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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: 2018/06/19 15:45 by sanjay