = 0.1020591610944255E+00;
w[18] = 0.9421380035591416E-01;
w[19] = 0.8504589431348521E-01;
w[20] = 0.7468414976565967E-01;
w[21] = 0.6327404632957484E-01;
w[22] = 0.5097582529714782E-01;
w[23] = 0.3796238329436282E-01;
w[24] = 0.2441785109263173E-01;
w[25] = 0.1055137261734304E-01;
}
else if ( n == 27 )
{
x[ 0] = -0.9961792628889886E+00;
x[ 1] = -0.9799234759615012E+00;
x[ 2] = -0.9509005578147051E+00;
x[ 3] = -0.9094823206774911E+00;
x[ 4] = -0.8562079080182945E+00;
x[ 5] = -0.7917716390705082E+00;
x[ 6] = -0.7170134737394237E+00;
x[ 7] = -0.6329079719464952E+00;
x[ 8] = -0.5405515645794569E+00;
x[ 9] = -0.4411482517500269E+00;
x[10] = -0.3359939036385089E+00;
x[11] = -0.2264593654395369E+00;
x[12] = -0.1139725856095300E+00;
x[13] = 0.0000000000000000E+00;
x[14] = 0.1139725856095300E+00;
x[15] = 0.2264593654395369E+00;
x[16] = 0.3359939036385089E+00;
x[17] = 0.4411482517500269E+00;
x[18] = 0.5405515645794569E+00;
x[19] = 0.6329079719464952E+00;
x[20] = 0.7170134737394237E+00;
x[21] = 0.7917716390705082E+00;
x[22] = 0.8562079080182945E+00;
x[23] = 0.9094823206774911E+00;
x[24] = 0.9509005578147051E+00;
x[25] = 0.9799234759615012E+00;
x[26] = 0.9961792628889886E+00;
w[ 0] = 0.9798996051294232E-02;
w[ 1] = 0.2268623159618062E-01;
w[ 2] = 0.3529705375741969E-01;
w[ 3] = 0.4744941252061504E-01;
w[ 4] = 0.5898353685983366E-01;
w[ 5] = 0.6974882376624561E-01;
w[ 6] = 0.7960486777305781E-01;
w[ 7] = 0.8842315854375689E-01;
w[ 8] = 0.9608872737002842E-01;
w[ 9] = 0.1025016378177459E+00;
w[10] = 0.1075782857885332E+00;
w[11] = 0.1112524883568452E+00;
w[12] = 0.1134763461089651E+00;
w[13] = 0.1142208673789570E+00;
w[14] = 0.1134763461089651E+00;
w[15] = 0.1112524883568452E+00;
w[16] = 0.1075782857885332E+00;
w[17] = 0.1025016378177459E+00;
w[18] = 0.9608872737002842E-01;
w[19] = 0.8842315854375689E-01;
w[20] = 0.7960486777305781E-01;
w[21] = 0.6974882376624561E-01;
w[22] = 0.5898353685983366E-01;
w[23] = 0.4744941252061504E-01;
w[24] = 0.3529705375741969E-01;
w[25] = 0.2268623159618062E-01;
w[26] = 0.9798996051294232E-02;
}
else if ( n == 28 )
{
x[ 0] = -0.9964424975739544E+00;
x[ 1] = -0.9813031653708728E+00;
x[ 2] = -0.9542592806289382E+00;
x[ 3] = -0.9156330263921321E+00;
x[ 4] = -0.8658925225743951E+00;
x[ 5] = -0.8056413709171791E+00;
x[ 6] = -0.7356108780136318E+00;
x[ 7] = -0.6566510940388650E+00;
x[ 8] = -0.5697204718114017E+00;
x[ 9] = -0.4758742249551183E+00;
x[10] = -0.3762515160890787E+00;
x[11] = -0.2720616276351780E+00;
x[12] = -0.1645692821333808E+00;
x[13] = -0.5507928988403427E-01;
x[14] = 0.5507928988403427E-01;
x[15] = 0.1645692821333808E+00;
x[16] = 0.2720616276351780E+00;
x[17] = 0.3762515160890787E+00;
x[18] = 0.4758742249551183E+00;
x[19] = 0.5697204718114017E+00;
x[20] = 0.6566510940388650E+00;
x[21] = 0.7356108780136318E+00;
x[22] = 0.8056413709171791E+00;
x[23] = 0.8658925225743951E+00;
x[24] = 0.9156330263921321E+00;
x[25] = 0.9542592806289382E+00;
x[26] = 0.9813031653708728E+00;
x[27] = 0.9964424975739544E+00;
w[ 0] = 0.9124282593094672E-02;
w[ 1] = 0.2113211259277118E-01;
w[ 2] = 0.3290142778230441E-01;
w[ 3] = 0.4427293475900429E-01;
w[ 4] = 0.5510734567571667E-01;
w[ 5] = 0.6527292396699959E-01;
w[ 6] = 0.7464621423456877E-01;
w[ 7] = 0.8311341722890127E-01;
w[ 8] = 0.9057174439303289E-01;
w[ 9] = 0.9693065799792999E-01;
w[10] = 0.1021129675780608E+00;
w[11] = 0.1060557659228464E+00;
w[12] = 0.1087111922582942E+00;
w[13] = 0.1100470130164752E+00;
w[14] = 0.1100470130164752E+00;
w[15] = 0.1087111922582942E+00;
w[16] = 0.1060557659228464E+00;
w[17] = 0.1021129675780608E+00;
w[18] = 0.9693065799792999E-01;
w[19] = 0.9057174439303289E-01;
w[20] = 0.8311341722890127E-01;
w[21] = 0.7464621423456877E-01;
w[22] = 0.6527292396699959E-01;
w[23] = 0.5510734567571667E-01;
w[24] = 0.4427293475900429E-01;
w[25] = 0.3290142778230441E-01;
w[26] = 0.2113211259277118E-01;
w[27] = 0.9124282593094672E-02;
}
else if ( n == 29 )
{
x[ 0] = -0.9966794422605966E+00;
x[ 1] = -0.9825455052614132E+00;
x[ 2] = -0.9572855957780877E+00;
x[ 3] = -0.9211802329530588E+00;
x[ 4] = -0.8746378049201028E+00;
x[ 5] = -0.8181854876152524E+00;
x[ 6] = -0.7524628517344771E+00;
x[ 7] = -0.6782145376026865E+00;
x[ 8] = -0.5962817971382278E+00;
x[ 9] = -0.5075929551242276E+00;
x[10] = -0.4131528881740087E+00;
x[11] = -0.3140316378676399E+00;
x[12] = -0.2113522861660011E+00;
x[13] = -0.1062782301326792E+00;
x[14] = 0.0000000000000000E+00;
x[15] = 0.1062782301326792E+00;
x[16] = 0.2113522861660011E+00;
x[17] = 0.3140316378676399E+00;
x[18] = 0.4131528881740087E+00;
x[19] = 0.5075929551242276E+00;
x[20] = 0.5962817971382278E+00;
x[21] = 0.6782145376026865E+00;
x[22] = 0.7524628517344771E+00;
x[23] = 0.8181854876152524E+00;
x[24] = 0.8746378049201028E+00;
x[25] = 0.9211802329530588E+00;
x[26] = 0.9572855957780877E+00;
x[27] = 0.9825455052614132E+00;
x[28] = 0.9966794422605966E+00;
w[ 0] = 0.8516903878746365E-02;
w[ 1] = 0.1973208505612276E-01;
w[ 2] = 0.3074049220209360E-01;
w[ 3] = 0.4140206251868281E-01;
w[ 4] = 0.5159482690249799E-01;
w[ 5] = 0.6120309065707916E-01;
w[ 6] = 0.7011793325505125E-01;
w[ 7] = 0.7823832713576385E-01;
w[ 8] = 0.8547225736617248E-01;
w[ 9] = 0.9173775713925882E-01;
w[10] = 0.9696383409440862E-01;
w[11] = 0.1010912737599150E+00;
w[12] = 0.1040733100777293E+00;
w[13] = 0.1058761550973210E+00;
w[14] = 0.1064793817183143E+00;
w[15] = 0.1058761550973210E+00;
w[16] = 0.1040733100777293E+00;
w[17] = 0.1010912737599150E+00;
w[18] = 0.9696383409440862E-01;
w[19] = 0.9173775713925882E-01;
w[20] = 0.8547225736617248E-01;
w[21] = 0.7823832713576385E-01;
w[22] = 0.7011793325505125E-01;
w[23] = 0.6120309065707916E-01;
w[24] = 0.5159482690249799E-01;
w[25] = 0.4140206251868281E-01;
w[26] = 0.3074049220209360E-01;
w[27] = 0.1973208505612276E-01;
w[28] = 0.8516903878746365E-02;
}
else if ( n == 30 )
{
x[ 0] = -0.9968934840746495E+00;
x[ 1] = -0.9836681232797472E+00;
x[ 2] = -0.9600218649683075E+00;
x[ 3] = -0.9262000474292743E+00;
x[ 4] = -0.8825605357920526E+00;
x[ 5] = -0.8295657623827684E+00;
x[ 6] = -0.7677774321048262E+00;
x[ 7] = -0.6978504947933158E+00;
x[ 8] = -0.6205261829892429E+00;
x[ 9] = -0.5366241481420199E+00;
x[10] = -0.4470337695380892E+00;
x[11] = -0.3527047255308781E+00;
x[12] = -0.2546369261678899E+00;
x[13] = -0.1538699136085835E+00;
x[14] = -0.5147184255531770E-01;
x[15] = 0.5147184255531770E-01;
x[16] = 0.1538699136085835E+00;
x[17] = 0.2546369261678899E+00;
x[18] = 0.3527047255308781E+00;
x[19] = 0.4470337695380892E+00;
x[20] = 0.5366241481420199E+00;
x[21] = 0.6205261829892429E+00;
x[22] = 0.6978504947933158E+00;
x[23] = 0.7677774321048262E+00;
x[24] = 0.8295657623827684E+00;
x[25] = 0.8825605357920526E+00;
x[26] = 0.9262000474292743E+00;
x[27] = 0.9600218649683075E+00;
x[28] = 0.9836681232797472E+00;
x[29] = 0.9968934840746495E+00;
w[ 0] = 0.7968192496166648E-02;
w[ 1] = 0.1846646831109099E-01;
w[ 2] = 0.2878470788332330E-01;
w[ 3] = 0.3879919256962704E-01;
w[ 4] = 0.4840267283059405E-01;
w[ 5] = 0.5749315621761905E-01;
w[ 6] = 0.6597422988218052E-01;
w[ 7] = 0.7375597473770516E-01;
w[ 8] = 0.8075589522942023E-01;
w[ 9] = 0.8689978720108314E-01;
w[10] = 0.9212252223778619E-01;
w[11] = 0.9636873717464424E-01;
w[12] = 0.9959342058679524E-01;
w[13] = 0.1017623897484056E+00;
w[14] = 0.1028526528935587E+00;
w[15] = 0.1028526528935587E+00;
w[16] = 0.1017623897484056E+00;
w[17] = 0.9959342058679524E-01;
w[18] = 0.9636873717464424E-01;
w[19] = 0.9212252223778619E-01;
w[20] = 0.8689978720108314E-01;
w[21] = 0.8075589522942023E-01;
w[22] = 0.7375597473770516E-01;
w[23] = 0.6597422988218052E-01;
w[24] = 0.5749315621761905E-01;
w[25] = 0.4840267283059405E-01;
w[26] = 0.3879919256962704E-01;
w[27] = 0.2878470788332330E-01;
w[28] = 0.1846646831109099E-01;
w[29] = 0.7968192496166648E-02;
}
else if ( n == 31 )
{
x[ 0] = -0.99708748181947707454263838179654;
x[ 1] = -0.98468590966515248400211329970113;
x[ 2] = -0.96250392509294966178905249675943;
x[ 3] = -0.93075699789664816495694576311725;
x[ 4] = -0.88976002994827104337419200908023;
x[ 5] = -0.83992032014626734008690453594388;
x[ 6] = -0.78173314841662494040636002019484;
x[ 7] = -0.71577678458685328390597086536649;
x[ 8] = -0.64270672292426034618441820323250;
x[ 9] = -0.56324916140714926272094492359516;
x[10] = -0.47819378204490248044059403935649;
x[11] = -0.38838590160823294306135146128752;
x[12] = -0.29471806998170161661790389767170;
x[13] = -0.19812119933557062877241299603283;
x[14] = -0.99555312152341520325174790118941E-01;
x[15] = 0.00000000000000000000000000000000;
x[16] = 0.99555312152341520325174790118941E-01;
x[17] = 0.19812119933557062877241299603283;
x[18] = 0.29471806998170161661790389767170;
x[19] = 0.38838590160823294306135146128752;
x[20] = 0.47819378204490248044059403935649;
x[21] = 0.56324916140714926272094492359516;
x[22] = 0.64270672292426034618441820323250;
x[23] = 0.71577678458685328390597086536649;
x[24] = 0.78173314841662494040636002019484;
x[25] = 0.83992032014626734008690453594388;
x[26] = 0.88976002994827104337419200908023;
x[27] = 0.93075699789664816495694576311725;
x[28] = 0.96250392509294966178905249675943;
x[29] = 0.98468590966515248400211329970113;
x[30] = 0.99708748181947707454263838179654;
w[ 0] = 0.74708315792487746093913218970494E-02;
w[ 1] = 0.17318620790310582463552990782414E-01;
w[ 2] = 0.27009019184979421800608642617676E-01;
w[ 3] = 0.36432273912385464024392008749009E-01;
w[ 4] = 0.45493707527201102902315857856518E-01;
w[ 5] = 0.54103082424916853711666259085477E-01;
w[ 6] = 0.62174786561028426910343543686657E-01;
w[ 7] = 0.69628583235410366167756126255124E-01;
w[ 8] = 0.76390386598776616426357674901331E-01;
w[ 9] = 0.82392991761589263903823367431962E-01;
w[10] = 0.87576740608477876126198069695333E-01;
w[11] = 0.91890113893641478215362871607150E-01;
w[12] = 0.95290242912319512807204197487597E-01;
w[13] = 0.97743335386328725093474010978997E-01;
w[14] = 0.99225011226672307874875514428615E-01;
w[15] = 0.99720544793426451427533833734349E-01;
w[16] = 0.99225011226672307874875514428615E-01;
w[17] = 0.97743335386328725093474010978997E-01;
w[18] = 0.95290242912319512807204197487597E-01;
w[19] = 0.91890113893641478215362871607150E-01;
w[20] = 0.87576740608477876126198069695333E-01;
w[21] = 0.82392991761589263903823367431962E-01;
w[22] = 0.76390386598776616426357674901331E-01;
w[23] = 0.69628583235410366167756126255124E-01;
w[24] = 0.62174786561028426910343543686657E-01;
w[25] = 0.54103082424916853711666259085477E-01;
w[26] = 0.45493707527201102902315857856518E-01;
w[27] = 0.36432273912385464024392008749009E-01;
w[28] = 0.27009019184979421800608642617676E-01;
w[29] = 0.17318620790310582463552990782414E-01;
w[30] = 0.74708315792487746093913218970494E-02;
}
else if ( n == 32 )
{
x[0] = - 0.997263861849481563544981128665;
x[1] = - 0.985611511545268335400175044631;
x[2] = - 0.964762255587506430773811928118;
x[3] = - 0.934906075937739689170919134835;
x[4] = - 0.896321155766052123965307243719;
x[5] = - 0.849367613732569970133693004968;
x[6] = - 0.794483795967942406963097298970;
x[7] = - 0.732182118740289680387426665091;
x[8] = - 0.663044266930215200975115168663;
x[9] = - 0.587715757240762329040745476402;
x[10] = - 0.506899908932229390023747474378;
x[11] = - 0.421351276130635345364119436172;
x[12] = - 0.331868602282127649779916805730;
x[13] = - 0.239287362252137074544603209166;
x[14] = - 0.144471961582796493485186373599;
x[15] = - 0.483076656877383162348125704405E-01;
x[16] = 0.483076656877383162348125704405E-01;
x[17] = 0.144471961582796493485186373599;
x[18] = 0.239287362252137074544603209166;
x[19] = 0.331868602282127649779916805730;
x[20] = 0.421351276130635345364119436172;
x[21] = 0.506899908932229390023747474378;
x[22] = 0.587715757240762329040745476402;
x[23] = 0.663044266930215200975115168663;
x[24] = 0.732182118740289680387426665091;
x[25] = 0.794483795967942406963097298970;
x[26] = 0.849367613732569970133693004968;
x[27] = 0.896321155766052123965307243719;
x[28] = 0.934906075937739689170919134835;
x[29] = 0.964762255587506430773811928118;
x[30] = 0.985611511545268335400175044631;
x[31] = 0.997263861849481563544981128665;
w[0] = 0.701861000947009660040706373885E-02;
w[1] = 0.162743947309056706051705622064E-01;
w[2] = 0.253920653092620594557525897892E-01;
w[3] = 0.342738629130214331026877322524E-01;
w[4] = 0.428358980222266806568786466061E-01;
w[5] = 0.509980592623761761961632446895E-01;
w[6] = 0.586840934785355471452836373002E-01;
w[7] = 0.658222227763618468376500637069E-01;
w[8] = 0.723457941088485062253993564785E-01;
w[9] = 0.781938957870703064717409188283E-01;
w[10] = 0.833119242269467552221990746043E-01;
w[11] = 0.876520930044038111427714627518E-01;
w[12] = 0.911738786957638847128685771116E-01;
w[13] = 0.938443990808045656391802376681E-01;
w[14] = 0.956387200792748594190820022041E-01;
w[15] = 0.965400885147278005667648300636E-01;
w[16] = 0.965400885147278005667648300636E-01;
w[17] = 0.956387200792748594190820022041E-01;
w[18] = 0.938443990808045656391802376681E-01;
w[19] = 0.911738786957638847128685771116E-01;
w[20] = 0.876520930044038111427714627518E-01;
w[21] = 0.833119242269467552221990746043E-01;
w[22] = 0.781938957870703064717409188283E-01;
w[23] = 0.723457941088485062253993564785E-01;
w[24] = 0.658222227763618468376500637069E-01;
w[25] = 0.586840934785355471452836373002E-01;
w[26] = 0.509980592623761761961632446895E-01;
w[27] = 0.428358980222266806568786466061E-01;
w[28] = 0.342738629130214331026877322524E-01;
w[29] = 0.253920653092620594557525897892E-01;
w[30] = 0.162743947309056706051705622064E-01;
w[31] = 0.701861000947009660040706373885E-02;
}
else if ( n == 33 )
{
x[ 0] = -0.9974246942464552;
x[ 1] = -0.9864557262306425;
x[ 2] = -0.9668229096899927;
x[ 3] = -0.9386943726111684;
x[ 4] = -0.9023167677434336;
x[ 5] = -0.8580096526765041;
x[ 6] = -0.8061623562741665;
x[ 7] = -0.7472304964495622;
x[ 8] = -0.6817319599697428;
x[ 9] = -0.6102423458363790;
x[10] = -0.5333899047863476;
x[11] = -0.4518500172724507;
x[12] = -0.3663392577480734;
x[13] = -0.2776090971524970;
x[14] = -0.1864392988279916;
x[15] = -0.09363106585473338;
x[16] = 0.000000000000000;
x[17] = 0.09363106585473338;
x[18] = 0.1864392988279916;
x[19] = 0.2776090971524970;
x[20] = 0.3663392577480734;
x[21] = 0.4518500172724507;
x[22] = 0.5333899047863476;
x[23] = 0.6102423458363790;
x[24] = 0.6817319599697428;
x[25] = 0.7472304964495622;
x[26] = 0.8061623562741665;
x[27] = 0.8580096526765041;
x[28] = 0.9023167677434336;
x[29] = 0.9386943726111684;
x[30] = 0.9668229096899927;
x[31] = 0.9864557262306425;
x[32] = 0.9974246942464552;
w[ 0] = 0.6606227847587558E-02;
w[ 1] = 0.1532170151293465E-01;
w[ 2] = 0.2391554810174960E-01;
w[ 3] = 0.3230035863232891E-01;
w[ 4] = 0.4040154133166965E-01;
w[ 5] = 0.4814774281871162E-01;
w[ 6] = 0.5547084663166357E-01;
w[ 7] = 0.6230648253031755E-01;
w[ 8] = 0.6859457281865676E-01;
w[ 9] = 0.7427985484395420E-01;
w[10] = 0.7931236479488685E-01;
w[11] = 0.8364787606703869E-01;
w[12] = 0.8724828761884425E-01;
w[13] = 0.9008195866063859E-01;
w[14] = 0.9212398664331678E-01;
w[15] = 0.9335642606559612E-01;
w[16] = 0.9376844616020999E-01;
w[17] = 0.9335642606559612E-01;
w[18] = 0.9212398664331678E-01;
w[19] = 0.9008195866063859E-01;
w[20] = 0.8724828761884425E-01;
w[21] = 0.8364787606703869E-01;
w[22] = 0.7931236479488685E-01;
w[23] = 0.7427985484395420E-01;
w[24] = 0.6859457281865676E-01;
w[25] = 0.6230648253031755E-01;
w[26] = 0.5547084663166357E-01;
w[27] = 0.4814774281871162E-01;
w[28] = 0.4040154133166965E-01;
w[29] = 0.3230035863232891E-01;
w[30] = 0.2391554810174960E-01;
w[31] = 0.1532170151293465E-01;
w[32] = 0.6606227847587558E-02;
}
else if ( n == 64 )
{
x[0] = - 0.999305041735772139456905624346;
x[1] = - 0.996340116771955279346924500676;
x[2] = - 0.991013371476744320739382383443;
x[3] = - 0.983336253884625956931299302157;
x[4] = - 0.973326827789910963741853507352;
x[5] = - 0.961008799652053718918614121897;
x[6] = - 0.946411374858402816062481491347;
x[7] = - 0.929569172131939575821490154559;
x[8] = - 0.910522137078502805756380668008;
x[9] = - 0.889315445995114105853404038273;
x[10] = - 0.865999398154092819760783385070;
x[11] = - 0.840629296252580362751691544696;
x[12] = - 0.813265315122797559741923338086;
x[13] = - 0.783972358943341407610220525214;
x[14] = - 0.752819907260531896611863774886;
x[15] = - 0.719881850171610826848940217832;
x[16] = - 0.685236313054233242563558371031;
x[17] = - 0.648965471254657339857761231993;
x[18] = - 0.611155355172393250248852971019;
x[19] = - 0.571895646202634034283878116659;
x[20] = - 0.531279464019894545658013903544;
x[21] = - 0.489403145707052957478526307022;
x[22] = - 0.446366017253464087984947714759;
x[23] = - 0.402270157963991603695766771260;
x[24] = - 0.357220158337668115950442615046;
x[25] = - 0.311322871990210956157512698560;
x[26] = - 0.264687162208767416373964172510;
x[27] = - 0.217423643740007084149648748989;
x[28] = - 0.169644420423992818037313629748;
x[29] = - 0.121462819296120554470376463492;
x[30] = - 0.729931217877990394495429419403E-01;
x[31] = - 0.243502926634244325089558428537E-01;
x[32] = 0.243502926634244325089558428537E-01;
x[33] = 0.729931217877990394495429419403E-01;
x[34] = 0.121462819296120554470376463492;
x[35] = 0.169644420423992818037313629748;
x[36] = 0.217423643740007084149648748989;
x[37] = 0.264687162208767416373964172510;
x[38] = 0.311322871990210956157512698560;
x[39] = 0.357220158337668115950442615046;
x[40] = 0.402270157963991603695766771260;
x[41] = 0.446366017253464087984947714759;
x[42] = 0.489403145707052957478526307022;
x[43] = 0.531279464019894545658013903544;
x[44] = 0.571895646202634034283878116659;
x[45] = 0.611155355172393250248852971019;
x[46] = 0.648965471254657339857761231993;
x[47] = 0.685236313054233242563558371031;
x[48] = 0.719881850171610826848940217832;
x[49] = 0.752819907260531896611863774886;
x[50] = 0.783972358943341407610220525214;
x[51] = 0.813265315122797559741923338086;
x[52] = 0.840629296252580362751691544696;
x[53] = 0.865999398154092819760783385070;
x[54] = 0.889315445995114105853404038273;
x[55] = 0.910522137078502805756380668008;
x[56] = 0.929569172131939575821490154559;
x[57] = 0.946411374858402816062481491347;
x[58] = 0.961008799652053718918614121897;
x[59] = 0.973326827789910963741853507352;
x[60] = 0.983336253884625956931299302157;
x[61] = 0.991013371476744320739382383443;
x[62] = 0.996340116771955279346924500676;
x[63] = 0.999305041735772139456905624346;
w[0] = 0.178328072169643294729607914497E-02;
w[1] = 0.414703326056246763528753572855E-02;
w[2] = 0.650445796897836285611736039998E-02;
w[3] = 0.884675982636394772303091465973E-02;
w[4] = 0.111681394601311288185904930192E-01;
w[5] = 0.134630478967186425980607666860E-01;
w[6] = 0.157260304760247193219659952975E-01;
w[7] = 0.179517157756973430850453020011E-01;
w[8] = 0.201348231535302093723403167285E-01;
w[9] = 0.222701738083832541592983303842E-01;
w[10] = 0.243527025687108733381775504091E-01;
w[11] = 0.263774697150546586716917926252E-01;
w[12] = 0.283396726142594832275113052002E-01;
w[13] = 0.302346570724024788679740598195E-01;
w[14] = 0.320579283548515535854675043479E-01;
w[15] = 0.338051618371416093915654821107E-01;
w[16] = 0.354722132568823838106931467152E-01;
w[17] = 0.370551285402400460404151018096E-01;
w[18] = 0.385501531786156291289624969468E-01;
w[19] = 0.399537411327203413866569261283E-01;
w[20] = 0.412625632426235286101562974736E-01;
w[21] = 0.424735151236535890073397679088E-01;
w[22] = 0.435837245293234533768278609737E-01;
w[23] = 0.445905581637565630601347100309E-01;
w[24] = 0.454916279274181444797709969713E-01;
w[25] = 0.462847965813144172959532492323E-01;
w[26] = 0.469681828162100173253262857546E-01;
w[27] = 0.475401657148303086622822069442E-01;
w[28] = 0.479993885964583077281261798713E-01;
w[29] = 0.483447622348029571697695271580E-01;
w[30] = 0.485754674415034269347990667840E-01;
w[31] = 0.486909570091397203833653907347E-01;
w[32] = 0.486909570091397203833653907347E-01;
w[33] = 0.485754674415034269347990667840E-01;
w[34] = 0.483447622348029571697695271580E-01;
w[35] = 0.479993885964583077281261798713E-01;
w[36] = 0.475401657148303086622822069442E-01;
w[37] = 0.469681828162100173253262857546E-01;
w[38] = 0.462847965813144172959532492323E-01;
w[39] = 0.454916279274181444797709969713E-01;
w[40] = 0.445905581637565630601347100309E-01;
w[41] = 0.435837245293234533768278609737E-01;
w[42] = 0.424735151236535890073397679088E-01;
w[43] = 0.412625632426235286101562974736E-01;
w[44] = 0.399537411327203413866569261283E-01;
w[45] = 0.385501531786156291289624969468E-01;
w[46] = 0.370551285402400460404151018096E-01;
w[47] = 0.354722132568823838106931467152E-01;
w[48] = 0.338051618371416093915654821107E-01;
w[49] = 0.320579283548515535854675043479E-01;
w[50] = 0.302346570724024788679740598195E-01;
w[51] = 0.283396726142594832275113052002E-01;
w[52] = 0.263774697150546586716917926252E-01;
w[53] = 0.243527025687108733381775504091E-01;
w[54] = 0.222701738083832541592983303842E-01;
w[55] = 0.201348231535302093723403167285E-01;
w[56] = 0.179517157756973430850453020011E-01;
w[57] = 0.157260304760247193219659952975E-01;
w[58] = 0.134630478967186425980607666860E-01;
w[59] = 0.111681394601311288185904930192E-01;
w[60] = 0.884675982636394772303091465973E-02;
w[61] = 0.650445796897836285611736039998E-02;
w[62] = 0.414703326056246763528753572855E-02;
w[63] = 0.178328072169643294729607914497E-02;
}
else if ( n == 65 )
{
x[ 0] = -0.9993260970754129;
x[ 1] = -0.9964509480618492;
x[ 2] = -0.9912852761768016;
x[ 3] = -0.9838398121870350;
x[ 4] = -0.9741315398335512;
x[ 5] = -0.9621827547180553;
x[ 6] = -0.9480209281684076;
x[ 7] = -0.9316786282287494;
x[ 8] = -0.9131934405428462;
x[ 9] = -0.8926078805047389;
x[10] = -0.8699692949264071;
x[11] = -0.8453297528999303;
x[12] = -0.8187459259226514;
x[13] = -0.7902789574921218;
x[14] = -0.7599943224419998;
x[15] = -0.7279616763294247;
x[16] = -0.6942546952139916;
x[17] = -0.6589509061936252;
x[18] = -0.6221315090854003;
x[19] = -0.5838811896604873;
x[20] = -0.5442879248622271;
x[21] = -0.5034427804550069;
x[22] = -0.4614397015691450;
x[23] = -0.4183752966234090;
x[24] = -0.3743486151220660;
x[25] = -0.3294609198374864;
x[26] = -0.2838154539022487;
x[27] = -0.2375172033464168;
x[28] = -0.1906726556261428;
x[29] = -0.1433895546989752;
x[30] = -0.9577665320919751E-01;
x[31] = -0.4794346235317186E-01;
x[32] = 0.000000000000000;
x[33] = 0.4794346235317186E-01;
x[34] = 0.9577665320919751E-01;
x[35] = 0.1433895546989752;
x[36] = 0.1906726556261428;
x[37] = 0.2375172033464168;
x[38] = 0.2838154539022487;
x[39] = 0.3294609198374864;
x[40] = 0.3743486151220660;
x[41] = 0.4183752966234090;
x[42] = 0.4614397015691450;
x[43] = 0.5034427804550069;
x[44] = 0.5442879248622271;
x[45] = 0.5838811896604873;
x[46] = 0.6221315090854003;
x[47] = 0.6589509061936252;
x[48] = 0.6942546952139916;
x[49] = 0.7279616763294247;
x[50] = 0.7599943224419998;
x[51] = 0.7902789574921218;
x[52] = 0.8187459259226514;
x[53] = 0.8453297528999303;
x[54] = 0.8699692949264071;
x[55] = 0.8926078805047389;
x[56] = 0.9131934405428462;
x[57] = 0.9316786282287494;
x[58] = 0.9480209281684076;
x[59] = 0.9621827547180553;
x[60] = 0.9741315398335512;
x[61] = 0.9838398121870350;
x[62] = 0.9912852761768016;
x[63] = 0.9964509480618492;
x[64] = 0.9993260970754129;
w[ 0] = 0.1729258251300218E-02;
w[ 1] = 0.4021524172003703E-02;
w[ 2] = 0.6307942578971821E-02;
w[ 3] = 0.8580148266881443E-02;
w[ 4] = 0.1083267878959798E-01;
w[ 5] = 0.1306031163999490E-01;
w[ 6] = 0.1525791214644825E-01;
w[ 7] = 0.1742042199767025E-01;
w[ 8] = 0.1954286583675005E-01;
w[ 9] = 0.2162036128493408E-01;
w[10] = 0.2364812969128723E-01;
w[11] = 0.2562150693803776E-01;
w[12] = 0.2753595408845034E-01;
w[13] = 0.2938706778931066E-01;
w[14] = 0.3117059038018911E-01;
w[15] = 0.3288241967636860E-01;
w[16] = 0.3451861839854901E-01;
w[17] = 0.3607542322556527E-01;
w[18] = 0.3754925344825770E-01;
w[19] = 0.3893671920405121E-01;
w[20] = 0.4023462927300549E-01;
w[21] = 0.4143999841724028E-01;
w[22] = 0.4255005424675579E-01;
w[23] = 0.4356224359580051E-01;
w[24] = 0.4447423839508296E-01;
w[25] = 0.4528394102630023E-01;
w[26] = 0.4598948914665173E-01;
w[27] = 0.4658925997223349E-01;
w[28] = 0.4708187401045461E-01;
w[29] = 0.4746619823288551E-01;
w[30] = 0.4774134868124067E-01;
w[31] = 0.4790669250049590E-01;
w[32] = 0.4796184939446662E-01;
w[33] = 0.4790669250049590E-01;
w[34] = 0.4774134868124067E-01;
w[35] = 0.4746619823288551E-01;
w[36] = 0.4708187401045461E-01;
w[37] = 0.4658925997223349E-01;
w[38] = 0.4598948914665173E-01;
w[39] = 0.4528394102630023E-01;
w[40] = 0.4447423839508296E-01;
w[41] = 0.4356224359580051E-01;
w[42] = 0.4255005424675579E-01;
w[43] = 0.4143999841724028E-01;
w[44] = 0.4023462927300549E-01;
w[45] = 0.3893671920405121E-01;
w[46] = 0.3754925344825770E-01;
w[47] = 0.3607542322556527E-01;
w[48] = 0.3451861839854901E-01;
w[49] = 0.3288241967636860E-01;
w[50] = 0.3117059038018911E-01;
w[51] = 0.2938706778931066E-01;
w[52] = 0.2753595408845034E-01;
w[53] = 0.2562150693803776E-01;
w[54] = 0.2364812969128723E-01;
w[55] = 0.2162036128493408E-01;
w[56] = 0.1954286583675005E-01;
w[57] = 0.1742042199767025E-01;
w[58] = 0.1525791214644825E-01;
w[59] = 0.1306031163999490E-01;
w[60] = 0.1083267878959798E-01;
w[61] = 0.8580148266881443E-02;
w[62] = 0.6307942578971821E-02;
w[63] = 0.4021524172003703E-02;
w[64] = 0.1729258251300218E-02;
}
else if ( n == 127 )
{
x[ 0] = -0.99982213041530614629963254927125E+00;
x[ 1] = -0.99906293435531189513828920479421E+00;
x[ 2] = -0.99769756618980462107441703193392E+00;
x[ 3] = -0.99572655135202722663543337085008E+00;
x[ 4] = -0.99315104925451714736113079489080E+00;
x[ 5] = -0.98997261459148415760778669967548E+00;
x[ 6] = -0.98619317401693166671043833175407E+00;
x[ 7] = -0.98181502080381411003346312451200E+00;
x[ 8] = -0.97684081234307032681744391886221E+00;
x[ 9] = -0.97127356816152919228894689830512E+00;
x[ 10] = -0.96511666794529212109082507703391E+00;
x[ 11] = -0.95837384942523877114910286998060E+00;
x[ 12] = -0.95104920607788031054790764659636E+00;
x[ 13] = -0.94314718462481482734544963026201E+00;
x[ 14] = -0.93467258232473796857363487794906E+00;
x[ 15] = -0.92563054405623384912746466814259E+00;
x[ 16] = -0.91602655919146580931308861741716E+00;
x[ 17] = -0.90586645826182138280246131760282E+00;
x[ 18] = -0.89515640941708370896904382642451E+00;
x[ 19] = -0.88390291468002656994525794802849E+00;
x[ 20] = -0.87211280599856071141963753428864E+00;
x[ 21] = -0.85979324109774080981203134414483E+00;
x[ 22] = -0.84695169913409759845333931085437E+00;
x[ 23] = -0.83359597615489951437955716480123E+00;
x[ 24] = -0.81973418036507867415511910167470E+00;
x[ 25] = -0.80537472720468021466656079404644E+00;
x[ 26] = -0.79052633423981379994544995252740E+00;
x[ 27] = -0.77519801587020238244496276354566E+00;
x[ 28] = -0.75939907785653667155666366659810E+00;
x[ 29] = -0.74313911167095451292056688997595E+00;
x[ 30] = -0.72642798867407268553569290153270E+00;
x[ 31] = -0.70927585412210456099944463906757E+00;
x[ 32] = -0.69169312100770067015644143286666E+00;
x[ 33] = -0.67369046373825048534668253831602E+00;
x[ 34] = -0.65527881165548263027676505156852E+00;
x[ 35] = -0.63646934240029724134760815684175E+00;
x[ 36] = -0.61727347512685828385763916340822E+00;
x[ 37] = -0.59770286357006522938441201887478E+00;
x[ 38] = -0.57776938897061258000325165713764E+00;
x[ 39] = -0.55748515286193223292186190687872E+00;
x[ 40] = -0.53686246972339756745816636353452E+00;
x[ 41] = -0.51591385950424935727727729906662E+00;
x[ 42] = -0.49465204002278211739494017368636E+00;
x[ 43] = -0.47308991924540524164509989939699E+00;
x[ 44] = -0.45124058745026622733189858020729E+00;
x[ 45] = -0.42911730928019337626254405355418E+00;
x[ 46] = -0.40673351568978256340867288124339E+00;
x[ 47] = -0.38410279579151693577907781452239E+00;
x[ 48] = -0.36123888860586970607092484346723E+00;
x[ 49] = -0.33815567472039850137600027657095E+00;
x[ 50] = -0.31486716786289498148601475374890E+00;
x[ 51] = -0.29138750639370562079451875284568E+00;
x[ 52] = -0.26773094472238862088834352027938E+00;
x[ 53] = -0.24391184465391785797071324453138E+00;
x[ 54] = -0.21994466666968754245452337866940E+00;
x[ 55] = -0.19584396114861085150428162519610E+00;
x[ 56] = -0.17162435953364216500834492248954E+00;
x[ 57] = -0.14730056544908566938932929319807E+00;
x[ 58] = -0.12288734577408297172603365288567E+00;
x[ 59] = -0.98399521677698970751091751509101E-01;
x[ 60] = -0.73851959621048545273440409360569E-01;
x[ 61] = -0.49259562331926630315379321821927E-01;
x[ 62] = -0.24637259757420944614897071846088E-01;
x[ 63] = 0.00000000000000000000000000000000E+00;
x[ 64] = 0.24637259757420944614897071846088E-01;
x[ 65] = 0.49259562331926630315379321821927E-01;
x[ 66] = 0.73851959621048545273440409360569E-01;
x[ 67] = 0.98399521677698970751091751509101E-01;
x[ 68] = 0.12288734577408297172603365288567E+00;
x[ 69] = 0.14730056544908566938932929319807E+00;
x[ 70] = 0.17162435953364216500834492248954E+00;
x[ 71] = 0.19584396114861085150428162519610E+00;
x[ 72] = 0.21994466666968754245452337866940E+00;
x[ 73] = 0.24391184465391785797071324453138E+00;
x[ 74] = 0.26773094472238862088834352027938E+00;
x[ 75] = 0.29138750639370562079451875284568E+00;
x[ 76] = 0.31486716786289498148601475374890E+00;
x[ 77] = 0.33815567472039850137600027657095E+00;
x[ 78] = 0.36123888860586970607092484346723E+00;
x[ 79] = 0.38410279579151693577907781452239E+00;
x[ 80] = 0.40673351568978256340867288124339E+00;
x[ 81] = 0.42911730928019337626254405355418E+00;
x[ 82] = 0.45124058745026622733189858020729E+00;
x[ 83] = 0.47308991924540524164509989939699E+00;
x[ 84] = 0.49465204002278211739494017368636E+00;
x[ 85] = 0.51591385950424935727727729906662E+00;
x[ 86] = 0.53686246972339756745816636353452E+00;
x[ 87] = 0.55748515286193223292186190687872E+00;
x[ 88] = 0.57776938897061258000325165713764E+00;
x[ 89] = 0.59770286357006522938441201887478E+00;
x[ 90] = 0.61727347512685828385763916340822E+00;
x[ 91] = 0.63646934240029724134760815684175E+00;
x[ 92] = 0.65527881165548263027676505156852E+00;
x[ 93] = 0.67369046373825048534668253831602E+00;
x[ 94] = 0.69169312100770067015644143286666E+00;
x[ 95] = 0.70927585412210456099944463906757E+00;
x[ 96] = 0.72642798867407268553569290153270E+00;
x[ 97] = 0.74313911167095451292056688997595E+00;
x[ 98] = 0.75939907785653667155666366659810E+00;
x[ 99] = 0.77519801587020238244496276354566E+00;
x[100] = 0.79052633423981379994544995252740E+00;
x[101] = 0.80537472720468021466656079404644E+00;
x[102] = 0.81973418036507867415511910167470E+00;
x[103] = 0.83359597615489951437955716480123E+00;
x[104] = 0.84695169913409759845333931085437E+00;
x[105] = 0.85979324109774080981203134414483E+00;
x[106] = 0.87211280599856071141963753428864E+00;
x[107] = 0.88390291468002656994525794802849E+00;
x[108] = 0.89515640941708370896904382642451E+00;
x[109] = 0.90586645826182138280246131760282E+00;
x[110] = 0.91602655919146580931308861741716E+00;
x[111] = 0.92563054405623384912746466814259E+00;
x[112] = 0.93467258232473796857363487794906E+00;
x[113] = 0.94314718462481482734544963026201E+00;
x[114] = 0.95104920607788031054790764659636E+00;
x[115] = 0.95837384942523877114910286998060E+00;
x[116] = 0.96511666794529212109082507703391E+00;
x[117] = 0.97127356816152919228894689830512E+00;
x[118] = 0.97684081234307032681744391886221E+00;
x[119] = 0.98181502080381411003346312451200E+00;
x[120] = 0.98619317401693166671043833175407E+00;
x[121] = 0.98997261459148415760778669967548E+00;
x[122] = 0.99315104925451714736113079489080E+00;
x[123] = 0.99572655135202722663543337085008E+00;
x[124] = 0.99769756618980462107441703193392E+00;
x[125] = 0.99906293435531189513828920479421E+00;
x[126] = 0.99982213041530614629963254927125E+00;
w[ 0] = 0.45645726109586654495731936146574E-03;
w[ 1] = 0.10622766869538486959954760554099E-02;
w[ 2] = 0.16683488125171936761028811985672E-02;
w[ 3] = 0.22734860707492547802810838362671E-02;
w[ 4] = 0.28772587656289004082883197417581E-02;
w[ 5] = 0.34792893810051465908910894094105E-02;
w[ 6] = 0.40792095178254605327114733456293E-02;
w[ 7] = 0.46766539777779034772638165662478E-02;
w[ 8] = 0.52712596565634400891303815906251E-02;
w[ 9] = 0.58626653903523901033648343751367E-02;
w[ 10] = 0.64505120486899171845442463868748E-02;
w[ 11] = 0.70344427036681608755685893032552E-02;
w[ 12] = 0.76141028256526859356393930849227E-02;
w[ 13] = 0.81891404887415730817235884718726E-02;
w[ 14] = 0.87592065795403145773316804234385E-02;
w[ 15] = 0.93239550065309714787536985834029E-02;
w[ 16] = 0.98830429087554914716648010899606E-02;
w[ 17] = 0.10436130863141005225673171997668E-01;
w[ 18] = 0.10982883090068975788799657376065E-01;
w[ 19] = 0.11522967656921087154811609734510E-01;
w[ 20] = 0.12056056679400848183529562144697E-01;
w[ 21] = 0.12581826520465013101514365424172E-01;
w[ 22] = 0.13099957986718627426172681912499E-01;
w[ 23] = 0.13610136522139249906034237533759E-01;
w[ 24] = 0.14112052399003395774044161633613E-01;
w[ 25] = 0.14605400905893418351737288078952E-01;
w[ 26] = 0.15089882532666922992635733981431E-01;
w[ 27] = 0.15565203152273955098532590262975E-01;
w[ 28] = 0.16031074199309941802254151842763E-01;
w[ 29] = 0.16487212845194879399346060358146E-01;
w[ 30] = 0.16933342169871654545878815295200E-01;
w[ 31] = 0.17369191329918731922164721250350E-01;
w[ 32] = 0.17794495722974774231027912900351E-01;
w[ 33] = 0.18208997148375106468721469154479E-01;
w[ 34] = 0.18612443963902310429440419898958E-01;
w[ 35] = 0.19004591238555646611148901044533E-01;
w[ 36] = 0.19385200901246454628112623489471E-01;
w[ 37] = 0.19754041885329183081815217323169E-01;
w[ 38] = 0.20110890268880247225644623956287E-01;
w[ 39] = 0.20455529410639508279497065713301E-01;
w[ 40] = 0.20787750081531811812652137291250E-01;
w[ 41] = 0.21107350591688713643523847921658E-01;
w[ 42] = 0.21414136912893259295449693233545E-01;
w[ 43] = 0.21707922796373466052301324695331E-01;
w[ 44] = 0.21988529885872983756478409758807E-01;
w[ 45] = 0.22255787825930280235631416460158E-01;
w[ 46] = 0.22509534365300608085694429903050E-01;
w[ 47] = 0.22749615455457959852242553240982E-01;
w[ 48] = 0.22975885344117206754377437838947E-01;
w[ 49] = 0.23188206663719640249922582981729E-01;
w[ 50] = 0.23386450514828194170722043496950E-01;
w[ 51] = 0.23570496544381716050033676844306E-01;
w[ 52] = 0.23740233018760777777714726703424E-01;
w[ 53] = 0.23895556891620665983864481754172E-01;
w[ 54] = 0.24036373866450369675132086026456E-01;
w[ 55] = 0.24162598453819584716522917710986E-01;
w[ 56] = 0.24274154023278979833195063936748E-01;
w[ 57] = 0.24370972849882214952813561907241E-01;
w[ 58] = 0.24452996155301467956140198471529E-01;
w[ 59] = 0.24520174143511508275183033290175E-01;
w[ 60] = 0.24572466031020653286354137335186E-01;
w[ 61] = 0.24609840071630254092545634003360E-01;
w[ 62] = 0.24632273575707679066033370218017E-01;
w[ 63] = 0.24639752923961094419579417477503E-01;
w[ 64] = 0.24632273575707679066033370218017E-01;
w[ 65] = 0.24609840071630254092545634003360E-01;
w[ 66] = 0.24572466031020653286354137335186E-01;
w[ 67] = 0.24520174143511508275183033290175E-01;
w[ 68] = 0.24452996155301467956140198471529E-01;
w[ 69] = 0.24370972849882214952813561907241E-01;
w[ 70] = 0.24274154023278979833195063936748E-01;
w[ 71] = 0.24162598453819584716522917710986E-01;
w[ 72] = 0.24036373866450369675132086026456E-01;
w[ 73] = 0.23895556891620665983864481754172E-01;
w[ 74] = 0.23740233018760777777714726703424E-01;
w[ 75] = 0.23570496544381716050033676844306E-01;
w[ 76] = 0.23386450514828194170722043496950E-01;
w[ 77] = 0.23188206663719640249922582981729E-01;
w[ 78] = 0.22975885344117206754377437838947E-01;
w[ 79] = 0.22749615455457959852242553240982E-01;
w[ 80] = 0.22509534365300608085694429903050E-01;
w[ 81] = 0.22255787825930280235631416460158E-01;
w[ 82] = 0.21988529885872983756478409758807E-01;
w[ 83] = 0.21707922796373466052301324695331E-01;
w[ 84] = 0.21414136912893259295449693233545E-01;
w[ 85] = 0.21107350591688713643523847921658E-01;
w[ 86] = 0.20787750081531811812652137291250E-01;
w[ 87] = 0.20455529410639508279497065713301E-01;
w[ 88] = 0.20110890268880247225644623956287E-01;
w[ 89] = 0.19754041885329183081815217323169E-01;
w[ 90] = 0.19385200901246454628112623489471E-01;
w[ 91] = 0.19004591238555646611148901044533E-01;
w[ 92] = 0.18612443963902310429440419898958E-01;
w[ 93] = 0.18208997148375106468721469154479E-01;
w[ 94] = 0.17794495722974774231027912900351E-01;
w[ 95] = 0.17369191329918731922164721250350E-01;
w[ 96] = 0.16933342169871654545878815295200E-01;
w[ 97] = 0.16487212845194879399346060358146E-01;
w[ 98] = 0.16031074199309941802254151842763E-01;
w[ 99] = 0.15565203152273955098532590262975E-01;
w[100] = 0.15089882532666922992635733981431E-01;
w[101] = 0.14605400905893418351737288078952E-01;
w[102] = 0.14112052399003395774044161633613E-01;
w[103] = 0.13610136522139249906034237533759E-01;
w[104] = 0.13099957986718627426172681912499E-01;
w[105] = 0.12581826520465013101514365424172E-01;
w[106] = 0.12056056679400848183529562144697E-01;
w[107] = 0.11522967656921087154811609734510E-01;
w[108] = 0.10982883090068975788799657376065E-01;
w[109] = 0.10436130863141005225673171997668E-01;
w[110] = 0.98830429087554914716648010899606E-02;
w[111] = 0.93239550065309714787536985834029E-02;
w[112] = 0.87592065795403145773316804234385E-02;
w[113] = 0.81891404887415730817235884718726E-02;
w[114] = 0.76141028256526859356393930849227E-02;
w[115] = 0.70344427036681608755685893032552E-02;
w[116] = 0.64505120486899171845442463868748E-02;
w[117] = 0.58626653903523901033648343751367E-02;
w[118] = 0.52712596565634400891303815906251E-02;
w[119] = 0.46766539777779034772638165662478E-02;
w[120] = 0.40792095178254605327114733456293E-02;
w[121] = 0.34792893810051465908910894094105E-02;
w[122] = 0.28772587656289004082883197417581E-02;
w[123] = 0.22734860707492547802810838362671E-02;
w[124] = 0.16683488125171936761028811985672E-02;
w[125] = 0.10622766869538486959954760554099E-02;
w[126] = 0.45645726109586654495731936146574E-03;
}
else if ( n == 255 )
{
x[ 0] = -0.9999557053175637;
x[ 1] = -0.9997666213120006;
x[ 2] = -0.99942647468017;
x[ 3] = -0.9989352412846546;
x[ 4] = -0.9982929861369679;
x[ 5] = -0.9974998041266158;
x[ 6] = -0.9965558144351986;
x[ 7] = -0.9954611594800263;
x[ 8] = -0.9942160046166302;
x[ 9] = -0.9928205380219891;
x[10] = -0.9912749706303856;
x[11] = -0.9895795360859201;
x[12] = -0.9877344906997324;
x[13] = -0.9857401134074193;
x[14] = -0.9835967057247763;
x[15] = -0.9813045917010171;
x[16] = -0.9788641178690681;
x[17] = -0.976275653192736;
x[18] = -0.9735395890106436;
x[19] = -0.9706563389768804;
x[20] = -0.9676263389983388;
x[21] = -0.9644500471687263;
x[22] = -0.9611279436992478;
x[23] = -0.957660530845962;
x[24] = -0.9540483328338163;
x[25] = -0.9502918957773683;
x[26] = -0.9463917875982043;
x[27] = -0.9423485979390644;
x[28] = -0.9381629380746873;
x[29] = -0.9338354408193861;
x[30] = -0.9293667604313699;
x[31] = -0.9247575725138244;
x[32] = -0.9200085739127664;
x[33] = -0.915120482611687;
x[34] = -0.9100940376230008;
x[35] = -0.904929998876315;
x[36] = -0.8996291471035368;
x[37] = -0.8941922837208367;
x[38] = -0.8886202307074841;
x[39] = -0.8829138304815741;
x[40] = -0.8770739457726654;
x[41] = -0.8711014594913465;
x[42] = -0.8649972745957512;
x[43] = -0.858762313955043;
x[44] = -0.8523975202098902;
x[45] = -0.8459038556299511;
x[46] = -0.839282301968391;
x[47] = -0.8325338603134556;
x[48] = -0.8256595509371186;
x[49] = -0.8186604131408319;
x[50] = -0.8115375050983958;
x[51] = -0.8042919036959787;
x[52] = -0.7969247043693057;
x[53] = -0.7894370209380444;
x[54] = -0.7818299854374094;
x[55] = -0.7741047479470157;
x[56] = -0.7662624764170006;
x[57] = -0.7583043564914468;
x[58] = -0.7502315913291283;
x[59] = -0.7420454014216102;
x[60] = -0.7337470244087263;
x[61] = -0.7253377148914649;
x[62] = -0.7168187442422908;
x[63] = -0.7081914004129306;
x[64] = -0.6994569877396524;
x[65] = -0.6906168267460676;
x[66] = -0.6816722539434864;
x[67] = -0.6726246216288551;
x[68] = -0.663475297680307;
x[69] = -0.6542256653503588;
x[70] = -0.6448771230567811;
x[71] = -0.6354310841711771;
x[72] = -0.6258889768052999;
x[73] = -0.6162522435951415;
x[74] = -0.6065223414828266;
x[75] = -0.5967007414963417;
x[76] = -0.5867889285271373;
x[77] = -0.5767884011056313;
x[78] = -0.5667006711746527;
x[79] = -0.5565272638608558;
x[80] = -0.5462697172441424;
x[81] = -0.5359295821251249;
x[82] = -0.5255084217906666;
x[83] = -0.5150078117775342;
x[84] = -0.5044293396341982;
x[85] = -0.493774604680817;
x[86] = -0.483045217767442;
x[87] = -0.4722428010304787;
x[88] = -0.4613689876474424;
x[89] = -0.4504254215900437;
x[90] = -0.4394137573756426;
x[91] = -0.4283356598171081;
x[92] = -0.4171928037711214;
x[93] = -0.4059868738849605;
x[94] = -0.3947195643418044;
x[95] = -0.3833925786045958;
x[96] = -0.3720076291585012;
x[97] = -0.3605664372520062;
x[98] = -0.3490707326366864;
x[99] = -0.3375222533056927;
x[100] = -0.3259227452309905;
x[101] = -0.3142739620993925;
x[102] = -0.3025776650474256;
x[103] = -0.2908356223950708;
x[104] = -0.2790496093784178;
x[105] = -0.2672214078812731;
x[106] = -0.2553528061657641;
x[107] = -0.243445598601978;
x[108] = -0.2315015853966777;
x[109] = -0.2195225723211354;
x[110] = -0.2075103704381242;
x[111] = -0.1954667958281108;
x[112] = -0.1833936693146885;
x[113] = -0.1712928161892939;
x[114] = -0.1591660659352477;
x[115] = -0.147015251951162;
x[116] = -0.1348422112737553;
x[117] = -0.1226487843001178;
x[118] = -0.1104368145094688;
x[119] = -0.09820814818444755;
x[120] = -0.08596463413198061;
x[121] = -0.07370812340376778;
x[122] = -0.06144046901642827;
x[123] = -0.04916352567134998;
x[124] = -0.03687914947428402;
x[125] = -0.02458919765472701;
x[126] = -0.01229552828513332;
x[127] = 0;
x[128] = 0.01229552828513332;
x[129] = 0.02458919765472701;
x[130] = 0.03687914947428402;
x[131] = 0.04916352567134998;
x[132] = 0.06144046901642827;
x[133] = 0.07370812340376778;
x[134] = 0.08596463413198061;
x[135] = 0.09820814818444755;
x[136] = 0.1104368145094688;
x[137] = 0.1226487843001178;
x[138] = 0.1348422112737553;
x[139] = 0.147015251951162;
x[140] = 0.1591660659352477;
x[141] = 0.1712928161892939;
x[142] = 0.1833936693146885;
x[143] = 0.1954667958281108;
x[144] = 0.2075103704381242;
x[145] = 0.2195225723211354;
x[146] = 0.2315015853966777;
x[147] = 0.243445598601978;
x[148] = 0.2553528061657641;
x[149] = 0.2672214078812731;
x[150] = 0.2790496093784178;
x[151] = 0.2908356223950708;
x[152] = 0.3025776650474256;
x[153] = 0.3142739620993925;
x[154] = 0.3259227452309905;
x[155] = 0.3375222533056927;
x[156] = 0.3490707326366864;
x[157] = 0.3605664372520062;
x[158] = 0.3720076291585012;
x[159] = 0.3833925786045958;
x[160] = 0.3947195643418044;
x[161] = 0.4059868738849605;
x[162] = 0.4171928037711214;
x[163] = 0.4283356598171081;
x[164] = 0.4394137573756426;
x[165] = 0.4504254215900437;
x[166] = 0.4613689876474424;
x[167] = 0.4722428010304787;
x[168] = 0.483045217767442;
x[169] = 0.493774604680817;
x[170] = 0.5044293396341982;
x[171] = 0.5150078117775342;
x[172] = 0.5255084217906666;
x[173] = 0.5359295821251249;
x[174] = 0.5462697172441424;
x[175] = 0.5565272638608558;
x[176] = 0.5667006711746527;
x[177] = 0.5767884011056313;
x[178] = 0.5867889285271373;
x[179] = 0.5967007414963417;
x[180] = 0.6065223414828266;
x[181] = 0.6162522435951415;
x[182] = 0.6258889768052999;
x[183] = 0.6354310841711771;
x[184] = 0.6448771230567811;
x[185] = 0.6542256653503588;
x[186] = 0.663475297680307;
x[187] = 0.6726246216288551;
x[188] = 0.6816722539434864;
x[189] = 0.6906168267460676;
x[190] = 0.6994569877396524;
x[191] = 0.7081914004129306;
x[192] = 0.7168187442422908;
x[193] = 0.7253377148914649;
x[194] = 0.7337470244087263;
x[195] = 0.7420454014216102;
x[196] = 0.7502315913291283;
x[197] = 0.7583043564914468;
x[198] = 0.7662624764170006;
x[199] = 0.7741047479470157;
x[200] = 0.7818299854374094;
x[201] = 0.7894370209380444;
x[202] = 0.7969247043693057;
x[203] = 0.8042919036959787;
x[204] = 0.8115375050983958;
x[205] = 0.8186604131408319;
x[206] = 0.8256595509371186;
x[207] = 0.8325338603134556;
x[208] = 0.839282301968391;
x[209] = 0.8459038556299511;
x[210] = 0.8523975202098902;
x[211] = 0.858762313955043;
x[212] = 0.8649972745957512;
x[213] = 0.8711014594913465;
x[214] = 0.8770739457726654;
x[215] = 0.8829138304815741;
x[216] = 0.8886202307074841;
x[217] = 0.8941922837208367;
x[218] = 0.8996291471035368;
x[219] = 0.904929998876315;
x[220] = 0.9100940376230008;
x[221] = 0.915120482611687;
x[222] = 0.9200085739127664;
x[223] = 0.9247575725138244;
x[224] = 0.9293667604313699;
x[225] = 0.9338354408193861;
x[226] = 0.9381629380746873;
x[227] = 0.9423485979390644;
x[228] = 0.9463917875982043;
x[229] = 0.9502918957773683;
x[230] = 0.9540483328338163;
x[231] = 0.957660530845962;
x[232] = 0.9611279436992478;
x[233] = 0.9644500471687263;
x[234] = 0.9676263389983388;
x[235] = 0.9706563389768804;
x[236] = 0.9735395890106436;
x[237] = 0.976275653192736;
x[238] = 0.9788641178690681;
x[239] = 0.9813045917010171;
x[240] = 0.9835967057247763;
x[241] = 0.9857401134074193;
x[242] = 0.9877344906997324;
x[243] = 0.9895795360859201;
x[244] = 0.9912749706303856;
x[245] = 0.9928205380219891;
x[246] = 0.9942160046166302;
x[247] = 0.9954611594800263;
x[248] = 0.9965558144351986;
x[249] = 0.9974998041266158;
x[250] = 0.9982929861369679;
x[251] = 0.9989352412846546;
x[252] = 0.99942647468017;
x[253] = 0.9997666213120006;
x[254] = 0.9999557053175637;
w[ 0] = 0.0001136736199914808;
w[ 1] = 0.0002645938711908564;
w[ 2] = 0.0004156976252681932;
w[ 3] = 0.0005667579456482639;
w[ 4] = 0.0007177364780061286;
w[ 5] = 0.0008686076661194581;
w[ 6] = 0.001019347976427318;
w[ 7] = 0.0011699343729388;
w[ 8] = 0.001320343990022177;
w[ 9] = 0.001470554042778403;
w[10] = 0.001620541799041545;
w[11] = 0.001770284570660304;
w[12] = 0.001919759711713187;
w[13] = 0.002068944619501569;
w[14] = 0.002217816736754017;
w[15] = 0.002366353554396287;
w[16] = 0.00251453261459971;
w[17] = 0.002662331513971696;
w[18] = 0.00280972790682046;
w[19] = 0.002956699508457498;
w[20] = 0.003103224098519095;
w[21] = 0.003249279524294296;
w[22] = 0.003394843704053401;
w[23] = 0.003539894630372244;
w[24] = 0.003684410373449933;
w[25] = 0.003828369084417135;
w[26] = 0.003971748998634907;
w[27] = 0.004114528438981242;
w[28] = 0.004256685819126112;
w[29] = 0.004398199646792759;
w[30] = 0.00453904852700618;
w[31] = 0.004679211165326077;
w[32] = 0.004818666371065699;
w[33] = 0.00495739306049505;
w[34] = 0.005095370260027839;
w[35] = 0.005232577109391968;
w[36] = 0.005368992864783177;
w[37] = 0.005504596902000804;
w[38] = 0.005639368719565862;
w[39] = 0.005773287941820301;
w[40] = 0.005906334322007422;
w[41] = 0.006038487745332765;
w[42] = 0.006169728232005295;
w[43] = 0.006300035940257733;
w[44] = 0.006429391169346602;
w[45] = 0.006557774362530328;
w[46] = 0.006685166110026254;
w[47] = 0.006811547151944815;
w[48] = 0.006936898381201466;
w[49] = 0.007061200846405536;
w[50] = 0.007184435754724984;
w[51] = 0.007306584474728122;
w[52] = 0.007427628539199977;
w[53] = 0.007547549647934514;
w[54] = 0.007666329670501377;
w[55] = 0.007783950648986801;
w[56] = 0.007900394800708624;
w[57] = 0.008015644520904983;
w[58] = 0.008129682385395602;
w[59] = 0.008242491153216323;
w[60] = 0.008354053769225508;
w[61] = 0.008464353366682819;
w[62] = 0.008573373269798925;
w[63] = 0.008681096996256795;
w[64] = 0.008787508259703609;
w[65] = 0.008892590972213036;
w[66] = 0.008996329246717397;
w[67] = 0.009098707399409718;
w[68] = 0.009199709952114802;
w[69] = 0.009299321634629343;
w[70] = 0.009397527387030594;
w[71] = 0.009494312361953241;
w[72] = 0.009589661926834022;
w[73] = 0.009683561666124043;
w[74] = 0.009775997383468165;
w[75] = 0.009866955103851452;
w[76] = 0.009956421075711706;
w[77] = 0.01004438177301882;
w[78] = 0.01013082389731963;
w[79] = 0.01021573437974821;
w[80] = 0.0102991003830022;
w[81] = 0.01038090930328312;
w[82] = 0.01046114877220228;
w[83] = 0.01053980665865038;
w[84] = 0.01061687107063194;
w[85] = 0.01069233035706287;
w[86] = 0.01076617310953212;
w[87] = 0.01083838816402652;
w[88] = 0.01090896460261843;
w[89] = 0.01097789175511656;
w[90] = 0.01104515920067912;
w[91] = 0.01111075676938929;
w[92] = 0.01117467454379268;
w[93] = 0.01123690286039691;
w[94] = 0.01129743231113249;
w[95] = 0.01135625374477508;
w[96] = 0.01141335826832922;
w[97] = 0.01146873724837283;
w[98] = 0.01152238231236217;
w[99] = 0.01157428534989815;
w[100] = 0.01162443851395193;
w[101] = 0.01167283422205182;
w[102] = 0.01171946515742932;
w[103] = 0.01176432427012535;
w[104] = 0.01180740477805627;
w[105] = 0.01184870016803913;
w[106] = 0.01188820419677619;
w[107] = 0.01192591089179929;
w[108] = 0.01196181455237226;
w[109] = 0.01199590975035326;
w[110] = 0.01202819133101508;
w[111] = 0.01205865441382472;
w[112] = 0.01208729439318107;
w[113] = 0.01211410693911137;
w[114] = 0.01213908799792579;
w[115] = 0.01216223379283022;
w[116] = 0.01218354082449738;
w[117] = 0.01220300587159574;
w[118] = 0.01222062599127671;
w[119] = 0.01223639851961942;
w[120] = 0.01225032107203351;
w[121] = 0.01226239154361966;
w[122] = 0.01227260810948789;
w[123] = 0.01228096922503318;
w[124] = 0.01228747362616942;
w[125] = 0.01229212032952021;
w[126] = 0.01229490863256759;
w[127] = 0.01229583811375833;
w[128] = 0.01229490863256759;
w[129] = 0.01229212032952021;
w[130] = 0.01228747362616942;
w[131] = 0.01228096922503318;
w[132] = 0.01227260810948789;
w[133] = 0.01226239154361966;
w[134] = 0.01225032107203351;
w[135] = 0.01223639851961942;
w[136] = 0.01222062599127671;
w[137] = 0.01220300587159574;
w[138] = 0.01218354082449738;
w[139] = 0.01216223379283022;
w[140] = 0.01213908799792579;
w[141] = 0.01211410693911137;
w[142] = 0.01208729439318107;
w[143] = 0.01205865441382472;
w[144] = 0.01202819133101508;
w[145] = 0.01199590975035326;
w[146] = 0.01196181455237226;
w[147] = 0.01192591089179929;
w[148] = 0.01188820419677619;
w[149] = 0.01184870016803913;
w[150] = 0.01180740477805627;
w[151] = 0.01176432427012535;
w[152] = 0.01171946515742932;
w[153] = 0.01167283422205182;
w[154] = 0.01162443851395193;
w[155] = 0.01157428534989815;
w[156] = 0.01152238231236217;
w[157] = 0.01146873724837283;
w[158] = 0.01141335826832922;
w[159] = 0.01135625374477508;
w[160] = 0.01129743231113249;
w[161] = 0.01123690286039691;
w[162] = 0.01117467454379268;
w[163] = 0.01111075676938929;
w[164] = 0.01104515920067912;
w[165] = 0.01097789175511656;
w[166] = 0.01090896460261843;
w[167] = 0.01083838816402652;
w[168] = 0.01076617310953212;
w[169] = 0.01069233035706287;
w[170] = 0.01061687107063194;
w[171] = 0.01053980665865038;
w[172] = 0.01046114877220228;
w[173] = 0.01038090930328312;
w[174] = 0.0102991003830022;
w[175] = 0.01021573437974821;
w[176] = 0.01013082389731963;
w[177] = 0.01004438177301882;
w[178] = 0.009956421075711706;
w[179] = 0.009866955103851452;
w[180] = 0.009775997383468165;
w[181] = 0.009683561666124043;
w[182] = 0.009589661926834022;
w[183] = 0.009494312361953241;
w[184] = 0.009397527387030594;
w[185] = 0.009299321634629343;
w[186] = 0.009199709952114802;
w[187] = 0.009098707399409718;
w[188] = 0.008996329246717397;
w[189] = 0.008892590972213036;
w[190] = 0.008787508259703609;
w[191] = 0.008681096996256795;
w[192] = 0.008573373269798925;
w[193] = 0.008464353366682819;
w[194] = 0.008354053769225508;
w[195] = 0.008242491153216323;
w[196] = 0.008129682385395602;
w[197] = 0.008015644520904983;
w[198] = 0.007900394800708624;
w[199] = 0.007783950648986801;
w[200] = 0.007666329670501377;
w[201] = 0.007547549647934514;
w[202] = 0.007427628539199977;
w[203] = 0.007306584474728122;
w[204] = 0.007184435754724984;
w[205] = 0.007061200846405536;
w[206] = 0.006936898381201466;
w[207] = 0.006811547151944815;
w[208] = 0.006685166110026254;
w[209] = 0.006557774362530328;
w[210] = 0.006429391169346602;
w[211] = 0.006300035940257733;
w[212] = 0.006169728232005295;
w[213] = 0.006038487745332765;
w[214] = 0.005906334322007422;
w[215] = 0.005773287941820301;
w[216] = 0.005639368719565862;
w[217] = 0.005504596902000804;
w[218] = 0.005368992864783177;
w[219] = 0.005232577109391968;
w[220] = 0.005095370260027839;
w[221] = 0.00495739306049505;
w[222] = 0.004818666371065699;
w[223] = 0.004679211165326077;
w[224] = 0.00453904852700618;
w[225] = 0.004398199646792759;
w[226] = 0.004256685819126112;
w[227] = 0.004114528438981242;
w[228] = 0.003971748998634907;
w[229] = 0.003828369084417135;
w[230] = 0.003684410373449933;
w[231] = 0.003539894630372244;
w[232] = 0.003394843704053401;
w[233] = 0.003249279524294296;
w[234] = 0.003103224098519095;
w[235] = 0.002956699508457498;
w[236] = 0.00280972790682046;
w[237] = 0.002662331513971696;
w[238] = 0.00251453261459971;
w[239] = 0.002366353554396287;
w[240] = 0.002217816736754017;
w[241] = 0.002068944619501569;
w[242] = 0.001919759711713187;
w[243] = 0.001770284570660304;
w[244] = 0.001620541799041545;
w[245] = 0.001470554042778403;
w[246] = 0.001320343990022177;
w[247] = 0.0011699343729388;
w[248] = 0.001019347976427318;
w[249] = 0.0008686076661194581;
w[250] = 0.0007177364780061286;
w[251] = 0.0005667579456482639;
w[252] = 0.0004156976252681932;
w[253] = 0.0002645938711908564;
w[254] = 0.0001136736199914808;
}
else
{
cerr << "\n";
cerr << "LEGENDRE_SET - Fatal error!\n";
cerr << " Illegal value of N = " << n << "\n";
cerr << " Legal values are 1 through 33, 63, 64, 65, 127 or 255.\n";
exit ( 1 );
}
return;
}
2D quadrature¶
The 2D quadrature for a unit triangle is implemented in the following code (ref theory: https://people.sc.fsu.edu/~jburkardt/datasets/quadrature_rules_tri/quadrature_rules_tri.html):
void triangle_unit_set ( int rule, double xtab[], double ytab[],
double weight[] )
//****************************************************************************80
//
// Purpose:
//
// TRIANGLE_UNIT_SET sets a quadrature rule in a unit triangle.
//
// Integration region:
//
// Points (X,Y) such that
//
// 0 <= X,
// 0 <= Y, and
// X + Y <= 1.
//
// Graph:
//
// ^
// 1 | *
// | ..
// Y | . .
// | . .
// 0 | *---*
// +------->
// 0 X 1
//
// Licensing:
//
// This code is distributed under the GNU LGPL license.
//
// Modified:
//
// 06 September 2005
//
// Author:
//
// John Burkardt
//
// References:
//
// H R Schwarz,
// Methode der Finiten Elemente,
// Teubner Studienbuecher, 1980.
//
// Strang and Fix,
// An Analysis of the Finite Element Method,
// Prentice Hall, 1973, page 184.
//
// Arthur H Stroud,
// Approximate Calculation of Multiple Integrals,
// Prentice Hall, 1971.
//
// O C Zienkiewicz,
// The Finite Element Method,
// McGraw Hill, Third Edition, 1977, page 201.
//
// Parameters:
//
// Input, int RULE, the index of the rule.
//
// 1, NORDER = 1, precision 1, Zienkiewicz #1.
// 2, NORDER = 3, precision 1, the "vertex rule".
// 3, NORDER = 3, precision 2, Strang and Fix formula #1.
// 4, NORDER = 3, precision 2, Strang and Fix formula #2, Zienkiewicz #2.
// 5, NORDER = 4, precision 3, Strang and Fix formula #3, Zienkiewicz #3.
// 6, NORDER = 6, precision 3, Strang and Fix formula #4.
// 7, NORDER = 6, precision 3, Stroud formula T2:3-1.
// 8, NORDER = 6, precision 4, Strang and Fix formula #5.
// 9, NORDER = 7, precision 4, Strang and Fix formula #6.
// 10, NORDER = 7, precision 5, Strang and Fix formula #7,
// Stroud formula T2:5-1, Zienkiewicz #4, Schwarz Table 2.2.
// 11, NORDER = 9, precision 6, Strang and Fix formula #8.
// 12, NORDER = 12, precision 6, Strang and Fix formula #9.
// 13, ORDER = 13, precision 7, Strang and Fix formula #10.
// Note that there is a typographical error in Strang and Fix
// which lists the value of the XSI(3) component of the
// last generator point as 0.4869... when it should be 0.04869...
// 14, ORDER = 7, precision ?.
// 15, ORDER = 16, precision 7, conical product Gauss, Stroud formula T2:7-1.
// 16, ORDER = 64, precision 15, triangular product Gauss rule.
// 17, ORDER = 19, precision 8, from CUBTRI, ACM TOMS #584.
// 18, ORDER = 19, precision 9, from TRIEX, Lyness and Jespersen.
// 19, ORDER = 28, precision 11, from TRIEX, Lyness and Jespersen.
// 20, ORDER = 37, precision 13, from ACM TOMS #706.
//
// Output, double XTAB[NORDER], YTAB[NORDER], the abscissas.
//
// Output, double WEIGHT[NORDER], the weights of the rule.
//
{
double a;
double b;
double c;
double d;
double e;
double f;
double g;
int i;
int j;
int k;
int order2;
double p;
double q;
double r;
double s;
double t;
double u;
double v;
double w;
double w1;
double w2;
double w3;
double w4;
double w5;
double w6;
double w7;
double w8;
double w9;
double weight1[8];
double weight2[8];
double wx;
double x;
double xtab1[8];
double xtab2[8];
double y;
double z;
//
// 1 point, precision 1.
//
if ( rule == 1 )
{
xtab[0] = 1.0 / 3.0;
ytab[0] = 1.0 / 3.0;
weight[0] = 1.0;
}
//
// 3 points, precision 1, the "vertex rule".
//
else if ( rule == 2 )
{
xtab[0] = 1.0;
xtab[1] = 0.0;
xtab[2] = 0.0;
ytab[0] = 0.0;
ytab[1] = 1.0;
ytab[2] = 0.0;
weight[0] = 1.0 / 3.0;
weight[1] = 1.0 / 3.0;
weight[2] = 1.0 / 3.0;
}
//
// 3 points, precision 2, Strang and Fix formula #1.
//
else if ( rule == 3 )
{
xtab[0] = 4.0 / 6.0;
xtab[1] = 1.0 / 6.0;
xtab[2] = 1.0 / 6.0;
ytab[0] = 1.0 / 6.0;
ytab[1] = 4.0 / 6.0;
ytab[2] = 1.0 / 6.0;
weight[0] = 1.0 / 3.0;
weight[1] = 1.0 / 3.0;
weight[2] = 1.0 / 3.0;
}
//
// 3 points, precision 2, Strang and Fix formula #2.
//
else if ( rule == 4 )
{
xtab[0] = 0.0;
xtab[1] = 1.0 / 2.0;
xtab[2] = 1.0 / 2.0;
ytab[0] = 1.0 / 2.0;
ytab[1] = 0.0;
ytab[2] = 1.0 / 2.0;
weight[0] = 1.0 / 3.0;
weight[1] = 1.0 / 3.0;
weight[2] = 1.0 / 3.0;
}
//
// 4 points, precision 3, Strang and Fix formula #3.
//
else if ( rule == 5 )
{
xtab[0] = 10.0 / 30.0;
xtab[1] = 18.0 / 30.0;
xtab[2] = 6.0 / 30.0;
xtab[3] = 6.0 / 30.0;
ytab[0] = 10.0 / 30.0;
ytab[1] = 6.0 / 30.0;
ytab[2] = 18.0 / 30.0;
ytab[3] = 6.0 / 30.0;
weight[0] = -27.0 / 48.0;
weight[1] = 25.0 / 48.0;
weight[2] = 25.0 / 48.0;
weight[3] = 25.0 / 48.0;
}
//
// 6 points, precision 3, Strang and Fix formula #4.
//
else if ( rule == 6 )
{
xtab[0] = 0.659027622374092;
xtab[1] = 0.659027622374092;
xtab[2] = 0.231933368553031;
xtab[3] = 0.231933368553031;
xtab[4] = 0.109039009072877;
xtab[5] = 0.109039009072877;
ytab[0] = 0.231933368553031;
ytab[1] = 0.109039009072877;
ytab[2] = 0.659027622374092;
ytab[3] = 0.109039009072877;
ytab[4] = 0.659027622374092;
ytab[5] = 0.231933368553031;
weight[0] = 1.0 / 6.0;
weight[1] = 1.0 / 6.0;
weight[2] = 1.0 / 6.0;
weight[3] = 1.0 / 6.0;
weight[4] = 1.0 / 6.0;
weight[5] = 1.0 / 6.0;
}
//
// 6 points, precision 3, Stroud T2:3-1.
//
else if ( rule == 7 )
{
xtab[0] = 0.0;
xtab[1] = 0.5;
xtab[2] = 0.5;
xtab[3] = 2.0 / 3.0;
xtab[4] = 1.0 / 6.0;
xtab[5] = 1.0 / 6.0;
ytab[0] = 0.5;
ytab[1] = 0.0;
ytab[2] = 0.5;
ytab[3] = 1.0 / 6.0;
ytab[4] = 2.0 / 3.0;
ytab[5] = 1.0 / 6.0;
weight[0] = 1.0 / 30.0;
weight[1] = 1.0 / 30.0;
weight[2] = 1.0 / 30.0;
weight[3] = 3.0 / 10.0;
weight[4] = 3.0 / 10.0;
weight[5] = 3.0 / 10.0;
}
//
// 6 points, precision 4, Strang and Fix, formula #5.
//
else if ( rule == 8 )
{
xtab[0] = 0.816847572980459;
xtab[1] = 0.091576213509771;
xtab[2] = 0.091576213509771;
xtab[3] = 0.108103018168070;
xtab[4] = 0.445948490915965;
xtab[5] = 0.445948490915965;
ytab[0] = 0.091576213509771;
ytab[1] = 0.816847572980459;
ytab[2] = 0.091576213509771;
ytab[3] = 0.445948490915965;
ytab[4] = 0.108103018168070;
ytab[5] = 0.445948490915965;
weight[0] = 0.109951743655322;
weight[1] = 0.109951743655322;
weight[2] = 0.109951743655322;
weight[3] = 0.223381589678011;
weight[4] = 0.223381589678011;
weight[5] = 0.223381589678011;
}
//
// 7 points, precision 4, Strang and Fix formula #6.
//
else if ( rule == 9 )
{
xtab[0] = 1.0 / 3.0;
xtab[1] = 0.736712498968435;
xtab[2] = 0.736712498968435;
xtab[3] = 0.237932366472434;
xtab[4] = 0.237932366472434;
xtab[5] = 0.025355134551932;
xtab[6] = 0.025355134551932;
ytab[0] = 1.0 / 3.0;
ytab[1] = 0.237932366472434;
ytab[2] = 0.025355134551932;
ytab[3] = 0.736712498968435;
ytab[4] = 0.025355134551932;
ytab[5] = 0.736712498968435;
ytab[6] = 0.237932366472434;
weight[0] = 0.375;
weight[1] = 0.1041666666666667;
weight[2] = 0.1041666666666667;
weight[3] = 0.1041666666666667;
weight[4] = 0.1041666666666667;
weight[5] = 0.1041666666666667;
weight[6] = 0.1041666666666667;
}
//
// 7 points, precision 5, Strang and Fix formula #7, Stroud T2:5-1
//
else if ( rule == 10 )
{
xtab[0] = 1.0 / 3.0;
xtab[1] = ( 9.0 + 2.0 * sqrt ( 15.0 ) ) / 21.0;
xtab[2] = ( 6.0 - sqrt ( 15.0 ) ) / 21.0;
xtab[3] = ( 6.0 - sqrt ( 15.0 ) ) / 21.0;
xtab[4] = ( 9.0 - 2.0 * sqrt ( 15.0 ) ) / 21.0;
xtab[5] = ( 6.0 + sqrt ( 15.0 ) ) / 21.0;
xtab[6] = ( 6.0 + sqrt ( 15.0 ) ) / 21.0;
ytab[0] = 1.0 / 3.0;
ytab[1] = ( 6.0 - sqrt ( 15.0 ) ) / 21.0;
ytab[2] = ( 9.0 + 2.0 * sqrt ( 15.0 ) ) / 21.0;
ytab[3] = ( 6.0 - sqrt ( 15.0 ) ) / 21.0;
ytab[4] = ( 6.0 + sqrt ( 15.0 ) ) / 21.0;
ytab[5] = ( 9.0 - 2.0 * sqrt ( 15.0 ) ) / 21.0;
ytab[6] = ( 6.0 + sqrt ( 15.0 ) ) / 21.0;
weight[0] = 0.225;
weight[1] = ( 155.0 - sqrt ( 15.0 ) ) / 1200.0;
weight[2] = ( 155.0 - sqrt ( 15.0 ) ) / 1200.0;
weight[3] = ( 155.0 - sqrt ( 15.0 ) ) / 1200.0;
weight[4] = ( 155.0 + sqrt ( 15.0 ) ) / 1200.0;
weight[5] = ( 155.0 + sqrt ( 15.0 ) ) / 1200.0;
weight[6] = ( 155.0 + sqrt ( 15.0 ) ) / 1200.0;
}
//
// 9 points, precision 6, Strang and Fix formula #8.
//
else if ( rule == 11 )
{
xtab[0] = 0.124949503233232;
xtab[1] = 0.437525248383384;
xtab[2] = 0.437525248383384;
xtab[3] = 0.797112651860071;
xtab[4] = 0.797112651860071;
xtab[5] = 0.165409927389841;
xtab[6] = 0.165409927389841;
xtab[7] = 0.037477420750088;
xtab[8] = 0.037477420750088;
ytab[0] = 0.437525248383384;
ytab[1] = 0.124949503233232;
ytab[2] = 0.437525248383384;
ytab[3] = 0.165409927389841;
ytab[4] = 0.037477420750088;
ytab[5] = 0.797112651860071;
ytab[6] = 0.037477420750088;
ytab[7] = 0.797112651860071;
ytab[8] = 0.165409927389841;
weight[0] = 0.205950504760887;
weight[1] = 0.205950504760887;
weight[2] = 0.205950504760887;
weight[3] = 0.063691414286223;
weight[4] = 0.063691414286223;
weight[5] = 0.063691414286223;
weight[6] = 0.063691414286223;
weight[7] = 0.063691414286223;
weight[8] = 0.063691414286223;
}
//
// 12 points, precision 6, Strang and Fix, formula #9.
//
else if ( rule == 12 )
{
xtab[0] = 0.873821971016996;
xtab[1] = 0.063089014491502;
xtab[2] = 0.063089014491502;
xtab[3] = 0.249286745170910;
xtab[4] = 0.501426509658179;
xtab[5] = 0.249286745170910;
xtab[6] = 0.636502499121399;
xtab[7] = 0.636502499121399;
xtab[8] = 0.310352451033785;
xtab[9] = 0.310352451033785;
xtab[10] = 0.053145049844816;
xtab[11] = 0.053145049844816;
ytab[0] = 0.063089014491502;
ytab[1] = 0.873821971016996;
ytab[2] = 0.063089014491502;
ytab[3] = 0.501426509658179;
ytab[4] = 0.249286745170910;
ytab[5] = 0.249286745170910;
ytab[6] = 0.310352451033785;
ytab[7] = 0.053145049844816;
ytab[8] = 0.636502499121399;
ytab[9] = 0.053145049844816;
ytab[10] = 0.636502499121399;
ytab[11] = 0.310352451033785;
weight[0] = 0.050844906370207;
weight[1] = 0.050844906370207;
weight[2] = 0.050844906370207;
weight[3] = 0.116786275726379;
weight[4] = 0.116786275726379;
weight[5] = 0.116786275726379;
weight[6] = 0.082851075618374;
weight[7] = 0.082851075618374;
weight[8] = 0.082851075618374;
weight[9] = 0.082851075618374;
weight[10] = 0.082851075618374;
weight[11] = 0.082851075618374;
}
//
// 13 points, precision 7, Strang and Fix, formula #10.
//
else if ( rule == 13 )
{
xtab[0] = 0.479308067841923;
xtab[1] = 0.260345966079038;
xtab[2] = 0.260345966079038;
xtab[3] = 0.869739794195568;
xtab[4] = 0.065130102902216;
xtab[5] = 0.065130102902216;
xtab[6] = 0.638444188569809;
xtab[7] = 0.638444188569809;
xtab[8] = 0.312865496004875;
xtab[9] = 0.312865496004875;
xtab[10] = 0.048690315425316;
xtab[11] = 0.048690315425316;
xtab[12] = 1.0 / 3.0;
ytab[0] = 0.260345966079038;
ytab[1] = 0.479308067841923;
ytab[2] = 0.260345966079038;
ytab[3] = 0.065130102902216;
ytab[4] = 0.869739794195568;
ytab[5] = 0.065130102902216;
ytab[6] = 0.312865496004875;
ytab[7] = 0.048690315425316;
ytab[8] = 0.638444188569809;
ytab[9] = 0.048690315425316;
ytab[10] = 0.638444188569809;
ytab[11] = 0.312865496004875;
ytab[12] = 1.0 / 3.0;
weight[0] = 0.175615257433204;
weight[1] = 0.175615257433204;
weight[2] = 0.175615257433204;
weight[3] = 0.053347235608839;
weight[4] = 0.053347235608839;
weight[5] = 0.053347235608839;
weight[6] = 0.077113760890257;
weight[7] = 0.077113760890257;
weight[8] = 0.077113760890257;
weight[9] = 0.077113760890257;
weight[10] = 0.077113760890257;
weight[11] = 0.077113760890257;
weight[12] = -0.149570044467670;
}
//
// 7 points, precision ?.
//
else if ( rule == 14 )
{
a = 1.0 / 3.0;
b = 1.0;
c = 0.5;
z = 0.0;
u = 27.0 / 60.0;
v = 3.0 / 60.0;
w = 8.0 / 60.0;
xtab[0] = a;
xtab[1] = b;
xtab[2] = z;
xtab[3] = z;
xtab[4] = z;
xtab[5] = c;
xtab[6] = c;
ytab[0] = a;
ytab[1] = z;
ytab[2] = b;
ytab[3] = z;
ytab[4] = c;
ytab[5] = z;
ytab[6] = c;
weight[0] = u;
weight[1] = v;
weight[2] = v;
weight[3] = v;
weight[4] = w;
weight[5] = w;
weight[6] = w;
weight[7] = w;
}
//
// 16 points.
//
else if ( rule == 15 )
{
//
// Legendre rule of order 4.
//
order2 = 4;
xtab1[0] = -0.861136311594052575223946488893;
xtab1[1] = -0.339981043584856264802665759103;
xtab1[2] = 0.339981043584856264802665759103;
xtab1[3] = 0.861136311594052575223946488893;
weight1[0] = 0.347854845137453857373063949222;
weight1[1] = 0.652145154862546142626936050778;
weight1[2] = 0.652145154862546142626936050778;
weight1[3] = 0.347854845137453857373063949222;
for ( i = 0; i < order2; i++ )
{
xtab1[i] = 0.5 * ( xtab1[i] + 1.0 );
}
weight2[0] = 0.1355069134;
weight2[1] = 0.2034645680;
weight2[2] = 0.1298475476;
weight2[3] = 0.0311809709;
xtab2[0] = 0.0571041961;
xtab2[1] = 0.2768430136;
xtab2[2] = 0.5835904324;
xtab2[3] = 0.8602401357;
k = 0;
for ( i = 0; i < order2; i++ )
{
for ( j = 0; j < order2; j++ )
{
xtab[k] = xtab2[j];
ytab[k] = xtab1[i] * ( 1.0 - xtab2[j] );
weight[k] = weight1[i] * weight2[j];
k = k + 1;
}
}
}
//
// 64 points, precision 15.
//
else if ( rule == 16 )
{
//
// Legendre rule of order 8.
//
order2 = 8;
xtab1[0] = -0.960289856497536231683560868569;
xtab1[1] = -0.796666477413626739591553936476;
xtab1[2] = -0.525532409916328985817739049189;
xtab1[3] = -0.183434642495649804939476142360;
xtab1[4] = 0.183434642495649804939476142360;
xtab1[5] = 0.525532409916328985817739049189;
xtab1[6] = 0.796666477413626739591553936476;
xtab1[7] = 0.960289856497536231683560868569;
weight1[0] = 0.101228536290376259152531354310;
weight1[1] = 0.222381034453374470544355994426;
weight1[2] = 0.313706645877887287337962201987;
weight1[3] = 0.362683783378361982965150449277;
weight1[4] = 0.362683783378361982965150449277;
weight1[5] = 0.313706645877887287337962201987;
weight1[6] = 0.222381034453374470544355994426;
weight1[7] = 0.101228536290376259152531354310;
weight2[0] = 0.00329519144;
weight2[1] = 0.01784290266;
weight2[2] = 0.04543931950;
weight2[3] = 0.07919959949;
weight2[4] = 0.10604735944;
weight2[5] = 0.11250579947;
weight2[6] = 0.09111902364;
weight2[7] = 0.04455080436;
xtab2[0] = 0.04463395529;
xtab2[1] = 0.14436625704;
xtab2[2] = 0.28682475714;
xtab2[3] = 0.45481331520;
xtab2[4] = 0.62806783542;
xtab2[5] = 0.78569152060;
xtab2[6] = 0.90867639210;
xtab2[7] = 0.98222008485;
k = 0;
for ( j = 0; j < order2; j++ )
{
for ( i = 0; i < order2; i++ )
{
xtab[k] = 1.0 - xtab2[j];
ytab[k] = 0.5 * ( 1.0 + xtab1[i] ) * xtab2[j];
weight[k] = weight1[i] * weight2[j];
k = k + 1;
}
}
}
//
// 19 points, precision 8.
//
else if ( rule == 17 )
{
a = 1.0 / 3.0;
b = ( 9.0 + 2.0 * sqrt ( 15.0 ) ) / 21.0;
c = ( 6.0 - sqrt ( 15.0 ) ) / 21.0;
d = ( 9.0 - 2.0 * sqrt ( 15.0 ) ) / 21.0;
e = ( 6.0 + sqrt ( 15.0 ) ) / 21.0;
f = ( 40.0 - 10.0 * sqrt ( 15.0 )
+ 10.0 * sqrt ( 7.0 ) + 2.0 * sqrt ( 105.0 ) ) / 90.0;
g = ( 25.0 + 5.0 * sqrt ( 15.0 )
- 5.0 * sqrt ( 7.0 ) - sqrt ( 105.0 ) ) / 90.0;
p = ( 40.0 + 10.0 * sqrt ( 15.0 )
+ 10.0 * sqrt ( 7.0 ) - 2.0 * sqrt ( 105.0 ) ) / 90.0;
q = ( 25.0 - 5.0 * sqrt ( 15.0 )
- 5.0 * sqrt ( 7.0 ) + sqrt ( 105.0 ) ) / 90.0;
r = ( 40.0 + 10.0 * sqrt ( 7.0 ) ) / 90.0;
s = ( 25.0 + 5.0 * sqrt ( 15.0 ) - 5.0 * sqrt ( 7.0 )
- sqrt ( 105.0 ) ) / 90.0;
t = ( 25.0 - 5.0 * sqrt ( 15.0 ) - 5.0 * sqrt ( 7.0 )
+ sqrt ( 105.0 ) ) / 90.0;
w1 = ( 7137.0 - 1800.0 * sqrt ( 7.0 ) ) / 62720.0;
w2 = -9301697.0 / 4695040.0 - 13517313.0 * sqrt ( 15.0 )
/ 23475200.0 + 764885.0 * sqrt ( 7.0 ) / 939008.0
+ 198763.0 * sqrt ( 105.0 ) / 939008.0;
w2 = w2 / 3.0;
w3 = -9301697.0 / 4695040.0 + 13517313.0 * sqrt ( 15.0 )
/ 23475200.0
+ 764885.0 * sqrt ( 7.0 ) / 939008.0
- 198763.0 * sqrt ( 105.0 ) / 939008.0;
w3 = w3 / 3.0;
w4 = ( 102791225.0 - 23876225.0 * sqrt ( 15.0 )
- 34500875.0 * sqrt ( 7.0 )
+ 9914825.0 * sqrt ( 105.0 ) ) / 59157504.0;
w4 = w4 / 3.0;
w5 = ( 102791225.0 + 23876225.0 * sqrt ( 15.0 )
- 34500875.0 * sqrt ( 7.0 )
- 9914825 * sqrt ( 105.0 ) ) / 59157504.0;
w5 = w5 / 3.0;
w6 = ( 11075.0 - 3500.0 * sqrt ( 7.0 ) ) / 8064.0;
w6 = w6 / 6.0;
xtab[0] = a;
xtab[1] = b;
xtab[2] = c;
xtab[3] = c;
xtab[4] = d;
xtab[5] = e;
xtab[6] = e;
xtab[7] = f;
xtab[8] = g;
xtab[9] = g;
xtab[10] = p;
xtab[11] = q;
xtab[12] = q;
xtab[13] = r;
xtab[14] = r;
xtab[15] = s;
xtab[16] = s;
xtab[17] = t;
xtab[18] = t;
ytab[0] = a;
ytab[1] = c;
ytab[2] = b;
ytab[3] = c;
ytab[4] = e;
ytab[5] = d;
ytab[6] = e;
ytab[7] = g;
ytab[8] = f;
ytab[9] = g;
ytab[10] = q;
ytab[11] = p;
ytab[12] = q;
ytab[13] = s;
ytab[14] = t;
ytab[15] = r;
ytab[16] = t;
ytab[17] = r;
ytab[18] = s;
weight[0] = w1;
weight[1] = w2;
weight[2] = w2;
weight[3] = w2;
weight[4] = w3;
weight[5] = w3;
weight[6] = w3;
weight[7] = w4;
weight[8] = w4;
weight[9] = w4;
weight[10] = w5;
weight[11] = w5;
weight[12] = w5;
weight[13] = w6;
weight[14] = w6;
weight[15] = w6;
weight[16] = w6;
weight[17] = w6;
weight[18] = w6;
}
//
// 19 points, precision 9.
//
else if ( rule == 18 )
{
a = 1.0 / 3.0;
b = 0.02063496160252593;
c = 0.4896825191987370;
d = 0.1258208170141290;
e = 0.4370895914929355;
f = 0.6235929287619356;
g = 0.1882035356190322;
r = 0.9105409732110941;
s = 0.04472951339445297;
t = 0.7411985987844980;
u = 0.03683841205473626;
v = 0.22196288916076574;
w1 = 0.09713579628279610;
w2 = 0.03133470022713983;
w3 = 0.07782754100477543;
w4 = 0.07964773892720910;
w5 = 0.02557767565869810;
w6 = 0.04328353937728940;
xtab[0] = a;
xtab[1] = b;
xtab[2] = c;
xtab[3] = c;
xtab[4] = d;
xtab[5] = e;
xtab[6] = e;
xtab[7] = f;
xtab[8] = g;
xtab[9] = g;
xtab[10] = r;
xtab[11] = s;
xtab[12] = s;
xtab[13] = t;
xtab[14] = t;
xtab[15] = u;
xtab[16] = u;
xtab[17] = v;
xtab[18] = v;
ytab[0] = a;
ytab[1] = c;
ytab[2] = b;
ytab[3] = c;
ytab[4] = e;
ytab[5] = d;
ytab[6] = e;
ytab[7] = g;
ytab[8] = f;
ytab[9] = g;
ytab[10] = s;
ytab[11] = r;
ytab[12] = s;
ytab[13] = u;
ytab[14] = v;
ytab[15] = t;
ytab[16] = v;
ytab[17] = t;
ytab[18] = u;
weight[0] = w1;
weight[1] = w2;
weight[2] = w2;
weight[3] = w2;
weight[4] = w3;
weight[5] = w3;
weight[6] = w3;
weight[7] = w4;
weight[8] = w4;
weight[9] = w4;
weight[10] = w5;
weight[11] = w5;
weight[12] = w5;
weight[13] = w6;
weight[14] = w6;
weight[15] = w6;
weight[16] = w6;
weight[17] = w6;
weight[18] = w6;
}
//
// 28 points, precision 11.
//
else if ( rule == 19 )
{
a = 1.0 / 3.0;
b = 0.9480217181434233;
c = 0.02598914092828833;
d = 0.8114249947041546;
e = 0.09428750264792270;
f = 0.01072644996557060;
g = 0.4946367750172147;
p = 0.5853132347709715;
q = 0.2073433826145142;
r = 0.1221843885990187;
s = 0.4389078057004907;
t = 0.6779376548825902;
u = 0.04484167758913055;
v = 0.27722066752827925;
w = 0.8588702812826364;
x = 0.0;
y = 0.1411297187173636;
w1 = 0.08797730116222190;
w2 = 0.008744311553736190;
w3 = 0.03808157199393533;
w4 = 0.01885544805613125;
w5 = 0.07215969754474100;
w6 = 0.06932913870553720;
w7 = 0.04105631542928860;
w8 = 0.007362383783300573;
xtab[0] = a;
xtab[1] = b;
xtab[2] = c;
xtab[3] = c;
xtab[4] = d;
xtab[5] = e;
xtab[6] = e;
xtab[7] = f;
xtab[8] = g;
xtab[9] = g;
xtab[10] = p;
xtab[11] = q;
xtab[12] = q;
xtab[13] = r;
xtab[14] = s;
xtab[15] = s;
xtab[16] = t;
xtab[17] = t;
xtab[18] = u;
xtab[19] = u;
xtab[20] = v;
xtab[21] = v;
xtab[22] = w;
xtab[23] = w;
xtab[24] = x;
xtab[25] = x;
xtab[26] = y;
xtab[27] = y;
ytab[0] = a;
ytab[1] = c;
ytab[2] = b;
ytab[3] = c;
ytab[4] = e;
ytab[5] = d;
ytab[6] = e;
ytab[7] = g;
ytab[8] = f;
ytab[9] = g;
ytab[10] = q;
ytab[11] = p;
ytab[12] = q;
ytab[13] = s;
ytab[14] = r;
ytab[15] = s;
ytab[16] = u;
ytab[17] = v;
ytab[18] = t;
ytab[19] = v;
ytab[20] = t;
ytab[21] = u;
ytab[22] = x;
ytab[23] = y;
ytab[24] = w;
ytab[25] = y;
ytab[26] = w;
ytab[27] = x;
weight[0] = w1;
weight[1] = w2;
weight[2] = w2;
weight[3] = w2;
weight[4] = w3;
weight[5] = w3;
weight[6] = w3;
weight[7] = w4;
weight[8] = w4;
weight[9] = w4;
weight[10] = w5;
weight[11] = w5;
weight[12] = w5;
weight[13] = w6;
weight[14] = w6;
weight[15] = w6;
weight[16] = w7;
weight[17] = w7;
weight[18] = w7;
weight[19] = w7;
weight[20] = w7;
weight[21] = w7;
weight[22] = w8;
weight[23] = w8;
weight[24] = w8;
weight[25] = w8;
weight[26] = w8;
weight[27] = w8;
}
//
// 37 points, precision 13.
//
else if ( rule == 20 )
{
a = 1.0 / 3.0;
b = 0.950275662924105565450352089520;
c = 0.024862168537947217274823955239;
d = 0.171614914923835347556304795551;
e = 0.414192542538082326221847602214;
f = 0.539412243677190440263092985511;
g = 0.230293878161404779868453507244;
w1 = 0.051739766065744133555179145422;
w2 = 0.008007799555564801597804123460;
w3 = 0.046868898981821644823226732071;
w4 = 0.046590940183976487960361770070;
w5 = 0.031016943313796381407646220131;
w6 = 0.010791612736631273623178240136;
w7 = 0.032195534242431618819414482205;
w8 = 0.015445834210701583817692900053;
w9 = 0.017822989923178661888748319485;
wx = 0.037038683681384627918546472190;
xtab[0] = a;
xtab[1] = b;
xtab[2] = c;
xtab[3] = c;
xtab[4] = d;
xtab[5] = e;
xtab[6] = e;
xtab[7] = f;
xtab[8] = g;
xtab[9] = g;
ytab[0] = a;
ytab[1] = c;
ytab[2] = b;
ytab[3] = c;
ytab[4] = e;
ytab[5] = d;
ytab[6] = e;
ytab[7] = g;
ytab[8] = f;
ytab[9] = g;
weight[0] = w1;
weight[1] = w2;
weight[2] = w2;
weight[3] = w2;
weight[4] = w3;
weight[5] = w3;
weight[6] = w3;
weight[7] = w4;
weight[8] = w4;
weight[9] = w4;
weight[10] = w5;
weight[11] = w5;
weight[12] = w5;
weight[13] = w6;
weight[14] = w6;
weight[15] = w6;
weight[16] = w7;
weight[17] = w7;
weight[18] = w7;
weight[19] = w8;
weight[20] = w8;
weight[21] = w8;
weight[22] = w8;
weight[23] = w8;
weight[24] = w8;
weight[25] = w9;
weight[26] = w9;
weight[27] = w9;
weight[28] = w9;
weight[29] = w9;
weight[30] = w9;
weight[31] = wx;
weight[32] = wx;
weight[33] = wx;
weight[34] = wx;
weight[35] = wx;
weight[36] = wx;
a = 0.772160036676532561750285570113;
b = 0.113919981661733719124857214943;
xtab[10] = a;
ytab[10] = b;
xtab[11] = b;
ytab[11] = a;
xtab[12] = b;
ytab[12] = b;
a = 0.009085399949835353883572964740;
b = 0.495457300025082323058213517632;
xtab[13] = a;
ytab[13] = b;
xtab[14] = b;
ytab[14] = a;
xtab[15] = b;
ytab[15] = b;
a = 0.062277290305886993497083640527;
b = 0.468861354847056503251458179727;
xtab[16] = a;
ytab[16] = b;
xtab[17] = b;
ytab[17] = a;
xtab[18] = b;
ytab[18] = b;
a = 0.022076289653624405142446876931;
b = 0.851306504174348550389457672223;
c = 1.0 - a - b;
xtab[19] = a;
ytab[19] = b;
xtab[20] = a;
ytab[20] = c;
xtab[21] = b;
ytab[21] = a;
xtab[22] = b;
ytab[22] = c;
xtab[23] = c;
ytab[23] = a;
xtab[24] = c;
ytab[24] = b;
a = 0.018620522802520968955913511549;
b = 0.689441970728591295496647976487;
c = 1.0 - a - b;
xtab[25] = a;
ytab[25] = b;
xtab[26] = a;
ytab[26] = c;
xtab[27] = b;
ytab[27] = a;
xtab[28] = b;
ytab[28] = c;
xtab[29] = c;
ytab[29] = a;
xtab[30] = c;
ytab[30] = b;
a = 0.096506481292159228736516560903;
b = 0.635867859433872768286976979827;
c = 1.0 - a - b;
xtab[31] = a;
ytab[31] = b;
xtab[32] = a;
ytab[32] = c;
xtab[33] = b;
ytab[33] = a;
xtab[34] = b;
ytab[34] = c;
xtab[35] = c;
ytab[35] = a;
xtab[36] = c;
ytab[36] = b;
}
else
{
cerr << "\n";
cerr << "TRIANGLE_UNIT_SET - Fatal error!\n";
cerr << " Illegal value of RULE = " << rule << "\n";
exit ( 1 );
}
return;
}
