LU Decomposition

LU decomposition is a method to solve matrix questions. By finding the "pivot", the original matrix can be transfer to an upper triangular matrix. The unknown matrix then can be easily solved by simple algebra calculation. Take the following matrix for example, the c3Z=C, b2Y+b3Z=B, a1X+a2Y+a3Z=A. Put Z=C/c3 into above equations, the X and Y can be easily solved.

Tcl/Tk Code:
## LU Decomposition. Find X if AX=B by LU decomposition. After calculation X=B.
for {set i 0} {$i <= 1} {incr i 1} {
for {set k [expr {$i + 1}]} {$k <= 2} {incr k 1} {
set temp [expr {double($A($k,$i)) / double($A($i,$i))}]
for {set l [expr {$i + 1}]} {$l <= 2} {incr l 1} {
set A($k,$l) [expr {$A($k,$l) - ($A($i,$l) * $temp)}]
}
set B($k) [expr {$B($k) - (($B($i)) * $temp)}]
}
}
set B(2) [expr {$B(2) / $A(2,2)}]
for {set i 1} {$i >= 0} {incr i -1} {
for {set k [expr {$i + 1}]} {$k <= 2} {incr k 1} {
set B($i) [expr {$B($i) - $A($i,$k) * $B($k)}]
}
set B($i) [expr {$B($i) / $A($i,$i)}]
}