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LU decomposition

From Rosetta Code
Task
LU decomposition
You are encouraged to solve this task according to the task description, using any language you may know.

Every square matrix can be decomposed into a product of a lower triangular matrix and a upper triangular matrix , as described in LU decomposition.

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:

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

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

Solving for the other and , we get the following equations:

and for :

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

and then for

We see in the second formula that to get the below the diagonal, we have to divide by the diagonal element (pivot) , so we get problems when is either 0 or very small, which leads to numerical instability.

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

Example:

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


Task description

The task is to implement a routine which will take a square nxn matrix and return a lower triangular matrix , a upper triangular matrix and a permutation matrix , 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

Ada[edit]

Works with: Ada 2005

decomposition.ads:

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;

decomposition.adb:

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 Ada.Text_IO;
with Decomposition;
procedure Decompose_Example is
package Real_Decomposition is new Decomposition
(Matrix => Ada.Numerics.Real_Arrays);
 
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;
Ada.Text_IO.New_Line;
end loop;
end Print;
 
Example_1 : constant Ada.Numerics.Real_Arrays.Real_Matrix :=
((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));
Example_2 : constant Ada.Numerics.Real_Arrays.Real_Matrix :=
((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);
Ada.Text_IO.Put_Line ("Example 1:");
Ada.Text_IO.Put_Line ("A:"); Print (Example_1);
Ada.Text_IO.Put_Line ("L:"); Print (L_1);
Ada.Text_IO.Put_Line ("U:"); Print (U_1);
Ada.Text_IO.Put_Line ("P:"); Print (P_1);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put_Line ("Example 2:");
Ada.Text_IO.Put_Line ("A:"); Print (Example_2);
Ada.Text_IO.Put_Line ("L:"); Print (L_2);
Ada.Text_IO.Put_Line ("U:"); Print (U_2);
Ada.Text_IO.Put_Line ("P:"); Print (P_2);
end Decompose_Example;
Output:
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[edit]

      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$(10)
NEXT i%
= a$
Output:
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[edit]

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[edit]

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)
(let* ((n (cadr (array-dimensions A)))
(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[edit]

Translation of: 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);
}
Output:
[[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[edit]

 
(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[edit]

 
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
 
Output:
 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[edit]

2D representation[edit]

Translation of: 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")
}
Output:
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[edit]

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/matrix[edit]

package main
 
import (
"fmt"
 
"github.com/gonum/matrix/mat64"
)
 
func main() {
showLU(mat64.NewDense(3, 3, []float64{
1, 3, 5,
2, 4, 7,
1, 1, 0,
}))
fmt.Println()
showLU(mat64.NewDense(4, 4, []float64{
11, 9, 24, 2,
1, 5, 2, 6,
3, 17, 18, 1,
2, 5, 7, 1,
}))
}
 
func showLU(a *mat64.Dense) {
fmt.Printf("a: %v\n\n", mat64.Formatted(a, mat64.Prefix(" ")))
var lu mat64.LU
lu.Factorize(a)
var l, u mat64.TriDense
l.LFrom(&lu)
u.UFrom(&lu)
fmt.Printf("l: %.5f\n\n", mat64.Formatted(&l, mat64.Prefix(" ")))
fmt.Printf("u: %.5f\n\n", mat64.Formatted(&u, mat64.Prefix(" ")))
fmt.Println("p:", lu.Pivot(nil))
}
Output:

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[edit]

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)
}
Output:
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}


Idris[edit]

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
 
Output:
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[edit]

Taken with slight modification from [1].

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 01 0 01 3 5
0 1 02 1 00 _2 _3
0 0 11 1 10 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 01 0 0 011 9 24 2
0 1 0 00.0909091 1 0 00 4.18182 _0.181818 5.81818
0 0 1 00.272727 3.47826 1 00 0 12.087 _19.7826
0 0 0 10.181818 0.804348 0.230216 10 0 0 0.510791
└───────┴─────────────────────────────┴─────────────────────────────┘
mp/> LUdecompose A
11 9 24 2
1 5 2 6
3 17 18 1
2 5 7 1

Java[edit]

Translation of Common Lisp via D

Works with: Java version 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[edit]

Works with: jq version 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)
 
Output:
 $ /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)
Output:
$ /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)
Output:
true

Julia[edit]

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

Output:
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])

Maple[edit]

 
A:=<<1.0|3.0|5.0>,<2.0|4.0|7.0>,<1.0|1.0|0.0>>:
 
LinearAlgebra:-LUDecomposition(A);
 
Output:
    [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);
 
Output:
    [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[edit]

(*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}
 
Output:

LUex1.png LUex2.png


MATLAB / Octave[edit]

LU decomposition is part of language

  A = [
1 3 5
2 4 7
1 1 0];
 
[L,U,P] = lu(A)
Output:
  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)
Output:
  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[edit]

 
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[edit]

/* 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[edit]

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 6[edit]

Works with: Rakudo version 2015-11-20

Translation of Ruby.

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;
}
Output:
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

PL/I[edit]

(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:

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[edit]

Translation of: D
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
Output:
[[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[edit]

Output:
> 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[edit]

 
#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[edit]

/*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 makeMat /*make the A matrix from the numbers.*/
call showMat 'A', N /*display the A matrix. */
call manPmat /*manufacture P (permutation). */
call showMat 'P', N /*display the P matrix. */
call multMat /*multiply the A and P matrices. */
call showMat 'PA', N /*display the PA matrix. */
do y=1 for N; call manUmat y /*manufacture U matrix, parts. */
call manLmat y /*manufacture L matrix, parts. */
end
call showMat 'L', N /*display the L matrix. */
call showMat 'U', N /*display the U matrix. */
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
er: say; say '***error!***'; say; say arg(1); say; exit 13
/*──────────────────────────────────────────────────────────────────────────────────────*/
makeMat: ?=words(x); do N=1 for ?; if N**2==? then leave; end /*N*/
if N**2\==? then call er 'not correct number of elements entered: ' ?
 
do r=1 for N /*build the "A" matrix from the input*/
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
/*──────────────────────────────────────────────────────────────────────────────────────*/
manLmat: parse arg ? /*manufacture L (lower) matrix.*/
do r=1 for N
do c=1 for N; if r==c then do; L.r.c=1; iterate; end
if c\==? | 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
/*──────────────────────────────────────────────────────────────────────────────────────*/
manPmat: c=N; do r=N by -1 for N /*manufacture P (permutation). */
P.r.c=1; c=c+1; if c>N then c=N%2; if c==N then c=1
end /*r*/
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
manUmat: parse arg ? /*manufacture U (upper) matrix.*/
do r=1 for N; if r\==? then iterate
do c=1 for N; if c<r then iterate
_=PA.r.c
do k=1 for r-1; _=_-U.k.c*L.r.k; end /*k*/
U.r.c=_/1
end /*c*/
end /*r*/
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
multMat: do i=1 for N /*multiply matrix P & A ──► PA */
do j=1 for N
do k=1 for N; pa.i.j=(pa.i.j + p.i.k * a.k.j) / 1
end /*k*/
end /*j*/
end /*i*/
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
showMat: parse arg mat,rows,cols; w=0; cols=word(cols rows,1); say
do r=1 for rows
do c=1 for cols; w=max(w, length( value( mat'.'r"."c ) ) )
end /*c*/
end /*r*/
say center(mat 'matrix',cols*(w+1)+7,"─")
do r=1 for rows; _=
do c=1 for cols; _=_ right(value(mat'.'r'.'c),w+1); end /*c*/
say _
end /*r*/
return

output   when using the input of:   1 3 5   2 4 7   1 1 0

──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

output   when using the input of:   11 9 24 2   1 5 2 6   3 17 18 1   2 5 7 1

─────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[edit]

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
 
def pretty_print(format, head=nil)
puts head if head
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")
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

Matrix has a lup_decomposition built-in method.

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.

Sidef[edit]

Translation of: Perl 6
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| [(i-1).of(0)..., 1, (n-i).of(0)...] } }
 
func pivotize(m) {
var size = m.len
var id = matrix_ident(size)
for i in ^size {
var max = m[i][i]
var row = i
for j in range(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 in (^a ~X ^b[0]) {
for i in ^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 in (^n ~X ^n) {
if (j >= i) {
U[i][j] = ([i][j] - (^i->map { U[_][j] * L[i][_] }.sum(0)))
} else {
L[i][j] = (([i][j] - (^j->map { U[_][j] * L[i][_] }.sum(0))) / 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),
]]
 
t.each { |test|
say_it('A Matrix', test);
for a in (['P Matrix', 'Aʼ Matrix', 'L Matrix', 'U Matrix'] ~Z lu(test)) {
say_it(a[0], a[1])
}
}
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

Tcl[edit]

package require Tcl 8.5
namespace eval matrix {
namespace path {::tcl::mathfunc ::tcl::mathop}
 
# Construct an identity matrix of the given size
proc identity {order} {
set m [lrepeat $order [lrepeat $order 0]]
for {set i 0} {$i < $order} {incr i} {
lset m $i $i 1
}
return $m
}
 
# Produce the pivot matrix for a given matrix
proc pivotize {matrix} {
set n [llength $matrix]
set p [identity $n]
for {set j 0} {$j < $n} {incr j} {
set max [lindex $matrix $j $j]
set row $j
for {set i $j} {$i < $n} {incr i} {
if {[lindex $matrix $i $j] > $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:

# Code adapted from Matrix_multiplication and Matrix_transposition tasks
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:

# 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}}
Output:
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 

zkl[edit]

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

var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
fcn luTask(A){
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
L,U:=luTask(A);
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);
L,U:=luTask(A);
println("L:\n",L.format(8,4),"\nU:\n",U.format(8,4));
Output:
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:

Translation of: Common Lisp
Translation of: D

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

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

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");
Output:
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

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).

Output:
  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