> For the complete documentation index, see [llms.txt](https://trizen.gitbook.io/perl6-rosettacode/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://trizen.gitbook.io/perl6-rosettacode/programming_tasks/g/gauss-jordan_matrix_inversion.md). # Gauss-Jordan matrix inversion Uses bits and pieces from other tasks, [Reduced row echelon form](https://rosettacode.org/wiki/Reduced_row_echelon_form#Raku) primarily. ```perl sub gauss-jordan-invert (@m where &is-square) { ^@m .map: { @m[$_].append: identity(+@m)[$_] }; @m.&rref[*]»[+@m .. *]; } sub is-square (@m) { so @m == all @m[*] } sub identity ($n) { [ 1, |(0 xx $n-1) ], *.rotate(-1).Array ... *.tail } # reduced row echelon form (from 'Gauss-Jordan elimination' task) sub rref (@m) { my ($lead, $rows, $cols) = 0, @m, @m[0]; for ^$rows -> $r { $lead < $cols or return @m; my $i = $r; until @m[$i;$lead] { ++$i == $rows or next; $i = $r; ++$lead == $cols and return @m; } @m[$i, $r] = @m[$r, $i] if $r != $i; @m[$r] »/=» $ = @m[$r;$lead]; for ^$rows -> $n { next if $n == $r; @m[$n] »-=» @m[$r] »×» (@m[$n;$lead] // 0); } ++$lead; } @m } sub rat-or-int ($num) { return $num unless $num ~~ Rat|FatRat; return $num.narrow if $num.narrow ~~ Int; $num.nude.join: '/'; } sub say_it ($message, @array) { my $max; @array.map: {$max max= max $_».&rat-or-int.comb(/\S+/)».chars}; say "\n$message"; $_».&rat-or-int.fmt(" %{$max}s").put for @array; } multi to-matrix ($str) { [$str.split(';').map(*.words.Array)] } multi to-matrix (@array) { @array } sub hilbert-matrix ($h) { [ (1..$h).map( -> $n { [ ($n ..^ $n + $h).map: { (1/$_).FatRat } ] } ) ] } my @tests = '1 2 3; 4 1 6; 7 8 9', '2 -1 0; -1 2 -1; 0 -1 2', '-1 -2 3 2; -4 -1 6 2; 7 -8 9 1; 1 -2 1 3', '1 2 3 4; 5 6 7 8; 9 33 11 12; 13 14 15 17', '3 1 8 9 6; 6 2 8 10 1; 5 7 2 10 3; 3 2 7 7 9; 3 5 6 1 1', # Test with a Hilbert matrix hilbert-matrix 10; @tests.map: { my @matrix = .&to-matrix; say_it( " {'=' x 20} Original Matrix: {'=' x 20}", @matrix ); say_it( ' Gauss-Jordan Inverted:', my @invert = gauss-jordan-invert @matrix ); say_it( ' Re-inverted:', gauss-jordan-invert @invert».Array ); } ``` #### Output: ``` ==================== Original Matrix: ==================== 1 2 3 4 1 6 7 8 9 Gauss-Jordan Inverted: -13/16 1/8 3/16 1/8 -1/4 1/8 25/48 1/8 -7/48 Re-inverted: 1 2 3 4 1 6 7 8 9 ==================== Original Matrix: ==================== 2 -1 0 -1 2 -1 0 -1 2 Gauss-Jordan Inverted: 3/4 1/2 1/4 1/2 1 1/2 1/4 1/2 3/4 Re-inverted: 2 -1 0 -1 2 -1 0 -1 2 ==================== Original Matrix: ==================== -1 -2 3 2 -4 -1 6 2 7 -8 9 1 1 -2 1 3 Gauss-Jordan Inverted: -21/23 17/69 13/138 19/46 -38/23 15/23 1/23 15/23 -16/23 25/69 11/138 9/46 -13/23 16/69 -2/69 13/23 Re-inverted: -1 -2 3 2 -4 -1 6 2 7 -8 9 1 1 -2 1 3 ==================== Original Matrix: ==================== 1 2 3 4 5 6 7 8 9 33 11 12 13 14 15 17 Gauss-Jordan Inverted: 19/184 -199/184 -1/46 1/2 1/23 -2/23 1/23 0 -441/184 813/184 -1/46 -3/2 2 -3 0 1 Re-inverted: 1 2 3 4 5 6 7 8 9 33 11 12 13 14 15 17 ==================== Original Matrix: ==================== 3 1 8 9 6 6 2 8 10 1 5 7 2 10 3 3 2 7 7 9 3 5 6 1 1 Gauss-Jordan Inverted: -4525/6238 2529/6238 -233/3119 1481/3119 -639/6238 1033/6238 -1075/6238 342/3119 -447/3119 871/6238 1299/6238 -289/6238 -204/3119 -390/3119 739/6238 782/3119 -222/3119 237/3119 -556/3119 -177/3119 -474/3119 -17/3119 -24/3119 688/3119 -140/3119 Re-inverted: 3 1 8 9 6 6 2 8 10 1 5 7 2 10 3 3 2 7 7 9 3 5 6 1 1 ==================== Original Matrix: ==================== 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/19 Gauss-Jordan Inverted: 100 -4950 79200 -600600 2522520 -6306300 9609600 -8751600 4375800 -923780 -4950 326700 -5880600 47567520 -208107900 535134600 -832431600 770140800 -389883780 83140200 79200 -5880600 112907520 -951350400 4281076800 -11237826600 17758540800 -16635041280 8506555200 -1829084400 -600600 47567520 -951350400 8245036800 -37875637800 101001700800 -161602721280 152907955200 -78843164400 17071454400 2522520 -208107900 4281076800 -37875637800 176752976400 -477233036280 771285715200 -735869534400 382086104400 -83223340200 -6306300 535134600 -11237826600 101001700800 -477233036280 1301544644400 -2121035716800 2037792556800 -1064382719400 233025352560 9609600 -832431600 17758540800 -161602721280 771285715200 -2121035716800 3480673996800 -3363975014400 1766086882560 -388375587600 -8751600 770140800 -16635041280 152907955200 -735869534400 2037792556800 -3363975014400 3267861442560 -1723286307600 380449555200 4375800 -389883780 8506555200 -78843164400 382086104400 -1064382719400 1766086882560 -1723286307600 912328045200 -202113826200 -923780 83140200 -1829084400 17071454400 -83223340200 233025352560 -388375587600 380449555200 -202113826200 44914183600 Re-inverted: 1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/6 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/7 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/8 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/9 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/10 1/11 1/12 1/13 1/14 1/15 1/16 1/17 1/18 1/19 ``` --- # Agent Instructions This documentation is published with GitBook. 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