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{{Task|Matrices}} Every square matrix can be decomposed into a product of a lower triangular matrix and a upper triangular matrix , as described in [[wp:LU decomposition|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:
:
\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 are set to 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
\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 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
{{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;
{{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)
(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
{{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]]
=={{header|MATLAB}} / {{header|Octave}}==
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
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")
```
{{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
{{libheader| ndarray}}
```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
# 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:
```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)
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));
```
{{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
```