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{{Task|Matrices}} Every square matrix $A$ can be decomposed into a product of a lower triangular matrix $L$ and a upper triangular matrix $U$, as described in [[wp:LU decomposition|LU decomposition]].

:$A = LU$

It is a modified form of Gaussian elimination. While the [[Cholesky decomposition]] only works for symmetric, positive definite matrices, the more general LU decomposition works for any square matrix.

There are several algorithms for calculating L and U. To derive ''Crout's algorithm'' for a 3x3 example, we have to solve the following system:

# :$A = \begin\left\{pmatrix\right\} a_\left\{11\right\} & a_\left\{12\right\} & a_\left\{13\right\}\ a_\left\{21\right\} & a_\left\{22\right\} & a_\left\{23\right\}\ a_\left\{31\right\} & a_\left\{32\right\} & a_\left\{33\right\}\ \end\left\{pmatrix\right\}$

\begin{pmatrix} l_{11} & 0 & 0 \ l_{21} & l_{22} & 0 \ l_{31} & l_{32} & l_{33}\ \end{pmatrix} \begin{pmatrix} u_{11} & u_{12} & u_{13} \ 0 & u_{22} & u_{23} \ 0 & 0 & u_{33} \end{pmatrix} = LU

We now would have to solve 9 equations with 12 unknowns. To make the system uniquely solvable, usually the diagonal elements of $L$ are set to 1

:$l_\left\{11\right\}=1$ :$l_\left\{22\right\}=1$ :$l_\left\{33\right\}=1$

so we get a solvable system of 9 unknowns and 9 equations.

# \begin{pmatrix} 1 & 0 & 0 \ l_{21} & 1 & 0 \ l_{31} & l_{32} & 1\ \end{pmatrix} \begin{pmatrix} u_{11} & u_{12} & u_{13} \ 0 & u_{22} & u_{23} \ 0 & 0 & u_{33} \end{pmatrix}

\begin{pmatrix} u_{11} & u_{12} & u_{13} \ u_{11}l_{21} & u_{12}l_{21}+u_{22} & u_{13}l_{21}+u_{23} \ u_{11}l_{31} & u_{12}l_{31}+u_{22}l_{32} & u_{13}l_{31} + u_{23}l_{32}+u_{33} \end{pmatrix} = LU

Solving for the other $l$ and $u$, we get the following equations:

:$u_\left\{11\right\}=a_\left\{11\right\}$ :$u_\left\{12\right\}=a_\left\{12\right\}$ :$u_\left\{13\right\}=a_\left\{13\right\}$

:$u_\left\{22\right\}=a_\left\{22\right\} - u_\left\{12\right\}l_\left\{21\right\}$ :$u_\left\{23\right\}=a_\left\{23\right\} - u_\left\{13\right\}l_\left\{21\right\}$

:$u_\left\{33\right\}=a_\left\{33\right\} - \left(u_\left\{13\right\}l_\left\{31\right\} + u_\left\{23\right\}l_\left\{32\right\}\right)$

and for $l$:

:$l_\left\{21\right\}=\frac\left\{1\right\}\left\{u_\left\{11\right\}\right\} a_\left\{21\right\}$ :$l_\left\{31\right\}=\frac\left\{1\right\}\left\{u_\left\{11\right\}\right\} a_\left\{31\right\}$

:$l_\left\{32\right\}=\frac\left\{1\right\}\left\{u_\left\{22\right\}\right\} \left(a_\left\{32\right\} - u_\left\{12\right\}l_\left\{31\right\}\right)$

We see that there is a calculation pattern, which can be expressed as the following formulas, first for $U$

:$u_\left\{ij\right\} = a_\left\{ij\right\} - \sum_\left\{k=1\right\}^\left\{i-1\right\} u_\left\{kj\right\}l_\left\{ik\right\}$

and then for $L$

:$l_\left\{ij\right\} = \frac\left\{1\right\}\left\{u_\left\{jj\right\}\right\} \left(a_\left\{ij\right\} - \sum_\left\{k=1\right\}^\left\{j-1\right\} u_\left\{kj\right\}l_\left\{ik\right\}\right)$

We see in the second formula that to get the $l_\left\{ij\right\}$ below the diagonal, we have to divide by the diagonal element (pivot) $u_\left\{jj\right\}$, so we get problems when $u_\left\{jj\right\}$ is either 0 or very small, which leads to numerical instability.

The solution to this problem is ''pivoting'' $A$, which means rearranging the rows of $A$, prior to the $LU$ decomposition, in a way that the largest element of each column gets onto the diagonal of $A$. Rearranging the rows means to multiply $A$ by a permutation matrix $P$:

:$PA \Rightarrow A\text{'}$

Example:

:$\begin\left\{pmatrix\right\} 0 & 1 \ 1 & 0 \end\left\{pmatrix\right\} \begin\left\{pmatrix\right\} 1 & 4 \ 2 & 3 \end\left\{pmatrix\right\} \Rightarrow \begin\left\{pmatrix\right\} 2 & 3 \ 1 & 4 \end\left\{pmatrix\right\}$

The decomposition algorithm is then applied on the rearranged matrix so that

:$PA = LU$

The task is to implement a routine which will take a square nxn matrix $A$ and return a lower triangular matrix $L$, a upper triangular matrix $U$ and a permutation matrix $P$, so that the above equation is fullfilled. You should then test it on the following two examples and include your output.

Example 1:


A

1   3   5
2   4   7
1   1   0

L

1.00000   0.00000   0.00000
0.50000   1.00000   0.00000
0.50000  -1.00000   1.00000

U

2.00000   4.00000   7.00000
0.00000   1.00000   1.50000
0.00000   0.00000  -2.00000

P

0   1   0
1   0   0
0   0   1



Example 2:


A

11    9   24    2
1    5    2    6
3   17   18    1
2    5    7    1

L

1.00000   0.00000   0.00000   0.00000
0.27273   1.00000   0.00000   0.00000
0.09091   0.28750   1.00000   0.00000
0.18182   0.23125   0.00360   1.00000

U

11.00000    9.00000   24.00000    2.00000
0.00000   14.54545   11.45455    0.45455
0.00000    0.00000   -3.47500    5.68750
0.00000    0.00000    0.00000    0.51079

P

1   0   0   0
0   0   1   0
0   1   0   0
0   0   0   1



with Ada.Numerics.Generic_Real_Arrays;
generic
with package Matrix is new Ada.Numerics.Generic_Real_Arrays (<>);
package Decomposition is

-- decompose a square matrix A by PA = LU
procedure Decompose (A : Matrix.Real_Matrix; P, L, U : out Matrix.Real_Matrix);

end Decomposition;


package body Decomposition is

procedure Swap_Rows (M : in out Matrix.Real_Matrix; From, To : Natural) is
Temporary : Matrix.Real;
begin
if From = To then
return;
end if;
for I in M'Range (2) loop
Temporary := M (M'First (1) + From, I);
M (M'First (1) + From, I) := M (M'First (1) + To, I);
M (M'First (1) + To, I) := Temporary;
end loop;
end Swap_Rows;

function Pivoting_Matrix
(M : Matrix.Real_Matrix)
return Matrix.Real_Matrix
is
use type Matrix.Real;
Order     : constant Positive := M'Length (1);
Result    : Matrix.Real_Matrix := Matrix.Unit_Matrix (Order);
Max       : Matrix.Real;
Row       : Natural;
begin
for J in 0 .. Order - 1 loop
Max := M (M'First (1) + J, M'First (2) + J);
Row := J;
for I in J .. Order - 1 loop
if M (M'First (1) + I, M'First (2) + J) > Max then
Max := M (M'First (1) + I, M'First (2) + J);
Row := I;
end if;
end loop;
if J /= Row then
-- swap rows J and Row
Swap_Rows (Result, J, Row);
end if;
end loop;
return Result;
end Pivoting_Matrix;

procedure Decompose (A : Matrix.Real_Matrix; P, L, U : out Matrix.Real_Matrix) is
use type Matrix.Real_Matrix, Matrix.Real;
Order : constant Positive := A'Length (1);
A2 : Matrix.Real_Matrix (A'Range (1), A'Range (2));
S : Matrix.Real;
begin
L := (others => (others => 0.0));
U := (others => (others => 0.0));
P := Pivoting_Matrix (A);
A2 := P * A;
for J in 0 .. Order - 1 loop
L (L'First (1) + J, L'First (2) + J) := 1.0;
for I in 0 .. J loop
S := 0.0;
for K in 0 .. I - 1 loop
S := S + U (U'First (1) + K, U'First (2) + J) *
L (L'First (1) + I, L'First (2) + K);
end loop;
U (U'First (1) + I, U'First (2) + J) :=
A2 (A2'First (1) + I, A2'First (2) + J) - S;
end loop;
for I in J + 1 .. Order - 1 loop
S := 0.0;
for K in 0 .. J loop
S := S + U (U'First (1) + K, U'First (2) + J) *
L (L'First (1) + I, L'First (2) + K);
end loop;
L (L'First (1) + I, L'First (2) + J) :=
(A2 (A2'First (1) + I, A2'First (2) + J) - S) /
U (U'First (1) + J, U'First (2) + J);
end loop;
end loop;
end Decompose;

end Decomposition;


Example usage:

with Ada.Numerics.Real_Arrays;
with Decomposition;
procedure Decompose_Example is
package Real_Decomposition is new Decomposition

package Real_IO is new Ada.Text_IO.Float_IO (Float);

procedure Print (M : Ada.Numerics.Real_Arrays.Real_Matrix) is
begin
for Row in M'Range (1) loop
for Col in M'Range (2) loop
Real_IO.Put (M (Row, Col), 3, 2, 0);
end loop;
end loop;
end Print;

((1.0, 3.0, 5.0),
(2.0, 4.0, 7.0),
(1.0, 1.0, 0.0));
P_1, L_1, U_1 : Ada.Numerics.Real_Arrays.Real_Matrix (Example_1'Range (1),
Example_1'Range (2));
((11.0, 9.0, 24.0, 2.0),
(1.0, 5.0, 2.0, 6.0),
(3.0, 17.0, 18.0, 1.0),
(2.0, 5.0, 7.0, 1.0));
P_2, L_2, U_2 : Ada.Numerics.Real_Arrays.Real_Matrix (Example_2'Range (1),
Example_2'Range (2));
begin
Real_Decomposition.Decompose (A => Example_1,
P => P_1,
L => L_1,
U => U_1);
Real_Decomposition.Decompose (A => Example_2,
P => P_2,
L => L_2,
U => U_2);
end Decompose_Example;


{{out}}

Example 1:
A:
1.00  3.00  5.00
2.00  4.00  7.00
1.00  1.00  0.00
L:
1.00  0.00  0.00
0.50  1.00  0.00
0.50 -1.00  1.00
U:
2.00  4.00  7.00
0.00  1.00  1.50
0.00  0.00 -2.00
P:
0.00  1.00  0.00
1.00  0.00  0.00
0.00  0.00  1.00

Example 2:
A:
11.00  9.00 24.00  2.00
1.00  5.00  2.00  6.00
3.00 17.00 18.00  1.00
2.00  5.00  7.00  1.00
L:
1.00  0.00  0.00  0.00
0.27  1.00  0.00  0.00
0.09  0.29  1.00  0.00
0.18  0.23  0.00  1.00
U:
11.00  9.00 24.00  2.00
0.00 14.55 11.45  0.45
0.00  0.00 -3.47  5.69
0.00  0.00  0.00  0.51
P:
1.00  0.00  0.00  0.00
0.00  0.00  1.00  0.00
0.00  1.00  0.00  0.00
0.00  0.00  0.00  1.00


## BBC BASIC

      DIM A1(2,2)
A1() = 1, 3, 5, 2, 4, 7, 1, 1, 0
PROCLUdecomposition(A1(), L1(), U1(), P1())
PRINT "L1:" ' FNshowmatrix(L1())
PRINT "U1:" ' FNshowmatrix(U1())
PRINT "P1:" ' FNshowmatrix(P1())

DIM A2(3,3)
A2() = 11, 9, 24, 2, 1, 5, 2, 6, 3, 17, 18, 1, 2, 5, 7, 1
PROCLUdecomposition(A2(), L2(), U2(), P2())
PRINT "L2:" ' FNshowmatrix(L2())
PRINT "U2:" ' FNshowmatrix(U2())
PRINT "P2:" ' FNshowmatrix(P2())
END

DEF PROCLUdecomposition(a(), RETURN l(), RETURN u(), RETURN p())
LOCAL i%, j%, k%, n%, s, b() : n% = DIM(a(),2)
DIM l(n%,n%), u(n%,n%), b(n%,n%)
PROCpivot(a(), p())
b() = p() . a()
FOR j% = 0 TO n%
l(j%,j%) = 1
FOR i% = 0 TO j%
s = 0
FOR k% = 0 TO i% : s += u(k%,j%) * l(i%,k%) : NEXT
u(i%,j%) = b(i%,j%) - s
NEXT
FOR i% = j% TO n%
s = 0
FOR k% = 0 TO j% : s += u(k%,j%) * l(i%,k%) : NEXT
IF i%<>j% l(i%,j%) = (b(i%,j%) - s) / u(j%,j%)
NEXT
NEXT j%
ENDPROC

DEF PROCpivot(a(), RETURN p())
LOCAL i%, j%, m%, n%, r% : n% = DIM(a(),2)
DIM p(n%,n%) : FOR i% = 0 TO n% : p(i%,i%) = 1 : NEXT
FOR i% = 0 TO n%
m% = a(i%,i%)
r% = i%
FOR j% = i% TO n%
IF a(j%,i%) > m% m% = a(j%,i%) : r% = j%
NEXT
IF i%<>r% THEN
FOR j% = 0 TO n% : SWAP p(i%,j%),p(r%,j%) : NEXT
ENDIF
NEXT i%
ENDPROC

DEF FNshowmatrix(a())
LOCAL @%, i%, j%, a$@% = &102050A FOR i% = 0 TO DIM(a(),1) FOR j% = 0 TO DIM(a(),2) a$ += STR$(a(i%,j%)) + ", " NEXT a$ = LEFT$(LEFT$(a$)) + CHR$(13) + CHR$(10) NEXT i% = a$


{{out}}


L1:
1.00000, 0.00000, 0.00000
0.50000, 1.00000, 0.00000
0.50000, -1.00000, 1.00000

U1:
2.00000, 4.00000, 7.00000
0.00000, 1.00000, 1.50000
0.00000, 0.00000, -2.00000

P1:
0.00000, 1.00000, 0.00000
1.00000, 0.00000, 0.00000
0.00000, 0.00000, 1.00000

L2:
1.00000, 0.00000, 0.00000, 0.00000
0.27273, 1.00000, 0.00000, 0.00000
0.09091, 0.28750, 1.00000, 0.00000
0.18182, 0.23125, 0.00360, 1.00000

U2:
11.00000, 9.00000, 24.00000, 2.00000
0.00000, 14.54545, 11.45455, 0.45455
0.00000, 0.00000, -3.47500, 5.68750
0.00000, 0.00000, 0.00000, 0.51079

P2:
1.00000, 0.00000, 0.00000, 0.00000
0.00000, 0.00000, 1.00000, 0.00000
0.00000, 1.00000, 0.00000, 0.00000
0.00000, 0.00000, 0.00000, 1.00000



## C

Compiled with gcc -std=gnu99 -Wall -lm -pedantic. Demonstrating how to do LU decomposition, and how (not) to use macros.

#include <stdio.h>
#include <stdlib.h>
#include <math.h>

#define foreach(a, b, c) for (int a = b; a < c; a++)
#define for_i foreach(i, 0, n)
#define for_j foreach(j, 0, n)
#define for_k foreach(k, 0, n)
#define for_ij for_i for_j
#define for_ijk for_ij for_k
#define _dim int n
#define _swap(x, y) { typeof(x) tmp = x; x = y; y = tmp; }
#define _sum_k(a, b, c, s) { s = 0; foreach(k, a, b) s+= c; }

typedef double **mat;

#define _zero(a) mat_zero(a, n)
void mat_zero(mat x, int n) { for_ij x[i][j] = 0; }

#define _new(a) a = mat_new(n)
mat mat_new(_dim)
{
mat x = malloc(sizeof(double*) * n);
x[0]  = malloc(sizeof(double) * n * n);

for_i x[i] = x[0] + n * i;
_zero(x);

return x;
}

#define _copy(a) mat_copy(a, n)
mat mat_copy(void *s, _dim)
{
mat x = mat_new(n);
for_ij x[i][j] = ((double (*)[n])s)[i][j];
return x;
}

#define _del(x) mat_del(x)
void mat_del(mat x) { free(x[0]); free(x); }

#define _QUOT(x) #x
#define QUOTE(x) _QUOT(x)
#define _show(a) printf(QUOTE(a)" =");mat_show(a, 0, n)
void mat_show(mat x, char *fmt, _dim)
{
if (!fmt) fmt = "%8.4g";
for_i {
printf(i ? "      " : " [ ");
for_j {
printf(fmt, x[i][j]);
printf(j < n - 1 ? "  " : i == n - 1 ? " ]\n" : "\n");
}
}
}

#define _mul(a, b) mat_mul(a, b, n)
mat mat_mul(mat a, mat b, _dim)
{
mat c = _new(c);
for_ijk c[i][j] += a[i][k] * b[k][j];
return c;
}

#define _pivot(a, b) mat_pivot(a, b, n)
void mat_pivot(mat a, mat p, _dim)
{
for_ij { p[i][j] = (i == j); }
for_i  {
int max_j = i;
foreach(j, i, n)
if (fabs(a[j][i]) > fabs(a[max_j][i])) max_j = j;

if (max_j != i)
for_k { _swap(p[i][k], p[max_j][k]); }
}
}

#define _LU(a, l, u, p) mat_LU(a, l, u, p, n)
void mat_LU(mat A, mat L, mat U, mat P, _dim)
{
_zero(L); _zero(U);
_pivot(A, P);

mat Aprime = _mul(P, A);

for_i  { L[i][i] = 1; }
for_ij {
double s;
if (j <= i) {
_sum_k(0, j, L[j][k] * U[k][i], s)
U[j][i] = Aprime[j][i] - s;
}
if (j >= i) {
_sum_k(0, i, L[j][k] * U[k][i], s);
L[j][i] = (Aprime[j][i] - s) / U[i][i];
}
}

_del(Aprime);
}

double A3[][3] = {{ 1, 3, 5 }, { 2, 4, 7 }, { 1, 1, 0 }};
double A4[][4] = {{11, 9, 24, 2}, {1, 5, 2, 6}, {3, 17, 18, 1}, {2, 5, 7, 1}};

int main()
{
int n = 3;
mat A, L, P, U;

_new(L); _new(P); _new(U);
A = _copy(A3);
_LU(A, L, U, P);
_show(A); _show(L); _show(U); _show(P);
_del(A);  _del(L);  _del(U);  _del(P);

printf("\n");

n = 4;

_new(L); _new(P); _new(U);
A = _copy(A4);
_LU(A, L, U, P);
_show(A); _show(L); _show(U); _show(P);
_del(A);  _del(L);  _del(U);  _del(P);

return 0;
}


## Common Lisp

Uses the routine (mmul A B) from [[Matrix multiplication]].

;; Creates a nxn identity matrix.
(defun eye (n)
(let ((I (make-array (,n ,n) :initial-element 0)))
(loop for j from 0 to (- n 1) do
(setf (aref I j j) 1))
I))

;; Swap two rows l and k of a mxn matrix A, which is a 2D array.
(defun swap-rows (A l k)
(row (make-array n :initial-element 0)))
(loop for j from 0 to (- n 1) do
(setf (aref row j) (aref A l j))
(setf (aref A l j) (aref A k j))
(setf (aref A k j) (aref row j)))))

;; Creates the pivoting matrix for A.
(defun pivotize (A)
(let* ((n (car (array-dimensions A)))
(P (eye n)))
(loop for j from 0 to (- n 1) do
(let ((max (aref A j j))
(row j))
(loop for i from j to (- n 1) do
(if (> (aref A i j) max)
(setq max (aref A i j)
row i)))
(if (not (= j row))
(swap-rows P j row))))

;; Return P.
P))

;; Decomposes a square matrix A by PA=LU and returns L, U and P.
(defun lu (A)
(let* ((n (car (array-dimensions A)))
(L (make-array (,n ,n) :initial-element 0))
(U (make-array (,n ,n) :initial-element 0))
(P (pivotize A))
(A (mmul P A)))

(loop for j from 0 to (- n 1) do
(setf (aref L j j) 1)
(loop for i from 0 to j do
(setf (aref U i j)
(- (aref A i j)
(loop for k from 0 to (- i 1)
sum (* (aref U k j)
(aref L i k))))))
(loop for i from j to (- n 1) do
(setf (aref L i j)
(/ (- (aref A i j)
(loop for k from 0 to (- j 1)
sum (* (aref U k j)
(aref L i k))))
(aref U j j)))))

;; Return L, U and P.
(values L U P)))


Example 1:

(setf g (make-array '(3 3) :initial-contents '((1 3 5) (2 4 7)(1 1 0))))
#2A((1 3 5) (2 4 7) (1 1 0))

(lu g)
#2A((1 0 0) (1/2 1 0) (1/2 -1 1))
#2A((2 4 7) (0 1 3/2) (0 0 -2))
#2A((0 1 0) (1 0 0) (0 0 1))


Example 2:

(setf h (make-array '(4 4) :initial-contents '((11 9 24 2)(1 5 2 6)(3 17 18 1)(2 5 7 1))))
#2A((11 9 24 2) (1 5 2 6) (3 17 18 1) (2 5 7 1))

(lup h)
#2A((1 0 0 0) (3/11 1 0 0) (1/11 23/80 1 0) (2/11 37/160 1/278 1))
#2A((11 9 24 2) (0 160/11 126/11 5/11) (0 0 -139/40 91/16) (0 0 0 71/139))
#2A((1 0 0 0) (0 0 1 0) (0 1 0 0) (0 0 0 1))


## D

{{trans|Common Lisp}}

import std.stdio, std.algorithm, std.typecons, std.numeric,
std.array, std.conv, std.string, std.range;

bool isRectangular(T)(in T[][] m) pure nothrow @nogc {
return m.all!(r => r.length == m[0].length);
}

bool isSquare(T)(in T[][] m) pure nothrow @nogc {
return m.isRectangular && m[0].length == m.length;
}

T[][] matrixMul(T)(in T[][] A, in T[][] B) pure nothrow
in {
assert(A.isRectangular && B.isRectangular &&
!A.empty && !B.empty && A[0].length == B.length);
} body {
auto result = new T[][](A.length, B[0].length);
auto aux = new T[B.length];

foreach (immutable j; 0 .. B[0].length) {
foreach (immutable k, const row; B)
aux[k] = row[j];
foreach (immutable i, const ai; A)
result[i][j] = dotProduct(ai, aux);
}

return result;
}

/// Creates the pivoting matrix for m.
T[][] pivotize(T)(immutable T[][] m) pure nothrow
in {
assert(m.isSquare);
} body {
immutable n = m.length;
auto id = iota(n)
.map!((in j) => n.iota.map!(i => T(i == j)).array)
.array;

foreach (immutable i; 0 .. n) {
// immutable row = iota(i, n).reduce!(max!(j => m[j][i]));
T maxm = m[i][i];
size_t row = i;
foreach (immutable j; i .. n)
if (m[j][i] > maxm) {
maxm = m[j][i];
row = j;
}

if (i != row)
swap(id[i], id[row]);
}

return id;
}

/// Decomposes a square matrix A by PA=LU and returns L, U and P.
Tuple!(T[][],"L", T[][],"U", const T[][],"P")
lu(T)(immutable T[][] A) pure nothrow
in {
assert(A.isSquare);
} body {
immutable n = A.length;
auto L = new T[][](n, n);
auto U = new T[][](n, n);
foreach (immutable i; 0 .. n) {
L[i][i .. $] = 0; U[i][0 .. i] = 0; } immutable P = A.pivotize!T; immutable A2 = matrixMul!T(P, A); foreach (immutable j; 0 .. n) { L[j][j] = 1; foreach (immutable i; 0 .. j+1) { T s1 = 0; foreach (immutable k; 0 .. i) s1 += U[k][j] * L[i][k]; U[i][j] = A2[i][j] - s1; } foreach (immutable i; j .. n) { T s2 = 0; foreach (immutable k; 0 .. j) s2 += U[k][j] * L[i][k]; L[i][j] = (A2[i][j] - s2) / U[j][j]; } } return typeof(return)(L, U, P); } void main() { immutable a = [[1.0, 3, 5], [2.0, 4, 7], [1.0, 1, 0]]; immutable b = [[11.0, 9, 24, 2], [1.0, 5, 2, 6], [3.0, 17, 18, 1], [2.0, 5, 7, 1]]; auto f = "[%([%(%.1f, %)],\n %)]]\n\n".replicate(3); foreach (immutable m; [a, b]) writefln(f, lu(m).tupleof); }  {{out}} [[1.0, 0.0, 0.0], [0.5, 1.0, 0.0], [0.5, -1.0, 1.0]] [[2.0, 4.0, 7.0], [0.0, 1.0, 1.5], [0.0, 0.0, -2.0]] [[0.0, 1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 1.0]] [[1.0, 0.0, 0.0, 0.0], [0.3, 1.0, 0.0, 0.0], [0.0, 0.3, 1.0, 0.0], [0.2, 0.2, 0.0, 1.0]] [[11.0, 9.0, 24.0, 2.0], [0.0, 14.5, 11.5, 0.5], [0.0, 0.0, -3.5, 5.7], [0.0, 0.0, 0.0, 0.5]] [[1.0, 0.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0], [0.0, 1.0, 0.0, 0.0], [0.0, 0.0, 0.0, 1.0]]  ## EchoLisp  (lib 'matrix) ;; the matrix library provides LU-decomposition (decimals 5) (define A (list->array' (1 3 5 2 4 7 1 1 0 ) 3 3)) (define PLU (matrix-lu-decompose A)) ;; -> list of three matrices, P, Lower, Upper (array-print (first PLU)) 0 1 0 1 0 0 0 0 1 (array-print (second PLU)) 1 0 0 0.5 1 0 0.5 -1 1 (array-print (caddr PLU)) 2 4 7 0 1 1.5 0 0 -2 (define A (list->array '(11 9 24 2 1 5 2 6 3 17 18 1 2 5 7 1 ) 4 4)) (define PLU (matrix-lu-decompose A)) ;; -> list of three matrices, P, Lower, Upper (array-print (first PLU)) 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 (array-print (second PLU)) 1 0 0 0 0.27273 1 0 0 0.09091 0.2875 1 0 0.18182 0.23125 0.0036 1 (array-print (caddr PLU)) 11 9 24 2 0 14.54545 11.45455 0.45455 0 0 -3.475 5.6875 0 0 0 0.51079  ## Fortran  program lu1 implicit none call check( reshape([real(8)::1,2,1,3,4,1,5,7,0 ],[3,3]) ) call check( reshape([real(8)::11,1,3,2,9,5,17,5,24,2,18,7,2,6,1,1],[4,4]) ) contains subroutine check(a) real(8), intent(in) :: a(:,:) integer :: i,j,n real(8), allocatable :: aa(:,:),l(:,:),u(:,:) integer, allocatable :: p(:,:) integer, allocatable :: ipiv(:) n = size(a,1) allocate(aa(n,n),l(n,n),u(n,n),p(n,n),ipiv(n)) forall (j=1:n,i=1:n) aa(i,j) = a(i,j) u (i,j) = 0d0 p (i,j) = merge(1 ,0 ,i.eq.j) l (i,j) = merge(1d0,0d0,i.eq.j) end forall call lu(aa, ipiv) do i = 1,n l(i, :i-1) = aa(ipiv(i), :i-1) u(i,i: ) = aa(ipiv(i),i: ) end do p(ipiv,:) = p call mat_print('a',a) call mat_print('p',p) call mat_print('l',l) call mat_print('u',u) print *, "residual" print *, "|| P.A - L.U || = ", maxval(abs(matmul(p,a)-matmul(l,u))) end subroutine subroutine lu(a,p) ! in situ decomposition, corresponds to LAPACK's dgebtrf real(8), intent(inout) :: a(:,:) integer, intent(out ) :: p(:) integer :: n, i,j,k,kmax n = size(a,1) p = [ ( i, i=1,n ) ] do k = 1,n-1 kmax = maxloc(abs(a(p(k:),k)),1) + k-1 if (kmax /= k ) p([k, kmax]) = p([kmax, k]) a(p(k+1:),k) = a(p(k+1:),k) / a(p(k),k) forall (j=k+1:n) a(p(k+1:),j) = a(p(k+1:),j) - a(p(k+1:),k) * a(p(k),j) end do end subroutine subroutine mat_print(amsg,a) character(*), intent(in) :: amsg class (*), intent(in) :: a(:,:) integer :: i print*,' ' print*,amsg do i=1,size(a,1) select type (a) type is (real(8)) ; print'(100f8.2)',a(i,:) type is (integer) ; print'(100i8 )',a(i,:) end select end do print*,' ' end subroutine end program  {{out}}  a 1.00 3.00 5.00 2.00 4.00 7.00 1.00 1.00 0.00 p 0.00 1.00 0.00 1.00 0.00 0.00 0.00 0.00 1.00 l 1.00 0.00 0.00 0.50 1.00 0.00 0.50 -1.00 1.00 u 2.00 4.00 7.00 0.00 1.00 1.50 0.00 0.00 -2.00 residual || P.A - L.U || = 0.0000000000000000 a 11.00 9.00 24.00 2.00 1.00 5.00 2.00 6.00 3.00 17.00 18.00 1.00 2.00 5.00 7.00 1.00 p 1.00 0.00 0.00 0.00 0.00 0.00 1.00 0.00 0.00 1.00 0.00 0.00 0.00 0.00 0.00 1.00 l 1.00 0.00 0.00 0.00 0.27 1.00 0.00 0.00 0.09 0.29 1.00 0.00 0.18 0.23 0.00 1.00 u 11.00 9.00 24.00 2.00 0.00 14.55 11.45 0.45 0.00 0.00 -3.47 5.69 0.00 0.00 0.00 0.51 residual || P.A - L.U || = 0.0000000000000000  ## Go ### 2D representation {{trans|Common Lisp}} package main import "fmt" type matrix [][]float64 func zero(n int) matrix { r := make([][]float64, n) a := make([]float64, n*n) for i := range r { r[i] = a[n*i : n*(i+1)] } return r } func eye(n int) matrix { r := zero(n) for i := range r { r[i][i] = 1 } return r } func (m matrix) print(label string) { if label > "" { fmt.Printf("%s:\n", label) } for _, r := range m { for _, e := range r { fmt.Printf(" %9.5f", e) } fmt.Println() } } func (a matrix) pivotize() matrix { p := eye(len(a)) for j, r := range a { max := r[j] row := j for i := j; i < len(a); i++ { if a[i][j] > max { max = a[i][j] row = i } } if j != row { // swap rows p[j], p[row] = p[row], p[j] } } return p } func (m1 matrix) mul(m2 matrix) matrix { r := zero(len(m1)) for i, r1 := range m1 { for j := range m2 { for k := range m1 { r[i][j] += r1[k] * m2[k][j] } } } return r } func (a matrix) lu() (l, u, p matrix) { l = zero(len(a)) u = zero(len(a)) p = a.pivotize() a = p.mul(a) for j := range a { l[j][j] = 1 for i := 0; i <= j; i++ { sum := 0. for k := 0; k < i; k++ { sum += u[k][j] * l[i][k] } u[i][j] = a[i][j] - sum } for i := j; i < len(a); i++ { sum := 0. for k := 0; k < j; k++ { sum += u[k][j] * l[i][k] } l[i][j] = (a[i][j] - sum) / u[j][j] } } return } func main() { showLU(matrix{ {1, 3, 5}, {2, 4, 7}, {1, 1, 0}}) showLU(matrix{ {11, 9, 24, 2}, {1, 5, 2, 6}, {3, 17, 18, 1}, {2, 5, 7, 1}}) } func showLU(a matrix) { a.print("\na") l, u, p := a.lu() l.print("l") u.print("u") p.print("p") }  {{out}}  a: 1.00000 3.00000 5.00000 2.00000 4.00000 7.00000 1.00000 1.00000 0.00000 l: 1.00000 0.00000 0.00000 0.50000 1.00000 0.00000 0.50000 -1.00000 1.00000 u: 2.00000 4.00000 7.00000 0.00000 1.00000 1.50000 0.00000 0.00000 -2.00000 p: 0.00000 1.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 1.00000 a: 11.00000 9.00000 24.00000 2.00000 1.00000 5.00000 2.00000 6.00000 3.00000 17.00000 18.00000 1.00000 2.00000 5.00000 7.00000 1.00000 l: 1.00000 0.00000 0.00000 0.00000 0.27273 1.00000 0.00000 0.00000 0.09091 0.28750 1.00000 0.00000 0.18182 0.23125 0.00360 1.00000 u: 11.00000 9.00000 24.00000 2.00000 0.00000 14.54545 11.45455 0.45455 0.00000 0.00000 -3.47500 5.68750 0.00000 0.00000 0.00000 0.51079 p: 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000 0.00000 0.00000 1.00000 0.00000 0.00000 0.00000 0.00000 0.00000 1.00000  ### Flat representation package main import "fmt" type matrix struct { stride int ele []float64 } func (m *matrix) print(heading string) { if heading > "" { fmt.Print("\n", heading, "\n") } for e := 0; e < len(m.ele); e += m.stride { fmt.Printf("%8.5f ", m.ele[e:e+m.stride]) fmt.Println() } } func (m1 *matrix) mul(m2 *matrix) (m3 *matrix, ok bool) { if m1.stride*m2.stride != len(m2.ele) { return nil, false } m3 = &matrix{m2.stride, make([]float64, (len(m1.ele)/m1.stride)*m2.stride)} for m1c0, m3x := 0, 0; m1c0 < len(m1.ele); m1c0 += m1.stride { for m2r0 := 0; m2r0 < m2.stride; m2r0++ { for m1x, m2x := m1c0, m2r0; m2x < len(m2.ele); m2x += m2.stride { m3.ele[m3x] += m1.ele[m1x] * m2.ele[m2x] m1x++ } m3x++ } } return m3, true } func zero(rows, cols int) *matrix { return &matrix{cols, make([]float64, rows*cols)} } func eye(n int) *matrix { m := zero(n, n) for ix := 0; ix < len(m.ele); ix += n + 1 { m.ele[ix] = 1 } return m } func (a *matrix) pivotize() *matrix { pv := make([]int, a.stride) for i := range pv { pv[i] = i } for j, dx := 0, 0; j < a.stride; j++ { row := j max := a.ele[dx] for i, ixcj := j, dx; i < a.stride; i++ { if a.ele[ixcj] > max { max = a.ele[ixcj] row = i } ixcj += a.stride } if j != row { pv[row], pv[j] = pv[j], pv[row] } dx += a.stride + 1 } p := zero(a.stride, a.stride) for r, c := range pv { p.ele[r*a.stride+c] = 1 } return p } func (a *matrix) lu() (l, u, p *matrix) { l = zero(a.stride, a.stride) u = zero(a.stride, a.stride) p = a.pivotize() a, _ = p.mul(a) for j, jxc0 := 0, 0; j < a.stride; j++ { l.ele[jxc0+j] = 1 for i, ixc0 := 0, 0; ixc0 <= jxc0; i++ { sum := 0. for k, kxcj := 0, j; k < i; k++ { sum += u.ele[kxcj] * l.ele[ixc0+k] kxcj += a.stride } u.ele[ixc0+j] = a.ele[ixc0+j] - sum ixc0 += a.stride } for ixc0 := jxc0; ixc0 < len(a.ele); ixc0 += a.stride { sum := 0. for k, kxcj := 0, j; k < j; k++ { sum += u.ele[kxcj] * l.ele[ixc0+k] kxcj += a.stride } l.ele[ixc0+j] = (a.ele[ixc0+j] - sum) / u.ele[jxc0+j] } jxc0 += a.stride } return } func main() { showLU(&matrix{3, []float64{ 1, 3, 5, 2, 4, 7, 1, 1, 0}}) showLU(&matrix{4, []float64{ 11, 9, 24, 2, 1, 5, 2, 6, 3, 17, 18, 1, 2, 5, 7, 1}}) } func showLU(a *matrix) { a.print("\na") l, u, p := a.lu() l.print("l") u.print("u") p.print("p") }  Output is same as from 2D solution. ### Library gonum/mat package main import ( "fmt" "gonum.org/v1/gonum/mat" ) func main() { showLU(mat.NewDense(3, 3, []float64{ 1, 3, 5, 2, 4, 7, 1, 1, 0, })) fmt.Println() showLU(mat.NewDense(4, 4, []float64{ 11, 9, 24, 2, 1, 5, 2, 6, 3, 17, 18, 1, 2, 5, 7, 1, })) } func showLU(a *mat.Dense) { fmt.Printf("a: %v\n\n", mat.Formatted(a, mat.Prefix(" "))) var lu mat.LU lu.Factorize(a) l := lu.LTo(nil) u := lu.UTo(nil) fmt.Printf("l: %.5f\n\n", mat.Formatted(l, mat.Prefix(" "))) fmt.Printf("u: %.5f\n\n", mat.Formatted(u, mat.Prefix(" "))) fmt.Println("p:", lu.Pivot(nil)) }  {{out}} Pivot format is a little different here. (But library solutions don't really meet task requirements anyway.)  a: ⎡1 3 5⎤ ⎢2 4 7⎥ ⎣1 1 0⎦ l: ⎡ 1.00000 0.00000 0.00000⎤ ⎢ 0.50000 1.00000 0.00000⎥ ⎣ 0.50000 -1.00000 1.00000⎦ u: ⎡ 2.00000 4.00000 7.00000⎤ ⎢ 0.00000 1.00000 1.50000⎥ ⎣ 0.00000 0.00000 -2.00000⎦ p: [1 0 2] a: ⎡11 9 24 2⎤ ⎢ 1 5 2 6⎥ ⎢ 3 17 18 1⎥ ⎣ 2 5 7 1⎦ l: ⎡1.00000 0.00000 0.00000 0.00000⎤ ⎢0.27273 1.00000 0.00000 0.00000⎥ ⎢0.09091 0.28750 1.00000 0.00000⎥ ⎣0.18182 0.23125 0.00360 1.00000⎦ u: ⎡11.00000 9.00000 24.00000 2.00000⎤ ⎢ 0.00000 14.54545 11.45455 0.45455⎥ ⎢ 0.00000 0.00000 -3.47500 5.68750⎥ ⎣ 0.00000 0.00000 0.00000 0.51079⎦ p: [0 2 1 3]  ### Library go.matrix package main import ( "fmt" mat "github.com/skelterjohn/go.matrix" ) func main() { showLU(mat.MakeDenseMatrixStacked([][]float64{ {1, 3, 5}, {2, 4, 7}, {1, 1, 0}})) showLU(mat.MakeDenseMatrixStacked([][]float64{ {11, 9, 24, 2}, {1, 5, 2, 6}, {3, 17, 18, 1}, {2, 5, 7, 1}})) } func showLU(a *mat.DenseMatrix) { fmt.Printf("\na:\n%v\n", a) l, u, p := a.LU() fmt.Printf("l:\n%v\n", l) fmt.Printf("u:\n%v\n", u) fmt.Printf("p:\n%v\n", p) }  {{out}}  a: {1, 3, 5, 2, 4, 7, 1, 1, 0} l: { 1, 0, 0, 0.5, 1, 0, 0.5, -1, 1} u: { 2, 4, 7, 0, 1, 1.5, 0, 0, -2} p: {0, 1, 0, 1, 0, 0, 0, 0, 1} a: {11, 9, 24, 2, 1, 5, 2, 6, 3, 17, 18, 1, 2, 5, 7, 1} l: { 1, 0, 0, 0, 0.272727, 1, 0, 0, 0.090909, 0.2875, 1, 0, 0.181818, 0.23125, 0.003597, 1} u: { 11, 9, 24, 2, 0, 14.545455, 11.454545, 0.454545, 0, 0, -3.475, 5.6875, 0, 0, 0, 0.510791} p: {1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1}  ## Haskell ''Without elem-at-index modifications; doesn't find maximum but any non-zero element''  import Data.List import Data.Maybe import Text.Printf -- a matrix is represented as a list of columns mmult :: Num a => [[a]] -> [[a]] -> [[a]] mmult a b = [ [ sum$ zipWith (*) ak bj | ak <- (transpose a) ] | bj <- b ]

nth mA i j = (mA !! j) !! i

idMatrixPart n m k = [ [if (i==j) then 1 else 0 | i <- [1..n]] | j <- [k..m]]
idMatrix n = idMatrixPart n n 1

permMatrix n ix1 ix2 =
[ [ if ((i==ix1 && j==ix2) || (i==ix2 && j==ix1) || (i==j && j /= ix1 && i /= ix2))
then 1 else 0| i <- [0..n-1]] | j <- [0..n-1]]
permMatrix_inv n ix1 ix2 = permMatrix n ix2 ix1

-- count k from zero
elimColumn :: Int -> [[Rational]] -> Int -> [Rational]
elimMatrix :: Int -> [[Rational]] -> Int -> [[Rational]]
elimMatrix_inv :: Int -> [[Rational]] -> Int -> [[Rational]]

elimColumn n mA k = [(let mAkk = (nth mA k k) in  if (i>k) then (-(nth mA i k)/mAkk)
else if (i==k) then 1 else 0) | i <- [0..n-1]]
elimMatrix n mA k = (idMatrixPart n k 1) ++ [elimColumn n mA k] ++ (idMatrixPart n n (k+2))
elimMatrix_inv n mA k = (idMatrixPart n k 1) ++ --mA is elimMatrix there
[let c = (mA!!k) in [if (i==k) then 1 else if (i<k) then 0 else (-(c!!i)) | i <- [0..n-1]]]
++ (idMatrixPart n n (k+2))

swapIndx :: [[Rational]] -> Int -> Int
swapIndx mA k = fromMaybe k (findIndex (>0) (drop k (mA!!k)))

-- LUP; lupStep returns [L:U:P]
paStep_recP :: Int -> [[Rational]] -> [[Rational]] -> [[Rational]] -> Int -> [[[Rational]]]
paStep_recM :: Int -> [[Rational]] -> [[Rational]] -> [[Rational]] -> Int -> [[[Rational]]]
lupStep :: Int -> [[Rational]] -> [[[Rational]]]

paStep_recP n mP mA mL cnt =
let mPt = permMatrix n cnt (swapIndx mA cnt) in
let mPtInv = permMatrix_inv n cnt (swapIndx mA cnt) in
if (cnt >= n) then [(mmult mP mL),mA,mP] else
(paStep_recM n (mmult mPt mP) (mmult mPt mA) (mmult mL mPtInv) cnt)

paStep_recM n mP mA mL cnt =
let mMt = elimMatrix n mA cnt in
let mMtInv = elimMatrix_inv n mMt cnt in
paStep_recP n mP (mmult mMt mA) (mmult mL mMtInv) (cnt + 1)

lupStep n mA = paStep_recP n (idMatrix n) mA (idMatrix n) 0

--IO
matrixFromRationalToString m = concat $intersperse "\n" (map (\x -> unwords$ printf "%8.4f" <$> (x::[Double])) (transpose (matrixFromRational m))) where matrixFromRational m = map (\x -> map fromRational x) m solveTask mY = let mLUP = lupStep (length mY) mY in putStrLn ("A: \n" ++ matrixFromRationalToString mY) >> putStrLn ("L: \n" ++ matrixFromRationalToString (mLUP!!0)) >> putStrLn ("U: \n" ++ matrixFromRationalToString (mLUP!!1)) >> putStrLn ("P: \n" ++ matrixFromRationalToString (mLUP!!2)) >> putStrLn ("Verify: PA\n" ++ matrixFromRationalToString (mmult (mLUP!!2) mY)) >> putStrLn ("Verify: LU\n" ++ matrixFromRationalToString (mmult (mLUP!!0) (mLUP!!1))) mY1 = [[1, 2, 1], [3, 4, 7], [5, 7, 0]] :: [[Rational]] mY2 = [[11, 1, 3, 2], [9, 5, 17, 5], [24, 2, 18, 7], [2, 6, 1, 1]] :: [[Rational]] main = putStrLn "Task1: \n" >> solveTask mY1 >> putStrLn "Task2: \n" >> solveTask mY2  {{out}}  Task1: A: 1.0000 3.0000 5.0000 2.0000 4.0000 7.0000 1.0000 7.0000 0.0000 L: 1.0000 0.0000 0.0000 2.0000 1.0000 0.0000 1.0000 -2.0000 1.0000 U: 1.0000 3.0000 5.0000 0.0000 -2.0000 -3.0000 0.0000 0.0000 -11.0000 P: 1.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.0000 Verify: PA 1.0000 3.0000 5.0000 2.0000 4.0000 7.0000 1.0000 7.0000 0.0000 Verify: LU 1.0000 3.0000 5.0000 2.0000 4.0000 7.0000 1.0000 7.0000 0.0000 Task2: A: 11.0000 9.0000 24.0000 2.0000 1.0000 5.0000 2.0000 6.0000 3.0000 17.0000 18.0000 1.0000 2.0000 5.0000 7.0000 1.0000 L: 1.0000 0.5556 0.2317 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 1.8889 1.0000 0.0000 0.0000 0.0909 0.0000 1.0000 U: 0.0081 0.0000 0.0000 0.5325 11.0000 9.0000 24.0000 2.0000 -17.7778 0.0000 -27.3333 -2.7778 0.0000 4.1818 -0.1818 5.8182 P: 0.0000 0.0000 0.0000 1.0000 1.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 1.0000 0.0000 0.0000 Verify: PA 2.0000 5.0000 7.0000 1.0000 11.0000 9.0000 24.0000 2.0000 3.0000 17.0000 18.0000 1.0000 1.0000 5.0000 2.0000 6.0000 Verify: LU 2.0000 5.0000 7.0000 1.0000 11.0000 9.0000 24.0000 2.0000 3.0000 17.0000 18.0000 1.0000 1.0000 5.0000 2.0000 6.0000  ## Idris '''works with Idris 0.10''' Uses The Method Of Partial Pivoting '''Solution:'''  module Main import Data.Vect Matrix : Nat -> Nat -> Type -> Type Matrix m n t = Vect m (Vect n t) -- Creates list from 0 to n (not including n) upTo : (m : Nat) -> Vect m (Fin m) upTo Z = [] upTo (S n) = 0 :: (map FS (upTo n)) -- Creates list from 0 to n-1 (not including n-1) upToM1 : (m : Nat) -> (sz ** Vect sz (Fin m)) upToM1 m = case (upTo m) of (y::ys) => (_ ** init(y::ys)) [] => (_ ** []) -- Creates list from i to n (not including n) fromUpTo : {n : Nat} -> Fin n -> (sz ** Vect sz (Fin n)) fromUpTo {n} m = filter (>= m) (upTo n) -- Creates list from i+1 to n (not including n) fromUpTo1 : {n : Nat} -> Fin n -> (sz ** Vect sz (Fin n)) fromUpTo1 {n} m with (fromUpTo m) | (_ ** xs) = case xs of (y::ys) => (_ ** ys) [] => (_ ** []) -- Create Zero Matrix of size m by n zeros : (m : Nat) -> (n : Nat) -> Matrix m n Double zeros m n = replicate m (replicate n 0.0) replaceAtM : (Fin m, Fin n) -> t -> Matrix m n t -> Matrix m n t replaceAtM (i, j) e a = replaceAt i (replaceAt j e (index i a)) a -- Create Identity Matrix of size m by m eye : (m : Nat) -> Matrix m m Double eye m = map create1Vec (upTo m) where set1 : Vect m Double -> Fin m -> Vect m Double set1 a n = replaceAt n 1.0 a create1Vec : Fin m -> Vect m Double create1Vec n = set1 (replicate m 0.0) n indexM : (Fin m, Fin n) -> Matrix m n t -> t indexM (i, j) a = index j (index i a) -- Obtain index for the row containing the -- largest absolute value for the given column colAbsMaxIndex : Fin m -> Fin m -> Matrix m m Double -> Fin m colAbsMaxIndex startRow col a {m} with (fromUpTo startRow) | (_ ** xs) = snd$ foldl (\(absMax, idx), curIdx =>
let curAbsVal = abs(indexM (curIdx, col) a) in
if (curAbsVal > absMax)
then (curAbsVal, curIdx)
else (absMax, idx)
) (0.0, startRow) xs

-- Swaps two rows in a given matrix
swapRows : Fin m -> Fin m -> Matrix m n t -> Matrix m n t
swapRows r1 r2 a = replaceAt r2 tempRow $replaceAt r1 (index r2 a) a where tempRow = index r1 a -- Swaps two individual values in a matrix swapValues : (Fin m, Fin m) -> (Fin m, Fin m) -> Matrix m m Double -> Matrix m m Double swapValues (i1, j1) (i2, j2) m = replaceAtM (i2, j2) v1$ replaceAtM (i1, j1) v2 m
where
v1 = indexM (i1, j1) m
v2 = indexM (i2, j2) m

-- Perform row Swap on Lower Triangular Matrix
lSwapRow : Fin m -> Fin m -> Matrix m m Double -> Matrix m m Double
lSwapRow row1 row2 l {m} with (filter (< row1) (upTo m))
| (_ ** xs) =  foldl (\l',col => swapValues (row1, col) (row2, col) l') l xs

rowSwap : Fin m -> (Matrix m m Double,  Matrix m m Double, Matrix m m Double) ->
(Matrix m m Double, Matrix m m Double, Matrix m m Double)
rowSwap col (l,u,p) = (lSwapRow col row l, swapRows col row u, swapRows col row p)
where row = colAbsMaxIndex col col u

calc : (Fin m) -> (Fin m) -> (Matrix m m Double, Matrix m m Double) ->
(Matrix m m Double, Matrix m m Double)
calc i j (l, u) {m} = (l', u')
where
l' : Matrix m m Double
l' = replaceAtM (j, i) ((indexM (j, i) u) / indexM (i, i) u) l

u'' : (Fin m) -> (Matrix m m Double) -> (Matrix m m Double)
u'' k u = replaceAtM (j, k) ((indexM (j, k) u) -
((indexM (j, i) l') * (indexM (i, k) u))) u

u' : (Matrix m m Double)
u' with (fromUpTo i) | (_ ** xs) = foldl (\curU, idx => u'' idx curU) u xs

-- Perform a single iteration of the algorithm for the given column
iteration : Fin m -> (Matrix m m Double, Matrix m m Double, Matrix m m Double) ->
(Matrix m m Double, Matrix m m Double, Matrix m m Double)
iteration i lup {m} = iterate' (rowSwap i lup)

where
modify : (Matrix m m Double, Matrix m m Double) ->
(Matrix m m Double, Matrix m m Double)
modify lu with (fromUpTo1 i) | (_ ** xs) =
foldl (\lu',j => calc i j lu') lu xs

iterate' : (Matrix m m Double, Matrix m m Double, Matrix m m Double) ->
(Matrix m m Double, Matrix m m Double, Matrix m m Double)
iterate' (l, u, p) with (modify (l, u)) | (l', u') = (l', u', p)

-- Generate L, U, P matricies from a given square matrix.
-- Where L * U = A, and P is the permutation matrix
luDecompose : Matrix m m Double -> (Matrix m m Double, Matrix m m Double, Matrix m m Double)
luDecompose a {m} with (upToM1 m)
| (_ ** xs) = foldl (\lup,idx => iteration idx lup) (eye m,a,eye m) xs

ex1 : (Matrix 3 3 Double, Matrix 3 3 Double, Matrix 3 3 Double)
ex1 = luDecompose [[1, 3, 5], [2, 4, 7], [1, 1, 0]]

ex2 : (Matrix 4 4 Double, Matrix 4 4 Double, Matrix 4 4 Double)
ex2 = luDecompose [[11, 9, 24, 2], [1, 5, 2, 6], [3, 17, 18, 1], [2, 5, 7, 1]]

printEx : (Matrix n n Double, Matrix n n Double, Matrix n n Double) -> IO ()
printEx (l, u, p) = do
putStr "l:"
print l
putStrLn "\n"

putStr "u:"
print u
putStrLn "\n"

putStr "p:"
print p
putStrLn "\n"

main : IO()
main = do
putStrLn "Solution 1:"
printEx ex1
putStrLn "Solution 2:"
printEx ex2



{{out}}


Solution 1:
l:[[1, 0, 0], [0.5, 1, 0], [0.5, -1, 1]]

u:[[2, 4, 7], [0, 1, 1.5], [0, 0, -2]]

p:[[0, 1, 0], [1, 0, 0], [0, 0, 1]]

Solution 2:
l:[[1, 0, 0, 0], [0.2727272727272727, 1, 0, 0], [0.09090909090909091, 0.2875, 1, 0], [0.1818181818181818, 0.23125, 0.003597122302158069, 1]]

u:[[11, 9, 24, 2], [0, 14.54545454545455, 11.45454545454546, 0.4545454545454546], [0, 0, -3.475, 5.6875], [0, 0, 0, 0.510791366906476]]

p:[[1, 0, 0, 0], [0, 0, 1, 0], [0, 1, 0, 0], [0, 0, 0, 1]]



## J

Taken with slight modification from [http://www.jsoftware.com/jwiki/Essays/LU%20Decomposition].

'''Solution:'''

mp=: +/ .*

LU=: 3 : 0
'm n'=. $A=. y if. 1=m do. p ; (=1) ; p{"1 A [ p=. C. (n-1);~.0,(0~:,A)i.1 else. m2=. >.m%2 'p1 L1 U1'=. LU m2{.A D=. (/:p1) {"1 m2}.A F=. m2 {."1 D E=. m2 {."1 U1 FE1=. F mp %. E G=. m2}."1 D - FE1 mp U1 'p2 L2 U2'=. LU G p3=. (i.m2),m2+p2 H=. (/:p3) {"1 U1 (p1{p3) ; (L1,FE1,.L2) ; H,(-n){."1 U2 end. ) permtomat=: 1 {.~"0 -@>:@:/: LUdecompose=: (permtomat&.>@{. , }.)@:LU  '''Example use:'''  A=:3 3$1 3 5 2 4 7 1 1 0
LUdecompose A
┌─────┬─────┬───────┐
│1 0 0│1 0 0│1  3  5│
│0 1 0│2 1 0│0 _2 _3│
│0 0 1│1 1 1│0  0 _2│
└─────┴─────┴───────┘
mp/> LUdecompose A
1 3 5
2 4 7
1 1 0

A=:4 4$11 9 24 2 1 5 2 6 3 17 18 1 2 5 7 1 LUdecompose A ┌───────┬─────────────────────────────┬─────────────────────────────┐ │1 0 0 0│ 1 0 0 0│11 9 24 2│ │0 1 0 0│0.0909091 1 0 0│ 0 4.18182 _0.181818 5.81818│ │0 0 1 0│ 0.272727 3.47826 1 0│ 0 0 12.087 _19.7826│ │0 0 0 1│ 0.181818 0.804348 0.230216 1│ 0 0 0 0.510791│ └───────┴─────────────────────────────┴─────────────────────────────┘ mp/> LUdecompose A 11 9 24 2 1 5 2 6 3 17 18 1 2 5 7 1  ## Java Translation of [[#Common_Lisp|Common Lisp]] via [[#D|D]] {{works with|Java|8}} import static java.util.Arrays.stream; import java.util.Locale; import static java.util.stream.IntStream.range; public class Test { static double dotProduct(double[] a, double[] b) { return range(0, a.length).mapToDouble(i -> a[i] * b[i]).sum(); } static double[][] matrixMul(double[][] A, double[][] B) { double[][] result = new double[A.length][B[0].length]; double[] aux = new double[B.length]; for (int j = 0; j < B[0].length; j++) { for (int k = 0; k < B.length; k++) aux[k] = B[k][j]; for (int i = 0; i < A.length; i++) result[i][j] = dotProduct(A[i], aux); } return result; } static double[][] pivotize(double[][] m) { int n = m.length; double[][] id = range(0, n).mapToObj(j -> range(0, n) .mapToDouble(i -> i == j ? 1 : 0).toArray()) .toArray(double[][]::new); for (int i = 0; i < n; i++) { double maxm = m[i][i]; int row = i; for (int j = i; j < n; j++) if (m[j][i] > maxm) { maxm = m[j][i]; row = j; } if (i != row) { double[] tmp = id[i]; id[i] = id[row]; id[row] = tmp; } } return id; } static double[][][] lu(double[][] A) { int n = A.length; double[][] L = new double[n][n]; double[][] U = new double[n][n]; double[][] P = pivotize(A); double[][] A2 = matrixMul(P, A); for (int j = 0; j < n; j++) { L[j][j] = 1; for (int i = 0; i < j + 1; i++) { double s1 = 0; for (int k = 0; k < i; k++) s1 += U[k][j] * L[i][k]; U[i][j] = A2[i][j] - s1; } for (int i = j; i < n; i++) { double s2 = 0; for (int k = 0; k < j; k++) s2 += U[k][j] * L[i][k]; L[i][j] = (A2[i][j] - s2) / U[j][j]; } } return new double[][][]{L, U, P}; } static void print(double[][] m) { stream(m).forEach(a -> { stream(a).forEach(n -> System.out.printf(Locale.US, "%5.1f ", n)); System.out.println(); }); System.out.println(); } public static void main(String[] args) { double[][] a = {{1.0, 3, 5}, {2.0, 4, 7}, {1.0, 1, 0}}; double[][] b = {{11.0, 9, 24, 2}, {1.0, 5, 2, 6}, {3.0, 17, 18, 1}, {2.0, 5, 7, 1}}; for (double[][] m : lu(a)) print(m); System.out.println(); for (double[][] m : lu(b)) print(m); } }   1.0 0.0 0.0 0.5 1.0 0.0 0.5 -1.0 1.0 2.0 4.0 7.0 0.0 1.0 1.5 0.0 0.0 -2.0 0.0 1.0 0.0 1.0 0.0 0.0 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0.3 1.0 0.0 0.0 0.1 0.3 1.0 0.0 0.2 0.2 0.0 1.0 11.0 9.0 24.0 2.0 0.0 14.5 11.5 0.5 0.0 0.0 -3.5 5.7 0.0 0.0 0.0 0.5 1.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 1.0  ## jq {{Works with|jq|1.4}} jq currently does not have builtin support for matrices and therefore some infrastructure is needed to make the following self-contained. Matrices here are represented as arrays of arrays in the usual way. '''Infrastructure''' # Create an m x n matrix def matrix(m; n; init): if m == 0 then [] elif m == 1 then [range(0;n)] | map(init) elif m > 0 then matrix(1;n;init) as$row
| [range(0;m)] | map( $row ) else error("matrix\(m);_;_) invalid") end ; def I(n): matrix(n;n;0) as$m
| reduce range(0;n) as $i ($m; . | setpath( [$i,$i]; 1));

def dot_product(a; b):
reduce range(0;a|length) as $i (0; . + (a[$i] * b[$i]) ); # transpose/0 expects its input to be a rectangular matrix def transpose: if (.[0] | length) == 0 then [] else [map(.[0])] + (map(.[1:]) | transpose) end ; # A and B should both be numeric matrices, A being m by n, and B being n by p. def multiply(A; B): (B[0]|length) as$p
| (B|transpose) as $BT | reduce range(0; A|length) as$i
([];
reduce range(0; $p) as$j
(.;
.[$i][$j] = dot_product( A[$i];$BT[$j] ) )); def swap_rows(i;j): if i == j then . else .[i] as$i | .[i] = .[j] | .[j] = $i end ; # Print a matrix neatly, each cell occupying n spaces, but without truncation def neatly(n): def right: tostring | ( " " * (n-length) + .); . as$in
| length as $length | reduce range (0;$length) as $i (""; . + reduce range(0;$length) as $j (""; "\(.) \($in[$i][$j] | right )" ) + "\n" ) ;


'''LU decomposition'''

# Create the pivot matrix for the input matrix.
# Use "range(0;$n) as$i" to handle ill-conditioned cases.
def pivotize:
def abs: if .<0 then -. else . end;
length as $n | . as$m
| reduce range(0;$n) as$j
(I($n); # state: [row; max] (reduce range(0;$n) as $i ([$j, $m[$j][$j]|abs ]; ($m[$i][$j]|abs) as $a | if$a > .[1] then [ $i,$a ] else . end) | .[0]) as $row | swap_rows($j; $row) ) ; # Decompose the input nxn matrix A by PA=LU and return [L, U, P]. def lup: def div(i;j): if j == 0 then if i==0 then 0 else error("\(i)/0") end else i/j end; . as$A
| length as $n | I($n) as $L # matrix($n; $n; 0.0) as$L
| matrix($n;$n; 0.0) as $U | ($A|pivotize) as $P | multiply($P;$A) as$A2
# state: [L, U]
| reduce range(0; $n) as$i ( [$L,$U];
reduce range(0; $n) as$j (.;
.[0] as $L | .[1] as$U
| if ($j >=$i) then
(reduce range(0;$i) as$k (0; . + ($U[$k][$j] *$L[$i][$k] ))) as $s1 | [$L, ($U| setpath([$i,$j]; ($A2[$i][$j] - $s1))) ] else (reduce range(0;$j) as $k (0; . + ($U[$k][$j] * $L[$i][$k]))) as$s2
| [ ($L | setpath([$i,$j]; div(($A2[$i][$j] - $s2) ;$U[$j][$j] ))), $U ] end )) | . + [$P ]
;



'''Example 1''':

def a: [[1, 3, 5], [2, 4, 7], [1, 1, 0]];
a | lup[] | neatly(4)



{{Out}}

 $/usr/local/bin/jq -M -n -r -f LU.jq 1 0 0 0.5 1 0 0.5 -1 1 2 4 7 0 1 1.5 0 0 -2 0 1 0 1 0 0 0 0 1  '''Example 2''': def b: [[11,9,24,2],[1,5,2,6],[3,17,18,1],[2,5,7,1]]; b | lup[] | neatly(21)  {{Out}} $ /usr/local/bin/jq -M -n -r -f LU.jq
1                     0                     0                     0
0.2727272727272727                     1                     0                     0
0.09090909090909091                0.2875                     1                     0
0.18181818181818182   0.23124999999999996 0.0035971223021580693                     1

11                     9                    24                     2
0    14.545454545454547    11.454545454545455    0.4545454545454546
0                     0   -3.4749999999999996                5.6875
0                     0                     0     0.510791366906476

1                     0                     0                     0
0                     0                     1                     0
0                     1                     0                     0
0                     0                     0                     1


'''Example 3''':


# A|lup|verify(A) should be true
def verify(A):
.[0] as $L | .[1] as$U | .[2] as $P | multiply($P; A) == multiply($L;$U);

def A:
[[1,  1,  1,  1],
[1,  1, -1, -1],
[1, -1,  0,  0],
[0,  0,  1, -1]];

A|lup|verify(A)


{{out}} true

## Julia

Julia has the predefined functions lu, lufact and lufact! in the standard library to compute the lu decomposition of a matrix. {{Out}}

julia> lu([1 3 5 ; 2 4 7 ; 1 1 0])
(
3x3 Array{Float64,2}:
1.0   0.0  0.0
0.5   1.0  0.0
0.5  -1.0  1.0,

3x3 Array{Float64,2}:
2.0  4.0   7.0
0.0  1.0   1.5
0.0  0.0  -2.0,

[2,1,3])


## Kotlin

// version 1.1.4-3

typealias Vector = DoubleArray
typealias Matrix = Array<Vector>

operator fun Matrix.times(other: Matrix): Matrix {
val rows1 = this.size
val cols1 = this[0].size
val rows2 = other.size
val cols2 = other[0].size
require(cols1 == rows2)
val result = Matrix(rows1) { Vector(cols2) }
for (i in 0 until rows1) {
for (j in 0 until cols2) {
for (k in 0 until rows2) {
result[i][j] += this[i][k] * other[k][j]
}
}
}
return result
}

fun pivotize(m: Matrix): Matrix {
val n = m.size
val im = Array(n) { Vector(n) }
for (i in 0 until n) im[i][i] = 1.0
for (i in 0 until n) {
var max = m[i][i]
var row = i
for (j in i until n) {
if (m[j][i] > max) {
max = m[j][i]
row = j
}
}
if (i != row) {
val t = im[i]
im[i] = im[row]
im[row] = t
}
}
return im
}

fun lu(a: Matrix): Array<Matrix> {
val n = a.size
val l = Array(n) { Vector(n) }
val u = Array(n) { Vector(n) }
val p = pivotize(a)
val a2 = p * a

for (j in 0 until n) {
l[j][j] = 1.0
for (i in 0 until j + 1) {
var sum = 0.0
for (k in 0 until i) sum += u[k][j] * l[i][k]
u[i][j] = a2[i][j] - sum
}
for (i in j until n) {
var sum2 = 0.0
for(k in 0 until j) sum2 += u[k][j] * l[i][k]
l[i][j] = (a2[i][j] - sum2) / u[j][j]
}
}
return arrayOf(l, u, p)
}

fun printMatrix(title: String, m: Matrix, f: String) {
val n = m.size
println("\n$title\n") for (i in 0 until n) { for (j in 0 until n) print("${f.format(m[i][j])}  ")
println()
}
}

fun main(args: Array<String>) {
val a1 = arrayOf(
doubleArrayOf( 1.0,  3.0,  5.0),
doubleArrayOf( 2.0,  4.0,  7.0),
doubleArrayOf( 1.0,  1.0,  0.0)
)
val (l1, u1, p1) = lu(a1)
println("EXAMPLE 1:-")
printMatrix("A:", a1, "%1.0f")
printMatrix("L:", l1, "% 7.5f")
printMatrix("U:", u1, "% 8.5f")
printMatrix("P:", p1, "%1.0f")

val a2 = arrayOf(
doubleArrayOf(11.0,  9.0, 24.0,  2.0),
doubleArrayOf( 1.0,  5.0,  2.0,  6.0),
doubleArrayOf( 3.0, 17.0, 18.0,  1.0),
doubleArrayOf( 2.0,  5.0,  7.0,  1.0)
)
val (l2, u2, p2) = lu(a2)
println("\nEXAMPLE 2:-")
printMatrix("A:", a2, "%2.0f")
printMatrix("L:", l2, "%7.5f")
printMatrix("U:", u2, "%8.5f")
printMatrix("P:", p2, "%1.0f")
}


{{out}}


EXAMPLE 1:-

A:

1  3  5
2  4  7
1  1  0

L:

1.00000   0.00000   0.00000
0.50000   1.00000   0.00000
0.50000  -1.00000   1.00000

U:

2.00000   4.00000   7.00000
0.00000   1.00000   1.50000
0.00000   0.00000  -2.00000

P:

0  1  0
1  0  0
0  0  1

EXAMPLE 2:-

A:

11   9  24   2
1   5   2   6
3  17  18   1
2   5   7   1

L:

1.00000  0.00000  0.00000  0.00000
0.27273  1.00000  0.00000  0.00000
0.09091  0.28750  1.00000  0.00000
0.18182  0.23125  0.00360  1.00000

U:

11.00000   9.00000  24.00000   2.00000
0.00000  14.54545  11.45455   0.45455
0.00000   0.00000  -3.47500   5.68750
0.00000   0.00000   0.00000   0.51079

P:

1  0  0  0
0  0  1  0
0  1  0  0
0  0  0  1



## Maple


A:=<<1.0|3.0|5.0>,<2.0|4.0|7.0>,<1.0|1.0|0.0>>:

LinearAlgebra:-LUDecomposition(A);



{{out}}


[0  1  0]  [              1.0   0.   0.]  [2.  4.                7.]
[       ]  [                           ]  [                        ]
[1  0  0], [0.500000000000000  1.0   0.], [0.  1.  1.50000000000000]
[       ]  [                           ]  [                        ]
[0  0  1]  [0.500000000000000  -1.  1.0]  [0.  0.               -2.]



A:=<<11.0|9.0|24.0|2.0>,<1.0|5.0|2.0|6.0>,
<3.0|17.0|18.0|1.0>,<2.0|5.0|7.0|1.0>>:

with(LinearAlgebra):

LUDecomposition(A);



{{out}}


[1  0  0  0]
[          ]
[0  0  1  0]
[          ],
[0  1  0  0]
[          ]
[0  0  0  1]

[               1.0                 0.                   0.   0.]
[                                                               ]
[ 0.272727272727273                1.0                   0.   0.]
[                                                               ],
[0.0909090909090909  0.287500000000000                  1.0   0.]
[                                                               ]
[ 0.181818181818182  0.231250000000000  0.00359712230215807  1.0]

[11.                9.                24.                 2.]
[                                                           ]
[ 0.  14.5454545454545   11.4545454545455  0.454545454545455]
[                                                           ]
[ 0.                0.  -3.47500000000000   5.68750000000000]
[                                                           ]
[ 0.                0.                 0.  0.510791366906476]



## Mathematica

(*Ex1*)a = {{1, 3, 5}, {2, 4, 7}, {1, 1, 0}};
{lu, p, c} = LUDecomposition[a];
l = LowerTriangularize[lu, -1] + IdentityMatrix[Length[p]];
u = UpperTriangularize[lu];
P = Part[IdentityMatrix[Length[p]], p] ;
MatrixForm /@ {P.a , P, l, u, l.u}

(*Ex2*)a = {{11, 9, 24, 2}, {1, 5, 2, 6}, {3, 17, 18, 1}, {2, 5, 7, 1}};
{lu, p, c} = LUDecomposition[a];
l = LowerTriangularize[lu, -1] + IdentityMatrix[Length[p]];
u = UpperTriangularize[lu];
P = Part[IdentityMatrix[Length[p]], p] ;
MatrixForm /@ {P.a , P, l, u, l.u}



{{out}} [[File:LUex1.png]] [[File:LUex2.png]]

LU decomposition is part of language

  A = [
1   3   5
2   4   7
1   1   0];

[L,U,P] = lu(A)


{{out}}


L =

1.00000   0.00000   0.00000
0.50000   1.00000   0.00000
0.50000  -1.00000   1.00000

U =

2.00000   4.00000   7.00000
0.00000   1.00000   1.50000
0.00000   0.00000  -2.00000

P =

0   1   0
1   0   0
0   0   1



2nd example:

  A = [
11    9   24    2
1    5    2    6
3   17   18    1
2    5    7    1 ];

[L,U,P] = lu(A)


{{out}}


L =

1.00000   0.00000   0.00000   0.00000
0.27273   1.00000   0.00000   0.00000
0.09091   0.28750   1.00000   0.00000
0.18182   0.23125   0.00360   1.00000

U =

11.00000    9.00000   24.00000    2.00000
0.00000   14.54545   11.45455    0.45455
0.00000    0.00000   -3.47500    5.68750
0.00000    0.00000    0.00000    0.51079

P =

1   0   0   0
0   0   1   0
0   1   0   0
0   0   0   1



### Creating a MATLAB function


function [ P, L, U ] = LUdecomposition(A)

% Ensures A is n by n
sz = size(A);
if sz(1)~=sz(2)
fprintf('A is not n by n\n');
clear x;
return;
end

n = sz(1);
L = eye(n);
P = eye(n);
U = A;

for i=1:sz(1)

% Row reducing
if U(i,i)==0
maximum = max(abs(U(i:end,1)));
for k=1:n
if maximum == abs(U(k,i))
temp = U(1,:);
U(1,:) = U(k,:);
U(k,:) = temp;

temp = P(:,1);
P(1,:) = P(k,:);
P(k,:) = temp;
end
end

end

if U(i,i)~=1
temp = eye(n);
temp(i,i)=U(i,i);
L = L * temp;
U(i,:) = U(i,:)/U(i,i); %Ensures the pivots are 1.
end

if i~=sz(1)

for j=i+1:length(U)
temp = eye(n);
temp(j,i) = U(j,i);
L = L * temp;
U(j,:) = U(j,:)-U(j,i)*U(i,:);

end
end

end
P = P';
end



## Maxima

/* LU decomposition is built-in */

a: hilbert_matrix(4)$/* LU in "packed" form */ lup: lu_factor(a); /* [matrix([1, 1/2, 1/3, 1/4 ], [1/2, 1/12, 1/12, 3/40 ], [1/3, 1, 1/180, 1/120 ], [1/4, 9/10, 3/2, 1/2800]), [1, 2, 3, 4], generalring] */ /* extract actual factors */ get_lu_factors(lup); /* [matrix([1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]), matrix([1, 0, 0, 0], [1/2, 1, 0, 0], [1/3, 1, 1, 0], [1/4, 9/10, 3/2, 1]), matrix([1, 1/2, 1/3, 1/4 ], [0, 1/12, 1/12, 3/40 ], [0, 0, 1/180, 1/120 ], [0, 0, 0, 1/2800]) ] */ /* solve for a given right-hand side */ lu_backsub(lup, transpose([1, 1, -1, -1])); /* matrix([-204], [2100], [-4740], [2940]) */  ## PARI/GP matlup(M) = { my (L = matid(#M), U = M, P = L); for (i = 1, #M-1, \\ pivoting p = M[z=i,i]; for (k = i, #M, if (M[k,i] > p, p = M[z=k,i])); if (i != z, \\ swap rows k = U[i,]; U[i,] = U[z,]; U[z,] = k; k = P[i,]; P[i,] = P[z,]; P[z,] = k; ); ); for (i = 1, #M-1, \\ decompose for (k = i+1, #M, L[k,i] = U[k,i] / U[i,i]; for (j = i, #M, U[k,j] -= L[k,i] * U[i,j]) ) ); [L,U,P] \\ return L,U,P triple matrix }  Output:  gp > [L,U,P] = matlup([1,3,5;2,4,7;1,1,0]); gp > L [ 1 0 0] [1/2 1 0] [1/2 -1 1] gp > U [2 4 7] [0 1 3/2] [0 0 -2] gp > P [0 1 0] [1 0 0] [0 0 1] gp > [L,U,P] = matlup([11,9,24,2;1,5,2,6;3,17,18,1;2,5,7,1]); gp > L [ 1 0 0 0] [3/11 1 0 0] [1/11 23/80 1 0] [2/11 37/160 1/278 1] gp > U [11 9 24 2] [ 0 160/11 126/11 5/11] [ 0 0 -139/40 91/16] [ 0 0 0 71/139] gp > P [1 0 0 0] [0 0 1 0] [0 1 0 0] [0 0 0 1]  ## Perl {{trans|Perl 6}} use List::Util qw(sum); for$test (
[[1, 3, 5],
[2, 4, 7],
[1, 1, 0]],

[[11,  9, 24,  2],
[ 1,  5,  2,  6],
[ 3, 17, 18,  1],
[ 2,  5,  7,  1]]
) {
my($P,$AP, $L,$U) = lu(@$test); say_it('A matrix', @$test);
say_it('P matrix',  @$P); say_it('AP matrix', @$AP);
say_it('L matrix',  @$L); say_it('U matrix', @$U);

}

sub lu {
my (@a) = @_;
my $n = +@a; my @P = pivotize(@a); my$AP = mmult(\@P, \@a);
my @L  = matrix_ident($n); my @U = matrix_zero($n);
for $i (0..$n-1) {
for $j (0..$n-1) {
if ($j >=$i) {
$U[$i][$j] = $$AP[i][j] - sum map { U[_][j] * L[i][_] } 0..i-1; } else { L[i][j] = ($$AP[$i][$j] - sum map {$U[$_][$j] * $L[$i][$_] } 0..$j-1) / $U[$j][$j]; } } } return \@P,$AP, \@L, \@U;
}

sub pivotize {
my(@m) = @_;
my $size = +@m; my @id = matrix_ident($size);
for $i (0..$size-1) {
my $max =$m[$i][$i];
my $row =$i;
for $j ($i .. $size-2) { if ($m[$j][$i] > $max) {$max = $m[$j][$i];$row = $j; } } ($id[$row],$id[$i]) = ($id[$i],$id[$row]) if$row != $i; } @id } sub matrix_zero { my($n) = @_; map { [ (0) x $n ] } 0..$n-1 }
sub matrix_ident { my($n) = @_; map { [ (0) x$_, 1, (0) x ($n-1 -$_) ] } 0..$n-1 } sub mmult { local *a = shift; local *b = shift; my @p = []; my$rows = @a;
my $cols = @{$b[0] };
my $n = @b - 1; for (my$r = 0 ; $r <$rows ; ++$r) { for (my$c = 0 ; $c <$cols ; ++$c) {$p[$r][$c] += $a[$r][$_] *$b[$_][$c] foreach 0 .. $n; } } return [@p]; } sub say_it { my($message, @array) = @_;
print "$message\n";$line = sprintf join("\n" => map join(" " => map(sprintf("%8.5f", $_), @$_)), @{+\@array})."\n";
$line =~ s/\.00000/ /g;$line =~ s/0000\b/    /g;
print "$line\n"; }  {{out}} A matrix 1 3 5 2 4 7 1 1 0 P matrix 0 1 0 1 0 0 0 0 1 AP matrix 2 4 7 1 3 5 1 1 0 L matrix 1 0 0 0.5 1 0 0.5 -1 1 U matrix 2 4 7 0 1 1.5 0 0 -2 A matrix 11 9 24 2 1 5 2 6 3 17 18 1 2 5 7 1 P matrix 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 AP matrix 11 9 24 2 3 17 18 1 1 5 2 6 2 5 7 1 L matrix 1 0 0 0 0.27273 1 0 0 0.09091 0.28750 1 0 0.18182 0.23125 0.00360 1 U matrix 11 9 24 2 0 14.54545 11.45455 0.45455 0 0 -3.47500 5.68750 0 0 0 0.51079  ## Perl 6 {{works with|Rakudo|2015-11-20}} Translation of Ruby. perl6 for ( [1, 3, 5], # Test Matrices [2, 4, 7], [1, 1, 0] ), ( [11, 9, 24, 2], [ 1, 5, 2, 6], [ 3, 17, 18, 1], [ 2, 5, 7, 1] ) -> @test { say-it 'A Matrix', @test; say-it($_[0], @($_[1]) ) for 'P Matrix', 'Aʼ Matrix', 'L Matrix', 'U Matrix' Z, lu @test; } sub lu (@a) { die unless @a.&is-square; my$n = +@a;
my @P = pivotize @a;
my @Aʼ = mmult @P, @a;
my @L = matrix-ident $n; my @U = matrix-zero$n;
for ^$n ->$i {
for ^$n ->$j {
if $j >=$i {
@U[$i][$j] =  @Aʼ[$i][$j] - [+] map { @U[$_][$j] * @L[$i][$_] }, ^$i } else { @L[$i][$j] = (@Aʼ[$i][$j] - [+] map { @U[$_][$j] * @L[$i][$_] }, ^$j) / @U[$j][$j];
}
}

}
return @P, @Aʼ, @L, @U;
}

sub pivotize (@m) {
my $size = +@m; my @id = matrix-ident$size;
for ^$size ->$i {
my $max = @m[$i][$i]; my$row = $i; for$i ..^ $size ->$j {
if @m[$j][$i] > $max {$max = @m[$j][$i];
$row =$j;
}
}
if $row !=$i {
@id[$row,$i] = @id[$i,$row]
}
}
@id
}

sub is-square (@m) { so @m == all @m[*] }

sub matrix-zero ($n,$m = $n) { map { [ flat 0 xx$n ] }, ^$m } sub matrix-ident ($n) { map { [ flat 0 xx $_, 1, 0 xx$n - 1 - $_ ] }, ^$n }

sub mmult(@a,@b) {
my @p;
for ^@a X ^@b[0] -> ($r,$c) {
@p[$r][$c] += @a[$r][$_] * @b[$_][$c] for ^@b;
}
@p
}

sub rat-int ($num) { return$num unless $num ~~ Rat; return$num.narrow if $num.narrow.WHAT ~~ Int;$num.nude.join: '/';
}

sub say-it ($message, @array) { say "\n$message";
$_».&rat-int.fmt("%7s").say for @array; }  {{out}} txt A Matrix 1 3 5 2 4 7 1 1 0 P Matrix 0 1 0 1 0 0 0 0 1 Aʼ Matrix 2 4 7 1 3 5 1 1 0 L Matrix 1 0 0 1/2 1 0 1/2 -1 1 U Matrix 2 4 7 0 1 3/2 0 0 -2 A Matrix 11 9 24 2 1 5 2 6 3 17 18 1 2 5 7 1 P Matrix 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 Aʼ Matrix 11 9 24 2 3 17 18 1 1 5 2 6 2 5 7 1 L Matrix 1 0 0 0 3/11 1 0 0 1/11 23/80 1 0 2/11 37/160 1/278 1 U Matrix 11 9 24 2 0 160/11 126/11 5/11 0 0 -139/40 91/16 0 0 0 71/139  ## Phix {{trans|Kotlin}} Phix function matrix_mul(sequence a, sequence b) sequence c if length(a[1]) != length(b) then return 0 else c = repeat(repeat(0,length(b[1])),length(a)) for i=1 to length(a) do for j=1 to length(b[1]) do for k=1 to length(a[1]) do c[i][j] += a[i][k]*b[k][j] end for end for end for return c end if end function function pivotize(sequence m) integer n = length(m) sequence im = repeat(repeat(0,n),n) for i=1 to n do im[i][i] = 1 end for for i=1 to n do atom mx = m[i][i] integer row = i for j=i to n do if m[j][i]>mx then mx = m[j][i] row = j end if end for if i!=row then {im[i],im[row]} = {im[row],im[i]} end if end for return im end function function lu(sequence a) integer n = length(a) sequence l = repeat(repeat(0,n),n), u = l, p = pivotize(a), a2 = matrix_mul(p,a) for j=1 to n do l[j][j] = 1.0 for i=1 to j do atom sum1 = 0.0 for k=1 to i do sum1 += u[k][j] * l[i][k] end for u[i][j] = a2[i][j] - sum1 end for for i=j+1 to n do atom sum2 = 0.0 for k=1 to j do sum2 += u[k][j] * l[i][k] end for l[i][j] = (a2[i][j] - sum2) / u[j][j] end for end for return {a, l, u, p} end function constant a = {{{1, 3, 5}, {2, 4, 7}, {1, 1, 0}}, {{11, 9,24, 2}, { 1, 5, 2, 6}, { 3,17,18, 1}, { 2, 5, 7, 1}}} for i=1 to length(a) do ?"== a,l,u,p: ==" pp(lu(a[i]),{pp_Nest,2,pp_Pause,0}) end for  {{out}} txt "== a,l,u,p: ==" {{{1,3,5}, {2,4,7}, {1,1,0}}, {{1,0,0}, {0.5,1,0}, {0.5,-1,1}}, {{2,4,7}, {0,1,1.5}, {0,0,-2}}, {{0,1,0}, {1,0,0}, {0,0,1}}} "== a,l,u,p: ==" {{{11,9,24,2}, {1,5,2,6}, {3,17,18,1}, {2,5,7,1}}, {{1,0,0,0}, {0.2727272727,1,0,0}, {0.09090909091,0.2875,1,0}, {0.1818181818,0.23125,0.003597122302,1}}, {{11,9,24,2}, {0,14.54545455,11.45454545,0.4545454545}, {0,0,-3.475,5.6875}, {0,0,0,0.5107913669}}, {{1,0,0,0}, {0,0,1,0}, {0,1,0,0}, {0,0,0,1}}}  ## PL/I PL/I (subscriptrange, fofl, size): /* 2 Nov. 2013 */ LU_Decomposition: procedure options (main); declare a1(3,3) float (18) initial ( 1, 3, 5, 2, 4, 7, 1, 1, 0); declare a2(4,4) float (18) initial (11, 9, 24, 2, 1, 5, 2, 6, 3, 17, 18, 1, 2, 5, 7, 1); call check(a1); call check(a2); /* In-situ decomposition */ LU: procedure(a, p); declare a(*,*) float (18); declare p(*) fixed binary; declare (maximum, rtemp) float (18); declare (n, i, j, k, ii, temp) fixed binary; n = hbound(a,1); do i = 1 to n; p(i) = i; end; do k = 1 to n-1; maximum = 0; ii = k; do i = k to n; if maximum < abs(a(p(i),k)) then do; maximum = abs(a(p(i),k)); ii = i; end; end; if ii ^= k then do; temp = p(k); p(k) = p(ii); p(ii) = temp; end; do i = k+1 to n; a(p(i),k) = a(p(i),k) / a(p(k),k); end; do j = k+1 to n; do i = k+1 to n; a(p(i),j) = a(p(i),j) - a(p(i),k) * a(p(k),j); end; end; end; end LU; CHECK: procedure(a); declare a(*,*) float (18) nonassignable; declare aa(hbound(a,1), hbound(a,2)) float (18); declare L(hbound(a,1), hbound(a,2)) float (18); declare U(hbound(a,1), hbound(a,2)) float (18); declare (p(hbound(a,1), hbound(a,2)), ipiv(hbound(a,1)) ) fixed binary; declare pp(hbound(a,1), hbound(a,2)) fixed binary; declare (i, j, n, temp(hbound(a,1))) fixed binary; n = hbound(a,1); aa = A; /* work with a copy */ P = 0; L = 0; U = 0; do i = 1 to n; p(i,i) = 1; L(i,i) = 1; /* convert permutation vector to a matrix */ end; call LU(aa, ipiv); do i = 1 to n; do j = 1 to i-1; L(i,j) = aa(ipiv(i),j); end; do j = i to n; U(i,j) = aa(ipiv(i),j); end; end; pp = p; do i = 1 to n; p(ipiv(i), *) = pp(i,*); end; put skip list ('A'); put edit (A) (skip, (n) f(10,5)); put skip list ('P'); put edit (P) (skip, (n) f(11)); put skip list ('L'); put edit (L) (skip, (n) f(10,5)); put skip list ('U'); put edit (U) (skip, (n) f(10,5)); end CHECK; end LU_Decomposition;  Derived from Fortran version above. Results: txt A 1.00000 3.00000 5.00000 2.00000 4.00000 7.00000 1.00000 1.00000 0.00000 P 0 1 0 1 0 0 0 0 1 L 1.00000 0.00000 0.00000 0.50000 1.00000 0.00000 0.50000 -1.00000 1.00000 U 2.00000 4.00000 7.00000 0.00000 1.00000 1.50000 0.00000 0.00000 -2.00000 A 11.00000 9.00000 24.00000 2.00000 1.00000 5.00000 2.00000 6.00000 3.00000 17.00000 18.00000 1.00000 2.00000 5.00000 7.00000 1.00000 P 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 L 1.00000 0.00000 0.00000 0.00000 0.27273 1.00000 0.00000 0.00000 0.09091 0.28750 1.00000 0.00000 0.18182 0.23125 0.00360 1.00000 U 11.00000 9.00000 24.00000 2.00000 0.00000 14.54545 11.45455 0.45455 0.00000 0.00000 -3.47500 5.68750 0.00000 0.00000 0.00000 0.51079  ## Python {{trans|D}} python from pprint import pprint def matrixMul(A, B): TB = zip(*B) return [[sum(ea*eb for ea,eb in zip(a,b)) for b in TB] for a in A] def pivotize(m): """Creates the pivoting matrix for m.""" n = len(m) ID = [[float(i == j) for i in xrange(n)] for j in xrange(n)] for j in xrange(n): row = max(xrange(j, n), key=lambda i: abs(m[i][j])) if j != row: ID[j], ID[row] = ID[row], ID[j] return ID def lu(A): """Decomposes a nxn matrix A by PA=LU and returns L, U and P.""" n = len(A) L = [[0.0] * n for i in xrange(n)] U = [[0.0] * n for i in xrange(n)] P = pivotize(A) A2 = matrixMul(P, A) for j in xrange(n): L[j][j] = 1.0 for i in xrange(j+1): s1 = sum(U[k][j] * L[i][k] for k in xrange(i)) U[i][j] = A2[i][j] - s1 for i in xrange(j, n): s2 = sum(U[k][j] * L[i][k] for k in xrange(j)) L[i][j] = (A2[i][j] - s2) / U[j][j] return (L, U, P) a = [[1, 3, 5], [2, 4, 7], [1, 1, 0]] for part in lu(a): pprint(part, width=19) print print b = [[11,9,24,2],[1,5,2,6],[3,17,18,1],[2,5,7,1]] for part in lu(b): pprint(part) print  {{out}} txt [[1.0, 0.0, 0.0], [0.5, 1.0, 0.0], [0.5, -1.0, 1.0]] [[2.0, 4.0, 7.0], [0.0, 1.0, 1.5], [0.0, 0.0, -2.0]] [[0.0, 1.0, 0.0], [1.0, 0.0, 0.0], [0.0, 0.0, 1.0]] [[1.0, 0.0, 0.0, 0.0], [0.27272727272727271, 1.0, 0.0, 0.0], [0.090909090909090912, 0.28749999999999998, 1.0, 0.0], [0.18181818181818182, 0.23124999999999996, 0.0035971223021580693, 1.0]] [[11.0, 9.0, 24.0, 2.0], [0.0, 14.545454545454547, 11.454545454545455, 0.45454545454545459], [0.0, 0.0, -3.4749999999999996, 5.6875], [0.0, 0.0, 0.0, 0.51079136690647597]] [[1.0, 0.0, 0.0, 0.0], [0.0, 0.0, 1.0, 0.0], [0.0, 1.0, 0.0, 0.0], [0.0, 0.0, 0.0, 1.0]]  ## R {{Out}} txt > A <- c(1, 2, 1, 3, 4, 1, 5, 7, 0) > dim(A) <- c(3, 3) > library(Matrix) > expand(lu(A))$L
3 x 3 Matrix of class "dtrMatrix" (unitriangular)
[,1] [,2] [,3]
[1,]  1.0    .    .
[2,]  0.5  1.0    .
[3,]  0.5 -1.0  1.0

$U 3 x 3 Matrix of class "dtrMatrix" [,1] [,2] [,3] [1,] 2.0 4.0 7.0 [2,] . 1.0 1.5 [3,] . . -2.0$P
3 x 3 sparse Matrix of class "pMatrix"

[1,] . | .
[2,] | . .
[3,] . . |



## Racket

racket

#lang racket
(require math)
(define A (matrix
[[1   3   5]
[2   4   7]
[1   1   0]]))

(matrix-lu A)
; result:
; (mutable-array #[#[1 0 0]
;                  #[2 1 0]
;                  #[1 1 1]])
; (mutable-array #[#[1 3 5]
;                  #[0 -2 -3]
;                  #[0 0 -2]])



## REXX

rexx
/*REXX program creates a  matrix  from console input, performs/shows  LU  decomposition.*/
#=0;     P.=0;     PA.=0;      L.=0;      U.=0   /*initialize some variables to zero.   */
parse arg x                                      /*obtain matrix elements from the C.L. */
call bldAMat;       call showMat 'A'    /*build and display A  matrix.*/
call bldPmat;       call showMat 'P'    /*  "    "     "    P     "   */
call multMat;       call showMat 'PA'   /*  "    "     "    PA    "   */
do y=1  for N;  call bldUmat;       call bldLmat        /*build     U  and  L     "   */
end   /*y*/
call showMat 'L';   call showMat 'U'    /*display   L  and  U     "   */
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
bldAMat: ?=words(x);   do N=1  for ?  until N**2>=?                 /*find matrix size. */
end  /*N*/
if N**2\==?  then do;  say '***error*** wrong # of elements entered:'  ?;  exit 9
end
do    r=1  for N                             /*build   A  matrix.*/
do c=1  for N;     #=# + 1;    _=word(x, #);        A.r.c=_
if \datatype(_, 'N')  then call er "element isn't numeric: " _
end   /*c*/
end      /*r*/;       return
/*──────────────────────────────────────────────────────────────────────────────────────*/
bldLmat:               do    r=1  for N                             /*build lower matrix*/
do c=1  for N;     if r==c  then do;  L.r.c=1;  iterate;     end
if c\==y | r==c | c>r  then iterate
_=PA.r.c
do k=1  for c-1;   _=_   -   U.k.c * L.r.k
end  /*k*/
L.r.c=_ / U.c.c
end   /*c*/
end      /*r*/;       return
/*──────────────────────────────────────────────────────────────────────────────────────*/
bldPmat: c=N;          do r=N  by -1  for N; P.r.c=1;    c=c+1      /*build perm. matrix*/
if c>N  then c=N%2;   if c==N  then c=1
end   /*r*/;          return
/*──────────────────────────────────────────────────────────────────────────────────────*/
bldUmat:               do    r=1  for N;     if r\==y  then iterate /*build upper matrix*/
do c=1  for N;     if c 1 3 5   2 4 7   1 1 0 }}

txt

──A matrix───
1  3  5
2  4  7
1  1  0

──P matrix───
0  1  0
1  0  0
0  0  1

──PA matrix──
2  4  7
1  3  5
1  1  0

─────L matrix──────
1    0    0
0.5    1    0
0.5   -1    1

─────U matrix──────
2    4    7
0    1  1.5
0    0   -2



{{out|output|text=  when using the input of:      11 9 24 2   1 5 2 6   3 17 18 1   2 5 7 1 }}

txt

─────A matrix──────
11   9  24   2
1   5   2   6
3  17  18   1
2   5   7   1

───P matrix────
1  0  0  0
0  0  1  0
0  1  0  0
0  0  0  1

─────PA matrix─────
11   9  24   2
3  17  18   1
1   5   2   6
2   5   7   1

───────────────────────────L matrix────────────────────────────
1              0              0              0
0.272727273              1              0              0
0.0909090909    0.287500001              1              0
0.181818182        0.23125  0.00359712804              1

───────────────────────U matrix────────────────────────
11            9           24            2
0   14.5454545   11.4545455   0.45454545
0            0  -3.47500002       5.6875
0            0            0  0.510791339



## Ruby

ruby
require 'matrix'

class Matrix
def lu_decomposition
p = get_pivot
tmp = p * self
u = Matrix.zero(row_size).to_a
l = Matrix.identity(row_size).to_a
(0 ... row_size).each do |i|
(0 ... row_size).each do |j|
if j >= i
# upper
u[i][j] = tmp[i,j] - (0 ... i).inject(0.0) {|sum, k| sum + u[k][j] * l[i][k]}
else
# lower
l[i][j] = (tmp[i,j] - (0 ... j).inject(0.0) {|sum, k| sum + u[k][j] * l[i][k]}) / u[j][j]
end
end
end
[ Matrix[*l], Matrix[*u], p ]
end

def get_pivot
raise ArgumentError, "must be square" unless square?
id = Matrix.identity(row_size).to_a
(0 ... row_size).each do |i|
max = self[i,i]
row = i
(i ... row_size).each do |j|
if self[j,i] > max
max = self[j,i]
row = j
end
end
id[i], id[row] = id[row], id[i]
end
Matrix[*id]
end

puts each_slice(column_size).map{|row| format*row_size % row}
end
end

puts "Example 1:"
a = Matrix[[1,  3,  5],
[2,  4,  7],
[1,  1,  0]]
a.pretty_print(" %2d", "A")
l, u, p = a.lu_decomposition
l.pretty_print(" %8.5f", "L")
u.pretty_print(" %8.5f", "U")
p.pretty_print(" %d",    "P")

puts "\nExample 2:"
a = Matrix[[11, 9,24,2],
[ 1, 5, 2,6],
[ 3,17,18,1],
[ 2, 5, 7,1]]
a.pretty_print(" %2d", "A")
l, u, p = a.lu_decomposition
l.pretty_print(" %8.5f", "L")
u.pretty_print(" %8.5f", "U")
p.pretty_print(" %d",    "P")


{{out}}
Example 1:
A
1  3  5
2  4  7
1  1  0
L
1.00000  0.00000  0.00000
0.50000  1.00000  0.00000
0.50000 -1.00000  1.00000
U
2.00000  4.00000  7.00000
0.00000  1.00000  1.50000
0.00000  0.00000 -2.00000
P
0 1 0
1 0 0
0 0 1

Example 2:
A
11  9 24  2
1  5  2  6
3 17 18  1
2  5  7  1
L
1.00000  0.00000  0.00000  0.00000
0.27273  1.00000  0.00000  0.00000
0.09091  0.28750  1.00000  0.00000
0.18182  0.23125  0.00360  1.00000
U
11.00000  9.00000 24.00000  2.00000
0.00000 14.54545 11.45455  0.45455
0.00000  0.00000 -3.47500  5.68750
0.00000  0.00000  0.00000  0.51079
P
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1



Matrix has a lup_decomposition built-in method.

ruby
l, u, p = a.lup_decomposition
l.pretty_print(" %8.5f", "L")
u.pretty_print(" %8.5f", "U")
p.pretty_print(" %d",    "P")


Output is the same.

## Rust

Rust

#![allow(non_snake_case)]
use ndarray::{Array, Axis, Array2, arr2, Zip, NdFloat, s};

fn main() {
println!("Example 1:");
let A: Array2 = arr2(&[
[1.0, 3.0, 5.0],
[2.0, 4.0, 7.0],
[1.0, 1.0, 0.0],
]);
println!("A \n {}", A);
let (L, U, P) = lu_decomp(A);
println!("L \n {}", L);
println!("U \n {}", U);
println!("P \n {}", P);

println!("\nExample 2:");
let A: Array2 = arr2(&[
[11.0, 9.0, 24.0, 2.0],
[1.0, 5.0, 2.0, 6.0],
[3.0, 17.0, 18.0, 1.0],
[2.0, 5.0, 7.0, 1.0],
]);
println!("A \n {}", A);
let (L, U, P) = lu_decomp(A);
println!("L \n {}", L);
println!("U \n {}", U);
println!("P \n {}", P);
}

fn pivot(A: &Array2) -> Array2
where T: NdFloat {
let matrix_dimension = A.rows();
let mut P: Array2 = Array::eye(matrix_dimension);
for (i, column) in A.axis_iter(Axis(1)).enumerate() {
// find idx of maximum value in column i
let mut max_pos = i;
for j in i..matrix_dimension {
if column[max_pos].abs() < column[j].abs() {
max_pos = j;
}
}
// swap rows of P if necessary
if max_pos != i {
swap_rows(&mut P, i, max_pos);
}
}
P
}

fn swap_rows(A: &mut Array2, idx_row1: usize, idx_row2: usize)
where T: NdFloat {
// to swap rows, get two ArrayViewMuts for the corresponding rows
// and apply swap elementwise using ndarray::Zip
let (.., mut matrix_rest) = A.view_mut().split_at(Axis(0), idx_row1);
let (row0, mut matrix_rest) = matrix_rest.view_mut().split_at(Axis(0), 1);
let (_matrix_helper, mut matrix_rest) = matrix_rest.view_mut().split_at(Axis(0), idx_row2 - idx_row1 - 1);
let (row1, ..) = matrix_rest.view_mut().split_at(Axis(0), 1);
Zip::from(row0).and(row1).apply(std::mem::swap);
}

fn lu_decomp(A: Array2) -> (Array2, Array2, Array2)
where T: NdFloat {

let matrix_dimension = A.rows();
assert_eq!(matrix_dimension, A.cols(), "Tried LU decomposition with a non-square matrix.");
let P = pivot(&A);
let pivotized_A = P.dot(&A);

let mut L: Array2 = Array::eye(matrix_dimension);
let mut U: Array2 = Array::zeros((matrix_dimension, matrix_dimension));
for idx_col in 0..matrix_dimension {
// fill U
for idx_row in 0..idx_col+1 {
U[[idx_row, idx_col]] = pivotized_A[[idx_row, idx_col]] -
U.slice(s![0..idx_row,idx_col]).dot(&L.slice(s![idx_row,0..idx_row]));
}
// fill L
for idx_row in idx_col+1..matrix_dimension {
L[[idx_row, idx_col]] = (pivotized_A[[idx_row, idx_col]] -
U.slice(s![0..idx_col,idx_col]).dot(&L.slice(s![idx_row,0..idx_col]))) /
U[[idx_col, idx_col]];
}
}
(L, U, P)
}



{{out}}
Example 1:
A
[[1, 3, 5],
[2, 4, 7],
[1, 1, 0]]
L
[[1, 0, 0],
[0.5, 1, 0],
[0.5, -1, 1]]
U
[[2, 4, 7],
[0, 1, 1.5],
[0, 0, -2]]
P
[[0, 1, 0],
[1, 0, 0],
[0, 0, 1]]

Example 2:
A
[[11, 9, 24, 2],
[1, 5, 2, 6],
[3, 17, 18, 1],
[2, 5, 7, 1]]
L
[[1, 0, 0, 0],
[0.2727272727272727, 1, 0, 0],
[0.09090909090909091, 0.2875, 1, 0],
[0.18181818181818182, 0.23124999999999996, 0.0035971223021580693, 1]]
U
[[11, 9, 24, 2],
[0, 14.545454545454547, 11.454545454545455, 0.4545454545454546],
[0, 0, -3.4749999999999996, 5.6875],
[0, 0, 0, 0.510791366906476]]
P
[[1, 0, 0, 0],
[0, 0, 1, 0],
[0, 1, 0, 0],
[0, 0, 0, 1]]



## Sidef

{{trans|Perl 6}}

ruby
func is_square(m) { m.all { .len == m.len } }
func matrix_zero(n, m=n) { m.of { n.of(0) } }
func matrix_ident(n) { n.of {|i| n.of {|j| i==j ? 1 : 0 } } }

func pivotize(m) {
var size = m.len
var id = matrix_ident(size)
for i (^size) {
var max = m[i][i]
var row = i
for j (i .. size-1) {
if (m[j][i] > max) {
max = m[j][i]
row = j
}
}
if (row != i) {
id.swap(row, i)
}
}
return id
}

func mmult(a, b) {
var p = []
for r,c (^a ~X ^b[0]) {
for i (^b) {
p[r][c] := 0 += (a[r][i] * b[i][c])
}
}
return p
}

func lu(a) {
is_square(a) || die "Defined only for square matrices!";
var n = a.len
var P = pivotize(a)
var Aʼ = mmult(P, a)
var L = matrix_ident(n)
var U = matrix_zero(n)
for i,j (^n ~X ^n) {
if (j >= i) {
U[i][j] = (Aʼ[i][j] - ({ U[_][j] * L[i][_] }.map(^i).sum))
} else {
L[i][j] = (Aʼ[i][j] - ({ U[_][j] * L[i][_] }.map(^j).sum))/U[j][j]
}
}
return [P, Aʼ, L, U]
}

func say_it(message, array) {
say "\n#{message}"
array.each { |row|
say row.map{"%7s" % .as_rat}.join(' ')
}
}

var t = [[
%n(1 3 5),
%n(2 4 7),
%n(1 1 0),
],[
%n(11  9 24  2),
%n( 1  5  2  6),
%n( 3 17 18  1),
%n( 2  5  7  1),
]]

for test (t) {
say_it('A Matrix', test);
for a,b (['P Matrix', 'Aʼ Matrix', 'L Matrix', 'U Matrix'] ~Z lu(test)) {
say_it(a, b)
}
}


A Matrix
1       3       5
2       4       7
1       1       0

P Matrix
0       1       0
1       0       0
0       0       1

Aʼ Matrix
2       4       7
1       3       5
1       1       0

L Matrix
1       0       0
1/2       1       0
1/2      -1       1

U Matrix
2       4       7
0       1     3/2
0       0      -2

A Matrix
11       9      24       2
1       5       2       6
3      17      18       1
2       5       7       1

P Matrix
1       0       0       0
0       0       1       0
0       1       0       0
0       0       0       1

Aʼ Matrix
11       9      24       2
3      17      18       1
1       5       2       6
2       5       7       1

L Matrix
1       0       0       0
3/11       1       0       0
1/11   23/80       1       0
2/11  37/160   1/278       1

U Matrix
11       9      24       2
0  160/11  126/11    5/11
0       0 -139/40   91/16
0       0       0  71/139



## Stata

###  Builtin LU decoposition

See [http://www.stata.com/help.cgi?mf_lud LU decomposition] in Stata help.

stata
mata
: lud(a=(1,3,5\2,4,7\1,1,0),l=.,u=.,p=.)

: a
1   2   3
+-------------+
1 |  1   3   5  |
2 |  2   4   7  |
3 |  1   1   0  |
+-------------+

: l
1    2    3
+----------------+
1 |   1    0    0  |
2 |  .5    1    0  |
3 |  .5   -1    1  |
+----------------+

: u
1     2     3
+-------------------+
1 |    2     4     7  |
2 |    0     1   1.5  |
3 |    0     0    -2  |
+-------------------+

: p
1
+-----+
1 |  2  |
2 |  1  |
3 |  3  |
+-----+


###  Implementation

stata
void ludec(real matrix a, real matrix l, real matrix u, real vector p) {
real scalar i,j,n,s
real vector js

l = a
n = rows(a)
p = 1::n
for (i=1; i $max} { set max [lindex$matrix $i$j]
set row $i } } if {$j != $row} { # Row swap inlined; too trivial to have separate procedure set tmp [lindex$p $j] lset p$j [lindex $p$row]
lset p $row$tmp
}
}
return $p } # Decompose a square matrix A by PA=LU and return L, U and P proc luDecompose {A} { set n [llength$A]
set L [lrepeat $n [lrepeat$n 0]]
set U $L set P [pivotize$A]
set A [multiply $P$A]

for {set j 0} {$j <$n} {incr j} {
lset L $j$j 1
for {set i 0} {$i <=$j} {incr i} {
lset U $i$j [- [lindex $A$i $j] [SumMul$L $U$i $j$i]]
}
for {set i $j} {$i < $n} {incr i} { set sum [SumMul$L $U$i $j$j]
lset L $i$j [/ [- [lindex $A$i $j]$sum] [lindex $U$j $j]] } } return [list$L $U$P]
}

# Helper that makes inner loop nicer; multiplies column and row,
# possibly partially...
proc SumMul {A B i j kmax} {
set s 0.0
for {set k 0} {$k <$kmax} {incr k} {
set s [+ $s [* [lindex$A $i$k] [lindex $B$k $j]]] } return$s
}
}


Support code:

tcl
namespace eval matrix {
# Get the size of a matrix; assumes that all rows are the same length, which
# is a basic well-formed-ness condition...
proc size {m} {
set rows [llength $m] set cols [llength [lindex$m 0]]
return [list $rows$cols]
}

# Matrix multiplication implementation
proc multiply {a b} {
lassign [size $a] a_rows a_cols lassign [size$b] b_rows b_cols
if {$a_cols !=$b_rows} {
error "incompatible sizes: a($a_rows,$a_cols), b($b_rows,$b_cols)"
}
set temp [lrepeat $a_rows [lrepeat$b_cols 0]]
for {set i 0} {$i <$a_rows} {incr i} {
for {set j 0} {$j <$b_cols} {incr j} {
lset temp $i$j [SumMul $a$b $i$j $a_cols] } } return$temp
}

# Pretty printer for matrices
proc print {matrix {fmt "%g"}} {
set max [Widest $matrix$fmt]
lassign [size $matrix] rows cols foreach row$matrix {
foreach val $row width$max {
puts -nonewline [format "%*s " $width [format$fmt $val]] } puts "" } } proc Widest {m fmt} { lassign [size$m] rows cols
set max [lrepeat $cols 0] foreach row$m {
for {set j 0} {$j <$cols} {incr j} {
set s [format $fmt [lindex$row $j]] lset max$j [max [lindex $max$j] [string length $s]] } } return$max
}
}


Demonstrating:

tcl
# This does the decomposition and prints it out nicely
proc demo {A} {
lassign [matrix::luDecompose $A] L U P foreach v {A L U P} { upvar 0$v matrix
puts "${v}:" matrix::print$matrix %.5g
if {\$v ne "P"} {puts "---------------------------------"}
}
}
demo {{1 3 5} {2 4 7} {1 1 0}}
puts "
### ===========================
"
demo {{11 9 24 2} {1 5 2 6} {3 17 18 1} {2 5 7 1}}


{{out}}

txt

A:
1 3 5
2 4 7
1 1 0
---------------------------------
L:
1  0 0
0.5  1 0
0.5 -1 1
---------------------------------
U:
2 4   7
0 1 1.5
0 0  -2
---------------------------------
P:
0 1 0
1 0 0
0 0 1

### ===========================

A:
11  9 24 2
1  5  2 6
3 17 18 1
2  5  7 1
---------------------------------
L:
1       0         0 0
0.27273       1         0 0
0.090909  0.2875         1 0
0.18182 0.23125 0.0035971 1
---------------------------------
U:
11      9     24       2
0 14.545 11.455 0.45455
0      0 -3.475  5.6875
0      0      0 0.51079
---------------------------------
P:
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1



## VBA

{{trans|Phix}}

vb
Option Base 1
Private Function pivotize(m As Variant) As Variant
Dim n As Integer: n = UBound(m)
Dim im() As Double
ReDim im(n, n)
For i = 1 To n
For j = 1 To n
im(i, j) = 0
Next j
im(i, i) = 1
Next i
For i = 1 To n
mx = m(i, i)
row_ = i
For j = i To n
If m(j, i) > mx Then
mx = m(j, i)
row_ = j
End If
Next j
If i <> Row Then
For j = 1 To n
tmp = im(i, j)
im(i, j) = im(row_, j)
im(row_, j) = tmp
Next j
End If
Next i
pivotize = im
End Function

Private Function lu(a As Variant) As Variant
Dim n As Integer: n = UBound(a)
Dim l() As Double
ReDim l(n, n)
For i = 1 To n
For j = 1 To n
l(i, j) = 0
Next j
Next i
u = l
p = pivotize(a)
a2 = WorksheetFunction.MMult(p, a)
For j = 1 To n
l(j, j) = 1#
For i = 1 To j
sum1 = 0#
For k = 1 To i
sum1 = sum1 + u(k, j) * l(i, k)
Next k
u(i, j) = a2(i, j) - sum1
Next i
For i = j + 1 To n
sum2 = 0#
For k = 1 To j
sum2 = sum2 + u(k, j) * l(i, k)
Next k
l(i, j) = (a2(i, j) - sum2) / u(j, j)
Next i
Next j
Dim res(4) As Variant
res(1) = a
res(2) = l
res(3) = u
res(4) = p
lu = res
End Function

Public Sub main()

a = [{1, 3, 5; 2, 4, 7; 1, 1, 0}]
Debug.Print "== a,l,u,p: =="
result = lu(a)
For i = 1 To 4
For j = 1 To UBound(result(1))
For k = 1 To UBound(result(1), 2)
Debug.Print result(i)(j, k),
Next k
Debug.Print
Next j
Debug.Print
Next i
a = [{11, 9,24, 2; 1, 5, 2, 6; 3,17,18, 1; 2, 5, 7, 1}]
Debug.Print "== a,l,u,p: =="
result = lu(a)
For i = 1 To 4
For j = 1 To UBound(result(1))
For k = 1 To UBound(result(1), 2)
Debug.Print Format(result(i)(j, k), "0.#####"),
Next k
Debug.Print
Next j
Debug.Print
Next i
End Sub

{{out}}

txt
== a,l,u,p: ==
1             3             5
2             4             7
1             1             0

1             0             0
0,5           1             0
0,5          -1             1

2             4             7
0             1             1,5
0             0            -2

0             1             0
1             0             0
0             0             1

== a,l,u,p: ==
11,           9,            24,           2,
1,            5,            2,            6,
3,            17,           18,           1,
2,            5,            7,            1,

1,            0,            0,            0,
0,27273       1,            0,            0,
0,09091       0,2875        1,            0,
0,18182       0,23125       0,0036        1,

11,           9,            24,           2,
0,            14,54545      11,45455      0,45455
0,            0,            -3,475        5,6875
0,            0,            0,            0,51079

1,            0,            0,            0,
0,            0,            1,            0,
0,            1,            0,            0,
0,            0,            0,            1,


## zkl

Using the GNU Scientific Library, which does the decomposition without returning the permutations:

zkl
var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
A.LUDecompose();	//  in place, contains L & U
L:=A.copy().lowerTriangle().setDiagonal(0,0,1);
U:=A.copy().upperTriangle();
return(L,U);
}

A:=GSL.Matrix(3,3).set(1,3,5,  2,4,7,   1,1,0);  // example 1
println("L:\n",L.format(),"\nU:\n",U.format());

A:=GSL.Matrix(4,4).set(11.0,  9.0, 24.0, 2.0,	// example 2
1.0,  5.0,  2.0, 6.0,
3.0, 17.0, 18.0, 1.0,
2.0,  5.0,  7.0, 1.0);
println("L:\n",L.format(8,4),"\nU:\n",U.format(8,4));


{{out}}

txt

L:
1.00,      0.00,      0.00
0.50,      1.00,      0.00
0.50,     -1.00,      1.00
U:
2.00,      4.00,      7.00
0.00,      1.00,      1.50
0.00,      0.00,     -2.00
L:
1.0000,  0.0000,  0.0000,  0.0000
0.2727,  1.0000,  0.0000,  0.0000
0.0909,  0.2875,  1.0000,  0.0000
0.1818,  0.2312,  0.0036,  1.0000
U:
11.0000,  9.0000, 24.0000,  2.0000
0.0000, 14.5455, 11.4545,  0.4545
0.0000,  0.0000, -3.4750,  5.6875
0.0000,  0.0000,  0.0000,  0.5108



Or, using lists:
{{trans|Common Lisp}}
{{trans|D}}

A matrix is a list of lists, ie list of rows in row major order.

zkl
fcn make_array(n,m,v){ (m).pump(List.createLong(m).write,v)*n }
fcn eye(n){ // Creates a nxn identity matrix.
I:=make_array(n,n,0.0);
foreach j in (n){ I[j][j]=1.0 }
I
}

// Creates the pivoting matrix for A.
fcn pivotize(A){
n:=A.len();	// rows
P:=eye(n);
foreach i in (n){
max,row:=A[i][i],i;
foreach j in ([i..n-1]){
if(A[j][i]>max) max,row=A[j][i],j;
}
if(i!=row) P.swap(i,row);
}
// Return P.
P
}

// Decomposes a square matrix A by PA=LU and returns L, U and P.
fcn lu(A){
n:=A.len();
L:=eye(n);
U:=make_array(n,n,0.0);
P:=pivotize(A);
A=matMult(P,A);

foreach j in (n){
foreach i in (j+1){
U[i][j]=A[i][j] - (i).reduce('wrap(s,k){ s + U[k][j]*L[i][k] },0.0);
}
foreach i in ([j..n-1]){
L[i][j]=( A[i][j] -
(j).reduce('wrap(s,k){ s + U[k][j]*L[i][k] },0.0) ) /
U[j][j];
}
}
// Return L, U and P.
return(L,U,P);
}

fcn matMult(a,b){
n,m,p:=a[0].len(),a.len(),b[0].len();
ans:=make_array(n,m,0.0);
foreach i,j,k in (m,p,n){ ans[i][j]+=a[i][k]*b[k][j]; }
ans
}


Example 1

zkl
g:=L(L(1.0,3.0,5.0),L(2.0,4.0,7.0),L(1.0,1.0,0.0));
lu(g).apply2("println");


{{out}}

txt

L(L(1,0,0),L(0.5,1,0),L(0.5,-1,1))
L(L(2,4,7),L(0,1,1.5),L(0,0,-2))
L(L(0,1,0),L(1,0,0),L(0,0,1))



Example 2

zkl
lu(L( L(11.0,  9.0, 24.0, 2.0),
L( 1.0,  5.0,  2.0, 6.0),
L( 3.0, 17.0, 18.0, 1.0),
L( 2.0,  5.0,  7.0, 1.0) )).apply2(T(printM,Console.writeln.fpM("-")));

fcn printM(m)  { m.pump(Console.println,rowFmt) }
fcn rowFmt(row){ ("%9.5f "*row.len()).fmt(row.xplode()) }


The list apply2 method is side effects only, it doesn't aggregate results. When given a list of actions, it applies the action and passes the result to the next action. The fpM method is partial application with a mask, "-" truncates the parameters at that point (in this case, no parameters, ie just print a blank line, not the result of printM).
{{out}}

txt

1.00000   0.00000   0.00000   0.00000
0.27273   1.00000   0.00000   0.00000
0.09091   0.28750   1.00000   0.00000
0.18182   0.23125   0.00360   1.00000

11.00000   9.00000  24.00000   2.00000
0.00000  14.54545  11.45455   0.45455
0.00000   0.00000  -3.47500   5.68750
0.00000   0.00000   0.00000   0.51079

1.00000   0.00000   0.00000   0.00000
0.00000   0.00000   1.00000   0.00000
0.00000   1.00000   0.00000   0.00000
0.00000   0.00000   0.00000   1.00000