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polylib.c File Reference

#include <stdlib.h>
#include <sys/types.h>
#include <stdio.h>
#include <float.h>
#include <math.h>
#include "polylib.h"

Go to the source code of this file.

Defines
#define	STOP   30
#define	EPS   100*DBL_EPSILON
#define	sign(a, b)   ((b)<0 ? -fabs(a) : fabs(a))
#define	jacobz(n, z, alpha, beta)   JacZeros(n,z,alpha,beta)
	zero determination using eigenvalues of tridiagaonl matrix
Functions
void	Jacobz (int n, double *z, double alpha, double beta)
	Calculate the n zeros, z, of the Jacobi polynomial, i.e. .
void	JacZeros (int n, double *a, double alpha, double beta)
	Zero determination through the eigenvalues of a tridiagonal matrix from teh three term recursion relationship.
void	TriQL (int n, double *d, double *e)
	QL algorithm for symmetric tridiagonal matrix.
double	gammaF (double)
	Calculate the Gamma function , , for integer values and halves.
void	zwgj (double *z, double *w, int np, double alpha, double beta)
	Gauss-Jacobi zeros and weights.
void	zwgrjm (double *z, double *w, int np, double alpha, double beta)
	Gauss-Radau-Jacobi zeros and weights with end point at z=-1.
void	zwgrjp (double *z, double *w, int np, double alpha, double beta)
	Gauss-Radau-Jacobi zeros and weights with end point at z=1.
void	zwglj (double *z, double *w, int np, double alpha, double beta)
	Gauss-Lobatto-Jacobi zeros and weights with end point at z=-1,1.
void	Dgj (double *D, double *Dt, double *z, int np, double alpha, double beta)
	Compute the Derivative Matrix and its transpose associated with the Gauss-Jacobi zeros.
void	Dgrjm (double *D, double *Dt, double *z, int np, double alpha, double beta)
	Compute the Derivative Matrix and its transpose associated with the Gauss-Radau-Jacobi zeros with a zero at z=-1.
void	Dgrjp (double *D, double *Dt, double *z, int np, double alpha, double beta)
	Compute the Derivative Matrix and its transpose associated with the Gauss-Radau-Jacobi zeros with a zero at z=1.
void	Dglj (double *D, double *Dt, double *z, int np, double alpha, double beta)
	Compute the Derivative Matrix and its transpose associated with the Gauss-Lobatto-Jacobi zeros.
double	hgj (int i, double z, double *zgj, int np, double alpha, double beta)
	Compute the value of the i th Lagrangian interpolant through the np Gauss-Jacobi points zgj at the arbitrary location z.
double	hgrjm (int i, double z, double *zgrj, int np, double alpha, double beta)
	Compute the value of the i th Lagrangian interpolant through the np Gauss-Radau-Jacobi points zgrj at the arbitrary location z. This routine assumes zgrj includes the point -1.
double	hgrjp (int i, double z, double *zgrj, int np, double alpha, double beta)
	Compute the value of the i th Lagrangian interpolant through the np Gauss-Radau-Jacobi points zgrj at the arbitrary location z. This routine assumes zgrj includes the point +1.
double	hglj (int i, double z, double *zglj, int np, double alpha, double beta)
	Compute the value of the i th Lagrangian interpolant through the np Gauss-Lobatto-Jacobi points zgrj at the arbitrary location z.
void	Imgj (double *im, double *zgj, double *zm, int nz, int mz, double alpha, double beta)
	Interpolation Operator from Gauss-Jacobi points to an arbitrary distrubtion at points zm.
void	Imgrjm (double *im, double *zgrj, double *zm, int nz, int mz, double alpha, double beta)
	Interpolation Operator from Gauss-Radau-Jacobi points (including z=-1) to an arbitrary distrubtion at points zm.
void	Imgrjp (double *im, double *zgrj, double *zm, int nz, int mz, double alpha, double beta)
	Interpolation Operator from Gauss-Radau-Jacobi points (including z=1) to an arbitrary distrubtion at points zm.
void	Imglj (double *im, double *zglj, double *zm, int nz, int mz, double alpha, double beta)
	Interpolation Operator from Gauss-Lobatto-Jacobi points to an arbitrary distrubtion at points zm.
void	jacobfd (int np, double *z, double *poly_in, double *polyd, int n, double alpha, double beta)
	Routine to calculate Jacobi polynomials, , and their first derivative, .
void	jacobd (int np, double *z, double *polyd, int n, double alpha, double beta)
	Calculate the derivative of Jacobi polynomials.

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Define Documentation

#define EPS   100*DBL_EPSILON

Definition at line 9 of file polylib.c. Referenced by hgj(), hglj(), hgrjm(), hgrjp(), and Jacobz().

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

zero determination using eigenvalues of tridiagaonl matrix Definition at line 164 of file polylib.c. Referenced by zwgj(), zwglj(), zwgrjm(), and zwgrjp().

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

Definition at line 10 of file polylib.c. Referenced by TriQL().

#define STOP   30

Definition at line 8 of file polylib.c. Referenced by Jacobz(), and TriQL().

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Function Documentation

void Dgj (  double *  D, double *  Dt, double *  z, int  np, double  alpha, double  beta )

Compute the Derivative Matrix and its transpose associated with the Gauss-Jacobi zeros. Compute the derivative matrix, d, and its transpose, dt, associated with the n_th order Lagrangian interpolants through the np Gauss-Jacobi points z such that d and dt are both square matrices. Definition at line 321 of file polylib.c. References jacobd().

void Dglj (  double *  D, double *  Dt, double *  z, int  np, double  alpha, double  beta )

Compute the Derivative Matrix and its transpose associated with the Gauss-Lobatto-Jacobi zeros. Compute the derivative matrix, d, and its transpose, dt, associated with the n_th order Lagrangian interpolants through the np Gauss-Lobatto-Jacobi points z such that d and dt are both square matrices. Definition at line 469 of file polylib.c. References gammaF(), and jacobd().

void Dgrjm (  double *  D, double *  Dt, double *  z, int  np, double  alpha, double  beta )

Compute the Derivative Matrix and its transpose associated with the Gauss-Radau-Jacobi zeros with a zero at z=-1. Compute the derivative matrix, d, and its transpose, dt, associated with the n_th order Lagrangian interpolants through the np Gauss-Radau-Jacobi points z such that d and dt are both square matrices. Definition at line 365 of file polylib.c. References gammaF(), and jacobd().

void Dgrjp (  double *  D, double *  Dt, double *  z, int  np, double  alpha, double  beta )

Compute the Derivative Matrix and its transpose associated with the Gauss-Radau-Jacobi zeros with a zero at z=1. Compute the derivative matrix, d, and its transpose, dt, associated with the n_th order Lagrangian interpolants through the np Gauss-Radau-Jacobi points z such that d and dt are both square matrices. Definition at line 417 of file polylib.c. References gammaF(), and jacobd().

double gammaF (  double  x  )  [static]

Calculate the Gamma function , , for integer values and halves. Determine the value of using: where Definition at line 947 of file polylib.c. Referenced by Dglj(), Dgrjm(), Dgrjp(), JacZeros(), zwgj(), zwglj(), zwgrjm(), and zwgrjp().

double hgj (  int  i, double  z, double *  zgj, int  np, double  alpha, double  beta )

Compute the value of the i th Lagrangian interpolant through the np Gauss-Jacobi points zgj at the arbitrary location z. Uses the defintion of the Lagrangian interpolant: Definition at line 532 of file polylib.c. References EPS, jacobd(), and jacobfd(). Referenced by Imgj().

double hglj (  int  i, double  z, double *  zglj, int  np, double  alpha, double  beta )

Compute the value of the i th Lagrangian interpolant through the np Gauss-Lobatto-Jacobi points zgrj at the arbitrary location z. Uses the defintion of the Lagrangian interpolant: Definition at line 651 of file polylib.c. References EPS, jacobd(), and jacobfd(). Referenced by Imglj().

double hgrjm (  int  i, double  z, double *  zgrj, int  np, double  alpha, double  beta )

Compute the value of the i th Lagrangian interpolant through the np Gauss-Radau-Jacobi points zgrj at the arbitrary location z. This routine assumes zgrj includes the point -1. Uses the defintion of the Lagrangian interpolant: Definition at line 569 of file polylib.c. References EPS, jacobd(), and jacobfd(). Referenced by Imgrjm().

double hgrjp (  int  i, double  z, double *  zgrj, int  np, double  alpha, double  beta )

Compute the value of the i th Lagrangian interpolant through the np Gauss-Radau-Jacobi points zgrj at the arbitrary location z. This routine assumes zgrj includes the point +1. Uses the defintion of the Lagrangian interpolant: Definition at line 610 of file polylib.c. References EPS, jacobd(), and jacobfd(). Referenced by Imgrjp().

void Imgj (  double *  im, double *  zgj, double *  zm, int  nz, int  mz, double  alpha, double  beta )

Interpolation Operator from Gauss-Jacobi points to an arbitrary distrubtion at points zm. Computes the one-dimensional interpolation matrix, im, to interpolate a function from at Gauss-Jacobi distribution of nz zeros zgrj to an arbitrary distribution of mz points zm, i.e. Definition at line 684 of file polylib.c. References hgj().

void Imglj (  double *  im, double *  zglj, double *  zm, int  nz, int  mz, double  alpha, double  beta )

Interpolation Operator from Gauss-Lobatto-Jacobi points to an arbitrary distrubtion at points zm. Computes the one-dimensional interpolation matrix, im, to interpolate a function from at Gauss-Lobatto-Jacobi distribution of nz zeros zgrj (where zgrj[0]=-1) to an arbitrary distribution of mz points zm, i.e. Definition at line 766 of file polylib.c. References hglj().

void Imgrjm (  double *  im, double *  zgrj, double *  zm, int  nz, int  mz, double  alpha, double  beta )

Interpolation Operator from Gauss-Radau-Jacobi points (including z=-1) to an arbitrary distrubtion at points zm. Computes the one-dimensional interpolation matrix, im, to interpolate a function from at Gauss-Radau-Jacobi distribution of nz zeros zgrj (where zgrj[0]=-1) to an arbitrary distribution of mz points zm, i.e. Definition at line 711 of file polylib.c. References hgrjm().

void Imgrjp (  double *  im, double *  zgrj, double *  zm, int  nz, int  mz, double  alpha, double  beta )

Interpolation Operator from Gauss-Radau-Jacobi points (including z=1) to an arbitrary distrubtion at points zm. Computes the one-dimensional interpolation matrix, im, to interpolate a function from at Gauss-Radau-Jacobi distribution of nz zeros zgrj (where zgrj[nz-1]=1) to an arbitrary distribution of mz points zm, i.e. Definition at line 738 of file polylib.c. References hgrjp().

void jacobd (  int  np, double *  z, double *  polyd, int  n, double  alpha, double  beta )

Calculate the derivative of Jacobi polynomials. Generates a vector poly of values of the derivative of the n th order Jacobi polynomial at the np points z. To do this we have used the relation This formulation is valid for Definition at line 921 of file polylib.c. References jacobfd(). Referenced by Dgj(), Dglj(), Dgrjm(), Dgrjp(), hgj(), hglj(), hgrjm(), hgrjp(), and zwgj().

void jacobfd (  int  np, double *  z, double *  poly_in, double *  polyd, int  n, double  alpha, double  beta )

Routine to calculate Jacobi polynomials, , and their first derivative, . This function returns the vectors poly_in and poly_d containing the value of the order Jacobi polynomial and its derivative at the np points in z[i] If poly_in = NULL then only calculate derivatice If polyd = NULL then only calculate polynomial To calculate the polynomial this routine uses the recursion relationship (see appendix A ref [4]) : To calculate the derivative of the polynomial this routine uses the relationship (see appendix A ref [4]) : Note the derivative from this routine is only valid for -1 < z < 1. Definition at line 821 of file polylib.c. Referenced by hgj(), hglj(), hgrjm(), hgrjp(), jacobd(), Jacobz(), zwglj(), zwgrjm(), and zwgrjp().

void Jacobz (  int  n, double *  z, double  alpha, double  beta )  [static]

Calculate the n zeros, z, of the Jacobi polynomial, i.e. . This routine is only value for and uses polynomial deflation in a Newton iteration Definition at line 984 of file polylib.c. References EPS, jacobfd(), and STOP.

void JacZeros (  int  n, double *  a, double  alpha, double  beta )  [static]

Zero determination through the eigenvalues of a tridiagonal matrix from teh three term recursion relationship. Set up a symmetric tridiagonal matrix Where the coefficients a[n], b[n] come from the recurrence relation where and are the Jacobi (normalized) orthogonal polynomials ( 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. Definition at line 1038 of file polylib.c. References gammaF(), and TriQL().

void TriQL (  int  n, double *  d, double *  e )  [static]

QL algorithm for symmetric tridiagonal matrix. This subroutine is a translation of an algol procedure, num. math. 12, 377-383(1968) by martin and wilkinson, as modified in num. math. 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-1 positions. e(n) is arbitrary; on output: d contains the eigenvalues in ascending order. e has been destroyed; Definition at line 1096 of file polylib.c. References sign, and STOP. Referenced by JacZeros().

void zwgj (  double *  z, double *  w, int  np, double  alpha, double  beta )

Gauss-Jacobi zeros and weights. Generate np Gauss Jacobi zeros, z, and weights,w, associated with the Jacobi polynomial , Exact for polynomials of order 2np-1 or less Definition at line 186 of file polylib.c. References gammaF(), jacobd(), and jacobz.

void zwglj (  double *  z, double *  w, int  np, double  alpha, double  beta )

Gauss-Lobatto-Jacobi zeros and weights with end point at z=-1,1. Generate np Gauss-Lobatto-Jacobi points, z, and weights, w, associated with polynomial Exact for polynomials of order 2np-3 or less Definition at line 282 of file polylib.c. References gammaF(), jacobfd(), and jacobz.

void zwgrjm (  double *  z, double *  w, int  np, double  alpha, double  beta )

Gauss-Radau-Jacobi zeros and weights with end point at z=-1. Generate np Gauss-Radau-Jacobi zeros, z, and weights,w, associated with the polynomial . Exact for polynomials of order 2np-2 or less Definition at line 212 of file polylib.c. References gammaF(), jacobfd(), and jacobz.

void zwgrjp (  double *  z, double *  w, int  np, double  alpha, double  beta )

Gauss-Radau-Jacobi zeros and weights with end point at z=1. Generate np Gauss-Radau-Jacobi zeros, z, and weights,w, associated with the polynomial . Exact for polynomials of order 2np-2 or less Definition at line 248 of file polylib.c. References gammaF(), jacobfd(), and jacobz.

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