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// Complete Elliptic Integral of the First Kind
//
// ellint_1 - K(m)
//
// K(m) = ∫₀^(π/2) dθ / √(1 - m*sin²(θ)), computed via AGM
//
// Inputs: 1 (parameter m)
// Outputs: 1 (K(m))
// Constants: 0
#include <math.h>
#include "ts.h"
ts_parameter_t const ts_ellint_1_p = { .n_inputs = 1, .n_outputs = 1, .n_constants = 0, .params_size = 0 };
uint_t ts_ellint_1_offset( ts_constant_t const *c )
{
( void )c;
return ( 0 );
}
int ts_ellint_1(
fp_t * const *y, uint_t y_n,
fp_t const * const *x, uint_t x_n,
uint_t n, ts_constant_t const *c
)
{
TS_PROLOGUE( ts_ellint_1_p );
(void)c;
/* Complete elliptic integral of the first kind K(m) */
for (uint_t i = 0; i < n; ++i)
{
fp_t m = x[0][i];
if (m <= 0.0)
{
y[0][i] = PI / 2.0;
}
else if (m >= 1.0)
{
y[0][i] = INFINITY;
}
else
{
/* Arithmetic-geometric mean method */
fp_t a = 1.0;
fp_t b = sqrt(1.0 - m);
fp_t cn = (a - b) / 2.0;
for (uint_t j = 0; j < 30; ++j)
{
fp_t a_new = (a + b) / 2.0;
fp_t b_new = sqrt(a * b);
fp_t cn_new = (a_new - b_new) / 2.0;
a = a_new;
b = b_new;
cn = cn_new;
if (cn / a < 1e-15) break;
}
y[0][i] = PI / (2.0 * a);
}
}
return 0;
}
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