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HeunC.impl.hpp
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HeunC.impl.hpp
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#if !defined(HEUNC_HPP_)
#error "This file should only be included through HeunC.hpp"
#endif
#ifndef HEUNC_IMPL_HPP_
#define HEUNC_IMPL_HPP_
// confluent Heun function, the first local solution of the equation
// HeunC''(z)+(gamma/z+delta/(z-1)+epsilon)*HeunC'(z)+(alpha*z-q)/(z*(z-1))*HeunC(z) = 0
// at z=0 such that HeunC(0)=1 and
// HeunC'(0)=-q/gamma when gamma is not equal to 0
// HeunC'(z)/log(z) -> -q as z->0 when gamma = 0
//
// computed by a consequence of power expansions with improvements near points z=1 and z=infty
//
// it is assumed that z does not belong to the branch-cut [1,infty)
//
// Usage:
// [val,dval,err,numb,wrnmsg,valwoexp,dvalwoexp,errwoexp] = HeunC(q,alpha,gamma,delta,epsilon,z)
//
// Returned parameters:
// val is the value of the Heun function
// dval is the value of z-derivative of the Heun function
// err is the estimated error
// numb is the number of power series terms needed for the evaluation
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 26 January 2018
//
inline HeunCvars HeunC::compute(std::complex<double> alpha_, std::complex<double> beta_, std::complex<double> gamma_,
std::complex<double> delta_, std::complex<double> eta_, double z)
{
HeunCvars result;
HeunCparams p;
p.q = 0.5*(alpha_*(1 + beta_) - beta_*(1+gamma_) - 2*eta_ - gamma_);
p.alpha = 0.5*alpha_*(2 + beta_ + gamma_) + delta_;
p.gamma = beta_ + 1;
p.delta = gamma_ + 1;
p.epsilon = alpha_;
if (z>=1){
throw std::invalid_argument("HeunC0: z belongs to the branch-cut [1,infty)");
}
else {
findR();
if ( (std::abs(p.epsilon)>1/2)&& (std::abs(p.q) < 2.5) && (std::abs(z)>Heun_proxcoinf_rel*R/(std::abs(eps)+std::abs(p.epsilon))) ) {
std::pair<HeunCvars, HeunCvars> vars1_vars = HeunCfaraway(p,z);
result = vars1_vars.second;
}
else {
result = HeunC0(p,z);
}
return result;
}
}
inline HeunCvars HeunC::compute_s(std::complex<double> alpha_, std::complex<double> beta_, std::complex<double> gamma_,
std::complex<double> delta_, std::complex<double> eta_, double z)
{
HeunCvars result;
HeunCparams p;
p.q = 0.5*(alpha_*(1 + beta_) - beta_*(1+gamma_) - 2*eta_ - gamma_);
p.alpha = 0.5*alpha_*(2 + beta_ + gamma_) + delta_;
p.gamma = beta_ + 1;
p.delta = gamma_ + 1;
p.epsilon = alpha_;
if (z>=1) {
throw std::invalid_argument("HeunCfaraway: z belongs to the branch-cut [1,infty)");
}
else {
findR();
if (( std::abs(p.epsilon)>1/2 )&&( std::abs(p.q)<2.5)&&(std::abs(z)>Heun_proxcoinf_rel*R/(std::abs(eps)+std::abs(p.epsilon)) )) {
std::pair<HeunCvars, HeunCvars> vars1_vars = HeunCfaraway(p,z);
result = vars1_vars.second;
}
else {
result = HeunCs0(p,z);
}
return result;
}
}
/* This computes the first local solution around z = 0
*/
// confluent Heun function, the first local solution of the ep.quation
// |z| should not exceed the convergency radius 1
//
// aux is an optional parameter, it only works for p.gamma = 0, -1, -2, ...
// if aux = "yes" then the function is computed as a combination with the second
// solution so as to satisfy
// HeunC00(p.q,p.alpha,p.gamma,p.delta,varp.epsilon;z)=
// exp(-\varp.epsilon z)[HeunC00(p.q-p.epsilon*p.gamma,p.alpha-varp.epsilon(p.gamma+p.delta),p.gamma,p.delta,-varp.epsilon;z)+
// + A * HeunCs00(p.q-p.epsilon*p.gamma,p.alpha-varp.epsilon(p.gamma+p.delta),p.gamma,p.delta,-varp.epsilon;z)]
//
// Returned parameters:
// val is the value of the Heun function
// dval is the value of z-derivative of the Heun function
// err is the estimated error
// numb is the number of power series terms needed for the evaluation
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 15 February 2018
//
inline HeunCvars HeunC::HeunC0(HeunCparams& p, double& z, bool aux){
HeunCvars result;
if (z>=1){
throw std::invalid_argument("HeunC0: z belongs to the branch-cut [1,infty)");
}
else {
bool expgrow = false;
HeunCparams p1 = p;
if (aux==false) {
expgrow = std::real(-p.epsilon*z)>0;
if (expgrow) {
p1.q = p.q - p.epsilon*p.gamma;
p1.alpha = p.alpha - p.epsilon * (p.gamma+p.delta);
p1.epsilon = -p.epsilon;
}
}
if (std::abs(z)<Heun_cont_coef) {
result = HeunC00(p1,z,aux);
}
else {
double z0 = Heun_cont_coef*z/std::abs(z);
HeunCvars result0 = HeunC00(p1,z0,aux);
HeunCvars result1 = HeunCconnect(p1,z,z0,result0.val,result0.dval,R);
result.numb = result0.numb + result1.numb;
result.err = result0.err + result1.err;
result.val = result1.val;
result.dval = result1.dval;
}
if (expgrow) {
result.val = result.val * exp(p1.epsilon*z);
result.dval = (p1.epsilon * result.val + result.dval) * exp(p1.epsilon*z);
result.err = result.err * std::abs(exp(p1.epsilon*z));
}
}
return result;
}
inline HeunCvars HeunC::HeunC00(HeunCparams& p, double& z, bool aux)
{
// define the result
HeunCvars result;
// check you are in the convergence range
if (std::abs(z)>=1){
throw std::invalid_argument("HeunC00: z is out of the convergence radius = 1");
}
else{
bool gamma_is_negative_integer = (std::imag(p.gamma)==0) && std::abs(std::ceil(std::real(p.gamma)-5*eps)+std::abs(p.gamma))<5*eps;
if (gamma_is_negative_integer) {
result = HeunC00log(p, z);
if (aux) {
std::complex<double> co = findcoef4HeunCs(p);
HeunCvars result_s = HeunCs00(p, z);
result.val = result.val - co * result_s.val;
result.dval = result.dval - co * result_s.dval;
result.err = result.err + std::abs(co) * result_s.err;
result.numb = result.numb + result_s.numb;
}
}
else{
result = HeunC00gen(p, z);
}
}
return result;
}
// confluent Heun function expansion for |z| < 1, gamma is not equal to 0, -1, -2, ...
inline HeunCvars HeunC::HeunC00gen(HeunCparams& p, double& z)
{
HeunCvars result;
if (z==0) {
result.val = 1;
result.dval = -p.q/p.gamma;
result.err = 0;
result.numb = 1;
}
else {
// recursion relation variables
std::complex<double> ckm2 = 1;
std::complex<double> ckm1 = -z*p.q/p.gamma;
result.val = ckm2+ckm1;
std::complex<double> vm1 = result.val;
std::complex<double> vm2;
result.dval = -p.q/p.gamma;
std::complex<double> dm1 = result.dval;
std::complex<double> dm2;
std::complex<double> ddval = 0;
int k = 2;
std::complex<double> ckm0 = 1;
// perform recursion
while ( (k<=Heun_klimit) && ( (vm2 != vm1) || (dm2 != dm1) || (std::abs(ckm0)>eps) ) ) {
ckm0 = (ckm1*z*(-p.q+(k-1)*(p.gamma-p.epsilon+p.delta+k-2)) + ckm2*(z*z)*((k-2)*p.epsilon+p.alpha))/(k*(p.gamma+k-1));
result.val = result.val + ckm0;
result.dval = dm1 + k*ckm0/z;
ddval = ddval + k*(k-1)*ckm0/(z*z);
ckm2 = ckm1;
ckm1 = ckm0;
vm2 = vm1;
vm1 = result.val;
dm2 = dm1;
dm1 = result.dval;
k += 1;
}
result.numb = k-1;
if ( std::isinf(std::abs(result.val)) || std::isinf(std::abs(result.dval)) || std::isnan(std::abs(result.val)) || std::isnan(std::abs(result.dval)) ) {
throw std::runtime_error("HeunC00: failed convergence of recurrence and summation");
}
else {
double err1;
std::complex<double> val2;
if (p.q-p.alpha*z != 0.0) {
val2 = ( z*(z-1)*ddval+(p.gamma*(z-1)+p.delta*z+p.epsilon*z*(z-1))*result.dval ) / (p.q-p.alpha*z);
err1 = std::abs(result.val-val2);
}
else {
err1 = INFINITY;
}
if (std::abs(p.q-p.alpha*z)<0.01) {
double err2;
err2 = std::abs(ckm0) * sqrt(result.numb) + eps * result.numb * std::abs(result.val);
result.err = std::min(err1,err2);
}
else {
result.err = err1;
}
}
}
return result;
}
// confluent Heun function, p.gamma = 0, -1, -2, ...
inline HeunCvars HeunC::HeunC00log(HeunCparams& p, double& z) {
HeunCvars result;
if (z==0) {
result.val = 1;
result.err = 0;
result.numb = 1;
if (std::abs(p.gamma)<eps) {
result.dval = INFINITY;
}
else {
result.dval = -p.q/p.gamma;
}
}
else {
int N = std::round(1-std::real(p.gamma));
// recursion relation variables
std::complex<double> L1 = 1, dL1 = 0, ddL1 = 0;
std::complex<double> L2 = 0, dL2 = 0, ddL2 = 0;
std::complex<double> L3, dL3, ddL3;
std::complex<double> ddval;
std::complex<double> ckm0 = 1, ckm1 = 1, ckm2 = 0;
std::complex<double> dm1, dm2, skm0, skm1, skm2 = 0;
std::complex<double> dsm1, dsm2;
for(int k=1; k<N; k++) {
ckm0 = (ckm1*z*(-p.q+(k-1)*(p.gamma-p.epsilon+p.delta+k-2)) + ckm2*(z*z)*((k-2)*p.epsilon+p.alpha))/(k*(p.gamma+k-1));
L1 = L1+ckm0;
dL1 = dL1+k*ckm0/z;
ddL1 = ddL1+k*(k-1)*ckm0/(z*z);
ckm2 = ckm1;
ckm1 = ckm0;
}
std::complex<double> sN = (ckm1*z*(p.q+p.gamma*(p.delta-p.epsilon-1)) + ckm2*(z*z)*(p.epsilon*(p.gamma+1)-p.alpha))/(p.gamma-1);
dm1 = dL2;
//dm2 = nan;
ckm1 = 0;
ckm2 = ckm0;
L3 = sN;
skm1 = sN;
dL3 = N*sN/z;
ddL3 = N*(N-1)*sN/(z*z);
dsm1 = dL3;
//dsm2 = nan;
//skm0 = nan;
int k = N+1;
while ( (k<=Heun_klimit) && ( (dsm2!=dsm1) || (std::abs(skm0)>eps) || (dm2!=dm1) || (std::abs(ckm0)>eps) ) ) {
skm0 = (skm1*z*(-p.q+(k-1)*(p.gamma-p.epsilon+p.delta+k-2)) + skm2*(z*z)*((k-2)*p.epsilon+p.alpha))/(k*(p.gamma+k-1));
ckm0 = (ckm1*z*(-p.q+(k-1)*(p.gamma-p.epsilon+p.delta+k-2)) + ckm2*(z*z)*((k-2)*p.epsilon+p.alpha))/(k*(p.gamma+k-1)) +
(-skm0*(p.gamma+2*k-1)+skm1*z*(p.gamma-p.epsilon+p.delta+2*k-3)+skm2*(z*z)*p.epsilon)/(k*(p.gamma+k-1));
L2 = L2+ckm0;
dL2 = dm1+k*ckm0/z;
ddL2 = ddL2+k*(k-1)*ckm0/(z*z);
ckm2 = ckm1;
ckm1 = ckm0;
dm2 = dm1;
dm1 = dL2;
L3 = L3+skm0; dL3 = dsm1+k*skm0/z; ddL3 = ddL3+k*(k-1)*skm0/(z*z);
skm2 = skm1; skm1 = skm0;
dsm2 = dsm1; dsm1 = dL3;
k++;
}
result.numb = k-1;
result.val = L1 + L2 + std::log(z) * L3;
result.dval = dL1 + dL2 + std::log(z) * dL3 + L3/z;
ddval = ddL1 + ddL2 - L3/(z*z) + 2*dL3/z + std::log(z) * ddL3;
if ( ( std::isinf(std::abs(result.val)) || std::isinf(std::abs(result.dval)) ) || ( std::isnan(std::abs(result.val)) || std::isnan(std::abs(result.dval)) ) ) {
throw std::runtime_error("HeunC00log: failed convergence of recurrence and summation");
}
else {
std::complex<double> val2, val3;
double err1, err2;
if (p.q-p.alpha*z != 0.0) {
val2 = ( z*(z-1)*ddval+(p.gamma*(z-1)+p.delta*z+p.epsilon*z*(z-1))*result.dval ) / (p.q-p.alpha*z);
val3 = ((dL3*p.epsilon+ddL3)*(z*z)*std::log(z)+(dL3*(p.gamma-p.epsilon+p.delta)-ddL3)*z*std::log(z)-dL3*p.gamma*std::log(z)+
(p.epsilon*(dL2+dL1)+ddL2+ddL1)*(z*z)+((dL1+dL2)*(p.gamma-p.epsilon+p.delta)+L3*p.epsilon-ddL2-ddL1+2*dL3)*z+
L3*(1-p.gamma)/z-(dL1+dL2)*p.gamma+L3*(p.gamma+p.delta-p.epsilon)-2*dL3-L3) / (p.q-p.alpha*z);
err1 = std::min(std::abs(result.val-val2),std::abs(result.val-val3));
}
else {
err1 = INFINITY;
}
if ((std::abs(p.q-p.alpha*z)<0.01)||(err1<eps)) {
err2 = std::abs(L1)*eps*N + std::abs(ckm0)*sqrt(result.numb-N+1) + std::abs(L2)*eps*(result.numb-N+1) +
std::abs(std::log(z)) * ( std::abs(skm0)*sqrt(result.numb-N+1) + std::abs(L3)*eps*(result.numb-N+1) );
result.err = std::min(err1,err2);
}
else {
result.err = err1;
}
}
}
return result;
}
// confluent Heun function,
// the second local solution
// with branch-cut (1,+\infinity)
// HeunCs(z) = z^(1-p.gamma)*h(z), where h(0)=1,
// h'(0)=(-q+(1-p.gamma)*(p.delta-p.epsilon))/(2-p.gamma) for p.gamma not equal to 1, 2
// h'(z)/log(z) -> -q+(1-p.gamma)*(p.delta-p.epsilon) as z\to0 for p.gamma=2
// and
// HeunCs(z) \sim log(z) - q * z * log(z) + as z\to0 for p.gamma=1
//
//
// Usage:
// [val,result.dval,err,numb,wrnmsg] = HeunCs00(q,p.alpha,p.gamma,p.delta,p.epsilon,z)
//
// Returned parameters:
// val is the value of the confluent Heun function
// dval is the value of z-derivative of the confluent Heun function
// err is the estimatedresult.error
// numb is the total result.number of power series terms needed for the evaluation
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 09 January 2018
//
// the second local solution at z=0 (see HeunCs00)
//
// computed by a consequence of power expansions
inline HeunCvars HeunC::HeunCs0(HeunCparams& p, double& z){
HeunCvars result;
if (z>=1){
throw std::invalid_argument("HeunC0: z belongs to the branch-cut [1,infty)");
}
else {
HeunCparams p1 = p;
bool expgrow = std::real(-p.epsilon*z)>0;
if (expgrow) {
p1.q = p.q - p.epsilon * p.gamma;
p1.alpha = p.alpha - p.epsilon * (p.gamma+p.delta);
p1.epsilon = -p.epsilon;
}
if (std::abs(z)<Heun_cont_coef){
result = HeunCs00(p1,z);
}
else {
double z0 = Heun_cont_coef*z/std::abs(z);
HeunCvars result0 = HeunCs00(p1,z0);
HeunCvars result1 = HeunCconnect(p1,z,z0,result0.val,result0.dval,R);
result.numb = result0.numb + result1.numb;
result.err = result0.err + result1.err;
}
if (expgrow) {
result.val = result.val * exp(p.epsilon*z);
result.dval = (p.epsilon*result.val + result.dval) * exp(p.epsilon*z);
result.err = result.err * std::abs(exp(p.epsilon*z));
}
return result;
}
}
// solution at z ~ 0
// |z| should not exceed the convergency radius 1
inline HeunCvars HeunC::HeunCs00(HeunCparams& p, double& z)
{
HeunCvars result;
if (std::abs(z)>=1){
throw std::invalid_argument("HeunCs00: z is outside the |z|<1 radius of convergence");
}
else {
if (std::abs(p.gamma-1)<eps){
if ( z==0 ){
result.val= INFINITY; result.dval = INFINITY;
result.err = INFINITY; result.numb = 1;
}
else {
result = HeunCs00gamma1(p,z);
}
}
else {
HeunCparams p1;
p1 = p;
p1.q = p.q + (p.gamma-1)*(p.delta-p.epsilon);
p1.alpha = p.alpha+p.epsilon*(1-p.gamma);
p1.gamma = 2-p.gamma;
std::cout << " using HeunC00 z = " << z << std::endl;
HeunCvars H0 = HeunC00(p1,z);
result.val= std::pow(z,(1-p.gamma)*H0.val);
result.dval = (1-p.gamma)*std::pow(z,(-p.gamma)*H0.val) + std::pow(z,(1-p.gamma)*H0.dval);
if ( std::isinf(std::abs(result.val)) || std::isinf(std::abs(result.dval)) ){
result.err = INFINITY;
}
else {
result.err = std::abs(std::pow(z,(1-p.gamma))*H0.err);
}
}
return result;
}
}
// confluent Heun function, second local solution at z=0, p.gamma = 1
//
inline HeunCvars HeunC::HeunCs00gamma1(HeunCparams& p,double& z)
{
HeunCvars result;
// declare iteration variables
std::complex<double> L1 = 0, dL1 = 0, ddL1 = 0, dm1 = 0, dm2, ckm0, ckm1 = 0, ckm2 = 0;
std::complex<double> L2 = 1, dL2 = 0, ddL2 = 0, skm2 = 0, skm1 = 1, dsm1 = 0, dsm2, skm0;
std::complex<double> ddval;
int k = 1;
while ( (k<=Heun_klimit) && ( (dsm2!=dsm1) || (std::abs(skm0)>eps) || (dm2!=dm1) || (std::abs(ckm0)>eps) ) ){
skm0 = (skm1*z*(-p.q+(k-1)*(-p.epsilon+p.delta+k-1)) + skm2*(z*z)*((k-2)*p.epsilon+p.alpha))/(k*k);
ckm0 = (ckm1*z*(-p.q+(k-1)*(-p.epsilon+p.delta+k-1)) + ckm2*(z*z)*((k-2)*p.epsilon+p.alpha))/(k*k)+
(skm0*2 + skm1*z*(p.epsilon/k-p.delta/k-2+2/k))/k + skm2*(z*z)*p.epsilon/(k*k);
L1 = L1+ckm0; dL1 = dm1+k*ckm0/z; ddL1 = ddL1+k*(k-1)*ckm0/(z*z);
ckm2 = ckm1; ckm1 = ckm0;
dm2 = dm1; dm1 = dL1;
L2 = L2+skm0; dL2 = dsm1+k*skm0/z; ddL2 = ddL2+k*(k-1)*skm0/(z*z);
skm2 = skm1; skm1 = skm0;
dsm2 = dsm1; dsm1 = dL2;
k += 1;
}
result.numb = k-1;
result.val = L1 + std::log(z) * L2;
result.dval = dL1 + std::log(z) * dL2 + L2/z;
ddval = ddL1 - L2/(z*z) + 2*dL2/z + std::log(z) * ddL2;
if ( ( std::isinf(std::abs(result.val)) || std::isinf(std::abs(result.dval)) ) || ( std::isnan(std::abs(result.val)) || std::isnan(std::abs(result.dval)) ) ){
throw std::runtime_error("HeunCs00gamma1: failed convergence of recurrence and summation");
}
else {
std::complex<double> val2, val3;
double err1;
if (p.q-p.alpha*z!= 0.0){
val2 = ( z*(z-1)*ddval+(z-1+p.delta*z+p.epsilon*z*(z-1))*result.dval) / (p.q-p.alpha*z);
val3 = ((dL2*p.epsilon+ddL2)*(z*z)*log(z)+(dL2*(-p.epsilon+p.delta+1)-ddL2)*z*log(z)-dL2*log(z)+
(dL1*p.epsilon+ddL1)*(z*z)+(dL1*(-p.epsilon+p.delta+1)+L2*p.epsilon-ddL1+2*dL2)*z-L2*p.epsilon+
L2*p.delta-2*dL2-dL1) / (p.q-p.alpha*z);
err1 = std::min(std::abs(result.val-val2),std::abs(result.val-val3));
}
else {
err1 = INFINITY;
}
if ((std::abs(p.q-p.alpha*z)<0.01)||(err1<eps)){
double err2 = std::abs(ckm0)*sqrt(result.numb) + std::abs(L1)*eps*result.numb +
std::abs(log(z)) * ( std::abs(skm0)*sqrt(result.numb) + std::abs(L2)*eps*result.numb );
result.err = err2+std::min(err1,err2);
}
else {
result.err = err1;
}
return result;
}
}
// confluent Heun function, a solution of the ep.quation
// HeunC""(z)+(p.gamma/z+p.delta/(z-1)+p.epsilon)*HeunC"(z)+(p.alpha*z-p.q)/(z*(z-1))*HeunC(z) = 0
// computed at z by power series about Z0 for the given values H(Z0)=H0, H"(Z0)=dH0
//
// it is assumed that z, Z0 are not equal to 0, 1 and |z-Z0| < std::min{|Z0|,|Z0-1|}
//
// Usage:
// [val,dval,err,numb] = HeunCfromZ0(p.q,p.alpha,p.gamma,p.delta,p.epsilon,z,Z0,H0,dH0)
//
// Returned parameters:
// val is the value of the confluent Heun function at point z
// dval is the value of z-derivative of the Heun function at point z
// err is the estimated error
// numb is the number of power series terms needed for the evaluation
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 09 January 2018
//
inline HeunCvars HeunC::HeunCfromZ0(HeunCparams& p,double& z,double& Z0,std::complex<double>& H0,std::complex<double>& dH0)
{
HeunCvars result;
R = std::min(std::abs(Z0),std::abs(Z0-1.0));
if (std::abs(z-Z0)>=R) {
throw std::invalid_argument("HeunCfromZ0: z is out of the convergence radius");
}
else if ((std::abs(z-1)<eps) || (std::abs(Z0-1)<eps)) {
throw std::invalid_argument("HeunCfromZ0: z or Z0 is too close to the singular points");
}
else if (z==Z0) {
result.val= H0; result.dval = dH0;
result.err= 0; result.numb = 0;
}
else {
double zeta = z-Z0;
// iteration variables
std::complex<double> ckm0, ckm1, ckm2, ckm3;
std::complex<double> dm1, dm2, vm1, vm2;
std::complex<double> ddval;
ckm3 = H0;
ckm2 = dH0*zeta;
// initialise with long recursion relation
ckm1 = (ckm2*zeta*(2-1)*(p.epsilon*(Z0*Z0)+(p.gamma-p.epsilon+p.delta)*Z0-p.gamma)+
ckm3*std::pow(zeta,2)*((p.alpha)*Z0-p.q)) / (Z0*(Z0-1)*(1-2)*2);
result.val= ckm3 + ckm2 + ckm1;
vm1 = result.val;
dm2 = dH0; dm1 = dH0 + 2*ckm1/zeta;
result.dval = dm1;
ddval = 2*ckm1/std::pow(zeta,2);
int k = 3;
ckm0 = 1;
while ( (k<=Heun_klimit) && ( ( vm2!=vm1 ) || ( dm2!=dm1 ) || (std::abs(ckm0)>eps) ) ) {
// long recursion relation
ckm0 = (ckm1*zeta*(k-1)*(p.epsilon*(Z0*Z0)+(p.gamma-p.epsilon+p.delta+2*(k-2))*Z0-p.gamma-k+2)+
ckm2*std::pow(zeta,2)*((2*(k-2)*p.epsilon+p.alpha)*Z0-p.q+(k-2)*(p.gamma-p.epsilon+p.delta+k-3))+
ckm3*std::pow(zeta,3)*((k-3)*p.epsilon+p.alpha)) / (Z0*(Z0-1)*(1-k)*k);
result.val += ckm0;
result.dval = dm1 + k*ckm0/zeta;
ddval += k*(k-1)*ckm0/std::pow(zeta,2);
ckm3 = ckm2;
ckm2 = ckm1;
ckm1 = ckm0;
vm2 = vm1; vm1 = result.val;
dm2 = dm1; dm1 = result.dval;
k++;
}
result.numb = k-1;
if ( std::isinf(std::abs(result.val)) || std::isinf(std::abs(result.dval)) || std::isnan(std::abs(result.val)) || std::isnan(std::abs(result.dval)) ) {
throw std::runtime_error("HeunCfromZ0: failed convergence of recurrence and summation");
}
else {
std::complex<double> val2;
double err1, err2;
if (p.q-p.alpha*z != 0.0) {
val2 = ( z*(z-1)*ddval+(p.gamma*(z-1)+p.delta*z+p.epsilon*z*(z-1))*result.dval ) / (p.q-p.alpha*z);
err1 = std::abs(result.val-val2);
}
else {
err1 = INFINITY;
}
if (std::abs(p.q-p.alpha*z)<0.01) {
err2 = std::abs(ckm0) * sqrt(result.numb) + std::abs(result.val) * eps * result.numb;
result.err = std::min(err1, err2);
}
else {
result.err = err1;
}
}
}
return result;
}
// HeunC''(z)+(gamma/z+delta/(z-1)+epsilon)*HeunC'(z)+(alpha*z-q)/(z*(z-1))*HeunC(z) = 0
// by analytic continuation from point z0, where HeunC(z0) = H0, HeunC'(z0) = dH0,
// to another point z, along the line [z0,z], using a consequence of power expansions
//
// Assumptions:
// z0, z are not 0 or 1
//
// R0 is an optional parameter, step size's guess
//
// R is the size of the last used step
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 09 January 2018
//
inline HeunCvars HeunC::HeunCconnect(HeunCparams& p, double& z, double& z0,std::complex<double>& H0,std::complex<double>& dH0, double R0, bool aux)
{
HeunCvars result;
if ((z==0)||(z==1)||(z0==0)||(z0==1)){
throw std::invalid_argument("HeunCconnect: assumed that z, z0 are not equal to 0, 1");
}
else {
bool insearch = true;
double Rmax;
if (R0>0) {
Rmax = R0;
}
else {
Rmax = 12*Heun_cont_coef/(1+std::abs(p.epsilon));
}
R = std::min(Rmax,std::min(std::abs(z0),std::abs(z0-1))*Heun_cont_coef);
// first set of iteration variables
bool Rtuned = false;
int iter = 1;
double z1; // step end point
int positivity = 2*(z >= z0) - 1;
while (Rtuned==false) {
if (std::abs(z-z0) <= R) {
z1 = z;
}
else {
z1 = z0 + R * positivity;
}
result = HeunCfromZ0(p,z1,z0,H0,dH0);
Rtuned = (result.err < 5*eps) && (result.numb < Heun_optserterms) || (iter>5) || (result.numb<=8);
if (Rtuned==false) {
R = R / std::max(result.err/(5*eps), static_cast<double>(result.numb)/Heun_optserterms);
}
insearch = !(Rtuned && (z==z1));
iter = iter+1;
}
//second set of iteration variables
double errsum;
int numbsum;
z0 = z1;
errsum = result.err;
numbsum = result.numb;
HeunCvars result_; // iteration version of result
H0 = result.val; dH0 = result.dval;
while (insearch) {
R = std::min(R, std::min(std::abs(z0),std::abs(z0-1))*Heun_cont_coef);
if (std::abs(z-z0) <= R) {
z1 = z; insearch = false;
}
else {
z1 = z0 + R * positivity;
}
result_ = HeunCfromZ0(p,z1,z0,H0,dH0);
H0 = result_.val;
dH0 = result_.dval;
errsum += result_.err;
numbsum += result_.numb;
if (insearch) {
R = Heun_optserterms * R / (result_.numb + eps);
}
z0 = z1;
}
result.numb = numbsum;
result.val = H0; result.dval = dH0; result.err = errsum;
}
return result;
}
/*
Depends on:
HeunCconnect
HeunCjoin0inf
HeunCinfA
HeunCinfB
*/
// confluent Heun function, a solution of the equation
// HeunC""(z)+(gamma/z+delta/(z-1)+epsilon)*HeunC"(z)+(alpha*z-q)/(z*(z-1))*HeunC(z) = 0
//
// computation for sufficiently large |z|, by analytic continuation from infinity
//
// computes both the first at z=0 local solution (see HeunC00) and the second at z=0 local solution (see HeunCs0)
//
// It is assumed that epsilon \neq 0 !
//
// Usage:
// [val1,dval1,result1.err,result2.val,result2.dval,result2.err,numb,,val1woexp,dval1woexp,result1.errwoexp,result2.valwoexp,result2.dvalwoexp,result2.errwoexp] = HeunCfaraway(p,z)
//
// Returned parameters:
// val1 is the value of the Heun function, growing from the first local solution at z=0
// dval1 is the value of z-derivative of the Heun function
// result1.err is the estimated error
// result2.val is the value of the Heun function, growing from the second local solution at z=0
// result2.dval is the value of z-derivative of the Heun function
// result2.err is the estimated error
// numb is the number of power series terms needed for the evaluation
//
// val1woexp = val1 * exp(epsilon*z)
// dval1woexp = dval1 * exp(epsilon*z)
// result1.errwoexp = result1.err * std::abs(exp(epsilon*z))
//
// result2.valwoexp = result2.val * exp(epsilon*z)
// result2.dvalwoexp = result2.dval * exp(epsilon*z)
// result2.errwoexp = result2.err * std::abs(exp(epsilon*z))
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 26 March 2018
//
//
// for large |z|
inline std::pair<HeunCvars, HeunCvars> HeunC::HeunCfaraway(HeunCparams& p, double& z)
{
HeunCvars result1, result2;
if (z>=1) {
throw std::invalid_argument("HeunCfaraway: z belongs to the branch-cut [1,infty)");
}
else {
HeunCparams pB = p;
pB.q = p.q-p.epsilon*p.gamma;
pB.alpha = p.alpha-p.epsilon*(p.gamma+p.delta);
pB.epsilon = -p.epsilon;
ConnectionVars CA = HeunCjoin0inf(p);
ConnectionVars CB = HeunCjoin0inf(pB);
findR();
double infpt = -std::max(1.0,R/(std::abs(eps)+std::abs(p.epsilon)));
HeunCvars varsA, varsB;
if (std::abs(z)>std::abs(infpt)) {
HeunCvars varsA = HeunCinfA(p,z);
HeunCvars varsB = HeunCinfA(pB,z);
result1.numb = CA.numb + CB.numb + varsA.numb + varsB.numb;
result1.err = CA.err + CB.err + varsA.err + varsB.err;
result2.numb = result1.numb;
result2.err = result1.err;
}
else {
HeunCvars varsinfA = HeunCinfA(p,z);
varsA = HeunCconnect(p,z,infpt,varsinfA.val,varsinfA.dval, R);
HeunCvars varsinfB = HeunCinfA(pB,z);
varsB = HeunCconnect(pB,z,infpt,varsinfA.val,varsinfA.dval, R);
result1.numb = varsinfA.numb + varsinfB.numb + varsA.numb + varsB.numb;
result1.err = varsinfA.err + varsinfB.err + varsA.err + varsB.err;
result2.err = result1.err;
}
Matrix<std::complex<double>> m;
m.init(CA.C10(0, 0),CA.C10(1, 0),CB.C10(0, 0),CB.C10(1, 0));
double co = m.cond();
result1.val = m(0, 0) * varsA.val + m(0, 1) * exp(-p.epsilon*z) * varsB.val;
result1.dval = m(0, 0) * varsA.dval + m(0, 1) * exp(-p.epsilon*z) * (-p.epsilon * varsB.val + varsB.dval);
result1.err = co * (std::abs(m(0, 0)) * varsA.err + std::abs(m(0, 1)) * std::abs(exp(-p.epsilon*z)) * varsB.err);
result2.val = m(1, 0) * varsA.val + m(1, 1) * exp(-p.epsilon*z) * varsB.val;
result2.dval = m(1, 0) * varsA.dval + m(1, 1) * exp(-p.epsilon*z) * (-p.epsilon * varsB.val + varsB.dval);
result2.err = co * (std::abs(m(1, 0)) * varsA.err + std::abs(m(1, 1)) * std::abs(exp(-p.epsilon*z)) * varsB.err);
std::pair<HeunCvars, HeunCvars> output(result1, result2);
return output;
}
}
// for confluent Heun function, a solution of the equation
// HeunC''(z)+(gamma/z+delta/(z-1)+epsilon)*HeunC'(z)+(alpha*z-q)/(z*(z-1))*HeunC(z) = 0
//
// HeunCjoin0inf finds connection coefficients C0, Cs, such that
// C0 * HeunC00(z) + Cs * HeunCs00(z) analytically continues to
// the first, power solution at infinity \exp(i\theta) infty
// (see HeunCinfA)
//
// Returned parameters:
// C10 is the matrix of connection coefficients
// err is the estimated error
// numb is the number of power series terms needed for the evaluation
//
// Oleg V. Motygin, copyright 2018, license: GNU GPL v3
//
// 15 March 2018
//
inline ConnectionVars HeunC::HeunCjoin0inf(HeunCparams& p,bool aux)
{
ConnectionVars result;
bool consts_known = false;
result = extrdatfromsav(p, savedata0inf, consts_known);
if (consts_known) {
result.numb = 0;
}
else {
HeunCvars varsinf, varsJinf, varsJ0, varsJs;
findR();
double R0, infpt, joinpt;
R0 = R/(std::abs(eps)+std::abs(p.epsilon));
infpt = -2 * R0;
joinpt = -std::min(1.0,R0);
varsinf = HeunCinfA(p,infpt); // value at "infinity" point
varsJinf = HeunCconnect(p,joinpt,infpt,varsinf.val,varsinf.dval); //
varsJ0 = HeunC0(p,joinpt,aux); // first solution at join point near zero
varsJs = HeunCs0(p,joinpt); // second solution at join point near zero
result.err = varsinf.err + varsJinf.err + varsJ0.err + varsJs.err;
result.numb = varsinf.numb + varsJinf.numb + varsJ0.numb + varsJs.numb;
Matrix<std::complex<double>> m, b;
m.init(varsJ0.val,varsJs.val, varsJ0.dval,varsJs.dval);
b.init(varsJinf.val, 0, varsJinf.dval, 0);
result.C10 = (m.inverse().dot(b));
savedataVars s;
s.p = p;
s.Cvars = result;
keepdattosav(s, savedata0inf);
}
return result;
}
inline ConnectionVars HeunC::extrdatfromsav(HeunCparams& p, std::vector<savedataVars>& savedata, bool& consts_known){
ConnectionVars result;
result.numb = 0;
savedataVars s;
if (savedata.size() !=0) {
for(int k=1; k < savedata.size(); k++){
s = savedata[k];
if (s.p.q == p.q && s.p.alpha == p.alpha && s.p.gamma == p.gamma && s.p.delta == p.delta && s.p.epsilon == p.epsilon) {
result = s.Cvars;
consts_known = "true";
break;
}
}
}
return result;
}
inline void HeunC::keepdattosav(savedataVars& s, std::vector<savedataVars>& savedata)
{
if (savedata.size() > Heun_memlimit)
{
savedata.erase(savedata.begin()); // remove first element
}
savedata.push_back(s); // store new element
}
// confluent Heun function
// asymptotic expansion at z=infinity
// the first, power solution
//
// Usage:
// [val,dval,err,numb] = HeunCinfA(p.q,p.alpha,p.gamma,p.delta,p.epsilon,z)
//
// Returned parameters:
// val is the value of the Heun function
// dval is the value of z-derivative of the Heun function
// err is the estimated error
// numb is the number of the summed series terms
//
// Oleg V. Motygin, copyright 2017-2018, license: GNU GPL v3
//
// 20 December 2017
//
inline HeunCvars HeunC::HeunCinfA(HeunCparams& p, double& z)
{
HeunCvars result;
result.val = 1;
result.dval = 0;
result.err = 0;
result.numb = 1;
// set up iteration variables
std::complex<double> cnm0, cnm1, cnm2 = 1, cnm3;
std::complex<double> dnm0, dnm1, dnm2 = 0, dnm3;
std::complex<double> vm0, vm1, vm2, vm3;
std::complex<double> dvm0, dvm1, dvm2, dvm3;
cnm1 = cnm2/(z*p.epsilon)*(1+(-p.q+p.alpha/p.epsilon*(2-p.gamma-p.delta-1+p.alpha/p.epsilon)+p.alpha-1));
dnm1 = -cnm1/z;
result.val = cnm2 + cnm1;
result.dval = dnm1;
vm0 = result.val;
dvm0 = result.dval;
result.numb = 2;
double small = sqrt(eps);
bool growcn = false, growdn = false, valstab = false, dvalstab = false;
int n;
while ( (result.numb<=Heun_asympt_klimit) && ((std::abs(cnm3)>small) || !(growcn||valstab) || !(growdn||dvalstab)) ){
n = result.numb;
cnm0 = cnm1*n/(z*p.epsilon)*(1+(-p.q+p.alpha/p.epsilon*(2*n-p.gamma-p.delta-1+p.alpha/p.epsilon)+
(p.gamma-p.epsilon+p.delta+1)*(1-n)+p.alpha-1)/(n*n)) + cnm2/((z*z)*p.epsilon)*
((n-2+p.alpha/p.epsilon)*(p.gamma-n+1-p.alpha/p.epsilon))/n;
dnm0 = -result.numb*cnm0/z;
result.val += cnm0;
result.dval = dnm0;
result.err = std::abs(cnm2);
result.numb += 1;
growcn = growcn || ((std::abs(cnm0)>std::abs(cnm1))&&(std::abs(cnm1)>std::abs(cnm2))&&(std::abs(cnm2)>std::abs(cnm3)));
valstab = valstab || ((vm3==vm2)&&(vm2==vm1)&&(vm1==result.val));
growdn = growdn || ((std::abs(dnm0)>std::abs(dnm1))&&(std::abs(dnm1)>std::abs(dnm2))&&(std::abs(dnm2)>std::abs(dnm3)));
dvalstab = dvalstab || ((dvm3==dvm2)&&(dvm2==dvm1)&&(dvm1==result.dval));
if ((std::abs(cnm2)>small) || !(growcn||valstab)){
cnm3 = cnm2; cnm2 = cnm1; cnm1 = cnm0;
vm3 = vm2; vm2 = vm1; vm1 = vm0; vm0 = result.val;
}
if ((std::abs(cnm2)>small) || !(growdn||dvalstab)){
dnm3 = dnm2; dnm2 = dnm1; dnm1 = dnm0;
dvm3 = dvm2; dvm2 = dvm1; dvm1 = dvm0; dvm0 = result.dval;
}
}
result.val = std::pow((-z),(-p.alpha/p.epsilon)) * vm3;
result.dval = std::pow((-z),(-p.alpha/p.epsilon)) * (dvm3-p.alpha/p.epsilon*vm3/z);
result.err = std::abs(std::pow(z,(-p.alpha/p.epsilon))) * result.err;
if ( std::isinf(std::abs(result.val)) || std::isinf(std::abs(result.dval)) || std::isnan(std::abs(result.val)) || std::isnan(std::abs(result.dval)) ){
throw std::runtime_error("HeunCinfA: failed convergence of recurrence and summation");
}
return result;
}