void FFT(const ColumnVector& U, const ColumnVector& V, ColumnVector& X, ColumnVector& Y) { // from Carl de Boor (1980), Siam J Sci Stat Comput, 1 173-8 // but first try Sande and Gentleman Tracer trace("FFT"); REPORT const int n = U.Nrows(); // length of arrays if (n != V.Nrows() || n == 0) Throw(ProgramException("Vector lengths unequal or zero", U, V)); if (n == 1) { REPORT X = U; Y = V; return; } // see if we can use the newfft routine if (!FFT_Controller::OnlyOldFFT && FFT_Controller::CanFactor(n)) { REPORT X = U; Y = V; if ( FFT_Controller::ar_1d_ft(n,X.Store(),Y.Store()) ) return; } ColumnVector B = V; ColumnVector A = U; X.ReSize(n); Y.ReSize(n); const int nextmx = 8; #ifndef ATandT int prime[8] = { 2,3,5,7,11,13,17,19 }; #else int prime[8]; prime[0]=2; prime[1]=3; prime[2]=5; prime[3]=7; prime[4]=11; prime[5]=13; prime[6]=17; prime[7]=19; #endif int after = 1; int before = n; int next = 0; bool inzee = true; int now = 0; int b1; // initialised to keep gnu happy do { for (;;) { if (next < nextmx) { REPORT now = prime[next]; } b1 = before / now; if (b1 * now == before) { REPORT break; } next++; now += 2; } before = b1; if (inzee) { REPORT fftstep(A, B, X, Y, after, now, before); } else { REPORT fftstep(X, Y, A, B, after, now, before); } inzee = !inzee; after *= now; }
static void fftstep(ColumnVector& A, ColumnVector& B, ColumnVector& X, ColumnVector& Y, int after, int now, int before) { REPORT Tracer trace("FFT(step)"); // const Real twopi = 6.2831853071795864769; const int gamma = after * before; const int delta = now * after; // const Real angle = twopi / delta; Real temp; // Real r_omega = cos(angle); Real i_omega = -sin(angle); Real r_arg = 1.0; Real i_arg = 0.0; Real* x = X.Store(); Real* y = Y.Store(); // pointers to array storage const int m = A.Nrows() - gamma; for (int j = 0; j < now; j++) { Real* a = A.Store(); Real* b = B.Store(); // pointers to array storage Real* x1 = x; Real* y1 = y; x += after; y += after; for (int ia = 0; ia < after; ia++) { // generate sins & cosines explicitly rather than iteratively // for more accuracy; but slower cossin(-(j*after+ia), delta, r_arg, i_arg); Real* a1 = a++; Real* b1 = b++; Real* x2 = x1++; Real* y2 = y1++; if (now==2) { REPORT int ib = before; if (ib) for (;;) { REPORT Real* a2 = m + a1; Real* b2 = m + b1; a1 += after; b1 += after; Real r_value = *a2; Real i_value = *b2; *x2 = r_value * r_arg - i_value * i_arg + *(a2-gamma); *y2 = r_value * i_arg + i_value * r_arg + *(b2-gamma); if (!(--ib)) break; x2 += delta; y2 += delta; } } else { REPORT int ib = before; if (ib) for (;;) { REPORT Real* a2 = m + a1; Real* b2 = m + b1; a1 += after; b1 += after; Real r_value = *a2; Real i_value = *b2; int in = now-1; while (in--) { // it should be possible to make this faster // hand code for now = 2,3,4,5,8 // use symmetry to halve number of operations a2 -= gamma; b2 -= gamma; Real temp = r_value; r_value = r_value * r_arg - i_value * i_arg + *a2; i_value = temp * i_arg + i_value * r_arg + *b2; } *x2 = r_value; *y2 = i_value; if (!(--ib)) break; x2 += delta; y2 += delta; } } // temp = r_arg; // r_arg = r_arg * r_omega - i_arg * i_omega; // i_arg = temp * i_omega + i_arg * r_omega; } } }