Functionals.md

@@ -12,4 +12,30 @@ The functional has the generic form:
     + [1-\alpha_{\downarrow}(n_{\uparrow},n_{\downarrow})]\dfrac{\bm{j}_{\downarrow}^2}{2n_{\downarrow}}
 \end{aligned}
 ```
-TODO
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+TODO
+# Stabilization of ASLDA functional
+In the case of calculations for trapped system term $`\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}`$ is source of numerical instabilities. Precisely, for small density regions we have:  
+* $`n_{\sigma}\rightarrow 0`$, 
+* $`\bm{j}_{\sigma}\rightarrow 0`$,
+* $`\frac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\rightarrow 0`$.  
+
+However, division of very small numbers is numerically not stable operations. For this reason, we introduce stabilization procedure:  
+```math
+\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}\longrightarrow f_{\textrm{reg.}}(n_{\sigma})\dfrac{\bm{j}_{\sigma}^2}{2n_{\sigma}}
+```
+Role of $`f_{\textrm{reg.}}`$ is to exclude from computation regions of small density. The function is controlled by two parameters in `predefines.h` file:
+```c
+/**
+ * Meaningful only in case of ASLDA.
+ * Paremeters defining stabilization procedure of ASLDA functional. 
+ * For regions with density smaller than ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 
+ * contribution from current term j^2/2n is assumed to be zero. 
+ * For regions with density above ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY 
+ * the contribution is assumed to be intact by stabilization procedure. 
+ * */
+#define ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY  1.0e-5
+#define ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY 1.0e-7
+```
+There meaning is presented on the figure below. 
+![stabilization](uploads/6528cf3b55ebb0d55a3e7f62a6124a83/stabilization.png)
+Smooth transition between densities `ASLDA_STABILIZATION_EXCLUDE_BELOW_DENISTY` and `ASLDA_STABILIZATION_RETAIN_ABOVE_DENSITY`
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