Polylib.cpp

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#include <stdlib.h>

#include <stdio.h>

#include <math.h>

#include "Polylib.h"

#include <float.h>

#include <complex>

#include <LibUtilities/BasicConst/NektarUnivTypeDefs.hpp>


/// Maximum number of iterations in polynomial defalation routine Jacobz

#define STOP  30 

/// Precision tolerance for two points to be similar 

#define EPS   100*DBL_EPSILON

/// return the sign(b)*a

#define sign(a,b) ((b)<0 ? -fabs(a) : fabs(a))



namespace Polylib {



    /// Define whether to use polynomial deflation (1)  or tridiagonal solver (0).

#define POLYNOMIAL_DEFLATION 0



#ifdef POLYNOMIAL_DEFLATION

    /// zero determination using Newton iteration with polynomial deflation

#define jacobz(n,z,alpha,beta) Jacobz(n,z,alpha,beta)

#else

    /// zero determination using eigenvalues of tridiagaonl matrix 

#define jacobz(n,z,alpha,beta) JacZeros(n,z,alpha,beta)

#endif







    /* local functions */

    static void   Jacobz   (const int n, double *z, const double alpha, 

        const double beta);

   // static void   JacZeros (const int n, double *a, const double alpha, 

    //    const double beta);

    //static void   TriQL    (const int n, double *d, double *e);

    static void TriQL(const int, double *,double *, double **);

    double gammaF (const double);

    static void RecCoeff(const int, double *, double *,const double, 

                  const double);

    void JKMatrix(int, double *, double *);

    void chri1(int,double*,double*,double*,double*,double);



    /**

    \brief  Gauss-Jacobi zeros and weights.



    \li Generate \a np Gauss Jacobi zeros, \a z, and weights,\a w,

    associated with the Jacobi polynomial \f$ P^{\alpha,\beta}_{np}(z)

    \f$,



    \li Exact for polynomials of order \a 2np-1 or less  

    */



    void zwgj (double *z, double *w, const int np, const double alpha, 

        const double beta)

    {

        register int i;

        double fac, one = 1.0, two = 2.0, apb = alpha + beta;



        jacobz (np,z,alpha,beta);

        jacobd (np,z,w,np,alpha,beta);



        fac  = pow(two,apb + one)*gammaF(alpha + np + one)*gammaF(beta + np + one);

        fac /= gammaF(np + one)*gammaF(apb + np + one);



        for(i = 0; i < np; ++i) w[i] = fac/(w[i]*w[i]*(one-z[i]*z[i]));



        return;

    }





    /** 

    \brief  Gauss-Radau-Jacobi zeros and weights with end point at \a z=-1.



    \li Generate \a np Gauss-Radau-Jacobi zeros, \a z, and weights,\a w,

    associated with the  polynomial \f$(1+z) P^{\alpha,\beta+1}_{np-1}(z)

    \f$.



    \li  Exact for polynomials of order \a 2np-2 or less    

    */



    void zwgrjm(double *z, double *w, const int np, const double alpha, 

        const double beta)

    {



        if(np == 1){

            z[0] = 0.0;

            w[0] = 2.0;

        }

        else{

            register int i;

            double fac, one = 1.0, two = 2.0, apb = alpha + beta;



            z[0] = -one;

            jacobz  (np-1,z+1,alpha,beta+1);

            jacobfd (np,z,w,NULL,np-1,alpha,beta);



            fac  = pow(two,apb)*gammaF(alpha + np)*gammaF(beta + np);

            fac /= gammaF(np)*(beta + np)*gammaF(apb + np + 1);



            for(i = 0; i < np; ++i) w[i] = fac*(1-z[i])/(w[i]*w[i]);

            w[0] *= (beta + one);

        }



        return;

    }





    /** 

    \brief  Gauss-Radau-Jacobi zeros and weights with end point at \a z=1





    \li Generate \a np Gauss-Radau-Jacobi zeros, \a z, and weights,\a w,

    associated with the  polynomial \f$(1-z) P^{\alpha+1,\beta}_{np-1}(z)

    \f$.



    \li Exact for polynomials of order \a 2np-2 or less    

    */



    void zwgrjp(double *z, double *w, const int np, const double alpha, 

        const double beta)

    {



        if(np == 1){

            z[0] = 0.0;

            w[0] = 2.0;

        }

        else{

            register int i;

            double fac, one = 1.0, two = 2.0, apb = alpha + beta;



            jacobz  (np-1,z,alpha+1,beta);

            z[np-1] = one;

            jacobfd (np,z,w,NULL,np-1,alpha,beta);



            fac  = pow(two,apb)*gammaF(alpha + np)*gammaF(beta + np);

            fac /= gammaF(np)*(alpha + np)*gammaF(apb + np + 1);



            for(i = 0; i < np; ++i) w[i] = fac*(1+z[i])/(w[i]*w[i]);

            w[np-1] *= (alpha + one);

        }



        return;

    }





    /** 

    \brief  Gauss-Lobatto-Jacobi zeros and weights with end point at \a z=-1,\a 1





    \li Generate \a np Gauss-Lobatto-Jacobi points, \a z, and weights, \a w,

    associated with polynomial \f$ (1-z)(1+z) P^{\alpha+1,\beta+1}_{np-2}(z) \f$

    \li Exact for polynomials of order \a 2np-3 or less

    */



    void zwglj(double *z, double *w, const int np, const double alpha, 

        const double beta)

    {



        if( np == 1 ){

            z[0] = 0.0;

            w[0] = 2.0;

        }

        else{

            register int i;

            double   fac, one = 1.0, apb = alpha + beta, two = 2.0;



            z[0]    = -one;

            z[np-1] =  one;

            jacobz  (np-2,z + 1,alpha + one,beta + one); 

            jacobfd (np,z,w,NULL,np-1,alpha,beta);



            fac  = pow(two,apb + 1)*gammaF(alpha + np)*gammaF(beta + np);

            fac /= (np-1)*gammaF(np)*gammaF(alpha + beta + np + one);



            for(i = 0; i < np; ++i) w[i] = fac/(w[i]*w[i]);

            w[0]    *= (beta  + one);

            w[np-1] *= (alpha + one);

        }



        return;

    }



    /**

    \brief  Gauss-Kronrod-Jacobi zeros and weights.



    \li Generate \a npt=2*np+1 Gauss-Kronrod Jacobi zeros, \a z, and weights,\a w,

    associated with the Jacobi polynomial \f$ P^{\alpha,\beta}_{np}(z)

    \f$,



    \li Exact for polynomials of order \a 3np+1 or less  

    */

    void zwgk(double *z, double *w, const int npt , const double alpha, 

        const double beta)

    {

        

        int np = (npt-1)/2; 

        

        int  i,j;

        

        // number of kronrod points associated with the np gauss rule

        int kpoints = 2*np + 1;



        // Define the number of required recurrence coefficents

        int ncoeffs = (int)floor(3.0*(np+1)/2);



        // Define arrays  for the recurrence coefficients

        // We will use these arrays for the Kronrod results too, hence the 

        // reason for the size of the arrays

        double *a = new double[kpoints];

        double *b = new double[kpoints];



        // Initialize a and b to zero

        for(i = 0; i < kpoints; i++)

        {

            a[i] = 0.0;

            b[i] = 0.0;

        }



        // Call the routine to calculate the recurrence coefficients

        RecCoeff(ncoeffs,a,b,alpha,beta);



        // Call the routine to caluclate the jacobi-Kronrod matrix

        JKMatrix(np,a,b);



        // Set up the identity matrix

        double** zmatrix = new double*[kpoints];

        for(i = 0; i < kpoints; i++)

        {

            zmatrix[i] = new double[kpoints];

            for(j = 0; j < kpoints; j++)

            {

                zmatrix[i][j] = 0.0;

            }

        }

        for(i = 0; i < kpoints; i++)

        {

            zmatrix[i][i] = 1.0;

        }



        // Calculte the points and weights

        TriQL(kpoints, a, b, zmatrix);



        for(i = 0; i < kpoints; i++)

        {

            z[i] = a[i];

            w[i] = b[i];

        }

        



    }



    /**

    \brief  Gauss-Radau-Kronrod-Jacobi zeros and weights.



    \li Generate \a npt=2*np Radau-Kronrod Jacobi zeros, \a z, and weights,\a w,

    associated with the Jacobi polynomial \f$ P^{\alpha,\beta}_{np}(z)

    \f$,

    */

    void zwrk(double* z, double* w, const int npt ,const double alpha, 

              const double beta)

    {

        

        int np = npt/2;



        if(np < 2)

        {

            fprintf(stderr,"too few points in formula\n");

            return;

        }



        double end0 = -1;



        int i,j;



        // number of kronrod points associated with the np gauss rule

        int kpoints = 2*np;



        // Define the number of required recurrence coefficents

        int ncoeffs = (int)ceil(3.0*np/2);



        // Define arrays  for the recurrence coefficients

        double *a = new double[ncoeffs+1];

        double *b = new double[ncoeffs+1];



        // Initialize a and b to zero

        for(i = 0; i < ncoeffs+1; i++)

        {

            a[i] = 0.0;

            b[i] = 0.0;

        }



        // Call the routine to calculate the recurrence coefficients

        RecCoeff(ncoeffs,a,b,alpha,beta);



        double* a0 = new double[ncoeffs];

        double* b0 = new double[ncoeffs];



        chri1(ncoeffs,a,b,a0,b0,end0);



        double s = b0[0]/fabs(b0[0]);

        b0[0] = s*b0[0];



        // Finding the 2*np-1 gauss-kronrod points

        double* z1 = new double[2*np-1];

        double* w1 = new double[2*np-1];

        for(i = 0; i < ncoeffs; i++)

        {

            z1[i] = a0[i];

            w1[i] = b0[i];

        }

        JKMatrix(np-1,z1,w1);

        // Set up the identity matrix

        double** zmatrix = new double*[2*np-1];

        for(i = 0; i < 2*np-1; i++)

        {

            zmatrix[i] = new double[2*np-1];

            for(j = 0; j < 2*np-1; j++)

            {

                zmatrix[i][j] = 0.0;

            }

        }

        for(i = 0; i < 2*np-1; i++)

        {

            zmatrix[i][i] = 1.0;

        }



        // Calculate the points and weights

        TriQL(2*np-1, z1, w1, zmatrix);



        double sumW1 = 0.0;

        for(i = 0; i < 2*np-1; i++)

        {

            w1[i] = s*w1[i]/(z1[i]-end0);

            sumW1 += w1[i];

        }

            

        z[0] = end0;

        w[0] = b[0]- sumW1;

        for(i = 1; i < kpoints; i++)

        {

            z[i] = z1[i-1];

            w[i] = w1[i-1];

        }

        



    }



    /**

    \brief  Gauss-Lobatto-Kronrod-Jacobi zeros and weights.



    \li Generate \a npt=2*np-1 Lobatto-Kronrod Jacobi zeros, \a z, and weights,\a w,

    associated with the Jacobi polynomial \f$ P^{\alpha,\beta}_{np}(z)

    \f$,

    */

    void zwlk(double* z, double* w, const int npt,  

              const double alpha, const double beta)

    {



        int np = (npt+1)/2;

        

        if(np < 4)

        {

            fprintf (stderr,"too few points in formula\n");

            return;

        }

        

        double endl = -1;

        double endr = 1;

        int i,j;



        // number of kronrod points associated with the np gauss rule

        int kpoints = 2*np-1;



        // Define the number of required recurrence coefficents

        int ncoeffs = (int)ceil(3.0*np/2)-1;



        // Define arrays  for the recurrence coefficients

        double *a = new double[ncoeffs+1];

        double *b = new double[ncoeffs+1];



        // Initialize a and b to zero

        for(i = 0; i < ncoeffs+1; i++)

        {

            a[i] = 0.0;

            b[i] = 0.0;

        }



        // Call the routine to calculate the recurrence coefficients

        RecCoeff(ncoeffs,a,b,alpha,beta);





        double* a0 = new double[ncoeffs];

        double* b0 = new double[ncoeffs];



        chri1(ncoeffs,a,b,a0,b0,endl);



        double* a1 = new double[ncoeffs-1];

        double* b1 = new double[ncoeffs-1];



        chri1(ncoeffs-1,a0,b0,a1,b1,endr);





        double s = b1[0]/fabs(b1[0]);

        b1[0] = s*b1[0];



        // Finding the 2*np-1 gauss-kronrod points

        double* z1 = new double[2*np-3];

        double* w1 = new double[2*np-3];

        for(i = 0; i < ncoeffs; i++)

        {

            z1[i] = a1[i];

            w1[i] = b1[i];

        }

        JKMatrix(np-2,z1,w1);

        // Set up the identity matrix

        double** zmatrix = new double*[2*np-3];

        for(i = 0; i < 2*np-3; i++)

        {

            zmatrix[i] = new double[2*np-3];

            for(j = 0; j < 2*np-3; j++)

            {

                zmatrix[i][j] = 0.0;

            }

        }

        for(i = 0; i < 2*np-3; i++)

        {

            zmatrix[i][i] = 1.0;

        }



        // Calculate the points and weights

        TriQL(2*np-3, z1, w1, zmatrix);



        double sumW1 = 0.0;

        double sumW1Z1 = 0.0;

        for(i = 0; i < 2*np-3; i++)

        {

            w1[i] = s*w1[i]/(z1[i]-endl)/(z1[i]-endr);

            sumW1 += w1[i];

            sumW1Z1 += z1[i]*w1[i];

        }



        double c0 = b[0]-sumW1;

        double c1 = a[0]*b[0]-sumW1Z1;

            

        z[0] = endl;

        z[2*np-2] = endr;

        w[0] = (c0*endr-c1)/(endr-endl);

        w[2*np-2] = (c1-c0*endl)/(endr-endl); 



        for(i = 1; i < kpoints-1; i++)

        {

            z[i] = z1[i-1];

            w[i] = w1[i-1];

        }

    }

    

    /** 

    \brief Compute the Derivative Matrix and its transpose associated

    with the Gauss-Jacobi zeros.



    \li Compute the derivative matrix, \a d, and its transpose, \a dt,

    associated with the n_th order Lagrangian interpolants through the

    \a np Gauss-Jacobi points \a z such that \n

    \f$  \frac{du}{dz}(z[i]) =  \sum_{j=0}^{np-1} D[i*np+j] u(z[j]) \f$



    */



    void Dgj(double *D,  const double *z, const int np, const double alpha,

        const double beta)

    {



        double one = 1.0, two = 2.0;



        if (np <= 0){

            D[0] = 0.0;

        }

        else{

            register int i,j; 

            double *pd;



            pd = (double *)malloc(np*sizeof(double));

            jacobd(np,z,pd,np,alpha,beta);



            for (i = 0; i < np; i++){

                for (j = 0; j < np; j++){



                    if (i != j) 

                        D[i*np+j] = pd[j]/(pd[i]*(z[j]-z[i]));

                    else    

                        D[i*np+j] = (alpha - beta + (alpha + beta + two)*z[j])/

                        (two*(one - z[j]*z[j]));

                }

            }

            free(pd);

        }

        return;

    }





    /** 

    \brief Compute the Derivative Matrix and its transpose associated

    with the Gauss-Radau-Jacobi zeros with a zero at \a z=-1.



    \li Compute the derivative matrix, \a d, associated with the n_th

    order Lagrangian interpolants through the \a np Gauss-Radau-Jacobi

    points \a z such that \n \f$ \frac{du}{dz}(z[i]) =

    \sum_{j=0}^{np-1} D[i*np+j] u(z[j]) \f$



    */



    void Dgrjm(double *D, const double *z, const int np, const double alpha, 

        const double beta)

    {



        if (np <= 0){

            D[0] = 0.0;

        }

        else{

            register int i, j; 

            double   one = 1.0, two = 2.0;

            double   *pd;



            pd  = (double *)malloc(np*sizeof(double));



            pd[0] = pow(-one,np-1)*gammaF(np+beta+one);

            pd[0] /= gammaF(np)*gammaF(beta+two);

            jacobd(np-1,z+1,pd+1,np-1,alpha,beta+1);

            for(i = 1; i < np; ++i) pd[i] *= (1+z[i]);



            for (i = 0; i < np; i++)

                for (j = 0; j < np; j++){

                    if (i != j) 

                        D[i*np+j] = pd[j]/(pd[i]*(z[j]-z[i]));

                    else { 

                        if(j == 0)

                            D[i*np+j] = -(np + alpha + beta + one)*(np - one)/

                            (two*(beta + two));

                        else

                            D[i*np+j] = (alpha - beta + one + (alpha + beta + one)*z[j])/

                            (two*(one - z[j]*z[j]));

                    }

                }

                free(pd);

        }



        return;

    }





    /** 

    \brief Compute the Derivative Matrix  associated with the

    Gauss-Radau-Jacobi zeros with a zero at \a z=1.



    \li Compute the derivative matrix, \a d, associated with the n_th

    order Lagrangian interpolants through the \a np Gauss-Radau-Jacobi

    points \a z such that \n \f$ \frac{du}{dz}(z[i]) =

    \sum_{j=0}^{np-1} D[i*np+j] u(z[j]) \f$

    */



    void Dgrjp(double *D, const double *z, const int np, const double alpha, 

        const double beta)

    {



        if (np <= 0){

            D[0] = 0.0;

        }

        else{

            register int i, j; 

            double   one = 1.0, two = 2.0;

            double   *pd;



            pd  = (double *)malloc(np*sizeof(double));





            jacobd(np-1,z,pd,np-1,alpha+1,beta);

            for(i = 0; i < np-1; ++i) pd[i] *= (1-z[i]);

            pd[np-1] = -gammaF(np+alpha+one);

            pd[np-1] /= gammaF(np)*gammaF(alpha+two);



            for (i = 0; i < np; i++)

                for (j = 0; j < np; j++){

                    if (i != j) 

                        D[i*np+j] = pd[j]/(pd[i]*(z[j]-z[i]));

                    else { 

                        if(j == np-1)

                            D[i*np+j] = (np + alpha + beta + one)*(np - one)/

                            (two*(alpha + two));

                        else

                            D[i*np+j] = (alpha - beta - one + (alpha + beta + one)*z[j])/

                            (two*(one - z[j]*z[j]));

                    }

                }

                free(pd);

        }



        return;

    }



    /** 

    \brief Compute the Derivative Matrix associated with the

    Gauss-Lobatto-Jacobi zeros.



    \li Compute the derivative matrix, \a d, associated with the n_th

    order Lagrange interpolants through the \a np

    Gauss-Lobatto-Jacobi points \a z such that \n \f$

    \frac{du}{dz}(z[i]) = \sum_{j=0}^{np-1} D[i*np+j] u(z[j]) \f$



    */



    void Dglj(double *D, const double *z, const int np, const double alpha,

        const double beta)

    {



        if (np <= 0){

            D[0] = 0.0;

        }

        else{

            register int i, j; 

            double   one = 1.0, two = 2.0;

            double   *pd;



            pd  = (double *)malloc(np*sizeof(double));



            pd[0]  = two*pow(-one,np)*gammaF(np + beta);

            pd[0] /= gammaF(np - one)*gammaF(beta + two);

            jacobd(np-2,z+1,pd+1,np-2,alpha+1,beta+1);

            for(i = 1; i < np-1; ++i) pd[i] *= (one-z[i]*z[i]);

            pd[np-1]  = -two*gammaF(np + alpha);

            pd[np-1] /= gammaF(np - one)*gammaF(alpha + two);



            for (i = 0; i < np; i++)

                for (j = 0; j < np; j++){

                    if (i != j) 

                        D[i*np+j] = pd[j]/(pd[i]*(z[j]-z[i]));

                    else { 

                        if (j == 0)

                            D[i*np+j] = (alpha - (np-1)*(np + alpha + beta))/(two*(beta+ two));

                        else if (j == np-1)

                            D[i*np+j] =-(beta - (np-1)*(np + alpha + beta))/(two*(alpha+ two));

                        else

                            D[i*np+j] = (alpha - beta + (alpha + beta)*z[j])/

                            (two*(one - z[j]*z[j]));

                    }

                }

                free(pd);

        }



        return;

    }





    /** 

    \brief Compute the value of the \a i th Lagrangian interpolant through  

    the \a np Gauss-Jacobi points \a zgj at the arbitrary location \a z.     



    \li \f$ -1 \leq z \leq 1 \f$



    \li Uses the defintion of the Lagrangian interpolant:\n

    %

    \f$ \begin{array}{rcl}

    h_j(z) =  \left\{ \begin{array}{ll}

    \displaystyle \frac{P_{np}^{\alpha,\beta}(z)}

    {[P_{np}^{\alpha,\beta}(z_j)]^\prime

    (z-z_j)} & \mbox{if $z \ne z_j$}\\ 

    & \\

    1 & \mbox{if $z=z_j$}

    \end{array}

    \right.

    \end{array}   \f$ 

    */



    double hgj (const int i, const double z, const double *zgj, 

        const int np, const double alpha, const double beta)

    {



        double zi, dz, p, pd, h;



        zi  = *(zgj+i);

        dz  = z - zi;

        if (fabs(dz) < EPS) return 1.0;



        jacobd (1, &zi, &pd , np, alpha, beta);

        jacobfd(1, &z , &p, NULL , np, alpha, beta);

        h = p/(pd*dz);



        return h;

    }



    /** 

    \brief Compute the value of the \a i th Lagrangian interpolant through the

    \a np Gauss-Radau-Jacobi points \a zgrj at the arbitrary location

    \a z. This routine assumes \a zgrj includes the point \a -1.



    \li \f$ -1 \leq z \leq 1 \f$



    \li Uses the defintion of the Lagrangian interpolant:\n

    %

    \f$ \begin{array}{rcl}

    h_j(z) = \left\{ \begin{array}{ll}

    \displaystyle \frac{(1+z) P_{np-1}^{\alpha,\beta+1}(z)}

    {((1+z_j) [P_{np-1}^{\alpha,\beta+1}(z_j)]^\prime +

    P_{np-1}^{\alpha,\beta+1}(z_j) ) (z-z_j)} & \mbox{if $z \ne z_j$}\\ 

    & \\

    1 & \mbox{if $z=z_j$}

    \end{array}

    \right.

    \end{array}   \f$ 

    */



    double hgrjm (const int i, const double z, const double *zgrj, const int np, 

        const double alpha, const double beta)

    {



        double zi, dz, p, pd, h;



        zi  = *(zgrj+i);

        dz  = z - zi;

        if (fabs(dz) < EPS) return 1.0;



        jacobfd (1, &zi, &p , NULL, np-1, alpha, beta + 1);

        // need to use this routine in caes zi = -1 or 1

        jacobd  (1, &zi, &pd, np-1, alpha, beta + 1);

        h = (1.0 + zi)*pd + p;

        jacobfd (1, &z, &p, NULL,  np-1, alpha, beta + 1);

        h = (1.0 + z )*p/(h*dz);



        return h;

    }





    /** 

    \brief Compute the value of the \a i th Lagrangian interpolant through the

    \a np Gauss-Radau-Jacobi points \a zgrj at the arbitrary location

    \a z. This routine assumes \a zgrj includes the point \a +1.



    \li \f$ -1 \leq z \leq 1 \f$



    \li Uses the defintion of the Lagrangian interpolant:\n

    %

    \f$ \begin{array}{rcl}

    h_j(z) = \left\{ \begin{array}{ll}

    \displaystyle \frac{(1-z) P_{np-1}^{\alpha+1,\beta}(z)}

    {((1-z_j) [P_{np-1}^{\alpha+1,\beta}(z_j)]^\prime -

    P_{np-1}^{\alpha+1,\beta}(z_j) ) (z-z_j)} & \mbox{if $z \ne z_j$}\\ 

    & \\

    1 & \mbox{if $z=z_j$}

    \end{array}

    \right.

    \end{array}   \f$ 

    */



    double hgrjp (const int i, const double z, const double *zgrj, const int np, 

        const double alpha, const double beta)

    {



        double zi, dz, p, pd, h;



        zi  = *(zgrj+i);

        dz  = z - zi;

        if (fabs(dz) < EPS) return 1.0;



        jacobfd (1, &zi, &p , NULL, np-1, alpha+1, beta );

        // need to use this routine in caes z = -1 or 1

        jacobd  (1, &zi, &pd, np-1, alpha+1, beta );

        h = (1.0 - zi)*pd - p;

        jacobfd (1, &z, &p, NULL,  np-1, alpha+1, beta);

        h = (1.0 - z )*p/(h*dz);



        return h;

    }





    /** 

    \brief Compute the value of the \a i th Lagrangian interpolant through the

    \a np Gauss-Lobatto-Jacobi points \a zgrj at the arbitrary location

    \a z. 



    \li \f$ -1 \leq z \leq 1 \f$



    \li Uses the defintion of the Lagrangian interpolant:\n

    %

    \f$ \begin{array}{rcl}

    h_j(z) = \left\{ \begin{array}{ll}

    \displaystyle \frac{(1-z^2) P_{np-2}^{\alpha+1,\beta+1}(z)}

    {((1-z^2_j) [P_{np-2}^{\alpha+1,\beta+1}(z_j)]^\prime -

    2 z_j P_{np-2}^{\alpha+1,\beta+1}(z_j) ) (z-z_j)}&\mbox{if $z \ne z_j$}\\ 

    & \\

    1 & \mbox{if $z=z_j$}

    \end{array}

    \right.

    \end{array}   \f$ 

    */



    double hglj (const int i, const double z, const double *zglj, const int np, 

        const double alpha, const double beta)

    {

        double one = 1., two = 2.;

        double zi, dz, p, pd, h;



        zi  = *(zglj+i);

        dz  = z - zi;

        if (fabs(dz) < EPS) return 1.0;



        jacobfd(1, &zi, &p , NULL, np-2, alpha + one, beta + one);

        // need to use this routine in caes z = -1 or 1

        jacobd (1, &zi, &pd, np-2, alpha + one, beta + one);

        h = (one - zi*zi)*pd - two*zi*p;

        jacobfd(1, &z, &p, NULL, np-2, alpha + one, beta + one);

        h = (one - z*z)*p/(h*dz);



        return h;

    }





    /** 

    \brief Interpolation Operator from Gauss-Jacobi points to an

    arbitrary distribution at points \a zm



    \li Computes the one-dimensional interpolation matrix, \a im, to

    interpolate a function from at Gauss-Jacobi distribution of \a nz

    zeros \a zgrj to an arbitrary distribution of \a mz points \a zm, i.e.\n

    \f$ 

    u(zm[i]) = \sum_{j=0}^{nz-1} im[i*nz+j] \ u(zgj[j]) 

    \f$



    */



    void Imgj(double *im, const double *zgj, const double *zm, const int nz, 

        const int mz,const double alpha, const double beta){

            double zp;

            register int i, j;



            for (i = 0; i < nz; ++i) {

                for (j = 0; j < mz; ++j)

                {

                    zp = zm[j];

                    im [i*mz+j] = hgj(i, zp, zgj, nz, alpha, beta);

                }

            }



            return;

    }



    /** 

    \brief Interpolation Operator from Gauss-Radau-Jacobi points

    (including \a z=-1) to an arbitrary distrubtion at points \a zm



    \li Computes the one-dimensional interpolation matrix, \a im, to

    interpolate a function from at Gauss-Radau-Jacobi distribution of

    \a nz zeros \a zgrj (where \a zgrj[0]=-1) to an arbitrary

    distribution of \a mz points \a zm, i.e.

    \n 

    \f$ u(zm[i]) =    \sum_{j=0}^{nz-1} im[i*nz+j] \ u(zgj[j]) \f$



    */



    void Imgrjm(double *im, const double *zgrj, const double *zm, const int nz,

        const int mz, const double alpha, const double beta)

    {

        double zp;

        register int i, j;



        for (i = 0; i < nz; i++) {

            for (j = 0; j < mz; j++)

            {

                zp = zm[j];

                im [i*mz+j] = hgrjm(i, zp, zgrj, nz, alpha, beta);

            }

        }



        return;

    }



    /** 

    \brief Interpolation Operator from Gauss-Radau-Jacobi points

    (including \a z=1) to an arbitrary distrubtion at points \a zm



    \li Computes the one-dimensional interpolation matrix, \a im, to

    interpolate a function from at Gauss-Radau-Jacobi distribution of

    \a nz zeros \a zgrj (where \a zgrj[nz-1]=1) to an arbitrary

    distribution of \a mz points \a zm, i.e.

    \n 

    \f$ u(zm[i]) =    \sum_{j=0}^{nz-1} im[i*nz+j] \ u(zgj[j]) \f$



    */



    void Imgrjp(double *im, const double *zgrj, const double *zm, const int nz, 

        const int mz,const double alpha, const double beta)

    {

            double zp;

            register int i, j;



            for (i = 0; i < nz; i++) {

                for (j = 0; j < mz; j++)

                {

                    zp = zm[j];

                    im [i*mz+j] = hgrjp(i, zp, zgrj, nz, alpha, beta);

                }

            }



            return;

    }





    /** 

    \brief Interpolation Operator from Gauss-Lobatto-Jacobi points

    to an arbitrary distrubtion at points \a zm



    \li Computes the one-dimensional interpolation matrix, \a im, to

    interpolate a function from at Gauss-Lobatto-Jacobi distribution of

    \a nz zeros \a zgrj (where \a zgrj[0]=-1) to an arbitrary

    distribution of \a mz points \a zm, i.e.

    \n 

    \f$ u(zm[i]) =    \sum_{j=0}^{nz-1} im[i*nz+j] \ u(zgj[j]) \f$



    */



    void Imglj(double *im, const double *zglj, const double *zm, const int nz, 

        const int mz, const double alpha, const double beta)

    {

        double zp;

        register int i, j;



        for (i = 0; i < nz; i++) {

            for (j = 0; j < mz; j++)

            {

                zp = zm[j];

                im[i*mz+j] = hglj(i, zp, zglj, nz, alpha, beta);

            }

        }



        return;

    }



    /** 

    \brief Routine to calculate Jacobi polynomials, \f$

    P^{\alpha,\beta}_n(z) \f$, and their first derivative, \f$

    \frac{d}{dz} P^{\alpha,\beta}_n(z) \f$.



    \li This function returns the vectors \a poly_in and \a poly_d

    containing the value of the \f$ n^th \f$ order Jacobi polynomial

    \f$ P^{\alpha,\beta}_n(z) \alpha > -1, \beta > -1 \f$ and its

    derivative at the \a np points in \a z[i]



    - If \a poly_in = NULL then only calculate derivatice



    - If \a polyd   = NULL then only calculate polynomial



    - To calculate the polynomial this routine uses the recursion

    relationship (see appendix A ref [4]) :

    \f$ \begin{array}{rcl}

    P^{\alpha,\beta}_0(z) &=& 1 \\

    P^{\alpha,\beta}_1(z) &=& \frac{1}{2} [ \alpha-\beta+(\alpha+\beta+2)z] \\

    a^1_n P^{\alpha,\beta}_{n+1}(z) &=& (a^2_n + a^3_n z) 

    P^{\alpha,\beta}_n(z) - a^4_n P^{\alpha,\beta}_{n-1}(z) \\

    a^1_n &=& 2(n+1)(n+\alpha + \beta + 1)(2n + \alpha + \beta) \\

    a^2_n &=& (2n + \alpha + \beta + 1)(\alpha^2 - \beta^2)  \\

    a^3_n &=& (2n + \alpha + \beta)(2n + \alpha + \beta + 1)

    (2n + \alpha + \beta + 2)  \\

    a^4_n &=& 2(n+\alpha)(n+\beta)(2n + \alpha + \beta + 2)

    \end{array} \f$



    - To calculate the derivative of the polynomial this routine uses

    the relationship (see appendix A ref [4]) :

    \f$ \begin{array}{rcl}

    b^1_n(z)\frac{d}{dz} P^{\alpha,\beta}_n(z)&=&b^2_n(z)P^{\alpha,\beta}_n(z)

    + b^3_n(z) P^{\alpha,\beta}_{n-1}(z) \hspace{2.2cm} \\

    b^1_n(z) &=& (2n+\alpha + \beta)(1-z^2) \\

    b^2_n(z) &=& n[\alpha - \beta - (2n+\alpha + \beta)z]\\

    b^3_n(z) &=& 2(n+\alpha)(n+\beta) 

    \end{array} \f$



    - Note the derivative from this routine is only valid for -1 < \a z < 1.

    */

    void jacobfd(const int np, const double *z, double *poly_in, double *polyd, 

        const int n, const double alpha, const double beta){

            register int i;

            double  zero = 0.0, one = 1.0, two = 2.0;



            if(!np)

                return;



            if(n == 0){

                if(poly_in)

                    for(i = 0; i < np; ++i) 

                        poly_in[i] = one;

                if(polyd)

                    for(i = 0; i < np; ++i) 

                        polyd[i] = zero; 

            }

            else if (n == 1){

                if(poly_in)

                    for(i = 0; i < np; ++i) 

                        poly_in[i] = 0.5*(alpha - beta + (alpha + beta + two)*z[i]);

                if(polyd)

                    for(i = 0; i < np; ++i) 

                        polyd[i] = 0.5*(alpha + beta + two);

            }

            else{

                register int k;

                double   a1,a2,a3,a4;

                double   two = 2.0, apb = alpha + beta;

                double   *poly, *polyn1,*polyn2;



                if(poly_in){ // switch for case of no poynomial function return

                    polyn1 = (double *)malloc(2*np*sizeof(double));

                    polyn2 = polyn1+np; 

                    poly   = poly_in;

                }

                else{

                    polyn1 = (double *)malloc(3*np*sizeof(double));

                    polyn2 = polyn1+np; 

                    poly   = polyn2+np;      

                }



                for(i = 0; i < np; ++i){

                    polyn2[i] = one;

                    polyn1[i] = 0.5*(alpha - beta + (alpha + beta + two)*z[i]);

                }



                for(k = 2; k <= n; ++k){

                    a1 =  two*k*(k + apb)*(two*k + apb - two);

                    a2 = (two*k + apb - one)*(alpha*alpha - beta*beta);

                    a3 = (two*k + apb - two)*(two*k + apb - one)*(two*k + apb);

                    a4 =  two*(k + alpha - one)*(k + beta - one)*(two*k + apb);



                    a2 /= a1;

                    a3 /= a1;

                    a4 /= a1;



                    for(i = 0; i < np; ++i){

                        poly  [i] = (a2 + a3*z[i])*polyn1[i] - a4*polyn2[i];

                        polyn2[i] = polyn1[i];

                        polyn1[i] = poly  [i];

                    }

                }



                if(polyd){

                    a1 = n*(alpha - beta);

                    a2 = n*(two*n + alpha + beta);

                    a3 = two*(n + alpha)*(n + beta);

                    a4 = (two*n + alpha + beta);

                    a1 /= a4;  a2 /= a4;   a3 /= a4;



                    // note polyn2 points to polyn1 at end of poly iterations

                    for(i = 0; i < np; ++i){

                        polyd[i]  = (a1- a2*z[i])*poly[i] + a3*polyn2[i];

                        polyd[i] /= (one - z[i]*z[i]);

                    }

                }



                free(polyn1);

            }



            return;

    }





    /**

    \brief Calculate the  derivative of Jacobi polynomials 



    \li Generates a vector \a poly of values of the derivative of the

    \a n th order Jacobi polynomial \f$ P^(\alpha,\beta)_n(z)\f$ at the

    \a np points \a z.



    \li To do this we have used the relation 

    \n

    \f$ \frac{d}{dz} P^{\alpha,\beta}_n(z) 

    = \frac{1}{2} (\alpha + \beta + n + 1)  P^{\alpha,\beta}_n(z) \f$



    \li This formulation is valid for \f$ -1 \leq z \leq 1 \f$ 



    */



    void jacobd(const int np, const double *z, double *polyd, const int n, 

        const double alpha, const double beta)

    {

        register int i;

        double one = 1.0;

        if(n == 0)

            for(i = 0; i < np; ++i) polyd[i] = 0.0;

        else{

            //jacobf(np,z,polyd,n-1,alpha+one,beta+one);

            jacobfd(np,z,polyd,NULL,n-1,alpha+one,beta+one);

            for(i = 0; i < np; ++i) polyd[i] *= 0.5*(alpha + beta + (double)n + one);

        }

        return;

    }





    /** 

    \brief Calculate the Gamma function , \f$ \Gamma(n)\f$, for integer

    values and halves.



    Determine the value of \f$\Gamma(n)\f$ using:



    \f$ \Gamma(n) = (n-1)!  \mbox{ or  }  \Gamma(n+1/2) = (n-1/2)\Gamma(n-1/2)\f$



    where \f$ \Gamma(1/2) = \sqrt(\pi)\f$

    */



    double gammaF(const double x){

        double gamma = 1.0;



        if     (x == -0.5) gamma = -2.0*sqrt(M_PI);

        else if (!x) return gamma;

        else if ((x-(int)x) == 0.5){ 

            int n = (int) x;

            double tmp = x;



            gamma = sqrt(M_PI);

            while(n--){

                tmp   -= 1.0;

                gamma *= tmp;

            }

        }

        else if ((x-(int)x) == 0.0){

            int n = (int) x;

            double tmp = x;



            while(--n){

                tmp   -= 1.0;

                gamma *= tmp;

            }

        }  

        else

            fprintf(stderr,"%lf is not of integer or half order\n",x);

        return gamma;

    }



    /** 

    \brief  Calculate the \a n zeros, \a z, of the Jacobi polynomial, i.e.

    \f$ P_n^{\alpha,\beta}(z) = 0 \f$



    This routine is only value for \f$( \alpha > -1, \beta > -1)\f$

    and uses polynomial deflation in a Newton iteration 

    */



    static void Jacobz(const int n, double *z, const double alpha, 

        const double beta){

            register int i,j,k;

            double   dth = M_PI/(2.0*(double)n);

            double   poly,pder,rlast=0.0;

            double   sum,delr,r;

            double one = 1.0, two = 2.0;



            if(!n)

                return;



            for(k = 0; k < n; ++k){

                r = -cos((two*(double)k + one) * dth);

                if(k) r = 0.5*(r + rlast);



                for(j = 1; j < STOP; ++j){

                    jacobfd(1,&r,&poly, &pder, n, alpha, beta);



                    for(i = 0, sum = 0.0; i < k; ++i) sum += one/(r - z[i]);



                    delr = -poly / (pder - sum * poly);

                    r   += delr;

                    if( fabs(delr) < EPS ) break;

                }

                z[k]  = r;

                rlast = r;

            }

            return;

    }





    /**

    \brief Zero and Weight determination through the eigenvalues and eigenvectors of a tridiagonal

    matrix from the three term recurrence relationship.



    Set up a symmetric tridiagonal matrix



    \f$ \left [  \begin{array}{ccccc}

    a[0] & b[0]   &        &        & \\

    b[0] & a[1]   & b[1]   &        & \\

    0   & \ddots & \ddots & \ddots &  \\

    &        & \ddots & \ddots & b[n-2] \\

    &        &        & b[n-2] & a[n-1] \end{array} \right ] \f$



    Where the coefficients a[n], b[n] come from the  recurrence relation



    \f$  b_j p_j(z) = (z - a_j ) p_{j-1}(z) - b_{j-1}   p_{j-2}(z) \f$



    where \f$ j=n+1\f$ and \f$p_j(z)\f$ are the Jacobi (normalized)

    orthogonal polynomials \f$ \alpha,\beta > -1\f$( integer values and

    halves). Since the polynomials are orthonormalized, the tridiagonal

    matrix is guaranteed to be symmetric. The eigenvalues of this

    matrix are the zeros of the Jacobi polynomial.

    */



    void JacZeros(const int n, double *a, double*b, const double alpha, 

        const double beta){

            

            int i,j;

            RecCoeff(n,a,b,alpha,beta);

            

            double **z = new double*[n];

            for(i = 0; i < n; i++)

            {

                z[i] = new double[n];

                for(j = 0; j < n; j++)

                {

                    z[i][j] = 0.0;

                }

            }

            for(i = 0; i < n; i++)

            {

                z[i][i] = 1.0;

            }

                    

            // find eigenvalues and eigenvectors

            TriQL(n, a, b,z);



            return;

    }



    /**

    \brief  The routine finds the recurrence coefficients \a a and

    \a b of the orthogonal polynomials

    */

    static void RecCoeff(const int n, double *a, double *b,const double alpha, 

              const double beta){



        int i;

        double apb, apbi,a2b2;

        

        if(!n)

            return;



        // generate normalised terms 

        apb  = alpha + beta;

        apbi = 2.0 + apb;



        b[0] = pow(2.0,apb+1.0)*gammaF(alpha+1.0)*gammaF(beta+1.0)/gammaF(apbi); //MuZero

        a[0]   = (beta-alpha)/apbi;

        b[1]   = (4.0*(1.0+alpha)*(1.0+beta)/((apbi+1.0)*apbi*apbi));

        

        a2b2 = beta*beta-alpha*alpha;

        

        for(i = 1; i < n-1; i++){

            apbi = 2.0*(i+1) + apb;

            a[i] = a2b2/((apbi-2.0)*apbi);

            b[i+1] = (4.0*(i+1)*(i+1+alpha)*(i+1+beta)*(i+1+apb)/

                 ((apbi*apbi-1)*apbi*apbi));

        }

        

        apbi   = 2.0*n + apb;

        a[n-1] = a2b2/((apbi-2.0)*apbi);

        

    }





    /** \brief QL algorithm for symmetric tridiagonal matrix 



    This subroutine is a translation of an algol procedure,

    num. math. \b 12, 377-383(1968) by martin and wilkinson, as modified

    in num. math. \b 15, 450(1970) by dubrulle.  Handbook for

    auto. comp., vol.ii-linear algebra, 241-248(1971).  This is a

    modified version from numerical recipes.



    This subroutine finds the eigenvalues and first components of the

    eigenvectors of a symmetric tridiagonal matrix by the implicit QL

    method.



    on input:

    - n is the order of the matrix;

    - d contains the diagonal elements of the input matrix;

    - e contains the subdiagonal elements of the input matrix

    in its first n-2 positions. 

    - z is the n by n identity matrix



    on output:



    - d contains the eigenvalues in ascending order.  

    - e contains the weight values - modifications of the first component

      of normalised eigenvectors

    */



    static void TriQL(const int n, double *d,double *e, double **z){

        int m,l,iter,i,k;

        double s,r,p,g,f,dd,c,b;

        

        double MuZero = e[0];

        

        // Renumber the elements of e

        for(i = 0; i < n-1; i++)

        {

            e[i] = sqrt(e[i+1]);

        }

        e[n-1] = 0.0;





        for (l=0;l<n;l++) {

            iter=0;

            do {

                for (m=l;m<n-1;m++) {

                    dd=fabs(d[m])+fabs(d[m+1]);

                    if (fabs(e[m])+dd == dd) break;

                }

                if (m != l) {

                    if (iter++ == STOP){

                        fprintf(stderr,"triQL: Too many iterations in TQLI");

                        exit(1);

                    }

                    g=(d[l+1]-d[l])/(2.0*e[l]);

                    r=sqrt((g*g)+1.0);

                    g=d[m]-d[l]+e[l]/(g+sign(r,g));

                    s=c=1.0;

                    p=0.0;

                    for (i=m-1;i>=l;i--) {

                        f=s*e[i];

                        b=c*e[i];

                        if (fabs(f) >= fabs(g)) {

                            c=g/f;

                            r=sqrt((c*c)+1.0);

                            e[i+1]=f*r;

                            c *= (s=1.0/r);

                        } else {

                            s=f/g;

                            r=sqrt((s*s)+1.0);

                            e[i+1]=g*r;

                            s *= (c=1.0/r);

                        }

                        g=d[i+1]-p;

                        r=(d[i]-g)*s+2.0*c*b;

                        p=s*r;

                        d[i+1]=g+p;

                        g=c*r-b;



                        // Calculate the eigenvectors

                        for(k = 0; k < n; k++)

                        {

                            f = z[k][i+1];

                            z[k][i+1] = s*z[k][i] + c*f;

                            z[k][i] = c*z[k][i] - s*f;

                        }

                    

                    }

                    d[l]=d[l]-p;

                    e[l]=g;

                    e[m]=0.0;

                }

            } while (m != l);

        }



        // order eigenvalues

        // Since we only need the first component of the eigenvectors

        // to calcualte the weight, we only swap the first components

        for(i = 0; i < n-1; ++i){ 

            k = i;

            p = d[i];

            for(l = i+1; l < n; ++l)

                if (d[l] < p) {

                    k = l;

                    p = d[l];

                }

            d[k] = d[i]; 

            d[i] = p;



            double temp = z[0][k];

            z[0][k] = z[0][i];

            z[0][i] = temp;

        }



        // Calculate the weights

        for(i =0 ; i < n; i++)

        {

            e[i] = MuZero*z[0][i]*z[0][i];

        }

    }



    /**

    \brief Calcualtes the Jacobi-kronrod matrix by determining the

    \a a and \b coefficients.

    

    The first \a 3n+1 coefficients are already known



    For more information refer to:

    "Dirk P. Laurie, Calcualtion of Gauss-Kronrod quadrature rules"

    */   

    void JKMatrix(int n, double *a, double *b)

    {

        int i,j,k,m;

        // Working storage

        int size = (int)floor(n/2.0)+2;

        double *s = new double[size];

        double *t = new double[size];



        // Initialize s and t to zero

        for(i = 0; i < size; i++)

        {

            s[i] = 0.0;

            t[i] = 0.0;

        }



        t[1] = b[n+1];

        for(m = 0; m <= n-2; m++)

        {

             double u = 0.0;

             for(k = (int)floor((m+1)/2.0); k >= 0; k--)

             {

                int l = m-k;

                u = u+(a[k+n+1]-a[l])*t[k+1] + b[k+n+1]*s[k] - b[l]*s[k+1];

                s[k+1] = u;

             }



             // Swap the contents of s and t

             double *hold = s;

             s = t;

             t  = hold;

        }

        

        

        for(j = (int)floor(n/2.0); j >= 0; j--)

        {

            s[j+1] = s[j];

        }



        for(m = n-1; m <= 2*n-3; m++)

        {

            double u = 0;

            for(k = m+1-n; k <= floor((m-1)/2.0); k++)

            {

                int l = m-k;

                j = n-1-l;

                u = u-(a[k+n+1]-a[l])*t[j+1] - b[k+n+1]*s[j+1] + b[l]*s[j+2];

                s[j+1] = u;

            }



            if(m%2 == 0)

            {

                k = m/2;

                a[k+n+1] = a[k] + (s[j+1]-b[k+n+1]*s[j+2])/t[j+2];

                

            }else

            {

                k = (m+1)/2;

                b[k+n+1] = s[j+1]/s[j+2];

            }

            

            

            // Swap the contents of s and t

            double  *hold = s;

            s = t;

            t  = hold;

        }



        a[2*n ] = a[n-1]-b[2*n]*s[1]/t[1];

        

    }



    /**

    \brief 

    Given a weight function \f$w(t)\f$ through the first \a n+1

    coefficients \a a and \a b of its orthogonal polynomials

    this routine generates the first \a n recurrence coefficients for the orthogonal

    polynomials relative to the modified weight function \f$(t-z)w(t)\f$.



    The result will be placed in the array \a a0 and \a b0.

    */

    

    void chri1(int n, double* a, double* b, double* a0,

           double* b0,double z)

    {

        

        double q = ceil(3.0*n/2);

        int size = (int)q+1;

        if(size < n+1)

        {

            fprintf(stderr,"input arrays a and b are too short\n");

        }

        double* r = new double[n+1];

        r[0] = z - a[0];

        r[1] = z - a[1] - b[1]/r[0];

        a0[0] = a[1] + r[1] - r[0];

        b0[0] = -r[0]*b[0];



        if(n == 1)

        {

            return;

        }

        int k = 0;

        for(k = 1; k < n; k++)

        {

            r[k+1] = z - a[k+1] - b[k+1]/r[k];

            a0[k] = a[k+1] + r[k+1] - r[k];

            b0[k] = b[k] * r[k]/r[k-1];

        }

            
    }

    /**

    \brief 

    Calcualte the bessel function of the first kind with complex double input y.
    Taken from Numerical Recipies in C 
    
    Returns a complex double
    */


    std::complex<Nektar::NekDouble> ImagBesselComp(int n,std::complex<Nektar::NekDouble> y)
    {
        std::complex<Nektar::NekDouble> z (1.0,0.0);
        std::complex<Nektar::NekDouble> zbes (1.0,0.0);
        std::complex<Nektar::NekDouble> zarg;
        Nektar::NekDouble tol = 1e-15;
        int maxit = 10000;
        int i = 1;

        zarg = -0.25*y*y;
        
        while (abs(z) > tol && i <= maxit){
            z = z*(1.0/i/(i+n)*zarg);
            if  (abs(z) <= tol) break;
            zbes = zbes + z;
            i++;
        }
        zarg = 0.5*y;
        for (i=1;i<=n;i++){
            zbes = zbes*zarg;
        }
        return zbes;

    }



} // end of namespace