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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT 257 10 "Function: " }{TEXT 262 
80 "RefPkg[XRef] - compute the row echelon form of a matrix and its LU
 factorization" }}{PARA 0 "" 0 "" {TEXT 256 17 "Calling Sequence:" }}
{PARA 0 "" 0 "" {TEXT 263 15 "RefPkg[XRef](A)" }}{PARA 0 "" 0 "" 
{TEXT 264 18 "RefPkg[XRef](A, m)" }}{PARA 259 "" 0 "" {TEXT -1 23 "Ref
Pkg[XRef](A, m, 'T')" }}{PARA 260 "" 0 "" {TEXT -1 28 "RefPkg[XRef](A,
 m, 'T', 'L')" }}{PARA 0 "" 0 "" {TEXT 268 33 "RefPkg[XRef](A, m, 'T',
 'L', 'P')" }}{PARA 0 "" 0 "" {TEXT 258 11 "Parameters:" }}{PARA 0 "" 
0 "" {TEXT 265 8 "A      -" }{TEXT -1 9 " a matrix" }}{PARA 0 "" 0 "" 
{TEXT 266 8 "m      -" }{TEXT -1 23 " a non-negative integer" }}{PARA 
0 "" 0 "" {TEXT 267 10 "T, L, P  -" }{TEXT -1 15 " variable names" }}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 259 12 "Description:" }}{PARA 256 "" 0 "
" {TEXT -1 135 "The call XRef(A) returns the row echelon form of the m
atrix A, where elimination stops after the last row/column, whichever \+
comes first" }}{PARA 257 "" 0 "" {TEXT -1 140 "The call XRef(A,m) retu
rns the row echelon form of the matrix A, where elimination stops in c
olumn m, unless it reaches the bottom row first" }}{PARA 261 "" 0 "" 
{TEXT -1 122 "optionally, T is assigned the transforming matrix, i.e.,
 T &* A = XRef(A, m), and L the inverse, i.e., A = L &* XRef(A, m)" }}
{PARA 262 "" 0 "" {TEXT -1 396 "if fifth argument P is given, a differ
ent method is performed in order to keep L lower triangular when row e
xchanges would have to be performed during the echelon form process.  \+
The process results in matrices such that A = L &* XRef(A,m) &* P wher
e P is a (column) permutation matrix.  The matrix xref(A,m) may not be
 in row echelon form but may have intervening 0 rows (in the first m c
olumns)." }}{PARA 258 "" 0 "" {TEXT -1 58 "if infolevel[RefPkg] > 2 a \+
trace of the process is printed" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT 260 
9 "Examples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "infolevel['R
efPkg']:=1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "with(RefPkg,
 XRef);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7#%%XRefG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "A := Matrix([ [1, 2, 1, 3], [3, -1, -3, -
1], [2, 3, 1, 4] ]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'RTABL
EG6$\"(%3OM-%'MATRIXG6#7%7&\"\"\"\"\"#F.\"\"$7&F0!\"\"!\"$F27&F/F0F.\"
\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "U := XRef(A, 3,'T','
L');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'RTABLEG6$\"(Oty$-%'MA
TRIXG6#7%7&\"\"\"\"\"#F.\"\"$7&\"\"!!\"(!\"'!#57&F2F2#!\"\"\"\"(#!\"%F
9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "T;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'RTABLEG6$\"(+\"3S-%'MATRIXG6#7%7%\"\"\"\"\"!F-7%!\"$
F,F-7%#!#6\"\"(#!\"\"F3F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 
"T . A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(K4:%-%'MATRI
XG6#7%7&\"\"\"\"\"#F,\"\"$7&\"\"!!\"(!\"'!#57&F0F0#!\"\"\"\"(#!\"%F7" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "L . U;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#-%'RTABLEG6$\"(C:n$-%'MATRIXG6#7%7&\"\"\"\"\"#F,\"\"$
7&F.!\"\"!\"$F07&F-F.F,\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 54 "B := Matrix(4,3,[[1,2,3], [0,0,0], [0,0,4], [0,1,5]]);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%'RTABLEG6$\"(_!=k-%'MATRIXG6#7&7%\"
\"\"\"\"#\"\"$7%\"\"!F2F27%F2F2\"\"%7%F2F.\"\"&" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "U2 := XRef(B, 3, 'T2', 'L2', 'P');" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#U2G-%'RTABLEG6$\"(;%3M-%'MATRIXG6#7&7%\"
\"\"\"\"$\"\"#7%\"\"!F2F27%F2\"\"%F27%F2F2F." }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 3 "L2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG
6$\"(+1<%-%'MATRIXG6#7&7&\"\"\"\"\"!F-F-7&F-F,F-F-7&F-F-F,F-7&F-F-#\"
\"&\"\"%F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "P;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(#fEY-%'MATRIXG6#7%7%\"\"\"\"
\"!F-7%F-F-F,7%F-F,F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "L2
 . U2 . P;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(W4C%-%'MA
TRIXG6#7&7%\"\"\"\"\"#\"\"$7%\"\"!F0F07%F0F0\"\"%7%F0F,\"\"&" }}}}
{SECT 0 {PARA 3 "" 0 "" {TEXT 261 9 "See Also:" }{HYPERLNK 17 "LinearA
lgebra[LUDecomposition]" 2 "LinearAlgebra[LUDecomposition]" "" }{TEXT 
269 2 ", " }{HYPERLNK 17 "RefPkg[Ref]" 2 "RefPkg[Ref]" "" }}}}{MARK "3
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