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Gauss-Jordan matrix inversion
{{Draft task}}[[Category:Matrices]]
;Task: Invert matrix '''A''' using Gauss-Jordan method.
'''A''' being an '''n''' by '''n''' matrix.
C#
using System; namespace Rosetta { internal class Vector { private double[] b; internal readonly int rows; internal Vector(int rows) { this.rows = rows; b = new double[rows]; } internal Vector(double[] initArray) { b = (double[])initArray.Clone(); rows = b.Length; } internal Vector Clone() { Vector v = new Vector(b); return v; } internal double this[int row] { get { return b[row]; } set { b[row] = value; } } internal void SwapRows(int r1, int r2) { if (r1 == r2) return; double tmp = b[r1]; b[r1] = b[r2]; b[r2] = tmp; } internal double norm(double[] weights) { double sum = 0; for (int i = 0; i < rows; i++) { double d = b[i] * weights[i]; sum += d*d; } return Math.Sqrt(sum); } internal void print() { for (int i = 0; i < rows; i++) Console.WriteLine(b[i]); Console.WriteLine(); } public static Vector operator-(Vector lhs, Vector rhs) { Vector v = new Vector(lhs.rows); for (int i = 0; i < lhs.rows; i++) v[i] = lhs[i] - rhs[i]; return v; } } class Matrix { private double[] b; internal readonly int rows, cols; internal Matrix(int rows, int cols) { this.rows = rows; this.cols = cols; b = new double[rows * cols]; } internal Matrix(int size) { this.rows = size; this.cols = size; b = new double[rows * cols]; for (int i = 0; i < size; i++) this[i, i] = 1; } internal Matrix(int rows, int cols, double[] initArray) { this.rows = rows; this.cols = cols; b = (double[])initArray.Clone(); if (b.Length != rows * cols) throw new Exception("bad init array"); } internal double this[int row, int col] { get { return b[row * cols + col]; } set { b[row * cols + col] = value; } } public static Vector operator*(Matrix lhs, Vector rhs) { if (lhs.cols != rhs.rows) throw new Exception("I can't multiply matrix by vector"); Vector v = new Vector(lhs.rows); for (int i = 0; i < lhs.rows; i++) { double sum = 0; for (int j = 0; j < rhs.rows; j++) sum += lhs[i,j]*rhs[j]; v[i] = sum; } return v; } internal void SwapRows(int r1, int r2) { if (r1 == r2) return; int firstR1 = r1 * cols; int firstR2 = r2 * cols; for (int i = 0; i < cols; i++) { double tmp = b[firstR1 + i]; b[firstR1 + i] = b[firstR2 + i]; b[firstR2 + i] = tmp; } } //with partial pivot internal bool InvPartial() { const double Eps = 1e-12; if (rows != cols) throw new Exception("rows != cols for Inv"); Matrix M = new Matrix(rows); //unitary for (int diag = 0; diag < rows; diag++) { int max_row = diag; double max_val = Math.Abs(this[diag, diag]); double d; for (int row = diag + 1; row < rows; row++) if ((d = Math.Abs(this[row, diag])) > max_val) { max_row = row; max_val = d; } if (max_val <= Eps) return false; SwapRows(diag, max_row); M.SwapRows(diag, max_row); double invd = 1 / this[diag, diag]; for (int col = diag; col < cols; col++) { this[diag, col] *= invd; } for (int col = 0; col < cols; col++) { M[diag, col] *= invd; } for (int row = 0; row < rows; row++) { d = this[row, diag]; if (row != diag) { for (int col = diag; col < this.cols; col++) { this[row, col] -= d * this[diag, col]; } for (int col = 0; col < this.cols; col++) { M[row, col] -= d * M[diag, col]; } } } } b = M.b; return true; } internal void print() { for (int i = 0; i < rows; i++) { for (int j = 0; j < cols; j++) Console.Write(this[i,j].ToString()+" "); Console.WriteLine(); } } } }
using System; namespace Rosetta { class Program { static void Main(string[] args) { Matrix M = new Matrix(4, 4, new double[] { -1, -2, 3, 2, -4, -1, 6, 2, 7, -8, 9, 1, 1, -2, 1, 3 }); M.InvPartial(); M.print(); } } }
{{out}}
-0.913043478260869 0.246376811594203 0.0942028985507246 0.413043478260869
-1.65217391304348 0.652173913043478 0.0434782608695652 0.652173913043478
-0.695652173913043 0.36231884057971 0.0797101449275362 0.195652173913043
-0.565217391304348 0.231884057971014 -0.0289855072463768 0.565217391304348
Go
{{trans|Kotlin}}
package main import "fmt" type vector = []float64 type matrix []vector func (m matrix) inverse() matrix { le := len(m) for _, v := range m { if len(v) != le { panic("Not a square matrix") } } aug := make(matrix, le) for i := 0; i < le; i++ { aug[i] = make(vector, 2*le) copy(aug[i], m[i]) // augment by identity matrix to right aug[i][i+le] = 1 } aug.toReducedRowEchelonForm() inv := make(matrix, le) // remove identity matrix to left for i := 0; i < le; i++ { inv[i] = make(vector, le) copy(inv[i], aug[i][le:]) } return inv } // note: this mutates the matrix in place func (m matrix) toReducedRowEchelonForm() { lead := 0 rowCount, colCount := len(m), len(m[0]) for r := 0; r < rowCount; r++ { if colCount <= lead { return } i := r for m[i][lead] == 0 { i++ if rowCount == i { i = r lead++ if colCount == lead { return } } } m[i], m[r] = m[r], m[i] if div := m[r][lead]; div != 0 { for j := 0; j < colCount; j++ { m[r][j] /= div } } for k := 0; k < rowCount; k++ { if k != r { mult := m[k][lead] for j := 0; j < colCount; j++ { m[k][j] -= m[r][j] * mult } } } lead++ } } func (m matrix) print(title string) { fmt.Println(title) for _, v := range m { fmt.Printf("% f\n", v) } fmt.Println() } func main() { a := matrix{{1, 2, 3}, {4, 1, 6}, {7, 8, 9}} a.inverse().print("Inverse of A is:\n") b := matrix{{2, -1, 0}, {-1, 2, -1}, {0, -1, 2}} b.inverse().print("Inverse of B is:\n") }
{{out}}
Inverse of A is:
[-0.812500 0.125000 0.187500]
[ 0.125000 -0.250000 0.125000]
[ 0.520833 0.125000 -0.145833]
Inverse of B is:
[ 0.750000 0.500000 0.250000]
[ 0.500000 1.000000 0.500000]
[ 0.250000 0.500000 0.750000]
J
'''Solution:'''
Uses Gauss-Jordan implementation (as described in [[Reduced_row_echelon_form#J]]) to find reduced row echelon form of the matrix after augmenting with an identity matrix.
require 'math/misc/linear'
augmentR_I1=: ,. e.@i.@# NB. augment matrix on the right with its Identity matrix
matrix_invGJ=: # }."1 [: gauss_jordan@augmentR_I1
'''Usage:'''
]A =: 1 2 3, 4 1 6,: 7 8 9
1 2 3
4 1 6
7 8 9
matrix_invGJ A
_0.8125 0.125 0.1875
0.125 _0.25 0.125
0.520833 0.125 _0.145833
Julia
{{works with|Julia|0.6}}
'''Built-in''' LAPACK-based linear solver uses partial-pivoted Gauss elimination):
A = [1 2 3; 4 1 6; 7 8 9] @show I / A @show inv(A)
'''Native implementation''':
function gaussjordan(A::Matrix) size(A, 1) == size(A, 2) || throw(ArgumentError("A must be squared")) n = size(A, 1) M = [convert(Matrix{float(eltype(A))}, A) I] i = 1 local tmp = Vector{eltype(M)}(2n) # forward while i ≤ n if M[i, i] ≈ 0.0 local j = i + 1 while j ≤ n && M[j, i] ≈ 0.0 j += 1 end if j ≤ n tmp .= M[i, :] M[i, :] .= M[j, :] M[j, :] .= tmp else throw(ArgumentError("matrix is singular, cannot compute the inverse")) end end for j in (i + 1):n M[j, :] .-= M[j, i] / M[i, i] .* M[i, :] end i += 1 end i = n # backward while i ≥ 1 if M[i, i] ≈ 0.0 local j = i - 1 while j ≥ 1 && M[j, i] ≈ 0.0 j -= 1 end if j ≥ 1 tmp .= M[i, :] M[i, :] .= M[j, :] M[j, :] .= tmp else throw(ArgumentError("matrix is singular, cannot compute the inverse")) end end for j in (i - 1):-1:1 M[j, :] .-= M[j, i] / M[i, i] .* M[i, :] end i -= 1 end M ./= diag(M) # normalize return M[:, n+1:2n] end @show gaussjordan(A) @assert gaussjordan(A) ≈ inv(A) A = rand(10, 10) @assert gaussjordan(A) ≈ inv(A)
{{out}}
I / A = [-0.8125 0.125 0.1875; 0.125 -0.25 0.125; 0.520833 0.125 -0.145833]
inv(A) = [-0.8125 0.125 0.1875; 0.125 -0.25 0.125; 0.520833 0.125 -0.145833]
gaussjordan(A) = [-0.8125 0.125 0.1875; 0.125 -0.25 0.125; 0.520833 0.125 -0.145833]
Kotlin
This follows the description of Gauss-Jordan elimination in Wikipedia whereby the original square matrix is first augmented to the right by its identity matrix, its reduced row echelon form is then found and finally the identity matrix to the left is removed to leave the inverse of the original square matrix.
// version 1.2.21 typealias Matrix = Array<DoubleArray> fun Matrix.inverse(): Matrix { val len = this.size require(this.all { it.size == len }) { "Not a square matrix" } val aug = Array(len) { DoubleArray(2 * len) } for (i in 0 until len) { for (j in 0 until len) aug[i][j] = this[i][j] // augment by identity matrix to right aug[i][i + len] = 1.0 } aug.toReducedRowEchelonForm() val inv = Array(len) { DoubleArray(len) } // remove identity matrix to left for (i in 0 until len) { for (j in len until 2 * len) inv[i][j - len] = aug[i][j] } return inv } fun Matrix.toReducedRowEchelonForm() { var lead = 0 val rowCount = this.size val colCount = this[0].size for (r in 0 until rowCount) { if (colCount <= lead) return var i = r while (this[i][lead] == 0.0) { i++ if (rowCount == i) { i = r lead++ if (colCount == lead) return } } val temp = this[i] this[i] = this[r] this[r] = temp if (this[r][lead] != 0.0) { val div = this[r][lead] for (j in 0 until colCount) this[r][j] /= div } for (k in 0 until rowCount) { if (k != r) { val mult = this[k][lead] for (j in 0 until colCount) this[k][j] -= this[r][j] * mult } } lead++ } } fun Matrix.printf(title: String) { println(title) val rowCount = this.size val colCount = this[0].size for (r in 0 until rowCount) { for (c in 0 until colCount) { if (this[r][c] == -0.0) this[r][c] = 0.0 // get rid of negative zeros print("${"% 10.6f".format(this[r][c])} ") } println() } println() } fun main(args: Array<String>) { val a = arrayOf( doubleArrayOf(1.0, 2.0, 3.0), doubleArrayOf(4.0, 1.0, 6.0), doubleArrayOf(7.0, 8.0, 9.0) ) a.inverse().printf("Inverse of A is :\n") val b = arrayOf( doubleArrayOf( 2.0, -1.0, 0.0), doubleArrayOf(-1.0, 2.0, -1.0), doubleArrayOf( 0.0, -1.0, 2.0) ) b.inverse().printf("Inverse of B is :\n") }
{{out}}
Inverse of A is :
-0.812500 0.125000 0.187500
0.125000 -0.250000 0.125000
0.520833 0.125000 -0.145833
Inverse of B is :
0.750000 0.500000 0.250000
0.500000 1.000000 0.500000
0.250000 0.500000 0.750000
Perl
Included code from [[Reduced_row_echelon_form#Perl|Reduced row echelon form]] task.
sub rref { our @m; local *m = shift; @m or return; my ($lead, $rows, $cols) = (0, scalar(@m), scalar(@{$m[0]})); foreach my $r (0 .. $rows - 1) { $lead < $cols or return; my $i = $r; until ($m[$i][$lead]) {++$i == $rows or next; $i = $r; ++$lead == $cols and return;} @m[$i, $r] = @m[$r, $i]; my $lv = $m[$r][$lead]; $_ /= $lv foreach @{ $m[$r] }; my @mr = @{ $m[$r] }; foreach my $i (0 .. $rows - 1) {$i == $r and next; ($lv, my $n) = ($m[$i][$lead], -1); $_ -= $lv * $mr[++$n] foreach @{ $m[$i] };} ++$lead;} } sub display { join("\n" => map join(" " => map(sprintf("%6.2f", $_), @$_)), @{+shift})."\n" } sub gauss_jordan_invert { my(@m) = @_; my $rows = @m; my @i = identity(scalar @m); push @{$m[$_]}, @{$i[$_]} for 0..$rows-1; rref(\@m); map { splice @$_, 0, $rows } @m; @m; } sub identity { my($n) = @_; map { [ (0) x $_, 1, (0) x ($n-1 - $_) ] } 0..$n-1 } my @tests = ( [ [ 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 ] ], ); for my $matrix (@tests) { print "Original Matrix:\n" . display(\@$matrix) . "\n"; my @gj = gauss_jordan_invert( @$matrix ); print "Gauss-Jordan Inverted Matrix:\n" . display(\@gj) . "\n"; my @rt = gauss_jordan_invert( @gj ); print "After round-trip:\n" . display(\@rt) . "\n";} . "\n" }
{{out}}
Original Matrix:
2.00 -1.00 0.00
-1.00 2.00 -1.00
0.00 -1.00 2.00
Gauss-Jordan Inverted Matrix:
0.75 0.50 0.25
0.50 1.00 0.50
0.25 0.50 0.75
After round-trip:
2.00 -1.00 0.00
-1.00 2.00 -1.00
0.00 -1.00 2.00
Original Matrix:
-1.00 -2.00 3.00 2.00
-4.00 -1.00 6.00 2.00
7.00 -8.00 9.00 1.00
1.00 -2.00 1.00 3.00
Gauss-Jordan Inverted Matrix:
-0.91 0.25 0.09 0.41
-1.65 0.65 0.04 0.65
-0.70 0.36 0.08 0.20
-0.57 0.23 -0.03 0.57
After round-trip:
-1.00 -2.00 3.00 2.00
-4.00 -1.00 6.00 2.00
7.00 -8.00 9.00 1.00
1.00 -2.00 1.00 3.00
Perl 6
{{works with|Rakudo|2019.03.1}} Uses bits and pieces from other tasks, [[Reduced_row_echelon_form#Perl_6|Reduced row echelon form]] primarily.
sub gauss-jordan-invert (@m where *.&is-square) {
^@m .map: { @m[$_].append: identity(+@m)[$_] };
@m.&rref[*]»[+@m .. *];
}
sub is-square (@m) { so @m == all @m[*] }
sub identity ($n) { [ 1, |(0 xx $n-1) ], *.rotate(-1) ... *.tail }
# reduced row echelon form (Gauss-Jordan elimination)
sub rref (@m) {
return unless @m;
my ($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;
my $lv = @m[$r;$lead];
@m[$r] »/=» $lv;
for ^$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;
return $num.narrow if $num.narrow.WHAT ~~ 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;
}
sub to-matrix ($str) { [$str.split(';').map(*.words.Array)] }
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',
'-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';
@tests.map: {
my @matrix = .&to-matrix;
say_it( 'Original Matrix:', @matrix );
say_it( 'Gauss-Jordan Inverted Matrix:', gauss-jordan-invert @matrix );
}
{{out}}
Original Matrix:
1 2 3
4 1 6
7 8 9
Gauss-Jordan Inverted Matrix:
-13/16 1/8 3/16
1/8 -1/4 1/8
25/48 1/8 -7/48
Original Matrix:
2 -1 0
-1 2 -1
0 -1 2
Gauss-Jordan Inverted Matrix:
3/4 1/2 1/4
1/2 1 1/2
1/4 1/2 3/4
Original Matrix:
-1 -2 3 2
-4 -1 6 2
7 -8 9 1
1 -2 1 3
Gauss-Jordan Inverted Matrix:
-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
Original Matrix:
1 2 3 4
5 6 7 8
9 33 11 12
13 14 15 17
Gauss-Jordan Inverted Matrix:
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
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 Matrix:
-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
Original Matrix:
-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
Gauss-Jordan Inverted 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
Phix
{{trans|Kotlin}} uses ToReducedRowEchelonForm() from [[Reduced_row_echelon_form#Phix]]
function inverse(sequence mat)
integer len = length(mat)
sequence aug = repeat(repeat(0,2*len),len)
for i=1 to len do
if length(mat[i])!=len then ?9/0 end if -- "Not a square matrix"
for j=1 to len do
aug[i][j] = mat[i][j]
end for
-- augment by identity matrix to right
aug[i][i + len] = 1
end for
aug = ToReducedRowEchelonForm(aug)
sequence inv = repeat(repeat(0,len),len)
-- remove identity matrix to left
for i=1 to len do
for j=len+1 to 2*len do
inv[i][j-len] = aug[i][j]
end for
end for
return inv
end function
constant test = {{ 2, -1, 0},
{-1, 2, -1},
{ 0, -1, 2}}
pp(inverse(test),{pp_Nest,1})
{{out}}
{{0.75,0.5,0.25},
{0.5,1,0.5},
{0.25,0.5,0.75}}
PowerShell
function gauss-jordan-inv([double[][]]$a) { $n = $a.count [double[][]]$b = 0..($n-1) | foreach{[double[]]$row = @(0) * $n; $row[$_] = 1; ,$row} for ($k = 0; $k -lt $n; $k++) { $lmax, $max = $k, [Math]::Abs($a[$k][$k]) for ($l = $k+1; $l -lt $n; $l++) { $tmp = [Math]::Abs($a[$l][$k]) if($max -lt $tmp) { $max, $lmax = $tmp, $l } } if ($k -ne $lmax) { $a[$k], $a[$lmax] = $a[$lmax], $a[$k] $b[$k], $b[$lmax] = $b[$lmax], $b[$k] } $akk = $a[$k][$k] if (0 -eq $akk) {throw "Irregular matrix"} for ($j = 0; $j -lt $n; $j++) { $a[$k][$j] /= $akk $b[$k][$j] /= $akk } for ($i = 0; $i -lt $n; $i++){ if ($i -ne $k) { $aik = $a[$i][$k] for ($j = 0; $j -lt $n; $j++) { $a[$i][$j] -= $a[$k][$j]*$aik $b[$i][$j] -= $b[$k][$j]*$aik } } } } $b } function show($a) { $a | foreach{ "$_"} } $a = @(@(@(1, 2, 3), @(4, 1, 6), @(7, 8, 9))) $inva = gauss-jordan-inv $a "a =" show $a "" "inv(a) =" show $inva "" $b = @(@(2, -1, 0), @(-1, 2, -1), @(0, -1, 2)) "b =" show $b "" $invb = gauss-jordan-inv $b "inv(b) =" show $invb
Output:
a =
1 2 3
4 1 6
7 8 9
inv(a) =
-0.8125 0.125 0.1875
0.125 -0.25 0.125
0.520833333333333 0.125 -0.145833333333333
b =
2 -1 0
-1 2 -1
0 -1 2
inv(b) =
0.75 0.5 0.25
0.5 1 0.5
0.25 0.5 0.75
Racket
Using the matrix library that comes with default Racket installation
#lang racket
(require math/matrix
math/array)
(define (inverse M)
(define dim (square-matrix-size M))
(define MI (matrix-augment (list M (identity-matrix dim))))
(submatrix (matrix-row-echelon MI #t #t) (::) (:: dim #f)))
(define A (matrix [[1 2 3] [4 1 6] [7 8 9]]))
(inverse A)
(matrix-inverse A)
{{out}}
(array #[#[-13/16 1/8 3/16] #[1/8 -1/4 1/8] #[25/48 1/8 -7/48]])
(array #[#[-13/16 1/8 3/16] #[1/8 -1/4 1/8] #[25/48 1/8 -7/48]])
REXX
/* REXX */
Parse Arg seed nn
If seed='' Then
seed=23345
If nn='' Then nn=5
If seed='?' Then Do
Say 'rexx gjmi seed n computes a random matrix with n rows and columns'
Say 'Default is 23345 5'
Exit
End
Numeric Digits 50
Call random 1,2,seed
a=''
Do i=1 To nn**2
a=a random(9)+1
End
n2=words(a)
Do n=2 To n2/2
If n**2=n2 Then
Leave
End
If n>n2/2 Then
Call exit 'Not a square matrix:' a '('n2 'elements).'
det=determinante(a,n)
If det=0 Then
Call exit 'Determinant is 0'
Do j=1 To n
Do i=1 To n
Parse Var A a.i.j a
aa.i.j=a.i.j
End
Do ii=1 To n
z=(ii=j)
iii=ii+n
a.iii.j=z
End
End
Call show 1,'The given matrix'
Do m=1 To n-1
If a.m.m=0 Then Do
Do j=m+1 To n
If a.m.j<>0 Then Leave
End
If j>n Then Do
Say 'No pivot>0 found in column' m
Exit
End
Do i=1 To n*2
temp=a.i.m
a.i.m=a.i.j
a.i.j=temp
End
End
Do j=m+1 To n
If a.m.j<>0 Then Do
jj=m
fact=divide(a.m.m,a.m.j)
Do i=1 To n*2
a.i.j=subtract(multiply(a.i.j,fact),a.i.jj)
End
End
End
Call show 2 m
End
Say 'Lower part has all zeros'
Say ''
Do j=1 To n
If denom(a.j.j)<0 Then Do
Do i=1 To 2*n
a.i.j=subtract(0,a.i.j)
End
End
End
Call show 3
Do m=n To 2 By -1
Do j=1 To m-1
jj=m
fact=divide(a.m.j,a.m.jj)
Do i=1 To n*2
a.i.j=subtract(a.i.j,multiply(a.i.jj,fact))
End
End
Call show 4 m
End
Say 'Upper half has all zeros'
Say ''
Do j=1 To n
If decimal(a.j.j)<>1 Then Do
z=a.j.j
Do i=1 To 2*n
a.i.j=divide(a.i.j,z)
End
End
End
Call show 5
Say 'Main diagonal has all ones'
Say ''
Do j=1 To n
Do i=1 To n
z=i+n
a.i.j=a.z.j
End
End
Call show 6,'The inverse matrix'
do i = 1 to n
do j = 1 to n
sum=0
Do k=1 To n
sum=add(sum,multiply(aa.i.k,a.k.j))
End
c.i.j = sum
end
End
Call showc 7,'The product of input and inverse matrix'
Exit
show:
Parse Arg num,text
Say 'show' arg(1) text
If arg(1)<>6 Then rows=n*2
Else rows=n
len=0
Do j=1 To n
Do i=1 To rows
len=max(len,length(a.i.j))
End
End
Do j=1 To n
ol=''
Do i=1 To rows
ol=ol||right(a.i.j,len+1)
End
Say ol
End
Say ''
Return
showc:
Parse Arg num,text
Say text
clen=0
Do j=1 To n
Do i=1 To n
clen=max(clen,length(c.i.j))
End
End
Do j=1 To n
ol=''
Do i=1 To n
ol=ol||right(c.i.j,clen+1)
End
Say ol
End
Say ''
Return
denom: Procedure
/* Return the denominator */
Parse Arg d '/' n
Return d
decimal: Procedure
/* compute the fraction's value */
Parse Arg a
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
Return ad/an
gcd: procedure
/**********************************************************************
* Greatest commn divisor
**********************************************************************/
Parse Arg a,b
If b = 0 Then Return abs(a)
Return gcd(b,a//b)
add: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
sum=divide(ad*bn+bd*an,an*bn)
Return sum
multiply: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
prd=divide(ad*bd,an*bn)
Return prd
subtract: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
div=divide(ad*bn-bd*an,an*bn)
Return div
divide: Procedure
Parse Arg a,b
If pos('/',a)=0 Then a=a'/1'; Parse Var a ad '/' an
If pos('/',b)=0 Then b=b'/1'; Parse Var b bd '/' bn
sd=ad*bn
sn=an*bd
g=gcd(sd,sn)
Select
When sd=0 Then res='0'
When abs(sn/g)=1 Then res=(sd/g)*sign(sn/g)
Otherwise Do
den=sd/g
nom=sn/g
If nom<0 Then Do
If den<0 Then den=abs(den)
Else den=-den
nom=abs(nom)
End
res=den'/'nom
End
End
Return res
determinante: Procedure
/* REXX ***************************************************************
* determinant.rex
* compute the determinant of the given square matrix
* Input: as: the representation of the matrix as vector (n**2 elements)
* 21.05.2013 Walter Pachl
**********************************************************************/
Parse Arg as,n
Do i=1 To n
Do j=1 To n
Parse Var as a.i.j as
End
End
Select
When n=2 Then det=subtract(multiply(a.1.1,a.2.2),multiply(a.1.2,a.2.1))
When n=3 Then Do
det=multiply(multiply(a.1.1,a.2.2),a.3.3)
det=add(det,multiply(multiply(a.1.2,a.2.3),a.3.1))
det=add(det,multiply(multiply(a.1.3,a.2.1),a.3.2))
det=subtract(det,multiply(multiply(a.1.3,a.2.2),a.3.1))
det=subtract(det,multiply(multiply(a.1.2,a.2.1),a.3.3))
det=subtract(det,multiply(multiply(a.1.1,a.2.3),a.3.2))
End
Otherwise Do
det=0
Do k=1 To n
sign=((-1)**(k+1))
If sign=1 Then
det=add(det,multiply(a.1.k,determinante(subm(k),n-1)))
Else
det=subtract(det,multiply(a.1.k,determinante(subm(k),n-1)))
End
End
End
Return det
subm: Procedure Expose a. n
/**********************************************************************
* compute the submatrix resulting when row 1 and column k are removed
* Input: a.*.*, k
* Output: bs the representation of the submatrix as vector
**********************************************************************/
Parse Arg k
bs=''
do i=2 To n
Do j=1 To n
If j=k Then Iterate
bs=bs a.i.j
End
End
Return bs
Exit: Say arg(1)
{{out}} Using the defaults for seed and n
show 1 The given matrix
10 3 8 6 3 1 0 0 0 0
5 7 8 8 2 0 1 0 0 0
4 10 5 4 7 0 0 1 0 0
9 4 5 3 3 0 0 0 1 0
6 3 3 3 7 0 0 0 0 1
show 2 1
10 3 8 6 3 1 0 0 0 0
0 11 8 10 1 -1 2 0 0 0
0 22 9/2 4 29/2 -1 0 5/2 0 0
0 13/9 -22/9 -8/3 1/3 -1 0 0 10/9 0
0 2 -3 -1 26/3 -1 0 0 0 5/3
show 2 2
10 3 8 6 3 1 0 0 0 0
0 11 8 10 1 -1 2 0 0 0
0 0 -23/4 -8 25/4 1/2 -2 5/4 0 0
0 0 -346/13 -394/13 20/13 -86/13 -2 0 110/13 0
0 0 -49/2 -31/2 140/3 -9/2 -2 0 0 55/6
show 2 3
10 3 8 6 3 1 0 0 0 0
0 11 8 10 1 -1 2 0 0 0
0 0 -23/4 -8 25/4 1/2 -2 5/4 0 0
0 0 0 1005/692 -4095/692 -1335/692 1085/692 -5/4 1265/692 0
0 0 0 855/196 395/84 -305/196 75/49 -5/4 0 1265/588
show 2 4
10 3 8 6 3 1 0 0 0 0
0 11 8 10 1 -1 2 0 0 0
0 0 -23/4 -8 25/4 1/2 -2 5/4 0 0
0 0 0 1005/692 -4095/692 -1335/692 1085/692 -5/4 1265/692 0
0 0 0 0 221375/29583 13915/9861 -13915/13148 16445/19722 -1265/692 84755/118332
Lower part has all zeros
show 3
10 3 8 6 3 1 0 0 0 0
0 11 8 10 1 -1 2 0 0 0
0 0 23/4 8 -25/4 -1/2 2 -5/4 0 0
0 0 0 1005/692 -4095/692 -1335/692 1085/692 -5/4 1265/692 0
0 0 0 0 221375/29583 13915/9861 -13915/13148 16445/19722 -1265/692 84755/118332
show 4 5
10 3 8 6 0 76/175 297/700 -117/350 513/700 -201/700
0 11 8 10 0 -208/175 1499/700 -39/350 171/700 -67/700
0 0 23/4 8 0 19/28 125/112 -31/56 -171/112 67/112
0 0 0 1005/692 0 -1407/1730 10117/13840 -4087/6920 5293/13840 7839/13840
0 0 0 0 221375/29583 13915/9861 -13915/13148 16445/19722 -1265/692 84755/118332
show 4 4
10 3 8 0 0 664/175 -1817/700 737/350 -593/700 -1839/700
0 11 8 0 0 772/175 -6073/2100 4153/1050 -5017/2100 -2797/700
0 0 23/4 0 0 3611/700 -24449/8400 11339/4200 -30521/8400 -7061/2800
0 0 0 1005/692 0 -1407/1730 10117/13840 -4087/6920 5293/13840 7839/13840
0 0 0 0 221375/29583 13915/9861 -13915/13148 16445/19722 -1265/692 84755/118332
show 4 3
10 3 0 0 0 -592/175 3053/2100 -1733/1050 8837/2100 617/700
0 11 0 0 0 -484/175 2431/2100 209/1050 5599/2100 -341/700
0 0 23/4 0 0 3611/700 -24449/8400 11339/4200 -30521/8400 -7061/2800
0 0 0 1005/692 0 -1407/1730 10117/13840 -4087/6920 5293/13840 7839/13840
0 0 0 0 221375/29583 13915/9861 -13915/13148 16445/19722 -1265/692 84755/118332
show 4 2
10 0 0 0 0 -92/35 239/210 -179/105 731/210 71/70
0 11 0 0 0 -484/175 2431/2100 209/1050 5599/2100 -341/700
0 0 23/4 0 0 3611/700 -24449/8400 11339/4200 -30521/8400 -7061/2800
0 0 0 1005/692 0 -1407/1730 10117/13840 -4087/6920 5293/13840 7839/13840
0 0 0 0 221375/29583 13915/9861 -13915/13148 16445/19722 -1265/692 84755/118332
Upper half has all zeros
show 5
1 0 0 0 0 -46/175 239/2100 -179/1050 731/2100 71/700
0 1 0 0 0 -44/175 221/2100 19/1050 509/2100 -31/700
0 0 1 0 0 157/175 -1063/2100 493/1050 -1327/2100 -307/700
0 0 0 1 0 -14/25 151/300 -61/150 79/300 39/100
0 0 0 0 1 33/175 -99/700 39/350 -171/700 67/700
Main diagonal has all ones
show 6 The inverse matrix
-46/175 239/2100 -179/1050 731/2100 71/700
-44/175 221/2100 19/1050 509/2100 -31/700
157/175 -1063/2100 493/1050 -1327/2100 -307/700
-14/25 151/300 -61/150 79/300 39/100
33/175 -99/700 39/350 -171/700 67/700
The product of input and inverse matrix
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
Sidef
Uses the '''rref(M)''' function from [https://rosettacode.org/wiki/Reduced_row_echelon_form#Sidef Reduced row echelon form]. {{trans|Perl 6}}
func gauss_jordan_invert (M) { var I = M.len.of {|i| M.len.of {|j| i == j ? 1 : 0 } } var A = gather { ^M -> each {|i| take(M[i] + I[i]) } } rref(A).map { .last(M.len) } } var A = [ [-1, -2, 3, 2], [-4, -1, 6, 2], [ 7, -8, 9, 1], [ 1, -2, 1, 3], ] say gauss_jordan_invert(A).map { .map { "%6s" % .as_rat }.join(" ") }.join("\n")
{{out}}
-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
VBA
{{trans|Phix}} Uses ToReducedRowEchelonForm() from [[Reduced_row_echelon_form#VBA]]
Private Function inverse(mat As Variant) As Variant
Dim len_ As Integer: len_ = UBound(mat)
Dim tmp() As Variant
ReDim tmp(2 * len_ + 1)
Dim aug As Variant
ReDim aug(len_)
For i = 0 To len_
If UBound(mat(i)) <> len_ Then Debug.Print 9 / 0 '-- "Not a square matrix"
aug(i) = tmp
For j = 0 To len_
aug(i)(j) = mat(i)(j)
Next j
'-- augment by identity matrix to right
aug(i)(i + len_ + 1) = 1
Next i
aug = ToReducedRowEchelonForm(aug)
Dim inv As Variant
inv = mat
'-- remove identity matrix to left
For i = 0 To len_
For j = len_ + 1 To 2 * len_ + 1
inv(i)(j - len_ - 1) = aug(i)(j)
Next j
Next i
inverse = inv
End Function
Public Sub main()
Dim test As Variant
test = inverse(Array( _
Array(2, -1, 0), _
Array(-1, 2, -1), _
Array(0, -1, 2)))
For i = LBound(test) To UBound(test)
For j = LBound(test(0)) To UBound(test(0))
Debug.Print test(i)(j),
Next j
Debug.Print
Next i
End Sub
{{out}}
0,75 0,5 0,25
0,5 1 0,5
0,25 0,5 0,75
zkl
This uses GSL to invert a matrix via LU decomposition, not Gauss-Jordan.
var [const] GSL=Import.lib("zklGSL"); // libGSL (GNU Scientific Library)
m:=GSL.Matrix(3,3).set(1,2,3, 4,1,6, 7,8,9);
i:=m.invert();
i.format(10,4).println("\n");
(m*i).format(10,4).println();
{{out}}
-0.8125, 0.1250, 0.1875
0.1250, -0.2500, 0.1250
0.5208, 0.1250, -0.1458
1.0000, 0.0000, 0.0000
-0.0000, 1.0000, 0.0000
-0.0000, 0.0000, 1.0000
m:=GSL.Matrix(3,3).set(2,-1,0, -1,2,-1, 0,-1,2);
m.invert().format(10,4).println("\n");
{{out}}
0.7500, 0.5000, 0.2500
0.5000, 1.0000, 0.5000
0.2500, 0.5000, 0.7500