QLA_D_Complex
QLA_D_cpow(QLA_D_Complex *a, QLA_D_Real b)
{
double _Complex ca, cr;
QLA_D_c99_eq_c(ca, *a);
cr = cpow(ca, b);
QLA_D_Complex r;
QLA_D_c_eq_c99(r, cr);
return r;
}
/*
void
traceless_herm_M_evalues(QLA_ColorMatrix *Q, double *u, double *w,
double *q1, double *q2, double *q3) {
QLA_Complex c0, c1;
QLA_ColorMatrix Q2;
QLA_M_eq_M_times_M(&Q2, Q, Q);
printf("Q^2 = \n"); printm(&Q2);
QLA_C_eq_det_M (&c0, Q); // c0 = det(Q)
QLA_C_eq_trace_M (&c1, &Q2); // c1 = tr(Q^2)
double athird = 1.0/3.0;
double cc0, cc1, cc0max;
cc0 = QLA_real(c0);
cc1 = 0.5 * QLA_real(c1);
cc0max = 2*sqrt(cc1 * athird)*(cc1 * athird); //c_0^max = 2 * (c1/3)^{3/2}
printf("c0 = %f\n", cc0);
printf("c1 = %f\n", cc1);
printf("c0_max = %f\n", cc0max);
double theta;
theta =acos(cc0/cc0max);
*u = sqrt(athird * cc1) * cos(athird * theta);
*w = sqrt(cc1) * sin(athird * theta);
*q1 = 2 * *u;
*q2 = -*u + *w;
*q3 = -*u - *w;
printf("u = %f, w = %f, q1 = %f, q2 = %f, q3 = %f\n", *u, *w, *q1, *q2, *q3);
}
void
get_f_coeffs(QLA_ColorMatrix *Q, double _Complex *f0, double _Complex *f1, double _Complex *f2){
double u, w, q1, q2, q3;
traceless_herm_M_evalues(Q, &u, &w, &q1, &q2, &q3);
printf("q1=\n"); printc99(&q1);
double _Complex e2iu, e_iu;
e2iu = cexp(2 * _Complex_I * u);
e_iu = cexp(-1.0 * _Complex_I * u);
double u2 = u*u;
double w2 = w*w;
double _Complex zeta0w;
if (fabs(w) > 0.05) {
zeta0w = sin(w)/w;
}
else {
zeta0w = 1 - w2/6. * (1-w2/20. * (1 - w2/42.));
}
double _Complex h0, h1, h2;
h0 = (u2 - w2) * e2iu + e_iu * ( 8 * u2 *cos(w) + 2*_Complex_I*u * (3*u2+w2)*zeta0w);
h1 = 2*u*e2iu - e_iu * (2 * u * cos(w) - _Complex_I * (3*u2-w2)*zeta0w);
h2 = e2iu - e_iu * ( cos(w) + 3*_Complex_I*u * zeta0w);
double fac = 1.0/(9*u2-w2);
*f0 = h0 * fac;
*f1 = h1 * fac;
*f2 = h2 * fac;
}
void
get_Bs(QLA_ColorMatrix *Q, QLA_ColorMatrix *Q2, QLA_ColorMatrix *B1, QLA_ColorMatrix *B2, double _Complex *f0, double _Complex *f1, double _Complex *f2) {
double u, w, q1, q2, q3;
traceless_herm_M_evalues(Q, &u, &w, &q1, &q2, &q3);
printf("q1=\n"); printc99(&q1);
double _Complex e2iu, e_iu;
e2iu = cexp(2 * _Complex_I * u);
e_iu = cexp(-1.0 * _Complex_I * u);
double u2 = u*u;
double w2 = w*w;
double _Complex zeta0w, zeta1w;
if (fabs(w) > 0.05) {
zeta0w = sin(w)/w;
zeta1w = (cos(w)-zeta0w)/w2;
}
else {
zeta0w = 1 - w2/6. * (1-w2/20. * (1 - w2/42.));
zeta1w = -(1 - w2/10. * (1 - w2/28.*(1 - w2/54.)))/3.0;
}
double _Complex h0, h1, h2;
h0 = (u2 - w2) * e2iu + e_iu * ( 8 * u2 *cos(w) + 2*_Complex_I*u * (3*u2+w2)*zeta0w);
h1 = 2*u*e2iu - e_iu * (2 * u * cos(w) - _Complex_I * (3*u2-w2)*zeta0w);
h2 = e2iu - e_iu * ( cos(w) + 3*_Complex_I*u * zeta0w);
double fac = 1.0/(9*u2-w2);
*f0 = h0 * fac;
*f1 = h1 * fac;
*f2 = h2 * fac;
double _Complex r01, r11, r21, r02, r12, r22, iu;
double cosw = cos(w);
iu = _Complex_I * u;
r01 = 2*(u + _Complex_I * (u2 - w2)) * e2iu
+ 2 * e_iu * ( 4*u*(2 - iu) * cosw + _Complex_I * (9 * u2 + w2 - iu * (3*u2 + w2))*zeta0w);
r11 = 2*(1 + 2*iu) * e2iu
+ e_iu * ( -2 * (1-iu) * cosw + _Complex_I * (6*u + _Complex_I * (w2 - 3*u2)) * zeta0w);
r21 = 2 * _Complex_I * e2iu + _Complex_I * e_iu * (cosw - 3*(1-iu)*zeta0w);
r02 = -2 * e2iu + 2 * iu * e_iu * (cosw + (1+4*iu) * zeta0w + 3 * u2 * zeta1w);
r12 = -_Complex_I * e_iu * ( cosw + (1+2*iu) * zeta0w - 3*u2 * zeta1w);
r22 = e_iu * (zeta0w - 3 * iu * zeta1w);
double _Complex b10, b11, b12, b20, b21, b22;
double fac1, fac2, fac3;
double mult = 0.5 * fac * fac;
fac1 = 2 * u;
fac2 = 3*u2 - w2;
fac3 = 2*(15*u2+w2);
b10 = fac1 * r01 + fac2 * r02 - fac3 * (*f0);
b10 *= mult;
b11 = fac1 * r11 + fac2 * r12 - fac3 * (*f1);
b11 *= mult;
b12 = fac1 * r21 + fac2 * r22 - fac3 * (*f2);
b12 *= mult;
fac2 = 3*u;
fac3 = 24*u;
b20 = r01 - fac2 * r02 - fac3 * (*f0);
b20 *= mult;
b21 = r11 - fac2 * r12 - fac3 * (*f1);
b21 *= mult;
b22 = r21 - fac2 * r22 - fac3 * (*f2);
b22 *= mult;
QLA_Complex qb10, qb11, qb12, qb20, qb21, qb22;
QLA_c_eq_c99(qb10, b10);
QLA_c_eq_c99(qb11, b11);
QLA_c_eq_c99(qb12, b12);
QLA_c_eq_c99(qb20, b20);
QLA_c_eq_c99(qb21, b21);
QLA_c_eq_c99(qb22, b22);
QLA_M_eq_c(B1, &qb10);
QLA_M_peq_c_times_M(B1, &qb11, Q);
QLA_M_peq_c_times_M(B1, &qb12, Q2);
QLA_M_eq_c(B2, &qb20);
QLA_M_peq_c_times_M(B2, &qb21, Q);
QLA_M_peq_c_times_M(B2, &qb22, Q2);
}
*/
int
main(void) {
QLA_ColorMatrix O, iQ, matI;
QLA_M_eq_zero(&matI);
for(int i=0; i<QLA_Nc; i++) {
QLA_c_eq_r_plus_ir(QLA_elem_M(matI,i,i), 1.0, 0.0);
}
printm(&matI);
//QLA_Complex tr;
//QLA_Real half = 0.5;
for(int i=0; i<QLA_Nc; i++) {
for(int j=0; j<QLA_Nc; j++) {
QLA_c_eq_r_plus_ir(QLA_elem_M(O,i,j), i+1, QLA_Nc*(j+1));
}
QLA_c_eq_r_plus_ir(QLA_elem_M(O,i,i), 2+1, 1);
}
printm(&O);
#if QDP_Colors == 3
QLA_ColorMatrix A;
QLA_M_eq_zero(&A);
for ( int m = 0; m < QLA_Nc; m++) {
for ( int n = 0; n < QLA_Nc; n++) {
QLA_c_eq_r_plus_ir(QLA_elem_M(A, m, n), 3+m, 2-n);
}
QLA_c_eq_r_plus_ir(QLA_elem_M(A,m,m), 2+1, 1);
}
QLA_M_eq_antiherm_M(&A, &A);
printm(&A);
QLA_M_eq_zero(&A);
QLA_c_eq_r_plus_ir(QLA_elem_M(A,0,0),0,-1);
QLA_c_eq_r_plus_ir(QLA_elem_M(A,1,1),0,0.4);
QLA_c_eq_r_plus_ir(QLA_elem_M(A,2,2),0,0.6);
printm(&A);
QLA_Complex minus_i;
QLA_c_eq_r_plus_ir(minus_i, 0, -1);
QLA_ColorMatrix Q, Q2, expiQ, qla_expA;
QLA_M_eq_C_times_M(&Q, &minus_i, &matI);
QLA_M_eq_M_times_M(&Q2, &Q, &Q);
printf("Q=\n"); printm(&Q);
double _Complex f0, f1, f2;
QLA_ColorMatrix B1, B2;
get_Bs(&Q, &Q2, &B1, &B2, &f0, &f1, &f2);
printf("f0, f1, f2=\n");
printc99(&f0);
printc99(&f1);
printc99(&f2);
QLA_Complex qf0, qf1, qf2;
QLA_c_eq_c99(qf0, f0);
QLA_c_eq_c99(qf1, f1);
QLA_c_eq_c99(qf2, f2);
QLA_M_eq_c(&expiQ, &qf0);
QLA_M_peq_c_times_M(&expiQ, &qf1, &Q);
QLA_M_peq_c_times_M(&expiQ, &qf2, &Q2);
QLA_M_eq_exp_M(&qla_expA, &matI);
printf("my expiQ = \n");
printm(&expiQ);
printf("qla expA = \n");
printm(&qla_expA);
// derivative
QLA_Complex trB1M, trB2M;
QLA_ColorMatrix prod, deriv;
//tr(B_1 M)
QLA_M_eq_M_times_M (&prod, &B1, &matI); //B_1 M
QLA_C_eq_trace_M (&trB1M, &prod);
//tr(B_2 M);
QLA_M_eq_M_times_M (&prod, &B2, &matI); //B_2 M
QLA_C_eq_trace_M (&trB2M, &prod);
// deriv = Tr(B_1 M) Q
QLA_M_eq_c_times_M (&deriv, &trB1M, &Q);
// deriv += Tr(B_2 M) Q^2
QLA_M_peq_c_times_M (&deriv, &trB2M, &Q2);
// deriv += f1 M
QLA_M_peq_c_times_M (&deriv, &qf1, &matI);
// deriv += f2 Q M
QLA_M_eq_M_times_M (&prod, &Q, &matI); // Q M
QLA_M_peq_c_times_M (&deriv, &qf2, &prod);
// deriv += f2 M Q
QLA_M_eq_M_times_M (&prod, &matI, &Q); // M Q
QLA_M_peq_c_times_M (&deriv, &qf2, &prod);
QLA_M_eq_c_times_M (&deriv, &minus_i, &deriv);
printf("M = \n");
printm(&matI);
printf("deriv = \n");
printm(&deriv);
QLA_M_meq_M(&deriv, &expiQ);
printf("diff = \n");
printm(&deriv);
/*
printf("2/3f0 = \n");
f0 *= 2.0/3;
printc99(&f0);
printc(&qf0);
printc(&qf1);
printc(&qf2);
*/
#endif
#if QDP_Colors == 2
QLA_ColorMatrix expO;
QLA_Complex Tr, det;
QLA_c_eq_c_times_c(det, QLA_elem_M(O,0,0),QLA_elem_M(O,1,1));
QLA_c_meq_c_times_c(det, QLA_elem_M(O,0,1), QLA_elem_M(O,1,0));
QLA_C_eq_trace_M(&Tr, &O);
QLA_Complex qs, qt;
QLA_c_eq_r_times_c(qs, 0.5, Tr); // s=TrA/2
QLA_Complex qs2;
QLA_c_eq_c_times_c(qs2, qs, qs); //s2 = s^2
QLA_c_meq_c(qs2, det); //s2 = s^2 - detA
double _Complex t = QLA_real(qs2) + QLA_imag(qs2) * _Complex_I;
t = csqrt(t); // sqrt(s^2 - det A)
QLA_c_eq_r_plus_ir(qt, creal(t), cimag(t)); // t = sqrt(s^2 - det A)
printf(" Matrix O = \n"); printm(&O);
printf("TrO = "); printc(&Tr);
printf("detO = "); printc(&det);
printf("s = "); printc(&qs);
printf("t^2 = "); printc(&qs2);
printf("t = "); printc(&qt);
//use the QLA exp function
QLA_ColorMatrix qla_exp;
QLA_M_eq_exp_M(&qla_exp, &O); //exp(O)
double _Complex exps, cosht, sinht, sinht_t;
double _Complex s = QLA_real(qs) + QLA_imag(qs) * _Complex_I;
exps = cexp(s);
if(creal(t) == 0 && cimag(t) == 0) {
cosht = 1;
sinht = 0;
sinht_t = 1;
} else {
cosht = ccosh(t);
sinht = csinh(t);
sinht_t = sinht/t;
}
double _Complex f0, f1;
f1 = exps * sinht_t;
f0 = exps * cosht - s * f1;;
//derivative of the exponential
double _Complex f0s, f1s, f1t, f0t2, f1t2, t2;
t2 = t*t;
f0s = f0 - f1;
f1s = f1;
if (cabs(t) > 0.05) {
f1t = ((f0-f1) + s*f1)/t;
f1t2 = f1t/t;
}
else { //when |t| < 0.05, the error is O(10^{-14})
f1t = exps * t/3 * (1+t2/10*(1+t2/28));
f1t2 = exps / 3 * (1+t2/10*(1+t2/28));
}
// f0t = t * f1 - s * f1t;
f0t2 = f1 - s * f1t2;
printf("f0 = \n"); printc99(&f0);
printf("f1 = \n"); printc99(&f1);
printf("f0s = \n"); printc99(&f0s);
printf("f1s = \n"); printc99(&f1s);
printf("f1t = \n"); printc99(&f1t);
printf("f0t2 = \n"); printc99(&f0t2);
printf("f1t2 = \n"); printc99(&f1t2);
QLA_Complex qf0, qf1;
QLA_c_eq_r_plus_ir(qf0, creal(f0), cimag(f0));
QLA_c_eq_r_plus_ir(qf1, creal(f1), cimag(f1));
QLA_M_eq_c_times_M(&expO, &qf1, &O);
QLA_M_peq_c(&expO, &qf0);
printf("QLA exp = \n"); printm(&qla_exp);
printf("my expO = \n"); printm(&expO);
/*
QLA_Complex qf0s, qf0t, qf1s, qf1t;
QLA_c_eq_r_plus_ir(qf0s, creal(f0s), cimag(f0s));
QLA_c_eq_r_plus_ir(qf0t, creal(f0t), cimag(f0t));
QLA_c_eq_r_plus_ir(qf1s, creal(f1s), cimag(f1s));
QLA_c_eq_r_plus_ir(qf1t, creal(f1t), cimag(f1t));
*/
QLA_ColorMatrix deriv;
QLA_M_eq_zero(&deriv);
QLA_ColorMatrix B, AB;
QLA_M_eq_M(&B, &matI);
//QLA_c_eq_r_plus_ir(QLA_elem_M(B,1,0), 0.1, 0.2);
//QLA_c_eq_r_plus_ir(QLA_elem_M(B,0,1), 0.2, 0.1);
printf("B=\n"); printm(&B);
QLA_M_eq_M_times_M(&AB, &O, &B);
printf("AB=\n"); printm(&AB);
QLA_M_eq_c_times_M(&deriv, &qf1, &B); //f1 * B
printf("f1 B = \n"); printm(&deriv);
QLA_Complex trB, trAB;
QLA_C_eq_trace_M(&trB, &B);
QLA_C_eq_trace_M(&trAB, &AB);
double _Complex ctrB = QLA_real(trB) + _Complex_I * QLA_imag(trB);
double _Complex ctrAB = QLA_real(trAB) + _Complex_I * QLA_imag(trAB);
double _Complex coeff;
coeff = (f0s - f0t2 * s) * ctrB;
coeff += (f1s - f1t2 * s) * ctrAB;
coeff *= 0.5;
printf("coeff = "); printc99(&coeff);
QLA_Complex qc;
QLA_D_c_eq_c99(qc, coeff);
printc(&qc);
QLA_M_peq_c_times_M(&deriv, &qc, &matI); // f1 * B + () I
printf("f1B+()I=\n"); printm(&deriv);
coeff = 0.5 * (f0t2 * ctrB + f1t2 * ctrAB);
QLA_D_c_eq_c99(qc, coeff);
printc(&qc);
QLA_M_peq_c_times_M(&deriv, &qc, &O);
printm(&deriv);
exp_deriv_site(&deriv, &expO, &O, &B);
printm(&deriv);
#endif
return 0;
}