Output of file : POLMISC.C contained in archive : CEPHES22.ZIP
/* square root, sine, and arctangent of polynomial
* See polyn.c for data structures and discussion.
*/

/* Highest degree of polynomial to be handled
* by the polyn.c subroutine package */
#define N 16

/* Taylor series coefficients for various functions
*/
double patan[N+1] = {
0.0, 1.0, 0.0, -1.0/3.0, 0.0,
1.0/5.0, 0.0, -1.0/7.0, 0.0, 1.0/9.0, 0.0, -1.0/11.0,
0.0, 1.0/13.0, 0.0, -1.0/15.0, 0.0 };

double psin[N+1] = {
0.0, 1.0, 0.0, -1.0/6.0, 0.0, 1.0/120.0, 0.0,
-1.0/5040.0, 0.0, 1.0/362880.0, 0.0, -1.0/39916800.0,
0.0, 1.0/6227020800.0, 0.0, -1.0/1.307674368e12, 0.0};

double pcos[N+1] = {
1.0, 0.0, -1.0/2.0, 0.0, 1.0/24.0, 0.0,
-1.0/720.0, 0.0, 1.0/40320.0, 0.0, -1.0/3628800.0, 0.0,
1.0/479001600.0, 0.0, -1.0/8.7179291e10, 0.0, 1.0/2.0922789888e13};

double pasin[N+1] = {
0.0, 1.0, 0.0, 1.0/6.0, 0.0,
3.0/40.0, 0.0, 15.0/336.0, 0.0, 105.0/3456.0, 0.0, 945.0/42240.0,
0.0, 10395.0/599040.0 , 0.0, 135135.0/9676800.0 , 0.0
};

static double polt[N+1];
static double polq[N+1];
static double polu[N+1];

/* Arctangent of the ratio num/den of two polynomials.
*/
polatn( den, num, ans )
double num[], den[], ans[];
{
double a, t;
int i;
double atan2();

/*
* arctan( a + b ) = arctan(a) + arctan( b/(1 + ab + a**2) )
*/
t = num[0];
a = den[0];
if( (t == 0.0) && (a == 0.0 ) )
{
t = num[1];
a = den[1];
}
t = atan2( a, t ); /* arctan(a) */
polclr( polq, N );
i = poldiv( den, N, num, N, polq );
a = polq[0]; /* a */
polq[0] = 0.0; /* b */
polmov( polq, N, polu ); /* b */
/*
* Form the polynomial
* 1 + ab + a**2
* where a is a scalar.
*/
for( i=0; i<=N; i++ )
polu[i] *= a;
polu[0] += 1.0 + a * a;
poldiv( polu, N, polq, N, polt ); /* divide into b */
polsbt( polt, N, patan, N, polu ); /* arctan(b) */
polu[0] += t; /* plus arctan(a) */
}

/* Square root of a polynomial.
* Assumes the lowest degree nonzero term is dominant
* and of even degree. An error message is given
* if the Newton iteration does not converge.
*/
polsqt( pol, ans )
double pol[], ans[];
{
double x[N+1], y[N+1], z[N+1];
double t, u;
int i, j, n0;
double sqrt(), fabs();

polmov( pol, N, x );
polclr( y, N );
/* Initial guess is square root of lowest order
* nonzero term
*/
t = 0.0;
for( i=0; i {
if( x[i] != 0.0 )
goto nzero;
}
polmov( y, N, ans );
return;

nzero:

if( i > 0 )
{
t = x[i];
i /= 2;
y[i] = sqrt( t );
}
else
{
t = x[0];
n0 = 0;
y[0] = 1.0;
for( i=1; i<=N; i++ )
x[i] /= t;
x[0] = 0.0;
/* series development sqrt(1+x) = 1 + x / 2 - x**2 / 8 + x**3 / 16
* assumes first (constant) term is greater than what follows
*/
polmov( x, N, z );
for( i=0; i<=N; i++ )
z[i] *= 0.5;
poladd( z, N, y, N, y );
polmul( x, N, x, N, z );
for( i=0; i<=N; i++ )
z[i] *= 0.25;
polsub( z, N, y, N, y );
polmul( x, N, z, N, z );
for( i=0; i<=N; i++ )
z[i] *= 0.5;
poladd( z, N, y, N, y );
t = sqrt( t );
for( i=0; i<=N; i++ )
y[i] *= t;
}
/* Newton iterations */
for( j=0; j<10; j++ )
{
poldiv( y, N, pol, N, z );
poladd( y, N, z, N, y );
for( i=0; i<=N; i++ )
y[i] *= 0.5;
for( i=0; i<=N; i++ )
{
u = fabs( y[i] - z[i] );
if( u > 1.0e-15 )
goto more;
}
goto done;
more: ;
}
printf( "square root did not converge\n" );
done:
polmov( y, N, ans );
}

/* Sine of a polynomial.
* The computation uses
* sin(a+b) = sin(a) cos(b) + cos(a) sin(b)
* where a is the constant term of the polynomial and
* b is the sum of the rest of the terms.
* Since sin(b) and cos(b) are computed by series expansions,
* the value of b should be small.
*/
polsin( x, y )
double x[], y[];
{
double w[N+1], c[N+1];
double a, sc;
int i;
double sin(), cos();

polmov( x, N, w );
polclr( c, N );
polclr( y, N );
a = w[0];
w[0] = 0.0;
polsbt( w, N, pcos, N, c );
sc = sin(a);
for( i=0; i<=N; i++ )
c[i] *= sc;
polsbt( w, N, psin, N, y );
sc = cos(a);
for( i=0; i<=N; i++ )
y[i] *= sc;
poladd( c, N, y, N, y );
}

### 3 Responses to “Category : C Source CodeArchive   : CEPHES22.ZIPFilename : POLMISC.C”

1. Very nice! Thank you for this wonderful archive. I wonder why I found it only now. Long live the BBS file archives!

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