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HaarFormulaX.m

Usage : HaarFormulaX(a1x,a2x,a3x,a4x,a1y,a2y,a3y,a4y,b1,b2,b3,b4)
   
Input : Haar wavelet coefficients corresponding to
   
  \( a_{1}^{x}\ldots a_{4}^{x} \) the \( (x,x) \) diffusion coefficient
   
  \( a_{1}^{y}\ldots a_{4}^{y} \) the \( (y,y) \) diffusion coefficient
   
  \( b_{1}\ldots b_{4} \) the \( (x,y) \) diffusion coefficient


The output is the \( (x,x) \) Haar homogenized diffusion coefficient.


See also HaarSmoothing (Paragraph A.7), HaarFormulaY (Paragraph A.5) and HaarFormulaZ (Paragraph A.6). The formula is

\begin{displaymath}
((a_{1}^{x})^{3}a_{1}^{y}-(a_{1}^{y}a_{3}^{x})^{2}+(b_{2}(a_...
...2}-2(a_{3}^{x}a_{3}^{y}a_{4}^{x}+a_{2}^{x}a_{4}^{x}b_{2}\ldots \end{displaymath}


\begin{displaymath}
-2a_{3}^{x}b_{1}b_{2}+a_{4}^{x}b_{2}^{2}+a_{3}^{x}a_{4}^{x}b...
...a_{3}^{x}(a_{3}^{y}+b_{3}))b_{4}^{2}+2a_{4}^{x}b_{4}^{3}\ldots \end{displaymath}


\begin{displaymath}
+b_{4}^{4}+2a_{1}^{y}a_{3}^{x}(-(a_{3}^{x}b_{1})+a_{4}^{x}(a...
...a_{1}^{y})^{2}-(a_{3}^{y})^{2}+2a_{1}^{y}b_{1}-b_{2}^{2}\ldots \end{displaymath}


\begin{displaymath}
-2a_{3}^{y}b_{3}-b_{3}^{2}-b_{4}^{2})+a_{1}^{x}(-a_{1}^{y}((...
...x})^{2}+2a_{2}^{x}b_{2}+2b_{2}^{2})+2b_{2}(-(b_{1}b_{2})\ldots \end{displaymath}


\begin{displaymath}
+a_{4}^{x}(a_{3}^{y}+b_{3}))+2(-(a_{1}^{y}a_{4}^{x})+a_{3}^{...
...}+2b_{2})(a_{3}^{y}+b_{3}))b_{4}-2(a_{1}^{y}+b_{1})b_{4}^{2}))/\end{displaymath}


\begin{displaymath}
((a_{1}^{x})^{2}a_{1}^{y}-a_{1}^{y}(a_{2}^{x}+b_{2})^{2}+2(a...
..._{1})b_{4}^{2}+a_{1}^{x}((a_{1}^{y})^{2}+2a_{1}^{y}b_{1}\ldots \end{displaymath}


\begin{displaymath}
-(a_{3}^{y}+b_{3})^{2}-b_{4}^{2}))\end{displaymath}



Yves Capdeboscq 2002-01-15