//bound integration error due to truncation at u
//Eq. 6, 7, 8 of Davies 1980
REAL8 truncation(qfvars *vars, REAL8 u, REAL8 tausq)
{
REAL8 sum1, sum2, prod1, prod2, prod3, x, y, err1, err2;
INT4 s;
//counter(vars);
(vars->count)++; //Increase counter
sum1 = 0.0; //Calculating N(u) = exp(-2u**2 sum_j(lambda_j**2 delta_j**2/(1+4u**2 lambda_j**2)))
prod2 = 0.0; //Calculating product (i)
prod3 = 0.0; //Calculating product (ii)
s = 0; //Sum of degrees of freedom
sum2 = (vars->sigsq + tausq) * u*u;
prod1 = 2.0 * sum2;
u *= 2.0; //This produces the factor of 4 in front of the products (U*lambda_j)**2 (i and ii) in Davies 1980
for (UINT4 ii=0; ii<vars->weights->length; ii++ ) {
x = (u * vars->weights->data[ii])*(u * vars->weights->data[ii]); //(2*U*lambda_j)**2
sum1 += vars->noncentrality->data[ii] * x / (1.0 + x); //Sum after eq 4 in Davies 1980
if (x > 1.0) {
prod2 += vars->dofs->data[ii] * log(x); //Logarithim of product (ii) produces sum of logorithms
prod3 += vars->dofs->data[ii] * gsl_sf_log_1plusx(x); //Logarithim of product (i) produces sum of logorithms
s += vars->dofs->data[ii]; //sum of degrees of freedom
}
else prod1 += vars->dofs->data[ii] * gsl_sf_log_1plusx(x);
} /* for ii < vars->weights->length */
sum1 *= 0.5; //Remove the extra prefactor of 2 before taking the exponential
prod2 += prod1;
prod3 += prod1;
x = exp1(-sum1 - 0.25*prod2)*LAL_1_PI; //Now remove logarithm by computing exponential (eq 6)
y = exp1(-sum1 - 0.25*prod3)*LAL_1_PI; //Now remove logarithm by computing exponential (eq 8)
if (s==0) err1 = 1.0;
else err1 = 2.0*x/s;
if (prod3>1.0) err2 = 2.5*y; //eq 8
else err2 = 1.0;
if (err2 < err1) err1 = err2;
x = 0.5 * sum2;
if (x<=y) err2 = 1.0;
else err2 = y/x;
if (err1<err2) return err1;
else return err2;
} /* truncation() */
//carry out integration with nterm terms, at stepsize interv. if (! mainx) multiply integrand by 1.0-exp(-0.5*tausq*u^2)
void integrate(qfvars *vars, INT4 nterm, REAL8 interv, REAL8 tausq, INT4 mainx)
{
REAL8 inpi, u, sum1, sum2, sum3, x, y, z;
inpi = interv*LAL_1_PI; //inpi = pi*(k + 1/2)
for (INT4 ii=nterm; ii>=0; ii--) {
u = (ii + 0.5)*interv; //First part of eq 3 in Davies 1980, eq 9 in Davies 1973
sum1 = -2.0*u*vars->c; //Third sum, eq 13 of Davies 1980, the u*c term, will divide by 2 at the end
sum2 = fabs(sum1); //Davies 1980 says that the sine term can be replaced by the sum of abs vals of the arguement
sum3 = -0.5*vars->sigsq * u*u; //First part of eq 13 Davies 1980 in the exponential
for (INT4 jj=vars->weights->length-1; jj>=0; jj--) {
x = 2.0 * vars->weights->data[jj] * u; //2 * lambda_j * u
y = x*x; //4 * lambda_j**2 * u**2
sum3 -= 0.25 * vars->dofs->data[jj] * gsl_sf_log_1plusx(y); //product in eq 13 of Davies 1980
y = vars->noncentrality->data[jj] * x / (1.0 + y); //First sum argument in eq 13 of Davies 1980
z = vars->dofs->data[jj] * atan(x) + y; //Third sum argument in eq 13 of Davies 1980
sum1 += z; //Third sum in eq 13
sum2 += fabs(z);
sum3 -= 0.5 * x * y; //Product
} /* for jj=vars->weights->length-1 --> 0 */
x = inpi * exp1(sum3) / u;
if ( !mainx ) x *= (1.0 - exp1(-0.5 * tausq * u*u)); //For auxillary integration, we multiply by this factor)
sum1 = sin(0.5 * sum1) * x; //Now compute the sine
sum2 *= 0.5*x;
vars->integrationValue += sum1; //integration value
vars->integrationError += sum2; //error on integration
} /* for ii=nterm --> 0 */
} /* integrate() */
static VALUE Log_log_1px(VALUE self, VALUE x) {
return rb_float_new(gsl_sf_log_1plusx(NUM2DBL(x)));
}
