WIP Ewald 2D in 3D general z != 0 constant factors.

Former-commit-id: 787689f357bd8670948ba8ce7d8dc1205ca77d0f

notes/ewald_23_z_nonzero.lyx

@@ -0,0 +1,245 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
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+
+\begin_body
+
+\begin_layout Standard
+
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+
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+
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+
+
+\begin_inset FormulaMacro
+\newcommand{\sgn}{\operatorname{sgn}}
+{\mathrm{sgn}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\pht}[2]{\mathfrak{\mathbb{H}}_{#1}#2}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\vect}[1]{\mathbf{#1}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\ud}{\mathrm{d}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\basis}[1]{\mathfrak{#1}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\dc}[1]{Ш_{#1}}
+\end_inset
+
+
+\begin_inset FormulaMacro
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+
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+
+
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+
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+
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+
+
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+\newcommand{\hgfr}{\mathbf{F}}
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+
+
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+\newcommand{\hgf}{F}
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+
+
+\begin_inset FormulaMacro
+\newcommand{\ghgf}[2]{\mbox{}_{#1}F_{#2}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\ghgfr}[2]{\mbox{}_{#1}\mathbf{F}_{#2}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\ph}{\mathrm{ph}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\kor}[1]{\underline{#1}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\koru}[1]{\utilde{#1}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\swv}{\mathscr{H}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\expint}{\mathrm{E}}
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+\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)!}\times\nonumber \\
+ &  & \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}\nonumber \\
+ & = & -\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\nonumber \\
+ &  & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\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}\nonumber \\
+ & = & -\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\times\nonumber \\
+ &  & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\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}\label{eq:2D Ewald in 3D long-range part}
+\end{eqnarray}
+
+\end_inset
+
+For 
+\begin_inset Formula $z\ne0$
+\end_inset
+
+
+\begin_inset Formula 
+\begin{align*}
+ & =-\frac{i^{n+1}}{k^{2}\mathscr{A}}\sqrt{\pi}2\left(\left(n-m\right)/2\right)!\left(\left(n+m\right)/2\right)!\\
+ & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}Y_{n}^{m}\left(\frac{\pi}{2},\phi_{\vect{\beta}_{pq}}\right)\sum_{j=0}^{n-\left|m\right|}\frac{\Delta_{npq}}{j!}\left(-1\right)^{j}\left(\gamma_{pq}\right)^{2j-1}\sum_{s\overset{*}{=}j}^{\min(2j,n-\left|m\right|)}\binom{j}{2j-s}\frac{\left(-\kappa z\right)^{2j-s}\left(\beta_{pq}/k\right)^{n-s}}{\left(\frac{1}{2}\left(n-m-s\right)\right)!\left(\frac{1}{2}\left(n+m-s\right)\right)!}
+\end{align*}
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document

notes/kambe_linton_dict.lyx

@@ -0,0 +1,375 @@
+#LyX 2.4 created this file. For more info see https://www.lyx.org/
+\lyxformat 584
+\begin_document
+\begin_header
+\save_transient_properties true
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+\language_package default
+\inputencoding utf8
+\fontencoding auto
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
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+\cite_engine_type default
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+\tablestyle default
+\tracking_changes false
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+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+
+\lang english
+\begin_inset FormulaMacro
+\newcommand{\vect}[1]{\mathbf{#1}}
+\end_inset
+
+
+\lang finnish
+
+\begin_inset FormulaMacro
+\newcommand{\Kambe}[1]{#1^{\mathrm{K}}}
+\end_inset
+
+
+\begin_inset FormulaMacro
+\newcommand{\Linton}[1]{#1^{\mathrm{L}}}
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Here and in Kambe's papers, 
+\begin_inset Formula $\kappa$
+\end_inset
+
+ is the wavenumber (
+\begin_inset Formula $k$
+\end_inset
+
+ in Linton).
+ Here 
+\begin_inset Formula $\vect K_{p}$
+\end_inset
+
+ is a point of the reciprocal lattice (
+\begin_inset Formula $\vect K_{p}=\Kambe{\vect K_{pt}}=\Linton{\vect{\beta}_{\mu}}$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Section
+\begin_inset Quotes eld
+\end_inset
+
+Gammas
+\begin_inset Quotes erd
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+For 
+\begin_inset Formula $\kappa$
+\end_inset
+
+ positive,
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\Kambe{\Gamma_{p}}\equiv\begin{cases}
+\sqrt{\kappa^{2}-\left|\vect K_{p}\right|^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}>0\\
+i\sqrt{\left|\vect K_{p}\right|^{2}-\kappa^{2}} & \kappa^{2}-\left|\vect K_{p}\right|^{2}<0
+\end{cases}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\Linton{\gamma_{\mu}}\equiv\begin{cases}
+\sqrt{\left(\frac{\vect K_{p}}{\kappa}\right)^{2}-1} & \kappa-\left|\vect K_{p}\right|\le0\\
+-i\sqrt{1-\left(\frac{\vect K_{p}}{\kappa}\right)^{2}} & \kappa-\left|\vect K_{p}\right|>0
+\end{cases}
+\]
+
+\end_inset
+
+hence 
+\begin_inset Formula 
+\[
+\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}},
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\Linton{\gamma_{\mu}}=i\frac{\Kambe{\Gamma_{p}}}{\kappa}.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+D vs sigma
+\end_layout
+
+\begin_layout Standard
+In-plane sums [Linton 2009, (4.5)], replacing 
+\begin_inset Formula $n,m\rightarrow L,M$
+\end_inset
+
+, 
+\begin_inset Formula $k\rightarrow\kappa$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+\begin_inset Formula 
+\begin{eqnarray*}
+\sigma_{L}^{M(1)} & = & -\frac{i^{L+1}}{2\kappa^{2}\mathscr{A}}\left(-1\right)^{\left(L+M\right)/2}\sqrt{\left(2L+1\right)\left(L-M\right)!\left(L+M\right)!}\times\\
+ &  & \times\sum_{\vect K_{pq}\in\Lambda^{*}}^{'}\sum_{j=0}^{\left[\left(L-\left|M\right|/2\right)\right]}\frac{\left(-1\right)^{j}\left(\beta_{pq}/2\kappa\right)^{L-2j}e^{iM\phi_{\vect{\beta}_{pq}}}\Gamma_{j,pq}}{j!\left(\frac{1}{2}\left(L-M\right)-j\right)!\left(\frac{1}{2}\left(L+M\right)-j\right)!}\left(\frac{\gamma_{pq}}{2}\right)^{2j-1}
+\end{eqnarray*}
+
+\end_inset
+
+[Kambe II, (3.17)], replacing 
+\lang finnish
+
+\begin_inset Formula $n\rightarrow j$
+\end_inset
+
+
+\lang english
+, 
+\lang finnish
+
+\begin_inset Formula $A\rightarrow\mathscr{A}$
+\end_inset
+
+, 
+\begin_inset Formula $\vect K_{pt}\to\vect K_{p}$
+\end_inset
+
+, 
+\begin_inset Formula $\Gamma\left(\frac{1}{2}-j,e^{-i\pi}\Gamma_{p}^{2}\omega/2\right)\to\Gamma_{j,p}$
+\end_inset
+
+ and performing little typographic modifications
+\lang english
+
+\begin_inset Formula 
+\begin{align*}
+D_{LM} & =-\frac{1}{\mathscr{A}\kappa}i^{\left|M\right|+1}2^{-L}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
+ & \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ijt}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(\Gamma_{p}/\kappa\right)^{2j-1}\left(K_{p}/\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}
+\end{align*}
+
+\end_inset
+
+Using the relations between 
+\begin_inset Formula $\Kambe{\Gamma_{p}}=-i\kappa\Linton{\gamma_{\mu}}$
+\end_inset
+
+, we have (also, we replace the 
+\begin_inset Formula $\mu$
+\end_inset
+
+ index with 
+\begin_inset Formula $p$
+\end_inset
+
+)
+\begin_inset Formula 
+\begin{align*}
+D_{LM} & =-\frac{1}{\mathscr{A}\kappa}i^{\left|M\right|+1}2^{-L}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
+ & \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ijt}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(-i\gamma_{p}\right)^{2j-1}\left(K_{p}/\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}
+\end{align*}
+
+\end_inset
+
+and now, trying to make the exponents look the same as in Linton, 
+\begin_inset Formula $2^{-1}2^{2j-L}2^{1-2j}=2^{-L}$
+\end_inset
+
+ (OK), 
+\begin_inset Formula $K_{p}^{L-2j}=K_{p}^{L-2j}$
+\end_inset
+
+ (OK), 
+\begin_inset Formula 
+\begin{align*}
+D_{LM} & =-\frac{1}{2\kappa\mathscr{A}}i^{\left|M\right|+1}\sqrt{\left(2L+1\right)\left(L+\left|M\right|\right)!\left(L-\left|M\right|\right)!}\times\\
+ & \quad\times\sum_{p}e^{i\vect K_{p}\cdot\vect c_{ij}}e^{-iM\phi_{K_{p}}}\sum_{j=0}^{\left(L-\left|M\right|\right)/2}\frac{\left(-i\right)^{2j-1}\left(K_{p}/2\kappa\right)^{L-2j}\Gamma_{j,p}}{j!\left(\frac{1}{2}\left(L-\left|M\right|\right)-j\right)!\left(\frac{1}{2}\left(L+\left|M\right|\right)-j\right)!}\left(\frac{\gamma_{p}}{2}\right)^{2j-1}
+\end{align*}
+
+\end_inset
+
+There are now these differences left:
+\end_layout
+
+\begin_layout Itemize
+
+\lang english
+Additional 
+\begin_inset Formula $\kappa$
+\end_inset
+
+ factor in 
+\begin_inset Formula $D_{LM}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\lang english
+\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}\left(-1\right)^{j}$
+\end_inset
+
+ vs.
+ 
+\begin_inset Formula $i^{\left|M\right|+1}\left(-i\right)^{2j-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+
+\lang english
+Opposite phase in the angular part.
+\end_layout
+
+\begin_layout Itemize
+
+\lang english
+Plane wave factor in 
+\begin_inset Formula $D_{LM}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+
+\lang english
+Let's look at the 
+\begin_inset Formula $i,-1$
+\end_inset
+
+ factors (note that 
+\begin_inset Formula $L+M$
+\end_inset
+
+ is odd): 
+\begin_inset Formula $\left(-i\right)^{2j}=\left(-1\right)^{j},$
+\end_inset
+
+ leaving 
+\begin_inset Formula $i^{L+1}\left(-1\right)^{\left(L+M\right)/2}$
+\end_inset
+
+ vs.
+ 
+\begin_inset Formula $i^{\left|M\right|+1}i$
+\end_inset
+
+.
+ So there is might be a phase difference due to different conventions, but
+ it does not depend on 
+\begin_inset Formula $j$
+\end_inset
+
+, so one should be able to transplant the 
+\begin_inset Formula $z\ne0$
+\end_inset
+
+ sum from Kambe without major problems.
+\end_layout
+
+\begin_layout Section
+Ewald parameter (integration limits)
+\end_layout
+
+\end_body
+\end_document

qpms/ewald.c

@@ -26,11 +26,12 @@
 #endif
 
 #ifndef M_SQRTPI
-#define M_SQRTPI 1.7724538509055160272981674833411452
+#define M_SQRTPI 1.7724538509055160272981674833411452L
 #endif
 
 
 // sloppy implementation of factorial
+// We prefer to avoid tgamma/lgamma, as their errors are about 4+ bits
 static inline double factorial(const int n) {
   assert(n >= 0);
   if (n < 0)
@@ -45,6 +46,44 @@ static inline double factorial(const int n) {
     return tgamma(n + 1); // hope it's precise and that overflow does not happen
 }
 
+// sloppy implementation of double factorial n!!
+static inline double double_factorial(int n) {
+  assert(n >= 0);
+  if (n <= 25) {
+    double fac = 1;
+    while (n > 0) {
+      fac *= n;
+      n -= 2;
+    }
+    return fac;
+  } else {
+    if (n % 2) { // odd, (2*k - 1)!! = 2**k * Γ(k + 0.5) / sqrt(п)
+      const int k = n / 2 + 1;
+      return pow(2, k) * tgamma(k + 0.5) / M_SQRTPI;
+    } else { // even, n!! = 2**(n/2) * (n/2)!
+      const int k = n/2;
+      return pow(2, k) * factorial(k);
+    }
+  }
+}
+
+// sloppy implementation of (n/2)! = Γ(n/2 + 1)
+// It is _slightly_ more precise than direct call of tgamma for small odd n
+static inline double factorial_of_half(const int n2) {
+  assert(n2 >= 0);
+  if (n2 % 2 == 0) return factorial(n2/2);
+  else {
+    if (n2 <= 50) { // odd, use (k - 0.5)! =  Γ(k + 0.5) = 2**(-k) (2*k - 1)!! sqrt(п) for small n2
+      const int k = n2 / 2 + 1;
+      double fac2 = 1;
+      for(int j = 2*k - 1; j > 0; j -= 2)
+        fac2 *= j;
+      return fac2 * pow(2, -k) * M_SQRTPI;
+    }
+    else return tgamma(1. + 0.5*n2);
+  }
+}
+
 static inline complex double csq(complex double x) { return x * x; }
 static inline double sq(double x) { return x * x; }
 

qpms/ewald.h

@@ -80,6 +80,34 @@ typedef struct qpms_ewald3_constants_t {
 	 * 2D sum with additional factor of 
 	 * \f$ \sqrt{pi} \kappa \gamma(\abs{\vect{k}+\vect{K}}/\kappa) \f$.
 	 */
+
+	///=============== NEW GENERATION GENERAL 2D-IN-3D, including z != 0 =========================
+	
+	// TODO indexing mechanisms
+	
+	/// The constant factors for the long range part of a 2D Ewald sum.
+	complex double **S1_constfacs; // indices [y][j] where j is same as in [1, (4.5)]
+	/* These are the actual numbers now: (in the EWALD32_CONSTANTS_AGNOSTIC version)
+	 * for m + n EVEN:
+	 *
+	 * s1_constfacs[y(m,n)][x(j,s)] = 
+	 *
+	 *   -2 * I**(n+1) * sqrt(π) * ((n-m)/2)! * ((n+m)/2)! * (-1)**j /   j    \
+	 *   ----------------------------------------------------------- |        |
+	 *              j! * ((n - m - s)/2)! * ((n + m - s)/2)!         \ 2j - s /
+	 *
+	 * for m + n ODD:
+	 *
+	 * s1_constfacs[y(m,n)][j] = 0
+	 */
+	complex double *S1_constfacs_base; ///< Internal pointer holding memory for the 2D Ewald sum constant factors.
+	/// The constant factors for the long range part of a 1D Ewald sum along the \a z axis.
+	/** If the summation points lie along a different direction, use the formula for
+	 * 2D sum with additional factor of 
+	 * \f$ \sqrt{pi} \kappa \gamma(\abs{\vect{k}+\vect{K}}/\kappa) \f$.
+	 */
+
+
 	complex double **s1_constfacs_1Dz; 
 	/* These are the actual numbers now:
 	 * s1_constfacs_1Dz[n][j] =