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tutorial_lud [2015/10/06 16:10]
sanjay
tutorial_lud [2016/12/12 10:12]
waruna Changed the latex notation to support new dokuwiki
Line 2: Line 2:
  
 The equation for LU Decomposition, derived from first principles using simple algebra in {{:foundations.pdf|Foundations}} (pg.3), is as follows: The equation for LU Decomposition, derived from first principles using simple algebra in {{:foundations.pdf|Foundations}} (pg.3), is as follows:
-<latex> + 
-    $U_{i,j}=\begin{cases}+/*<latex>*/ 
 +$
 +    U_{i,j}=\begin{cases}
 1=i\le j & A_{i,j}\\ 1=i\le j & A_{i,j}\\
 1<i\le j & A_{i,j}-\sum_{k=1}^{i-1}L_{i,k}U_{k,j} 1<i\le j & A_{i,j}-\sum_{k=1}^{i-1}L_{i,k}U_{k,j}
-\end{cases}+\end{cases}\\
  
 L_{i,j}=\begin{cases} L_{i,j}=\begin{cases}
 1 = i\le j & \frac{A_{i,j}}{U_{j,j}}\\ 1 = i\le j & \frac{A_{i,j}}{U_{j,j}}\\
 1< i\le j & \frac{1}{U_{j,j}}(A_{i,j}-\sum_{k=1}^{j-1}L_{i,k}U_{k,j}) 1< i\le j & \frac{1}{U_{j,j}}(A_{i,j}-\sum_{k=1}^{j-1}L_{i,k}U_{k,j})
-\end{cases}$ +\end{cases} 
-</latex>+$
 +/*<\latex>*/ 
  
 [Temp note due to : in the last case of L, the condition is "1 < j <= i"] [Temp note due to : in the last case of L, the condition is "1 < j <= i"]
Line 143: Line 147:
 L = case L = case
    {i,j|1==j} : (A / (i,j->j,j)@U);    {i,j|1==j} : (A / (i,j->j,j)@U);
-   {i,j|1<i} : (A - reduce(+, (i,j,k->i,j), (i,j,k->i,k)@L*(i,j,k->k,j)@U))/(i,j->i,i)@U;+   {i,j|1<i} : (A - reduce(+, (i,j,k->i,j), (i,j,k->i,k)@L*(i,j,k->k,j)@U))/(i,j->j,j)@U;
 esac; esac;
 </sxh> </sxh>
tutorial_lud.txt · Last modified: 2019/04/05 09:02 by sanjay