Fix spherical harmonics for negative m in ewald.c.

Former-commit-id: 960dd721a009e2b2f82c986677c3faa65d36394c

notes/ewald.lyx

@@ -3104,9 +3104,12 @@ Y_{n}^{m}\left(\frac{\pi}{2},\phi\right) & = & \left(-1\right)^{m}\sqrt{\frac{\l
  Let us substitute this into [LT(4.5)]
 \begin_inset Formula 
 \begin{eqnarray*}
-\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\\
+\sigma_{n}^{m(1)} & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\left(-1\right)^{\left(n+m\right)/2}\sqrt{\left(2n+1\right)\left(n-m\right)!\left(n+m\right)!}\times\\
- & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\sqrt{\pi}2^{n+1}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!Y_{n}^{m}\left(0,\phi_{\vect{\beta}_{pq}}\right)\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\\
+ &  & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}e^{im\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\\
- & = & -\frac{i^{n+1}}{k\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!Y_{n}^{m}\left(0,\phi_{\vect{\beta}_{pq}}\right)\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1},
+ & = & -\frac{i^{n+1}}{2k^{2}\mathscr{A}}\sqrt{\pi}2^{n+1}\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\\
+ &  & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(0,\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}\\
+ & = & -\frac{i^{n+1}}{k\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\\
+ &  & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(0,\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{\left[\left(n-\left|m\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/k\right)^{n-2j}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(n-m\right)-j\right)!\left(\frac{1}{2}\left(n+m\right)-j\right)!}\left(\gamma_{pq}\right)^{2j-1}
 \end{eqnarray*}
 
 \end_inset

qpms/ewald.c

@@ -208,7 +208,7 @@ int ewald32_sigma_long_shiftedpoints (
         assert((n-abs(m))/2 == c->s1_jMaxes[y]);
         for(qpms_l_t j = 0; j <= c->s1_jMaxes[y]/*(n-abs(m))/2*/; ++j) { // FIXME </<= ?
           complex double summand = pow(rbeta_pq/k, n-2*j) 
-            * e_imalpha_pq  * c->legendre0[gsl_sf_legendre_array_index(n,abs(m))] // This line can actually go outside j-loop
+            * e_imalpha_pq  * c->legendre0[gsl_sf_legendre_array_index(n,abs(m))] * min1pow_m_neg(m) // This line can actually go outside j-loop
             * cpow(gamma_pq, 2*j-1) // * Gamma_pq[j] bellow (GGG) after error computation
             * c->s1_constfacs[y][j];
           if(err) {
@@ -322,7 +322,7 @@ int ewald32_sigma_short_shiftedpoints(
           kahanadd(err + y, err_c + y, cabs(leg * (prefacn / I) * R_pq_pown
               * interr)); // TODO include also other errors
         ckahanadd(target + y, target_c + y,
-            prefacn * R_pq_pown * leg * intres * e_beta_Rpq * e_imf);
+            prefacn * R_pq_pown * leg * intres * e_beta_Rpq * e_imf * min1pow_m_neg(m));
       }
 
     }

qpms/tiny_inlines.h

@@ -4,6 +4,13 @@
 
 static inline int min1pow(int pow) { return (pow % 2) ? -1 : 1; }
 
+
+// This is useful for calculating spherical harmonics with negative m
+// if spharm-normalised legendre functions for positive m are available.
+static inline int min1pow_m_neg(int m) {
+	return (m < 0) ? min1pow(m) : 1;
+}
+
 // this has shitty precision:
 // static inline complex double ipow(int x) { return cpow(I, x); }
 

tests/ewalds.c

@@ -175,7 +175,7 @@ ewaldtest_triang_results *ewaldtest_triang(const ewaldtest_triang_params p) {
         complex double eimf = cexp(I*m*arg_pq);
         results->regsigmas_416[y] +=
           4*M_PI*ipow(n)/p.k/A 
-          * eimf * legendres[gsl_sf_legendre_array_index(n,abs(m))]
+          * eimf * legendres[gsl_sf_legendre_array_index(n,abs(m))] * min1pow_m_neg(m)
           / denom;
       }
     }