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/* revers.c
*
* Reversion of power series
*
*
*
* SYNOPSIS:
*
* extern int MAXPOL;
* int n;
* double x[n+1], y[n+1];
*
* polini(n);
* revers( y, x, n );
*
* Note, polini() initializes the polynomial arithmetic subroutines;
* see polyn.c.
*
*
* DESCRIPTION:
*
* If
*
* inf
* - i
* y(x) = > a x
* - i
* i=1
*
* then
*
* inf
* - j
* x(y) = > A y ,
* - j
* j=1
*
* where
* 1
* A = ---
* 1 a
* 1
*
* etc. The coefficients of x(y) are found by expanding
*
* inf inf
* - - i
* x(y) = > A > a x
* - j - i
* j=1 i=1
*
* and setting each coefficient of x , higher than the first,
* to zero.
*
*
*
* RESTRICTIONS:
*
* y[0] must be zero, and y[1] must be nonzero.
*
*/
/*
Cephes Math Library Release 2.8: June, 2000
Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
*/
#include <math.h>
extern int MAXPOL; /* initialized by polini() */
#ifdef ANSIPROT
/* See polyn.c. */
void polmov ( double *, int, double * );
void polclr ( double *, int );
void poladd ( double *, int, double *, int, double * );
void polmul ( double *, int, double *, int, double * );
void * malloc ( long );
void free ( void * );
#else
void polmov(), polclr(), poladd(), polmul();
void * malloc();
void free ();
#endif
void revers( y, x, n)
double y[], x[];
int n;
{
double *yn, *yp, *ysum;
int j;
if( y[1] == 0.0 )
mtherr( "revers", DOMAIN );
/* printf( "revers: y[1] = 0\n" );*/
j = (MAXPOL + 1) * sizeof(double);
yn = (double *)malloc(j);
yp = (double *)malloc(j);
ysum = (double *)malloc(j);
polmov( y, n, yn );
polclr( ysum, n );
x[0] = 0.0;
x[1] = 1.0/y[1];
for( j=2; j<=n; j++ )
{
/* A_(j-1) times the expansion of y^(j-1) */
polmul( &x[j-1], 0, yn, n, yp );
/* The expansion of the sum of A_k y^k up to k=j-1 */
poladd( yp, n, ysum, n, ysum );
/* The expansion of y^j */
polmul( yn, n, y, n, yn );
/* The coefficient A_j to make the sum up to k=j equal to zero */
x[j] = -ysum[j]/yn[j];
}
free(yn);
free(yp);
free(ysum);
}
#if 0
/* Demonstration program
*/
#define N 10
double y[N], x[N];
double fac();
main()
{
double a, odd;
int i;
polini( N-1 );
a = 1.0;
y[0] = 0.0;
odd = 1.0;
for( i=1; i<N; i++ )
{
/* sin(x) */
/*
if( i & 1 )
{
y[i] = odd/fac(i);
odd = -odd;
}
else
y[i] = 0.0;
*/
y[i] = 1.0/fac(i);
}
revers( y, x, N-1 );
for( i=0; i<N; i++ )
printf( "%2d %.10e %.10e\n", i, x[i], y[i] );
}
#endif
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