Gauss-Jordan matrix inversion: Difference between revisions
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=={{header|VBA}}== |
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{{trans|Phix}} |
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Uses ToReducedRowEchelonForm() from [[Reduced_row_echelon_form#VBA]]<lang vb>Private Function inverse(mat As Variant) As Variant |
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Dim len_ As Integer: len_ = UBound(mat) |
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Dim tmp() As Variant |
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ReDim tmp(2 * len_ + 1) |
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Dim aug As Variant |
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ReDim aug(len_) |
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For i = 0 To len_ |
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If UBound(mat(i)) <> len_ Then Debug.Print 9 / 0 '-- "Not a square matrix" |
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aug(i) = tmp |
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For j = 0 To len_ |
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aug(i)(j) = mat(i)(j) |
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Next j |
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'-- augment by identity matrix to right |
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aug(i)(i + len_ + 1) = 1 |
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Next i |
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aug = ToReducedRowEchelonForm(aug) |
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Dim inv As Variant |
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inv = mat |
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'-- remove identity matrix to left |
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For i = 0 To len_ |
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For j = len_ + 1 To 2 * len_ + 1 |
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inv(i)(j - len_ - 1) = aug(i)(j) |
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Next j |
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Next i |
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inverse = inv |
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End Function |
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Public Sub main() |
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Dim test As Variant |
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test = inverse(Array( _ |
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Array(2, -1, 0), _ |
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Array(-1, 2, -1), _ |
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Array(0, -1, 2))) |
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For i = LBound(test) To UBound(test) |
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For j = LBound(test(0)) To UBound(test(0)) |
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Debug.Print test(i)(j), |
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Next j |
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Debug.Print |
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Next i |
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End Sub</lang>{{out}} |
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<pre> 0,75 0,5 0,25 |
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0,5 1 0,5 |
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0,25 0,5 0,75 </pre> |
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=={{header|zkl}}== |
=={{header|zkl}}== |
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This uses GSL to invert a matrix via LU decomposition, not Gauss-Jordan. |
This uses GSL to invert a matrix via LU decomposition, not Gauss-Jordan. |
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Revision as of 12:19, 1 March 2019
Gauss-Jordan matrix inversion
Gauss-Jordan matrix inversion is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
- Task
Invert matrix A using Gauss-Jordan method.
A being an n by n matrix.
C#
<lang csharp> 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();
}
}
}
} </lang> <lang csharp> 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();
}
}
} </lang>
- Output:
-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
<lang go>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")
}</lang>
- Output:
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. <lang j>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</lang>
Usage: <lang j> ]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</lang>
Julia
Built-in LAPACK-based linear solver uses partial-pivoted Gauss elimination): <lang julia>A = [1 2 3; 4 1 6; 7 8 9] @show I / A @show inv(A)</lang>
Native implementation: <lang julia>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)</lang>
- Output:
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. <lang scala>// 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("${"% 10.6f".format(this[r][c])} ")
}
println()
}
println()
}
fun main(args: Array<String>) {
val a = arrayOf(
doubleArrayOf(1.0, 2.0, 3.0),
doubleArrayOf(4.0, 1.0, 6.0),
doubleArrayOf(7.0, 8.0, 9.0)
)
a.inverse().printf("Inverse of A is :\n")
val b = arrayOf(
doubleArrayOf( 2.0, -1.0, 0.0),
doubleArrayOf(-1.0, 2.0, -1.0),
doubleArrayOf( 0.0, -1.0, 2.0)
)
b.inverse().printf("Inverse of B is :\n")
}</lang>
- Output:
Inverse of A is : -0.812500 0.125000 0.187500 0.125000 -0.250000 0.125000 0.520833 0.125000 -0.145833 Inverse of B is : 0.750000 0.500000 0.250000 0.500000 1.000000 0.500000 0.250000 0.500000 0.750000
Perl
Included code from Reduced row echelon form task. <lang perl>sub rref {
our @m; local *m = shift;
@m or return;
my ($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( @$matrix ); print "Gauss-Jordan Inverted Matrix:\n" . display(\@gj) . "\n"; my @rt = gauss_jordan_invert( @gj ); print "After round-trip:\n" . display(\@rt) . "\n";} . "\n"
}</lang>
- Output:
Original Matrix: 2.00 -1.00 0.00 -1.00 2.00 -1.00 0.00 -1.00 2.00 Gauss-Jordan Inverted Matrix: 0.75 0.50 0.25 0.50 1.00 0.50 0.25 0.50 0.75 After round-trip: 2.00 -1.00 0.00 -1.00 2.00 -1.00 0.00 -1.00 2.00 Original Matrix: -1.00 -2.00 3.00 2.00 -4.00 -1.00 6.00 2.00 7.00 -8.00 9.00 1.00 1.00 -2.00 1.00 3.00 Gauss-Jordan Inverted Matrix: -0.91 0.25 0.09 0.41 -1.65 0.65 0.04 0.65 -0.70 0.36 0.08 0.20 -0.57 0.23 -0.03 0.57 After round-trip: -1.00 -2.00 3.00 2.00 -4.00 -1.00 6.00 2.00 7.00 -8.00 9.00 1.00 1.00 -2.00 1.00 3.00
Perl 6
Uses bits and pieces from other tasks, Reduced row echelon form primarily.
<lang perl6>sub gauss-jordan-invert (@m where *.&is-square) {
^@m .map: { @m[$_].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 );
}</lang>
- Output:
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
uses ToReducedRowEchelonForm() from Reduced_row_echelon_form#Phix <lang 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})</lang>
- Output:
{{0.75,0.5,0.25},
{0.5,1,0.5},
{0.25,0.5,0.75}}
REXX
<lang 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)</lang>
- Output:
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 Reduced row echelon form.
<lang ruby>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")</lang>
- Output:
-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
Uses ToReducedRowEchelonForm() from Reduced_row_echelon_form#VBA<lang vb>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</lang>
- Output:
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. <lang zkl>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();</lang>
- Output:
-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
<lang zkl>m:=GSL.Matrix(3,3).set(2,-1,0, -1,2,-1, 0,-1,2); m.invert().format(10,4).println("\n");</lang>
- Output:
0.7500, 0.5000, 0.2500
0.5000, 1.0000, 0.5000
0.2500, 0.5000, 0.7500