math.cern

Class Polynomial

java.lang.Object

math.cern.Polynomial

public final class Polynomial
extends Object

Polynomial functions.

Method Summary

Modifier and Type	Method and Description
static double	chbevl(double x, double[] coef, int N) Evaluates the series of Chebyshev polynomials Ti at argument x/2.
static double	evalChebyStar(double[] coef, int N, double x) Evaluates a series of shifted Chebyshev polynomials T_j^* at x over the basic interval [0, 1] using the method of Clenshaw.
static double	p1evl(double x, double[] coef, int N) Evaluates the given polynomial of degree N at x, assuming coefficient of N is 1.0.
static double	polevl(double x, double[] coef, int N) Evaluates the given polynomial of degree N at x.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

p1evl

public static double p1evl(double x,
                           double[] coef,
                           int N)

Evaluates the given polynomial of degree N at x, assuming coefficient of N is 1.0. Otherwise same as polevl().

                     2          N
 y  =  C  + C x + C x  +...+ C x
        0    1     2          N
 
 where C  = 1 and hence is omitted from the array.
        N
 
 Coefficients are stored in reverse order:
 
 coef[0] = C  , ..., coef[N-1] = C  .
            N-1                   0
 
 Calling arguments are otherwise the same as polevl().
 

In the interest of speed, there are no checks for out of bounds arithmetic.

Parameters:

x - argument to the polynomial

coef - the coefficients of the polynomial

N - the degree of the polynomial

Returns:

the value of the polynomial for x

polevl

public static double polevl(double x,
                            double[] coef,
                            int N)

Evaluates the given polynomial of degree N at x.

                     2          N
 y  =  C  + C x + C x  +...+ C x
        0    1     2          N
 
 Coefficients are stored in reverse order:
 
 coef[0] = C  , ..., coef[N] = C  .
            N                   0
 

In the interest of speed, there are no checks for out of bounds arithmetic.

Parameters:

x - argument to the polynomial

coef - the coefficients of the polynomial

N - the degree of the polynomial

Returns:

the value of the polynomial for x

chbevl

public static double chbevl(double x,
                            double[] coef,
                            int N)

Evaluates the series of Chebyshev polynomials T_i at argument x/2. The series is given by

        N-1
         - '
  y  =   >   coef[i] T (x/2)
         -            i
        i=0
 

Coefficients are stored in reverse order, i.e. the zero order term is last in the array. Note: N is the number of coefficients, not the order.

If coefficients are for the interval a to b, x must have been transformed to x -> 2(2x - b - a)/(b-a) before entering the routine. This maps x from (a, b) to (-1, 1), over which the Chebyshev polynomials are defined.

If the coefficients are for the inverted interval, in which (a, b) is mapped to (1/b, 1/a), the transformation required is x -> 2(2ab/x - b - a)/(b-a). If b is infinity, this becomes x -> 4a/x - 1.

SPEED:

Taking advantage of the recurrence properties of the Chebyshev polynomials, the routine requires one more addition per loop than evaluating a nested polynomial of the same degree.

Parameters:

x - argument to the polynomial

coef - the coefficients of the polynomial

N - the number of coefficients

Returns:

the value of the series for x

evalChebyStar

public static double evalChebyStar(double[] coef,
                                   int N,
                                   double x)

Evaluates a series of shifted Chebyshev polynomials T_j^* at x over the basic interval [0, 1] using the method of Clenshaw.

Parameters:

coef - coefficients of the polynomials

N - largest degree of polynomials

x - the parameter of the T_j^* functions

Returns:

the value of a series of Chebyshev polynomials T_j^*