# Cholesky decomposition

Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose:

Cholesky decomposition
You are encouraged to solve this task according to the task description, using any language you may know.
${\displaystyle A=LL^{T}}$

${\displaystyle L}$ is called the Cholesky factor of ${\displaystyle A}$, and can be interpreted as a generalized square root of ${\displaystyle A}$, as described in Cholesky decomposition.

In a 3x3 example, we have to solve the following system of equations:

{\displaystyle {\begin{aligned}A&={\begin{pmatrix}a_{11}&a_{21}&a_{31}\\a_{21}&a_{22}&a_{32}\\a_{31}&a_{32}&a_{33}\\\end{pmatrix}}\\&={\begin{pmatrix}l_{11}&0&0\\l_{21}&l_{22}&0\\l_{31}&l_{32}&l_{33}\\\end{pmatrix}}{\begin{pmatrix}l_{11}&l_{21}&l_{31}\\0&l_{22}&l_{32}\\0&0&l_{33}\end{pmatrix}}\equiv LL^{T}\\&={\begin{pmatrix}l_{11}^{2}&l_{21}l_{11}&l_{31}l_{11}\\l_{21}l_{11}&l_{21}^{2}+l_{22}^{2}&l_{31}l_{21}+l_{32}l_{22}\\l_{31}l_{11}&l_{31}l_{21}+l_{32}l_{22}&l_{31}^{2}+l_{32}^{2}+l_{33}^{2}\end{pmatrix}}\end{aligned}}}

We can see that for the diagonal elements (${\displaystyle l_{kk}}$) of ${\displaystyle L}$ there is a calculation pattern:

${\displaystyle l_{11}={\sqrt {a_{11}}}}$
${\displaystyle l_{22}={\sqrt {a_{22}-l_{21}^{2}}}}$
${\displaystyle l_{33}={\sqrt {a_{33}-(l_{31}^{2}+l_{32}^{2})}}}$

or in general:

${\displaystyle l_{kk}={\sqrt {a_{kk}-\sum _{j=1}^{k-1}l_{kj}^{2}}}}$

For the elements below the diagonal (${\displaystyle l_{ik}}$, where ${\displaystyle i>k}$) there is also a calculation pattern:

${\displaystyle l_{21}={\frac {1}{l_{11}}}a_{21}}$
${\displaystyle l_{31}={\frac {1}{l_{11}}}a_{31}}$
${\displaystyle l_{32}={\frac {1}{l_{22}}}(a_{32}-l_{31}l_{21})}$

which can also be expressed in a general formula:

${\displaystyle l_{ik}={\frac {1}{l_{kk}}}\left(a_{ik}-\sum _{j=1}^{k-1}l_{ij}l_{kj}\right)}$

The task is to implement a routine which will return a lower Cholesky factor ${\displaystyle L}$ for every given symmetric, positive definite nxn matrix ${\displaystyle A}$. You should then test it on the following two examples and include your output.

Example 1:

25  15  -5                 5   0   0
15  18   0         -->     3   3   0
-5   0  11                -1   1   3


Example 2:

18  22   54   42           4.24264    0.00000    0.00000    0.00000
22  70   86   62   -->     5.18545    6.56591    0.00000    0.00000
54  86  174  134          12.72792    3.04604    1.64974    0.00000
42  62  134  106           9.89949    1.62455    1.84971    1.39262


Note
1. The Cholesky decomposition of a Pascal upper-triangle matrix is the Identity matrix of the same size.
2. The Cholesky decomposition of a Pascal symmetric matrix is the Pascal lower-triangle matrix of the same size.

## 11l

Translation of: Python
F cholesky(A)
V l = [[0.0] * A.len] * A.len
L(i) 0 .< A.len
L(j) 0 .. i
V s = sum((0 .< j).map(k -> @l[@i][k] * @l[@j][k]))
l[i][j] = I (i == j) {sqrt(A[i][i] - s)} E (1.0 / l[j][j] * (A[i][j] - s))
R l

F pprint(m)
print(‘[’)
L(row) m
print(row)
print(‘]’)

V m1 = [[25, 15, -5],
[15, 18,  0],
[-5,  0, 11]]
print(cholesky(m1))
print()

V m2 = [[18, 22,  54,  42],
[22, 70,  86,  62],
[54, 86, 174, 134],
[42, 62, 134, 106]]
pprint(cholesky(m2))
Output:
[[5, 0, 0], [3, 3, 0], [-1, 1, 3]]

[
[4.24264, 0, 0, 0]
[5.18545, 6.56591, 0, 0]
[12.7279, 3.04604, 1.64974, 0]
[9.89949, 1.62455, 1.84971, 1.39262]
]


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 A = L * Transpose (L)
procedure Decompose (A : Matrix.Real_Matrix; L : out Matrix.Real_Matrix);

end Decomposition;


with Ada.Numerics.Generic_Elementary_Functions;

package body Decomposition is
(Matrix.Real);

procedure Decompose (A : Matrix.Real_Matrix; L : out Matrix.Real_Matrix) is
use type Matrix.Real_Matrix, Matrix.Real;
Order : constant Positive := A'Length (1);
S     : Matrix.Real;
begin
L := (others => (others => 0.0));
for I in 0 .. Order - 1 loop
for K in 0 .. I loop
S := 0.0;
for J in 0 .. K - 1 loop
S := S +
L (L'First (1) + I, L'First (2) + J) *
L (L'First (1) + K, L'First (2) + J);
end loop;
-- diagonals
if K = I then
L (L'First (1) + K, L'First (2) + K) :=
Math.Sqrt (A (A'First (1) + K, A'First (2) + K) - S);
else
L (L'First (1) + I, L'First (2) + K) :=
1.0 / L (L'First (1) + K, L'First (2) + K) *
(A (A'First (1) + I, A'First (2) + K) - S);
end if;
end loop;
end loop;
end Decompose;
end Decomposition;


Example usage:

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

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

procedure Print (M : Ada.Numerics.Real_Arrays.Real_Matrix) is
begin
for Row in M'Range (1) loop
for Col in M'Range (2) loop
Real_IO.Put (M (Row, Col), 4, 3, 0);
end loop;
end loop;
end Print;

((25.0, 15.0, -5.0),
(15.0, 18.0, 0.0),
(-5.0, 0.0, 11.0));
Example_1'Range (2));
((18.0, 22.0, 54.0, 42.0),
(22.0, 70.0, 86.0, 62.0),
(54.0, 86.0, 174.0, 134.0),
(42.0, 62.0, 134.0, 106.0));
Example_2'Range (2));
begin
Real_Decomposition.Decompose (A => Example_1,
L => L_1);
Real_Decomposition.Decompose (A => Example_2,
L => L_2);
end Decompose_Example;

Output:
Example 1:
A:
25.000  15.000  -5.000
15.000  18.000   0.000
-5.000   0.000  11.000
L:
5.000   0.000   0.000
3.000   3.000   0.000
-1.000   1.000   3.000

Example 2:
A:
18.000  22.000  54.000  42.000
22.000  70.000  86.000  62.000
54.000  86.000 174.000 134.000
42.000  62.000 134.000 106.000
L:
4.243   0.000   0.000   0.000
5.185   6.566   0.000   0.000
12.728   3.046   1.650   0.000
9.899   1.625   1.850   1.393

## ALGOL 68

Translation of: C
Note: This specimen retains the original C coding style. diff
Works with: ALGOL 68 version Revision 1 - no extensions to language used.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny.
#!/usr/local/bin/a68g --script #

MODE FIELD=LONG REAL;
PROC (FIELD)FIELD field sqrt = long sqrt;
INT field prec = 5;
FORMAT field fmt = $g(-(2+1+field prec),field prec)$;

MODE MAT = [0,0]FIELD;

PROC cholesky = (MAT a) MAT:(
[UPB a, 2 UPB a]FIELD l;

FOR i FROM LWB a TO UPB a DO
FOR j FROM 2 LWB a TO i DO
FIELD s := 0;
FOR k FROM 2 LWB a TO j-1 DO
s +:= l[i,k] * l[j,k]
OD;
l[i,j] := IF i = j
THEN field sqrt(a[i,i] - s)
ELSE 1.0 / l[j,j] * (a[i,j] - s) FI
OD;
FOR j FROM i+1 TO 2 UPB a DO
l[i,j]:=0 # Not required if matrix is declared as triangular #
OD
OD;
l
);

PROC print matrix v1 =(MAT a)VOID:(
FOR i FROM LWB a TO UPB a DO
FOR j FROM 2 LWB a TO 2 UPB a DO
printf(($g(-(2+1+field prec),field prec)$, a[i,j]))
OD;
printf($l$)
OD
);

PROC print matrix =(MAT a)VOID:(
FORMAT vector fmt = $"("f(field fmt)n(2 UPB a-2 LWB a)(", " f(field fmt))")"$;
FORMAT matrix fmt = $"("f(vector fmt)n( UPB a- LWB a)(","lxf(vector fmt))")"$;
printf((matrix fmt, a))
);

main: (
MAT m1 = ((25, 15, -5),
(15, 18,  0),
(-5,  0, 11));
MAT c1 = cholesky(m1);
print matrix(c1);
printf($l$);

MAT m2 = ((18, 22,  54,  42),
(22, 70,  86,  62),
(54, 86, 174, 134),
(42, 62, 134, 106));
MAT c2 = cholesky(m2);
print matrix(c2)
)
Output:
(( 5.00000,  0.00000,  0.00000),
( 3.00000,  3.00000,  0.00000),
(-1.00000,  1.00000,  3.00000))
(( 4.24264,  0.00000,  0.00000,  0.00000),
( 5.18545,  6.56591,  0.00000,  0.00000),
(12.72792,  3.04604,  1.64974,  0.00000),
( 9.89949,  1.62455,  1.84971,  1.39262))


## Arturo

cholesky: function [m][
result: array.of: @[size m, size m] 0.0

loop 0..dec size m\0 'i [
loop 0..i 'j [
s: 0.0
loop 0..j 'k ->
s: s + result\[i]\[k] * result\[j]\[k]

result\[i]\[j]: (i = j)? -> sqrt m\[i]\[i] - s
-> (1.0 // result\[j]\[j]) * (m\[i]\[j] - s)
]
]
return result
]

printMatrix: function [a]->
loop a 'b ->
print to [:string] .format:"8.5f" b

m1: @[
@[25.0, 15.0, neg 5.0]
@[15.0, 18.0,  0.0]
@[neg 5.0,  0.0, 11.0]
]
printMatrix cholesky m1

print ""

m2: [
[18.0, 22.0,  54.0,  42.0]
[22.0, 70.0,  86.0,  62.0]
[54.0, 86.0, 174.0, 134.0]
[42.0, 62.0, 134.0, 106.0]
]
printMatrix cholesky m2

Output:
 5.00000  0.00000  0.00000
3.00000  3.00000  0.00000
-1.00000  1.00000  3.00000

4.24264  0.00000  0.00000  0.00000
5.18545  6.56591  0.00000  0.00000
12.72792  3.04604  1.64974  0.00000
9.89949  1.62455  1.84971  1.39262

## ATS

%{^
#include <math.h>
#include <float.h>
%}

macdef NAN = g0f2f ($extval (float, "NAN")) macdef Zero = g0i2f 0 macdef One = g0i2f 1 (* The sqrt(3) function made part of the ‘g0float’ typekind series. (The ats2-xprelude package will do this for you, but it is easy to do if you are not using a lot of math functions. *) extern fn {tk : tkind} g0float_sqrt : g0float tk -<> g0float tk overload sqrt with g0float_sqrt implement g0float_sqrt<fltknd> x =$extfcall (float, "sqrtf", x)
implement g0float_sqrt<dblknd> x = $extfcall (double, "sqrt", x) implement g0float_sqrt<ldblknd> x =$extfcall (ldouble, "sqrtl", x)

(*------------------------------------------------------------------*)
(* A "very little matrix library"                                   *)

typedef Matrix_Index_Map (m1 : int, n1 : int, m0 : int, n0 : int) =
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(int i1, int j1) -<cloref0>
[i0, j0 : pos | i0 <= m0; j0 <= n0]
@(int i0, int j0)

datatype Real_Matrix (tk : tkind,
m1 : int, n1 : int,
m0 : int, n0 : int) =
| Real_Matrix of (matrixref (g0float tk, m0, n0),
int m1, int n1, int m0, int n0,
Matrix_Index_Map (m1, n1, m0, n0))
typedef Real_Matrix (tk : tkind, m1 : int, n1 : int) =
[m0, n0 : pos] Real_Matrix (tk, m1, n1, m0, n0)
typedef Real_Vector (tk : tkind, m1 : int, n1 : int) =
[m1 == 1 || n1 == 1] Real_Matrix (tk, m1, n1)
typedef Real_Row (tk : tkind, n1 : int) = Real_Vector (tk, 1, n1)
typedef Real_Column (tk : tkind, m1 : int) = Real_Vector (tk, m1, 1)

extern fn {tk : tkind}
Real_Matrix_make_elt :
{m0, n0 : pos}
(int m0, int n0, g0float tk) -< !wrt >
Real_Matrix (tk, m0, n0, m0, n0)

extern fn {tk : tkind}
Real_Matrix_copy :
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -< !refwrt > Real_Matrix (tk, m1, n1)

extern fn {tk : tkind}
Real_Matrix_copy_to :
{m1, n1 : pos}
(Real_Matrix (tk, m1, n1),    (* destination *)
Real_Matrix (tk, m1, n1)) -< !refwrt >
void

extern fn {}
Real_Matrix_dimension :
{tk : tkind}
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -<> @(int m1, int n1)

extern fn {tk : tkind}
Real_Matrix_get_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1) -< !ref > g0float tk

extern fn {tk : tkind}
Real_Matrix_set_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1, g0float tk) -< !refwrt >
void

extern fn {}
Real_Matrix_reflect_lower_triangle :
(* This operation makes every It is a change in how INDEXING
works. All the storage is still in the lower triangle. *)
{tk     : tkind}
{n1     : pos}
{m0, n0 : pos}
Real_Matrix (tk, n1, n1, m0, n0) -<>
Real_Matrix (tk, n1, n1, m0, n0)

extern fn {tk : tkind}
Real_Matrix_fprint :
{m, n : pos}
(FILEref, Real_Matrix (tk, m, n)) -<1> void

Real_Matrix_reflect_lower_triangle

(*------------------------------------------------------------------*)
(* Implementation of the "very little matrix library"               *)

implement {tk}
Real_Matrix_make_elt (m0, n0, elt) =
Real_Matrix (matrixref_make_elt<g0float tk> (i2sz m0, i2sz n0, elt),
m0, n0, m0, n0, lam (i1, j1) => @(i1, j1))

implement {}
Real_Matrix_dimension A =
case+ A of Real_Matrix (_, m1, n1, _, _, _) => @(m1, n1)

implement {tk}
Real_Matrix_get_at (A, i1, j1) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_get_at<g0float tk> (storage, pred i0, n0, pred j0)
end

implement {tk}
Real_Matrix_set_at (A, i1, j1, x) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_set_at<g0float tk> (storage, pred i0, n0, pred j0, x)
end

implement {}
Real_Matrix_reflect_lower_triangle {..} {n1} A =
let
typedef t = intBtwe (1, n1)
val+ Real_Matrix (storage, n1, _, m0, n0, index_map) = A
in
Real_Matrix (storage, n1, n1, m0, n0,
lam (i, j) =>
index_map ((if j <= i then i else j) : t,
(if j <= i then j else i) : t))
end

implement {tk}
Real_Matrix_copy A =
let
val @(m1, n1) = dimension A
val C = Real_Matrix_make_elt<tk> (m1, n1, A[1, 1])
val () = copy_to<tk> (C, A)
in
C
end

implement {tk}
Real_Matrix_copy_to (Dst, Src) =
let
val @(m1, n1) = dimension Src
prval [m1 : int] EQINT () = eqint_make_gint m1
prval [n1 : int] EQINT () = eqint_make_gint n1

var i : intGte 1
in
for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m1; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n1; j := succ j)
Dst[i, j] := Src[i, j]
end
end

implement {tk}
Real_Matrix_fprint {m, n} (outf, A) =
let
val @(m, n) = dimension A
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
typedef FILEstar = $extype"FILE *" extern castfn FILEref2star : FILEref -<> FILEstar val _ =$extfcall (int, "fprintf", FILEref2star outf,
"%16.6g", A[i, j])
in
end;
fprintln! (outf)
end
end

(*------------------------------------------------------------------*)
(* Cholesky-Banachiewicz, in place. See
https://en.wikipedia.org/w/index.php?title=Cholesky_decomposition&oldid=1149960985#The_Cholesky%E2%80%93Banachiewicz_and_Cholesky%E2%80%93Crout_algorithms

I would use Cholesky-Crout if my matrices were stored in column
major order. But it makes little difference. *)

extern fn {tk : tkind}
Real_Matrix_cholesky_decomposition :
(* Only the lower triangle is considered. *)
{n : pos}
Real_Matrix (tk, n, n) -< !refwrt > void

Real_Matrix_cholesky_decomposition

implement {tk}
Real_Matrix_cholesky_decomposition {n} A =
let
val @(n, _) = dimension A

(* I arrange the nested loops somewhat differently from how it is
done in the Wikipedia article's C snippet. *)
fun
repeat {i, j : pos | j <= i; i <= n + 1} (* <-- allowed values *)
.<(n + 1) - i, i - j>. (* <-- proof of termination *)
(i : int i, j : int j) :<!refwrt> void =
if i = n + 1 then
()                      (* All done. *)
else
let
fun
_sum {k : pos | k <= j} .<j - k>.
(x : g0float tk, k : int k) :<!refwrt> g0float tk =
if k = j then
x
else
_sum (x + (A[i, k] * A[j, k]), succ k)

val sum = _sum (Zero, 1)
in
if j = i then
begin
A[i, j] := sqrt (A[i, i] - sum);
repeat (succ i, 1)
end
else
begin
A[i, j] := (One / A[j, j]) * (A[i, j] - sum);
repeat (i, succ j)
end
end
in
repeat (1, 1)
end

(*------------------------------------------------------------------*)

fn {tk : tkind}           (* We like Fortran, so COLUMN major here. *)
column_major_list_to_square_matrix
{n   : pos}
(n   : int n,
lst : list (g0float tk, n * n))
: Real_Matrix (tk, n, n) =
let
#define :: list_cons
prval () = mul_gte_gte_gte {n, n} ()
val A = Real_Matrix_make_elt (n, n, NAN)
val lstref : ref (List0 (g0float tk)) = ref lst
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
var i : intGte 1
in
for* {i : pos | i <= n + 1} .<(n + 1) - i>.
(i : int i) =>
(i := 1; i <> succ n; i := succ i)
case- !lstref of
| hd :: tl =>
begin
A[i, j] := hd;
!lstref := tl
end
end;
A
end

implement
main0 () =
let
val _A =
column_major_list_to_square_matrix
(3, $list (25.0, 15.0, ~5.0, 0.0, 18.0, 0.0, 0.0, 0.0, 11.0)) val A = reflect_lower_triangle _A and B = copy _A val () = begin cholesky_decomposition B; print! ("\nThe Cholesky decomposition of\n\n"); Real_Matrix_fprint (stdout_ref, A); print! ("is\n"); Real_Matrix_fprint (stdout_ref, B) end val _A = column_major_list_to_square_matrix (4,$list (18.0, 22.0, 54.0, 42.0,
0.0, 70.0, 86.0, 62.0,
0.0, 0.0, 174.0, 134.0,
0.0, 0.0, 0.0, 106.0))
val A = reflect_lower_triangle _A
and B = copy _A
val () =
begin
cholesky_decomposition B;
print! ("\nThe Cholesky decomposition of\n\n");
Real_Matrix_fprint (stdout_ref, A);
print! ("is\n");
Real_Matrix_fprint (stdout_ref, B)
end
in
println! ()
end

(*------------------------------------------------------------------*)
Output:
$patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW cholesky_decomposition_task.dats -lgc -lm && ./a.out The Cholesky decomposition of 25 15 -5 15 18 0 -5 0 11 is 5 0 0 3 3 0 -1 1 3 The Cholesky decomposition of 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 is 4.24264 0 0 0 5.18545 6.56591 0 0 12.7279 3.04604 1.64974 0 9.89949 1.62455 1.84971 1.39262  ## AutoHotkey Cholesky_Decomposition(A){ L := [], n := A.Count() L[1,1] := Sqrt(A[1,1]) loop % n { k := A_Index loop % n-1 { i := A_Index+1 Sigma := 0, j := 0 while (++j <= k-1) Sigma += L[i, j] * L[k, j] L[i, k] := (A[i, k] - Sigma) / L[k, k] Sigma := 0, j := 0 while (++j <= k-1) Sigma += (L[k, j])**2 L[k, k] := Sqrt(A[k, k] - Sigma) } } loop % n{ k := A_Index loop % n L[k, A_Index] := L[k, A_Index] ? L[k, A_Index] : 0 } return L } ShowMatrix(L){ for r, obj in L{ row := "" for c, v in obj row .= Format("{:.3f}", v) ", " output .= "[" trim(row, ", ") "]n," } return "[" Trim(output, "n,") "]" }  Examples: A := [[25, 15, -5] , [15, 18, 0] , [-5, 0 , 11]] L1 := Cholesky_Decomposition(A) A := [[18, 22, 54, 42] , [22, 70, 86, 62] , [54, 86, 174, 134] , [42, 62, 134, 106]] L2 := Cholesky_Decomposition(A) MsgBox % Result := ShowMatrix(L1) "n----n" ShowMatrix(L2) "n----" return  Output: [[5.000, 0.000, 0.000] ,[3.000, 3.000, 0.000] ,[-1.000, 1.000, 3.000]] ---- [[4.243, 0.000, 0.000, 0.000] ,[5.185, 6.566, 0.000, 0.000] ,[12.728, 3.046, 1.650, 0.000] ,[9.899, 1.625, 1.850, 1.393]] ---- ## BBC BASIC  DIM m1(2,2) m1() = 25, 15, -5, \ \ 15, 18, 0, \ \ -5, 0, 11 PROCcholesky(m1()) PROCprint(m1()) PRINT @% = &2050A DIM m2(3,3) m2() = 18, 22, 54, 42, \ \ 22, 70, 86, 62, \ \ 54, 86, 174, 134, \ \ 42, 62, 134, 106 PROCcholesky(m2()) PROCprint(m2()) END DEF PROCcholesky(a()) LOCAL i%, j%, k%, l(), s DIM l(DIM(a(),1),DIM(a(),2)) FOR i% = 0 TO DIM(a(),1) FOR j% = 0 TO i% s = 0 FOR k% = 0 TO j%-1 s += l(i%,k%) * l(j%,k%) NEXT IF i% = j% THEN l(i%,j%) = SQR(a(i%,i%) - s) ELSE l(i%,j%) = (a(i%,j%) - s) / l(j%,j%) ENDIF NEXT j% NEXT i% a() = l() ENDPROC DEF PROCprint(a()) LOCAL row%, col% FOR row% = 0 TO DIM(a(),1) FOR col% = 0 TO DIM(a(),2) PRINT a(row%,col%); NEXT PRINT NEXT row% ENDPROC  Output:  5 0 0 3 3 0 -1 1 3 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262  ## C #include <stdio.h> #include <stdlib.h> #include <math.h> double *cholesky(double *A, int n) { double *L = (double*)calloc(n * n, sizeof(double)); if (L == NULL) exit(EXIT_FAILURE); for (int i = 0; i < n; i++) for (int j = 0; j < (i+1); j++) { double s = 0; for (int k = 0; k < j; k++) s += L[i * n + k] * L[j * n + k]; L[i * n + j] = (i == j) ? sqrt(A[i * n + i] - s) : (1.0 / L[j * n + j] * (A[i * n + j] - s)); } return L; } void show_matrix(double *A, int n) { for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) printf("%2.5f ", A[i * n + j]); printf("\n"); } } int main() { int n = 3; double m1[] = {25, 15, -5, 15, 18, 0, -5, 0, 11}; double *c1 = cholesky(m1, n); show_matrix(c1, n); printf("\n"); free(c1); n = 4; double m2[] = {18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106}; double *c2 = cholesky(m2, n); show_matrix(c2, n); free(c2); return 0; }  Output: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262 ## C# using System; using System.Collections.Generic; using System.Linq; using System.Text; namespace Cholesky { class Program { /// <summary> /// This is example is written in C#, and compiles with .NET Framework 4.0 /// </summary> /// <param name="args"></param> static void Main(string[] args) { double[,] test1 = new double[,] { {25, 15, -5}, {15, 18, 0}, {-5, 0, 11}, }; double[,] test2 = new double[,] { {18, 22, 54, 42}, {22, 70, 86, 62}, {54, 86, 174, 134}, {42, 62, 134, 106}, }; double[,] chol1 = Cholesky(test1); double[,] chol2 = Cholesky(test2); Console.WriteLine("Test 1: "); Print(test1); Console.WriteLine(""); Console.WriteLine("Lower Cholesky 1: "); Print(chol1); Console.WriteLine(""); Console.WriteLine("Test 2: "); Print(test2); Console.WriteLine(""); Console.WriteLine("Lower Cholesky 2: "); Print(chol2); } public static void Print(double[,] a) { int n = (int)Math.Sqrt(a.Length); StringBuilder sb = new StringBuilder(); for (int r = 0; r < n; r++) { string s = ""; for (int c = 0; c < n; c++) { s += a[r, c].ToString("f5").PadLeft(9) + ","; } sb.AppendLine(s); } Console.WriteLine(sb.ToString()); } /// <summary> /// Returns the lower Cholesky Factor, L, of input matrix A. /// Satisfies the equation: L*L^T = A. /// </summary> /// <param name="a">Input matrix must be square, symmetric, /// and positive definite. This method does not check for these properties, /// and may produce unexpected results of those properties are not met.</param> /// <returns></returns> public static double[,] Cholesky(double[,] a) { int n = (int)Math.Sqrt(a.Length); double[,] ret = new double[n, n]; for (int r = 0; r < n; r++) for (int c = 0; c <= r; c++) { if (c == r) { double sum = 0; for (int j = 0; j < c; j++) { sum += ret[c, j] * ret[c, j]; } ret[c, c] = Math.Sqrt(a[c, c] - sum); } else { double sum = 0; for (int j = 0; j < c; j++) sum += ret[r, j] * ret[c, j]; ret[r, c] = 1.0 / ret[c, c] * (a[r, c] - sum); } } return ret; } } }  Output: Test 1: 25.00000, 15.00000, -5.00000, 15.00000, 18.00000, 0.00000, -5.00000, 0.00000, 11.00000,  Lower Cholesky 1:  5.00000, 0.00000, 0.00000, 3.00000, 3.00000, 0.00000, -1.00000, 1.00000, 3.00000,  Test 2: 18.00000, 22.00000, 54.00000, 42.00000, 22.00000, 70.00000, 86.00000, 62.00000, 54.00000, 86.00000,174.00000,134.00000, 42.00000, 62.00000,134.00000,106.00000,  Lower Cholesky 2:  4.24264, 0.00000, 0.00000, 0.00000, 5.18545, 6.56591, 0.00000, 0.00000, 12.72792, 3.04604, 1.64974, 0.00000, 9.89949, 1.62455, 1.84971, 1.39262,  ## C++ #include <cassert> #include <cmath> #include <iomanip> #include <iostream> #include <vector> template <typename scalar_type> class matrix { public: matrix(size_t rows, size_t columns) : rows_(rows), columns_(columns), elements_(rows * columns) {} matrix(size_t rows, size_t columns, scalar_type value) : rows_(rows), columns_(columns), elements_(rows * columns, value) {} matrix(size_t rows, size_t columns, const std::initializer_list<std::initializer_list<scalar_type>>& values) : rows_(rows), columns_(columns), elements_(rows * columns) { assert(values.size() <= rows_); size_t i = 0; for (const auto& row : values) { assert(row.size() <= columns_); std::copy(begin(row), end(row), &elements_[i]); i += columns_; } } size_t rows() const { return rows_; } size_t columns() const { return columns_; } const scalar_type& operator()(size_t row, size_t column) const { assert(row < rows_); assert(column < columns_); return elements_[row * columns_ + column]; } scalar_type& operator()(size_t row, size_t column) { assert(row < rows_); assert(column < columns_); return elements_[row * columns_ + column]; } private: size_t rows_; size_t columns_; std::vector<scalar_type> elements_; }; template <typename scalar_type> void print(std::ostream& out, const matrix<scalar_type>& a) { size_t rows = a.rows(), columns = a.columns(); out << std::fixed << std::setprecision(5); for (size_t row = 0; row < rows; ++row) { for (size_t column = 0; column < columns; ++column) { if (column > 0) out << ' '; out << std::setw(9) << a(row, column); } out << '\n'; } } template <typename scalar_type> matrix<scalar_type> cholesky_factor(const matrix<scalar_type>& input) { assert(input.rows() == input.columns()); size_t n = input.rows(); matrix<scalar_type> result(n, n); for (size_t i = 0; i < n; ++i) { for (size_t k = 0; k < i; ++k) { scalar_type value = input(i, k); for (size_t j = 0; j < k; ++j) value -= result(i, j) * result(k, j); result(i, k) = value/result(k, k); } scalar_type value = input(i, i); for (size_t j = 0; j < i; ++j) value -= result(i, j) * result(i, j); result(i, i) = std::sqrt(value); } return result; } void print_cholesky_factor(const matrix<double>& matrix) { std::cout << "Matrix:\n"; print(std::cout, matrix); std::cout << "Cholesky factor:\n"; print(std::cout, cholesky_factor(matrix)); } int main() { matrix<double> matrix1(3, 3, {{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}); print_cholesky_factor(matrix1); matrix<double> matrix2(4, 4, {{18, 22, 54, 42}, {22, 70, 86, 62}, {54, 86, 174, 134}, {42, 62, 134, 106}}); print_cholesky_factor(matrix2); return 0; }  Output: Matrix: 25.00000 15.00000 -5.00000 15.00000 18.00000 0.00000 -5.00000 0.00000 11.00000 Cholesky factor: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 Matrix: 18.00000 22.00000 54.00000 42.00000 22.00000 70.00000 86.00000 62.00000 54.00000 86.00000 174.00000 134.00000 42.00000 62.00000 134.00000 106.00000 Cholesky factor: 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262  ## Clojure Translation of: Python (defn cholesky [matrix] (let [n (count matrix) A (to-array-2d matrix) L (make-array Double/TYPE n n)] (doseq [i (range n) j (range (inc i))] (let [s (reduce + (for [k (range j)] (* (aget L i k) (aget L j k))))] (aset L i j (if (= i j) (Math/sqrt (- (aget A i i) s)) (* (/ 1.0 (aget L j j)) (- (aget A i j) s)))))) (vec (map vec L))))  Example: (cholesky [[25 15 -5] [15 18 0] [-5 0 11]]) ;=> [[ 5.0 0.0 0.0] ; [ 3.0 3.0 0.0] ; [-1.0 1.0 3.0]] (cholesky [[18 22 54 42] [22 70 86 62] [54 86 174 134] [42 62 134 106]]) ;=> [[ 4.242640687119285 0.0 0.0 0.0 ] ; [ 5.185449728701349 6.565905201197403 0.0 0.0 ] ; [12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 ] ; [ 9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026]]  ## Common Lisp ;; Calculates the Cholesky decomposition matrix L ;; for a positive-definite, symmetric nxn matrix A. (defun chol (A) (let* ((n (car (array-dimensions A))) (L (make-array (,n ,n) :initial-element 0))) (do ((k 0 (incf k))) ((> k (- n 1)) nil) ;; First, calculate diagonal elements L_kk. (setf (aref L k k) (sqrt (- (aref A k k) (do* ((j 0 (incf j)) (sum (expt (aref L k j) 2) (incf sum (expt (aref L k j) 2)))) ((> j (- k 1)) sum))))) ;; Then, all elements below a diagonal element, L_ik, i=k+1..n. (do ((i (+ k 1) (incf i))) ((> i (- n 1)) nil) (setf (aref L i k) (/ (- (aref A i k) (do* ((j 0 (incf j)) (sum (* (aref L i j) (aref L k j)) (incf sum (* (aref L i j) (aref L k j))))) ((> j (- k 1)) sum))) (aref L k k))))) ;; Return the calculated matrix L. L))  ;; Example 1: (setf A (make-array '(3 3) :initial-contents '((25 15 -5) (15 18 0) (-5 0 11)))) (chol A) #2A((5.0 0 0) (3.0 3.0 0) (-1.0 1.0 3.0))  ;; Example 2: (setf B (make-array '(4 4) :initial-contents '((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106)))) (chol B) #2A((4.2426405 0 0 0) (5.18545 6.565905 0 0) (12.727922 3.0460374 1.6497375 0) (9.899495 1.6245536 1.849715 1.3926151))  ;; case of matrix stored as a list of lists (inner lists are rows of matrix) ;; as above, returns the Cholesky decomposition matrix of a square positive-definite, symmetric matrix (defun cholesky (m) (let ((l (list (list (sqrt (caar m))))) x (j 0) i) (dolist (cm (cdr m) (mapcar #'(lambda (x) (nconc x (make-list (- (length m) (length x)) :initial-element 0))) l)) (setq x (list (/ (car cm) (caar l))) i 0) (dolist (cl (cdr l)) (setf (cdr (last x)) (list (/ (- (elt cm (incf i)) (*v x cl)) (car (last cl)))))) (setf (cdr (last l)) (list (nconc x (list (sqrt (- (elt cm (incf j)) (*v x x)))))))))) ;; where *v is the scalar product defined as (defun *v (v1 v2) (reduce #'+ (mapcar #'* v1 v2)))  ;; example 1 CL-USER> (setf a '((25 15 -5) (15 18 0) (-5 0 11))) ((25 15 -5) (15 18 0) (-5 0 11)) CL-USER> (cholesky a) ((5 0 0) (3 3 0) (-1 1 3)) CL-USER> (format t "~{~{~5d~}~%~}" (cholesky a)) 5 0 0 3 3 0 -1 1 3 NIL  ;; example 2 CL-USER> (setf a '((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106))) ((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106)) CL-USER> (cholesky a) ((4.2426405 0 0 0) (5.18545 6.565905 0 0) (12.727922 3.0460374 1.6497375 0) (9.899495 1.6245536 1.849715 1.3926151)) CL-USER> (format t "~{~{~10,5f~}~%~}" (cholesky a)) 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89950 1.62455 1.84971 1.39262 NIL  ## D import std.stdio, std.math, std.numeric; T[][] cholesky(T)(in T[][] A) pure nothrow /*@safe*/ { auto L = new T[][](A.length, A.length); foreach (immutable r, row; L) row[r + 1 ..$] = 0;
foreach (immutable i; 0 .. A.length)
foreach (immutable j; 0 .. i + 1) {
auto t = dotProduct(L[i][0 .. j], L[j][0 .. j]);
L[i][j] = (i == j) ? (A[i][i] - t) ^^ 0.5 :
(1.0 / L[j][j] * (A[i][j] - t));
}
return L;
}

void main() {
immutable double[][] m1 = [[25, 15, -5],
[15, 18,  0],
[-5,  0, 11]];
writefln("%(%(%2.0f %)\n%)\n", m1.cholesky);

immutable double[][] m2 = [[18, 22,  54,  42],
[22, 70,  86,  62],
[54, 86, 174, 134],
[42, 62, 134, 106]];
writefln("%(%(%2.3f %)\n%)", m2.cholesky);
}

Output:
 5  0  0
3  3  0
-1  1  3

4.243 0.000 0.000 0.000
5.185 6.566 0.000 0.000
12.728 3.046 1.650 0.000
9.899 1.625 1.850 1.393

See Pascal.

## DWScript

Translation of: C
function Cholesky(a : array of Float) : array of Float;
var
i, j, k, n : Integer;
s : Float;
begin
n:=Round(Sqrt(a.Length));
Result:=new Float[n*n];
for i:=0 to n-1 do begin
for j:=0 to i do begin
s:=0 ;
for k:=0 to j-1 do
s+=Result[i*n+k] * Result[j*n+k];
if i=j then
Result[i*n+j]:=Sqrt(a[i*n+i]-s)
else Result[i*n+j]:=1/Result[j*n+j]*(a[i*n+j]-s);
end;
end;
end;

procedure ShowMatrix(a : array of Float);
var
i, j, n : Integer;
begin
n:=Round(Sqrt(a.Length));
for i:=0 to n-1 do begin
for j:=0 to n-1 do
Print(Format('%2.5f ', [a[i*n+j]]));
PrintLn('');
end;
end;

var m1 := new Float[9];
m1 := [ 25.0, 15.0, -5.0,
15.0, 18.0,  0.0,
-5.0,  0.0, 11.0 ];
var c1 := Cholesky(m1);
ShowMatrix(c1);

PrintLn('');

var m2 : array of Float := [ 18.0, 22.0,  54.0,  42.0,
22.0, 70.0,  86.0,  62.0,
54.0, 86.0, 174.0, 134.0,
42.0, 62.0, 134.0, 106.0 ];
var c2 := Cholesky(m2);
ShowMatrix(c2);


## F#

open Microsoft.FSharp.Collections

let cholesky a =
let calc (a: float[,]) (l: float[,]) i j =
let c1 j =
let sum = List.sumBy (fun k -> l.[j, k] ** 2.0) [0..j - 1]
sqrt (a.[j, j] - sum)
let c2 i j =
let sum = List.sumBy (fun k -> l.[i, k] * l.[j, k]) [0..j - 1]
(1.0 / l.[j, j]) * (a.[i, j] - sum)
if j > i then 0.0 else
if i = j
then c1 j
else c2 i j
let l = Array2D.zeroCreate (Array2D.length1 a) (Array2D.length2 a)
Array2D.iteri (fun i j _ -> l.[i, j] <- calc a l i j) l
l

let printMat a =
let arrow = (Array2D.length2 a |> float) / 2.0 |> int
let c = cholesky a
for row in 0..(Array2D.length1 a) - 1 do
for col in 0..(Array2D.length2 a) - 1 do
printf "%.5f,\t" a.[row, col]
printf (if arrow = row then "--> \t" else "\t\t")
for col in 0..(Array2D.length2 c) - 1 do
printf "%.5f,\t" c.[row, col]
printfn ""

let ex1 = array2D [
[25.0; 15.0; -5.0];
[15.0; 18.0; 0.0];
[-5.0; 0.0; 11.0]]

let ex2 = array2D [
[18.0; 22.0; 54.0; 42.0];
[22.0; 70.0; 86.0; 62.0];
[54.0; 86.0; 174.0; 134.0];
[42.0; 62.0; 134.0; 106.0]]

printfn "ex1:"
printMat ex1

printfn "ex2:"
printMat ex2

Output:
ex1:
25.00000,	15.00000,	-5.00000,		5.00000,	0.00000,	0.00000,
15.00000,	18.00000,	0.00000,	--> 	3.00000,	3.00000,	0.00000,
-5.00000,	0.00000,	11.00000,		-1.00000,	1.00000,	3.00000,
ex2:
18.00000,	22.00000,	54.00000,	42.00000,		4.24264,	0.00000,	0.00000,	0.00000,
22.00000,	70.00000,	86.00000,	62.00000,		5.18545,	6.56591,	0.00000,	0.00000,
54.00000,	86.00000,	174.00000,	134.00000,	--> 	12.72792,	3.04604,	1.64974,	0.00000,
42.00000,	62.00000,	134.00000,	106.00000,		9.89949,	1.62455,	1.84971,	1.39262,


## Fantom

**
** Cholesky decomposition
**

class Main
{
// create an array of Floats, initialised to 0.0
Float[][] makeArray (Int i, Int j)
{
Float[][] result := [,]
i.times |Int x|
{
j.times
{
}
}
return result
}

// perform the Cholesky decomposition
Float[][] cholesky (Float[][] array)
{
m := array.size
Float[][] l := makeArray (m, m)
m.times |Int i|
{
(i+1).times |Int k|
{
Float sum := (0..<k).toList.reduce (0f) |Float a, Int j -> Float|
{
a + l[i][j] * l[k][j]
}
if (i == k)
l[i][k] = (array[i][i]-sum).sqrt
else
l[i][k] = (1.0f / l[k][k]) * (array[i][k] - sum)
}
}
return l
}

Void runTest (Float[][] array)
{
echo (array)
echo (cholesky (array))
}

Void main ()
{
runTest ([[25f,15f,-5f],[15f,18f,0f],[-5f,0f,11f]])
runTest ([[18f,22f,54f,42f],[22f,70f,86f,62f],[54f,86f,174f,134f],[42f,62f,134f,106f]])
}
}
Output:
[[25.0, 15.0, -5.0], [15.0, 18.0, 0.0], [-5.0, 0.0, 11.0]]
[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]]
[[18.0, 22.0, 54.0, 42.0], [22.0, 70.0, 86.0, 62.0], [54.0, 86.0, 174.0, 134.0], [42.0, 62.0, 134.0, 106.0]]
[[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]


## Fortran

Program Cholesky_decomp
! *************************************************!
! LBH @ ULPGC 06/03/2014
! Compute the Cholesky decomposition for a matrix A
! after the attached
! http://rosettacode.org/wiki/Cholesky_decomposition
! note that the matrix A is complex since there might
! be values, where the sqrt has complex solutions.
! Here, only the real values are taken into account
!*************************************************!
implicit none

INTEGER, PARAMETER :: m=3 !rows
INTEGER, PARAMETER :: n=3 !cols
COMPLEX, DIMENSION(m,n) :: A
REAL, DIMENSION(m,n) :: L
REAL :: sum1, sum2
INTEGER i,j,k

! Assign values to the matrix
A(1,:)=(/ 25,  15,  -5 /)
A(2,:)=(/ 15,  18,   0 /)
A(3,:)=(/ -5,   0,  11 /)
! !!!!!!!!!!!another example!!!!!!!
! A(1,:) = (/ 18,  22,   54,   42 /)
! A(2,:) = (/ 22,  70,   86,   62 /)
! A(3,:) = (/ 54,  86,  174,  134 /)
! A(4,:) = (/ 42,  62,  134,  106 /)

! Initialize values
L(1,1)=real(sqrt(A(1,1)))
L(2,1)=A(2,1)/L(1,1)
L(2,2)=real(sqrt(A(2,2)-L(2,1)*L(2,1)))
L(3,1)=A(3,1)/L(1,1)
! for greater order than m,n=3 add initial row value
! for instance if m,n=4 then add the following line
! L(4,1)=A(4,1)/L(1,1)

do i=1,n
do k=1,i
sum1=0
sum2=0
do j=1,k-1
if (i==k) then
sum1=sum1+(L(k,j)*L(k,j))
L(k,k)=real(sqrt(A(k,k)-sum1))
elseif (i > k) then
sum2=sum2+(L(i,j)*L(k,j))
L(i,k)=(1/L(k,k))*(A(i,k)-sum2)
else
L(i,k)=0
end if
end do
end do
end do

! write output
do i=1,m
print "(3(1X,F6.1))",L(i,:)
end do

End program Cholesky_decomp

Output:
   5.0   0.0   0.0
3.0   3.0   0.0
-1.0   1.0   3.0


## FreeBASIC

Translation of: BBC BASIC
' version 18-01-2017
' compile with: fbc -s console

Sub Cholesky_decomp(array() As Double)

Dim As Integer i, j, k
Dim As Double s, l(UBound(array), UBound(array, 2))

For i = 0 To UBound(array)
For j = 0 To i
s = 0
For k = 0 To j -1
s += l(i, k) * l(j, k)
Next
If i = j Then
l(i, j) = Sqr(array(i, i) - s)
Else
l(i, j) = (array(i, j) - s) / l(j, j)
End If
Next
Next

For i = 0 To UBound(array)
For j = 0 To UBound(array, 2)
Swap array(i, j), l(i, j)
Next
Next

End Sub

Sub Print_(array() As Double)

Dim As Integer i, j

For i = 0 To UBound(array)
For j = 0 To UBound(array, 2)
Print Using "###.#####";array(i,j);
Next
Print
Next

End Sub

' ------=< MAIN >=------

Dim  m1(2,2) As Double  => {{25, 15, -5}, _
{15, 18,  0}, _
{-5,  0, 11}}

Dim m2(3, 3) As Double => {{18, 22,  54,  42}, _
{22, 70,  86,  62}, _
{54, 86, 174, 134}, _
{42, 62, 134, 106}}

Cholesky_decomp(m1())
Print_(m1())

Print
Cholesky_decomp(m2())
Print_(m2())

' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
  5.00000  0.00000  0.00000
3.00000  3.00000  0.00000
-1.00000  1.00000  3.00000

4.24264  0.00000  0.00000  0.00000
5.18545  6.56591  0.00000  0.00000
12.72792  3.04604  1.64974  0.00000
9.89949  1.62455  1.84971  1.39262

## Frink

Frink's package Matrix.frink contains routines for Cholesky-Crout decomposition of a square Hermitian matrix (which can be real or complex.) This code is adapted from that more powerful class to work on raw 2-dimensional arrays. This also demonstrates Frink's layout routines.

Cholesky[array] :=
{
n = length[array]
L = new array[[n,n], 0]

for j = 0 to n-1
{
sum = 0
for k = 0 to j-1
sum = sum + (L@j@k)^2

L@j@j = sqrt[array@j@j - sum]

for i = j+1 to n-1
{
sum = 0
for k = 0 to j-1
sum = sum + L@i@k * L@j@k

L@i@j = (1 / L@j@j * (array@i@j -sum))
}
}

return L
}

A = [[  25, 15, -5],
[  15, 18,  0],
[  -5,  0, 11]]

println[formatTable[[[formatMatrix[A], "->", formatMatrix[Cholesky[A]]]]]]

B = [[18,  22,  54,  42],
[22,  70,  86,  62],
[54,  86, 174, 134],
[42,  62, 134, 106]]

println[formatTable[[[formatMatrix[B], "->", formatMatrix[formatFix[Cholesky[B], 1, 5]]]]]]
Output:
┌          ┐    ┌        ┐
│25  15  -5│    │ 5  0  0│
│          │    │        │
│15  18   0│ -> │ 3  3  0│
│          │    │        │
│-5   0  11│    │-1  1  3│
└          ┘    └        ┘
┌                ┐    ┌                                   ┐
│18  22   54   42│    │ 4.24264  0.00000  0.00000  0.00000│
│                │    │                                   │
│22  70   86   62│    │ 5.18545  6.56591  0.00000  0.00000│
│                │ -> │                                   │
│54  86  174  134│    │12.72792  3.04604  1.64974  0.00000│
│                │    │                                   │
│42  62  134  106│    │ 9.89949  1.62455  1.84971  1.39262│
└                ┘    └                                   ┘


## Go

### Real

This version works with real matrices, like most other solutions on the page. The representation is packed, however, storing only the lower triange of the input symetric matrix and the output lower matrix. The decomposition algorithm computes rows in order from top to bottom but is a little different thatn Cholesky–Banachiewicz.

package main

import (
"fmt"
"math"
)

// symmetric and lower use a packed representation that stores only
// the lower triangle.

type symmetric struct {
order int
ele   []float64
}

type lower struct {
order int
ele   []float64
}

// symmetric.print prints a square matrix from the packed representation,
// printing the upper triange as a transpose of the lower.
func (s *symmetric) print() {
const eleFmt = "%10.5f "
row, diag := 1, 0
for i, e := range s.ele {
fmt.Printf(eleFmt, e)
if i == diag {
for j, col := diag+row, row; col < s.order; j += col {
fmt.Printf(eleFmt, s.ele[j])
col++
}
fmt.Println()
row++
diag += row
}
}
}

// lower.print prints a square matrix from the packed representation,
// printing the upper triangle as all zeros.
func (l *lower) print() {
const eleFmt = "%10.5f "
row, diag := 1, 0
for i, e := range l.ele {
fmt.Printf(eleFmt, e)
if i == diag {
for j := row; j < l.order; j++ {
fmt.Printf(eleFmt, 0.)
}
fmt.Println()
row++
diag += row
}
}
}

// choleskyLower returns the cholesky decomposition of a symmetric real
// matrix.  The matrix must be positive definite but this is not checked.
func (a *symmetric) choleskyLower() *lower {
l := &lower{a.order, make([]float64, len(a.ele))}
row, col := 1, 1
dr := 0 // index of diagonal element at end of row
dc := 0 // index of diagonal element at top of column
for i, e := range a.ele {
if i < dr {
d := (e - l.ele[i]) / l.ele[dc]
l.ele[i] = d
ci, cx := col, dc
for j := i + 1; j <= dr; j++ {
cx += ci
ci++
l.ele[j] += d * l.ele[cx]
}
col++
dc += col
} else {
l.ele[i] = math.Sqrt(e - l.ele[i])
row++
dr += row
col = 1
dc = 0
}
}
return l
}

func main() {
demo(&symmetric{3, []float64{
25,
15, 18,
-5, 0, 11}})
demo(&symmetric{4, []float64{
18,
22, 70,
54, 86, 174,
42, 62, 134, 106}})
}

func demo(a *symmetric) {
fmt.Println("A:")
a.print()
fmt.Println("L:")
a.choleskyLower().print()
}

Output:
A:
25.00000   15.00000   -5.00000
15.00000   18.00000    0.00000
-5.00000    0.00000   11.00000
L:
5.00000    0.00000    0.00000
3.00000    3.00000    0.00000
-1.00000    1.00000    3.00000
A:
18.00000   22.00000   54.00000   42.00000
22.00000   70.00000   86.00000   62.00000
54.00000   86.00000  174.00000  134.00000
42.00000   62.00000  134.00000  106.00000
L:
4.24264    0.00000    0.00000    0.00000
5.18545    6.56591    0.00000    0.00000
12.72792    3.04604    1.64974    0.00000
9.89949    1.62455    1.84971    1.39262


### Hermitian

This version handles complex Hermitian matricies as described on the WP page. The matrix representation is flat, and storage is allocated for all elements, not just the lower triangles. The decomposition algorithm is Cholesky–Banachiewicz.

package main

import (
"fmt"
"math/cmplx"
)

type matrix struct {
stride int
ele    []complex128
}

func like(a *matrix) *matrix {
return &matrix{a.stride, make([]complex128, len(a.ele))}
}

func (m *matrix) print(heading string) {
}
for e := 0; e < len(m.ele); e += m.stride {
fmt.Printf("%7.2f ", m.ele[e:e+m.stride])
fmt.Println()
}
}

func (a *matrix) choleskyDecomp() *matrix {
l := like(a)
// Cholesky-Banachiewicz algorithm
for r, rxc0 := 0, 0; r < a.stride; r++ {
// calculate elements along row, up to diagonal
x := rxc0
for c, cxc0 := 0, 0; c < r; c++ {
sum := a.ele[x]
for k := 0; k < c; k++ {
sum -= l.ele[rxc0+k] * cmplx.Conj(l.ele[cxc0+k])
}
l.ele[x] = sum / l.ele[cxc0+c]
x++
cxc0 += a.stride
}
// calcualate diagonal element
sum := a.ele[x]
for k := 0; k < r; k++ {
sum -= l.ele[rxc0+k] * cmplx.Conj(l.ele[rxc0+k])
}
l.ele[x] = cmplx.Sqrt(sum)
rxc0 += a.stride
}
return l
}

func main() {
demo("A:", &matrix{3, []complex128{
25, 15, -5,
15, 18, 0,
-5, 0, 11,
}})
demo("A:", &matrix{4, []complex128{
18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106,
}})
// one more example, from the Numpy manual, with a non-real
demo("A:", &matrix{2, []complex128{
1, -2i,
2i, 5,
}})
}

func demo(heading string, a *matrix) {
a.choleskyDecomp().print("Cholesky factor L:")
}

Output:
A:
[(  25.00  +0.00i) (  15.00  +0.00i) (  -5.00  +0.00i)]
[(  15.00  +0.00i) (  18.00  +0.00i) (   0.00  +0.00i)]
[(  -5.00  +0.00i) (   0.00  +0.00i) (  11.00  +0.00i)]

Cholesky factor L:
[(   5.00  +0.00i) (   0.00  +0.00i) (   0.00  +0.00i)]
[(   3.00  +0.00i) (   3.00  +0.00i) (   0.00  +0.00i)]
[(  -1.00  +0.00i) (   1.00  +0.00i) (   3.00  +0.00i)]

A:
[(  18.00  +0.00i) (  22.00  +0.00i) (  54.00  +0.00i) (  42.00  +0.00i)]
[(  22.00  +0.00i) (  70.00  +0.00i) (  86.00  +0.00i) (  62.00  +0.00i)]
[(  54.00  +0.00i) (  86.00  +0.00i) ( 174.00  +0.00i) ( 134.00  +0.00i)]
[(  42.00  +0.00i) (  62.00  +0.00i) ( 134.00  +0.00i) ( 106.00  +0.00i)]

Cholesky factor L:
[(   4.24  +0.00i) (   0.00  +0.00i) (   0.00  +0.00i) (   0.00  +0.00i)]
[(   5.19  +0.00i) (   6.57  +0.00i) (   0.00  +0.00i) (   0.00  +0.00i)]
[(  12.73  +0.00i) (   3.05  +0.00i) (   1.65  +0.00i) (   0.00  +0.00i)]
[(   9.90  +0.00i) (   1.62  +0.00i) (   1.85  +0.00i) (   1.39  +0.00i)]

A:
[(   1.00  +0.00i) (   0.00  -2.00i)]
[(   0.00  +2.00i) (   5.00  +0.00i)]

Cholesky factor L:
[(   1.00  +0.00i) (   0.00  +0.00i)]
[(   0.00  +2.00i) (   1.00  +0.00i)]


### Library gonum/mat

package main

import (
"fmt"

"gonum.org/v1/gonum/mat"
)

func cholesky(order int, elements []float64) fmt.Formatter {
var c mat.Cholesky
c.Factorize(mat.NewSymDense(order, elements))
return mat.Formatted(c.LTo(nil))
}

func main() {
fmt.Println(cholesky(3, []float64{
25, 15, -5,
15, 18, 0,
-5, 0, 11,
}))
fmt.Printf("\n%.5f\n", cholesky(4, []float64{
18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106,
}))
}

Output:
⎡ 5   0   0⎤
⎢ 3   3   0⎥
⎣-1   1   3⎦

⎡ 4.24264   0.00000   0.00000   0.00000⎤
⎢ 5.18545   6.56591   0.00000   0.00000⎥
⎢12.72792   3.04604   1.64974   0.00000⎥
⎣ 9.89949   1.62455   1.84971   1.39262⎦


### Library go.matrix

package main

import (
"fmt"

mat "github.com/skelterjohn/go.matrix"
)

func main() {
demo(mat.MakeDenseMatrix([]float64{
25, 15, -5,
15, 18, 0,
-5, 0, 11,
}, 3, 3))
demo(mat.MakeDenseMatrix([]float64{
18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106,
}, 4, 4))
}

func demo(m *mat.DenseMatrix) {
fmt.Println("A:")
fmt.Println(m)
l, err := m.Cholesky()
if err != nil {
fmt.Println(err)
return
}
fmt.Println("L:")
fmt.Println(l)
}


Output:

A:
{25, 15, -5,
15, 18,  0,
-5,  0, 11}
L:
{ 5,  0,  0,
3,  3,  0,
-1,  1,  3}
A:
{ 18,  22,  54,  42,
22,  70,  86,  62,
54,  86, 174, 134,
42,  62, 134, 106}
L:
{ 4.242641,         0,         0,         0,
5.18545,  6.565905,         0,         0,
12.727922,  3.046038,  1.649742,         0,
9.899495,  1.624554,  1.849711,  1.392621}


## Groovy

Translation of: Java
def decompose = { a ->
assert a.size > 0 && a[0].size == a.size
def m = a.size
def l = [].withEagerDefault { [].withEagerDefault { 0 } }
(0..<m).each { i ->
(0..i).each { k ->
Number s = (0..<k).sum { j -> l[i][j] * l[k][j] } ?: 0
l[i][k] = (i == k)
? Math.sqrt(a[i][i] - s)
: (1.0 / l[k][k] * (a[i][k] - s))
}
}
l
}


Test:

def test1 = [[25, 15, -5],
[15, 18,  0],
[-5,  0, 11]]

def test2 = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]];

[test1,test2]. each { test ->
println()
decompose(test).each { println it[0..<(test.size)] }
}

Output:
[5.0, 0, 0]
[3.0, 3.0, 0]
[-1.0, 1.0, 3.0]

[4.242640687119285, 0, 0, 0]
[5.185449728701349, 6.565905201197403, 0, 0]
[12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0]
[9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]

We use the Cholesky–Banachiewicz algorithm described in the Wikipedia article.

For more serious numerical analysis there is a Cholesky decomposition function in the hmatrix package.

The Cholesky module:

module Cholesky (Arr, cholesky) where

import Data.Array.IArray
import Data.Array.MArray
import Data.Array.Unboxed
import Data.Array.ST

type Idx = (Int,Int)
type Arr = UArray Idx Double

-- Return the (i,j) element of the lower triangular matrix.  (We assume the
-- lower array bound is (0,0).)
get :: Arr -> Arr -> Idx -> Double
get a l (i,j) | i == j = sqrt $a!(j,j) - dot | i > j = (a!(i,j) - dot) / l!(j,j) | otherwise = 0 where dot = sum [l!(i,k) * l!(j,k) | k <- [0..j-1]] -- Return the lower triangular matrix of a Cholesky decomposition. We assume -- the input is a real, symmetric, positive-definite matrix, with lower array -- bounds of (0,0). cholesky :: Arr -> Arr cholesky a = let n = maxBnd a in runSTUArray$ do
l <- thaw a
mapM_ (update a l) [(i,j) | i <- [0..n], j <- [0..n]]
return l
where maxBnd = fst . snd . bounds
update a l i = unsafeFreeze l >>= \l' -> writeArray l i (get a l' i)


The main module:

import Data.Array.IArray
import Data.List
import Cholesky

fm _ [] = ""
fm _ [x] = fst x
fm width ((a,b):xs) = a ++ (take (width - b) $cycle " ") ++ (fm width xs) fmt width row (xs,[]) = fm width xs fmt width row (xs,ys) = fm width xs ++ "\n" ++ fmt width row (splitAt row ys) showMatrice row xs = ys where vs = map (\s -> let sh = show s in (sh,length sh)) xs width = (maximum$ snd $unzip vs) + 1 ys = fmt width row (splitAt row vs) ex1, ex2 :: Arr ex1 = listArray ((0,0),(2,2)) [25, 15, -5, 15, 18, 0, -5, 0, 11] ex2 = listArray ((0,0),(3,3)) [18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106] main :: IO () main = do putStrLn$ showMatrice 3 $elems$ cholesky ex1
print sb
print $chol sb print$ tr $chol sb  Output: Herm (3><3) [ 25.0, 15.0, -5.0 , 15.0, 18.0, 0.0 , -5.0, 0.0, 11.0 ] (3><3) [ 5.0, 3.0, -1.0 , 0.0, 3.0, 1.0 , 0.0, 0.0, 3.0 ] Herm (4><4) [ 18.0, 22.0, 54.0, 42.0 , 22.0, 70.0, 86.0, 62.0 , 54.0, 86.0, 174.0, 134.0 , 42.0, 62.0, 134.0, 106.0 ] (4><4) [ 4.242640687119285, 5.185449728701349, 12.727922061357857, 9.899494936611665 , 0.0, 6.565905201197403, 3.0460384954008553, 1.6245538642137891 , 0.0, 0.0, 1.6497422479090704, 1.849711005231382 , 0.0, 0.0, 0.0, 1.3926212476455904 ] (4><4) [ 4.242640687119285, 0.0, 0.0, 0.0 , 5.185449728701349, 6.565905201197403, 0.0, 0.0 , 12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0 , 9.899494936611665, 1.6245538642137891, 1.849711005231382, 1.3926212476455904 ] ## Icon and Unicon procedure cholesky (array) result := make_square_array (*array) every (i := 1 to *array) do { every (k := 1 to i) do { sum := 0 every (j := 1 to (k-1)) do { sum +:= result[i][j] * result[k][j] } if (i = k) then result[i][k] := sqrt(array[i][i] - sum) else result[i][k] := 1.0 / result[k][k] * (array[i][k] - sum) } } return result end procedure make_square_array (n) result := [] every (1 to n) do push (result, list(n, 0)) return result end procedure print_array (array) every (row := !array) do { every writes (!row || " ") write () } end procedure do_cholesky (array) write ("Input:") print_array (array) result := cholesky (array) write ("Result:") print_array (result) end procedure main () do_cholesky ([[25,15,-5],[15,18,0],[-5,0,11]]) do_cholesky ([[18,22,54,42],[22,70,86,62],[54,86,174,134],[42,62,134,106]]) end  Output: Input: 25 15 -5 15 18 0 -5 0 11 Result: 5.0 0 0 3.0 3.0 0 -1.0 1.0 3.0 Input: 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 Result: 4.242640687 0 0 0 5.185449729 6.565905201 0 0 12.72792206 3.046038495 1.649742248 0 9.899494937 1.624553864 1.849711005 1.392621248  ## Idris works with Idris 0.10 Solution: module Main import Data.Vect Matrix : Nat -> Nat -> Type -> Type Matrix m n t = Vect m (Vect n t) zeros : (m : Nat) -> (n : Nat) -> Matrix m n Double zeros m n = replicate m (replicate n 0.0) indexM : (Fin m, Fin n) -> Matrix m n t -> t indexM (i, j) a = index j (index i a) 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 get : Matrix m m Double -> Matrix m m Double -> (Fin m, Fin m) -> Double get a l (i, j) {m} = if i == j then sqrt$ indexM (j, j) a - dot
else if i > j  then (indexM (i, j) a - dot) / indexM (j, j) l
else 0.0

where
-- Obtain indicies 0 to j -1
ks : List (Fin m)
ks = case (findIndices (\_ => True) a) of
[] => []
(x::xs) => init (x::xs)

dot : Double
dot = sum [(indexM (i, k) l) * (indexM (j, k) l) | k <- ks]

updateL : Matrix m m Double -> Matrix m m Double -> (Fin m, Fin m) -> Matrix m m Double
updateL a l idx = replaceAtM idx (get a l idx) l

cholesky : Matrix m m Double -> Matrix m m Double
cholesky a {m} =
foldl (\l',i =>
foldl (\l'',j => updateL a l'' (i, j)) l' (js i))
l is
where  l = zeros m m

is : List (Fin m)
is = findIndices (\_ => True) a

js : Fin m -> List (Fin m)
js n = filter (<= n) is

ex1 : Matrix 3 3 Double
ex1 = cholesky [[25.0, 15.0, -5.0], [15.0, 18.0, 0.0], [-5.0, 0.0, 11.0]]

ex2 : Matrix 4 4 Double
ex2 = cholesky [[18.0, 22.0, 54.0, 42.0], [22.0, 70.0, 86.0, 62.0],
[54.0, 86.0, 174.0, 134.0], [42.0, 62.0, 134.0, 106.0]]

main : IO ()
main = do
print ex1
putStrLn "\n"
print ex2
putStrLn "\n"

Output:
[[5, 0, 0], [3, 3, 0], [-1, 1, 3]]

[[4.242640687119285, 0, 0, 0], [5.185449728701349, 6.565905201197403, 0, 0], [12.72792206135786, 3.046038495400855, 1.64974224790907, 0], [9.899494936611665, 1.624553864213789, 1.849711005231382, 1.392621247645587]]


## J

Solution:

mp=: +/ . *  NB. matrix product
h =: +@|:    NB. conjugate transpose

cholesky=: 3 : 0
n=. #A=. y
if. 1>:n do.
assert. (A=|A)>0=A  NB. check for positive definite
%:A
else.
'X Y t Z'=. , (;~n$(>.-:n){.1) <;.1 A L0=. cholesky X L1=. cholesky Z-(T=.(h Y) mp %.X) mp Y L0,(T mp L0),.L1 end. )  See Cholesky Decomposition essay on the J Wiki. Examples:  eg1=: 25 15 _5 , 15 18 0 ,: _5 0 11 eg2=: 18 22 54 42 , 22 70 86 62 , 54 86 174 134 ,: 42 62 134 106 cholesky eg1 5 0 0 3 3 0 _1 1 3 cholesky eg2 4.24264 0 0 0 5.18545 6.56591 0 0 12.7279 3.04604 1.64974 0 9.89949 1.62455 1.84971 1.39262  Using math/lapack addon  load 'math/lapack' load 'math/lapack/potrf' potrf_jlapack_ eg1 5 0 0 3 3 0 _1 1 3 potrf_jlapack_ eg2 4.24264 0 0 0 5.18545 6.56591 0 0 12.7279 3.04604 1.64974 0 9.89949 1.62455 1.84971 1.39262  ## Java Works with: Java version 1.5+ import java.util.Arrays; public class Cholesky { public static double[][] chol(double[][] a){ int m = a.length; double[][] l = new double[m][m]; //automatically initialzed to 0's for(int i = 0; i< m;i++){ for(int k = 0; k < (i+1); k++){ double sum = 0; for(int j = 0; j < k; j++){ sum += l[i][j] * l[k][j]; } l[i][k] = (i == k) ? Math.sqrt(a[i][i] - sum) : (1.0 / l[k][k] * (a[i][k] - sum)); } } return l; } public static void main(String[] args){ double[][] test1 = {{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}; System.out.println(Arrays.deepToString(chol(test1))); double[][] test2 = {{18, 22, 54, 42}, {22, 70, 86, 62}, {54, 86, 174, 134}, {42, 62, 134, 106}}; System.out.println(Arrays.deepToString(chol(test2))); } } Output: [[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]] ## JavaScript const cholesky = function (array) { const zeros = [...Array(array.length)].map( _ => Array(array.length).fill(0)); const L = zeros.map((row, r, xL) => row.map((v, c) => { const sum = row.reduce((s, _, i) => i < c ? s + xL[r][i] * xL[c][i] : s, 0); return xL[r][c] = c < r + 1 ? r === c ? Math.sqrt(array[r][r] - sum) : (array[r][c] - sum) / xL[c][c] : v; })); return L; } let arr3 = [[25, 15, -5], [15, 18, 0], [-5, 0, 11]]; console.log(cholesky(arr3)); let arr4 = [[18, 22, 54, 42], [22, 70, 86, 62], [54, 86, 174, 134], [42, 62, 134, 106]]; console.log(cholesky(arr4));  Output: 0: (3) [5, 0, 0] 1: (3) [3, 3, 0] 2: (3) [-1, 1, 3] 0: (4) [4.242640687119285, 0, 0, 0] 1: (4) [5.185449728701349, 6.565905201197403, 0, 0] 2: (4) [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0] 3: (4) [9.899494936611665, 1.6245538642137891, 1.849711005231382, 1.3926212476455924]  ## jq Works with: jq version 1.4 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 ; # Print a matrix neatly, each cell ideally 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" ) ;

def is_square:
type == "array" and (map(type == "array") | all) and
length == 0 or ( (.[0]|length) as $l | map(length ==$l) | all) ;

# This implementation of is_symmetric/0 uses a helper function that circumvents
# limitations of jq 1.4:
def is_symmetric:
# [matrix, i,j, len]
def test:
if .[1] > .[3] then true
elif .[1] == .[2] then [ .[0], .[1] + 1, 0, .[3]] | test
elif .[0][.[1]][.[2]] == .[0][.[2]][.[1]]
then [ .[0], .[1], .[2]+1, .[3]] | test
else false
end;
if is_square|not then false
else [ ., 0, 0, length ] | test
end ;
Cholesky Decomposition:
def cholesky_factor:
if is_symmetric then
length as $length | . as$self
| reduce range(0; $length) as$k
( matrix(length; length; 0); # the matrix that will hold the answer
reduce range(0; $length) as$i
(.;
if $i ==$k
then (. as $lower | reduce range(0;$k) as $j (0; . + ($lower[$k][$j] | .*.) )) as $sum | .[$k][$k] = (($self[$k][$k] - $sum) | sqrt) elif$i > $k then (. as$lower
| reduce range(0; $k) as$j
(0; . + $lower[$i][$j] *$lower[$k][$j])) as $sum | .[$i][$k] = (($self[$k][$i] - $sum) / .[$k][$k] ) else . end )) else error( "cholesky_factor: matrix is not symmetric" ) end ; Task 1: [[25,15,-5],[15,18,0],[-5,0,11]] | cholesky_factor  Output: [[5,0,0],[3,3,0],[-1,1,3]]  Task 2: [[18, 22, 54, 42], [22, 70, 86, 62], [54, 86, 174, 134], [42, 62, 134, 106]] | cholesky_factor | neatly(20)  Output:  4.242640687119285 0 0 0 5.185449728701349 6.565905201197403 0 0 12.727922061357857 3.0460384954008553 1.6497422479090704 0 9.899494936611665 1.6245538642137891 1.849711005231382 1.3926212476455924 ## Julia Julia's strong linear algebra support includes Cholesky decomposition. a = [25 15 5; 15 18 0; -5 0 11] b = [18 22 54 22; 22 70 86 62; 54 86 174 134; 42 62 134 106] println(a, "\n => \n", chol(a, :L)) println(b, "\n => \n", chol(b, :L))  Output: [25 15 5 15 18 0 -5 0 11] => [5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0] [18 22 54 22 22 70 86 62 54 86 174 134 42 62 134 106] => [4.242640687119285 0.0 0.0 0.0 5.185449728701349 6.565905201197403 0.0 0.0 12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026]  ## Kotlin Translation of: C // version 1.0.6 fun cholesky(a: DoubleArray): DoubleArray { val n = Math.sqrt(a.size.toDouble()).toInt() val l = DoubleArray(a.size) var s: Double for (i in 0 until n) for (j in 0 .. i) { s = 0.0 for (k in 0 until j) s += l[i * n + k] * l[j * n + k] l[i * n + j] = when { (i == j) -> Math.sqrt(a[i * n + i] - s) else -> 1.0 / l[j * n + j] * (a[i * n + j] - s) } } return l } fun showMatrix(a: DoubleArray) { val n = Math.sqrt(a.size.toDouble()).toInt() for (i in 0 until n) { for (j in 0 until n) print("%8.5f ".format(a[i * n + j])) println() } } fun main(args: Array<String>) { val m1 = doubleArrayOf(25.0, 15.0, -5.0, 15.0, 18.0, 0.0, -5.0, 0.0, 11.0) val c1 = cholesky(m1) showMatrix(c1) println() val m2 = doubleArrayOf(18.0, 22.0, 54.0, 42.0, 22.0, 70.0, 86.0, 62.0, 54.0, 86.0, 174.0, 134.0, 42.0, 62.0, 134.0, 106.0) val c2 = cholesky(m2) showMatrix(c2) }  Output:  5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262  ## Lobster Translation of: Go Translated from the Go Real version: This version works with real matrices, like most other solutions on the page. The representation is packed, however, storing only the lower triange of the input symetric matrix and the output lower matrix. The decomposition algorithm computes rows in order from top to bottom but is a little different than Cholesky–Banachiewicz. import std // choleskyLower returns the cholesky decomposition of a symmetric real // matrix. The matrix must be positive definite but this is not checked def choleskyLower(order, a) -> [float]: let l = map(a.length): 0.0 var row, col = 1, 1 var dr = 0 // index of diagonal element at end of row var dc = 0 // index of diagonal element at top of column for(a) e, i: if i < dr: let d = (e - l[i]) / l[dc] l[i] = d var ci, cx = col, dc var j = i + 1 while j <= dr: cx += ci ci += 1 l[j] += d * l[cx] j += 1 col += 1 dc += col else: l[i] = sqrt(e - l[i]) row += 1 dr += row col = 1 dc = 0 return l // symmetric.print prints a square matrix from the packed representation, // printing the upper triange as a transpose of the lower def print_symmetric(order, s): //const eleFmt = "%10.5f " var str = "" var row, diag = 1, 0 for(s) e, i: str += e + " " // format? if i == diag: var j, col = diag+row, row while col < order: str += s[j] + " " // format? col++ j += col print(str); str = "" row += 1 diag += row // lower.print prints a square matrix from the packed representation, // printing the upper triangle as all zeros. def print_lower(order, l): //const eleFmt = "%10.5f " var str = "" var row, diag = 1, 0 for(l) e, i: str += e + " " // format? if i == diag: var j = row while j < order: str += 0.0 + " " // format? j += 1 print(str); str = "" row += 1 diag += row def demo(order, a): print("A:") print_symmetric(order, a) print("L:") print_lower(order, choleskyLower(order, a)) demo(3, [25.0, 15.0, 18.0, -5.0, 0.0, 11.0]) demo(4, [18.0, 22.0, 70.0, 54.0, 86.0, 174.0, 42.0, 62.0, 134.0, 106.0]) Output: A: 25.0 15.0 -5.0 15.0 18.0 0.0 -5.0 0.0 11.0 L: 5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0 A: 18.0 22.0 54.0 42.0 22.0 70.0 86.0 62.0 54.0 86.0 174.0 134.0 42.0 62.0 134.0 106.0 L: 4.242640687119 0.0 0.0 0.0 5.185449728701 6.565905201197 0.0 0.0 12.72792206135 3.046038495401 1.649742247909 0.0 9.899494936612 1.624553864214 1.849711005231 1.392621247646  ## Maple The Cholesky decomposition is obtained by passing the method = Cholesky' option to the LUDecomposition procedure in the LinearAlgebra pacakge. This is illustrated below for the two requested examples. The first is computed exactly; the second is also, but the subsequent application of evalf' to the result produces a matrix with floating point entries which can be compared with the expected output in the problem statement. > A := << 25, 15, -5; 15, 18, 0; -5, 0, 11 >>; [25 15 -5] [ ] A := [15 18 0] [ ] [-5 0 11] > B := << 18, 22, 54, 42; 22, 70, 86, 62; 54, 86, 174, 134; 42, 62, 134, 106>>; [18 22 54 42] [ ] [22 70 86 62] B := [ ] [54 86 174 134] [ ] [42 62 134 106] > use LinearAlgebra in > LUDecomposition( A, method = Cholesky ); > LUDecomposition( B, method = Cholesky ); > evalf( % ); > end use; [ 5 0 0] [ ] [ 3 3 0] [ ] [-1 1 3] [ 1/2 ] [3 2 0 0 0 ] [ ] [ 1/2 1/2 ] [11 2 2 97 ] [------- ------- 0 0 ] [ 3 3 ] [ ] [ 1/2 1/2 ] [ 1/2 30 97 2 6402 ] [9 2 -------- --------- 0 ] [ 97 97 ] [ ] [ 1/2 1/2 1/2] [ 1/2 16 97 74 6402 8 33 ] [7 2 -------- ---------- -------] [ 97 3201 33 ] [4.242640686 0. 0. 0. ] [ ] [5.185449728 6.565905202 0. 0. ] [ ] [12.72792206 3.046038495 1.649742248 0. ] [ ] [9.899494934 1.624553864 1.849711006 1.392621248] ## Mathematica / Wolfram Language CholeskyDecomposition[{{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}]  Without the use of built-in functions, making use of memoization: chol[A_] := Module[{L}, L[k_, k_] := L[k, k] = Sqrt[A[[k, k]] - Sum[L[k, j]^2, {j, 1, k-1}]]; L[i_, k_] := L[i, k] = L[k, k]^-1 (A[[i, k]] - Sum[L[i, j] L[k, j], {j, 1, k-1}]); PadRight[Table[L[i, j], {i, Length[A]}, {j, i}]] ]  ## MATLAB / Octave The cholesky decomposition chol() is an internal function  A = [ 25 15 -5 15 18 0 -5 0 11 ]; B = [ 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 ]; [L] = chol(A,'lower') [L] = chol(B,'lower')  Output:  > [L] = chol(A,'lower') L = 5 0 0 3 3 0 -1 1 3 > [L] = chol(B,'lower') L = 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262  ## Maxima /* Cholesky decomposition is built-in */ a: hilbert_matrix(4)$

b: cholesky(a);
/* matrix([1,   0,             0,             0             ],
[1/2, 1/(2*sqrt(3)), 0,             0             ],
[1/3, 1/(2*sqrt(3)), 1/(6*sqrt(5)), 0             ],
[1/4, 3^(3/2)/20,    1/(4*sqrt(5)), 1/(20*sqrt(7))]) */

b . transpose(b) - a;
matrix([0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0])


## Nim

Translation of: C
import math, strutils, strformat

type Matrix[N: static int, T: SomeFloat] = array[N, array[N, T]]

proc cholesky[Matrix](a: Matrix): Matrix =
for i in 0 ..< a[0].len:
for j in 0 .. i:
var s = 0.0
for k in 0 ..< j:
s += result[i][k] * result[j][k]
result[i][j] = if i == j: sqrt(a[i][i]-s)
else: 1.0 / result[j][j] * (a[i][j] - s)

proc $(a: Matrix): string = result = "" for b in a: var line = "" for c in b: line.addSep(" ", 0) line.add fmt"{c:8.5f}" result.add line & '\n' let m1 = [[25.0, 15.0, -5.0], [15.0, 18.0, 0.0], [-5.0, 0.0, 11.0]] echo cholesky(m1) let m2 = [[18.0, 22.0, 54.0, 42.0], [22.0, 70.0, 86.0, 62.0], [54.0, 86.0, 174.0, 134.0], [42.0, 62.0, 134.0, 106.0]] echo cholesky(m2)  Output:  5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262 ## Objeck Translation of: C class Cholesky { function : Main(args : String[]) ~ Nil { n := 3; m1 := [25.0, 15.0, -5.0, 15.0, 18.0, 0.0, -5.0, 0.0, 11.0]; c1 := Cholesky(m1, n); ShowMatrix(c1, n); IO.Console->PrintLine(); n := 4; m2 := [18.0, 22.0, 54.0, 42.0, 22.0, 70.0, 86.0, 62.0, 54.0, 86.0, 174.0, 134.0, 42.0, 62.0, 134.0, 106.0]; c2 := Cholesky(m2, n); ShowMatrix(c2, n); } function : ShowMatrix(A : Float[], n : Int) ~ Nil { for (i := 0; i < n; i+=1;) { for (j := 0; j < n; j+=1;) { IO.Console->Print(A[i * n + j])->Print('\t'); }; IO.Console->PrintLine(); }; } function : Cholesky(A : Float[], n : Int) ~ Float[] { L := Float->New[n * n]; for (i := 0; i < n; i+=1;) { for (j := 0; j < (i+1); j+=1;) { s := 0.0; for (k := 0; k < j; k+=1;) { s += L[i * n + k] * L[j * n + k]; }; L[i * n + j] := (i = j) ? (A[i * n + i] - s)->SquareRoot() : (1.0 / L[j * n + j] * (A[i * n + j] - s)); }; }; return L; } } 5 0 0 3 3 0 -1 1 3 4.24264069 0 0 0 5.18544973 6.5659052 0 0 12.7279221 3.0460385 1.64974225 0 9.89949494 1.62455386 1.84971101 1.39262125  ## OCaml let cholesky inp = let n = Array.length inp in let res = Array.make_matrix n n 0.0 in let factor i k = let rec sum j = if j = k then 0.0 else res.(i).(j) *. res.(k).(j) +. sum (j+1) in inp.(i).(k) -. sum 0 in for col = 0 to n-1 do res.(col).(col) <- sqrt (factor col col); for row = col+1 to n-1 do res.(row).(col) <- (factor row col) /. res.(col).(col) done done; res let pr_vec v = Array.iter (Printf.printf " %9.5f") v; print_newline() let show = Array.iter pr_vec let test a = print_endline "\nin:"; show a; print_endline "out:"; show (cholesky a) let _ = test [| [|25.0; 15.0; -5.0|]; [|15.0; 18.0; 0.0|]; [|-5.0; 0.0; 11.0|] |]; test [| [|18.0; 22.0; 54.0; 42.0|]; [|22.0; 70.0; 86.0; 62.0|]; [|54.0; 86.0; 174.0; 134.0|]; [|42.0; 62.0; 134.0; 106.0|] |];  Output: in: 25.00000 15.00000 -5.00000 15.00000 18.00000 0.00000 -5.00000 0.00000 11.00000 out: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 in: 18.00000 22.00000 54.00000 42.00000 22.00000 70.00000 86.00000 62.00000 54.00000 86.00000 174.00000 134.00000 42.00000 62.00000 134.00000 106.00000 out: 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262 ## ooRexx Translation of: REXX /*REXX program performs the Cholesky decomposition on a square matrix. */ niner = '25 15 -5' , /*define a 3x3 matrix. */ '15 18 0' , '-5 0 11' call Cholesky niner hexer = 18 22 54 42, /*define a 4x4 matrix. */ 22 70 86 62, 54 86 174 134, 42 62 134 106 call Cholesky hexer exit /*stick a fork in it, we're all done. */ /*----------------------------------------------------------------------------*/ Cholesky: procedure; parse arg mat; say; say; call tell 'input matrix',mat do r=1 for ord do c=1 for r; d=0; do i=1 for c-1; d=d+!.r.i*!.c.i; end /*i*/ if r=c then !.r.r=sqrt(!.r.r-d) else !.r.c=1/!.c.c*(a.r.c-d) end /*c*/ end /*r*/ call tell 'Cholesky factor',,!.,'-' return /*----------------------------------------------------------------------------*/ err: say; say; say '***error***!'; say; say arg(1); say; say; exit 13 /*----------------------------------------------------------------------------*/ tell: parse arg hdr,x,y,sep; n=0; if sep=='' then sep='-' dPlaces= 5 /*n decimal places past the decimal point*/ width =10 /*width of field used to display elements*/ if y=='' then !.=0 else do row=1 for ord; do col=1 for ord; x=x !.row.col; end; end w=words(x) do ord=1 until ord**2>=w; end /*a fast way to find matrix's order*/ say if ord**2\==w then call err "matrix elements don't form a square matrix." say center(hdr, ((width+1)*w)%ord, sep) say do row=1 for ord; z='' do col=1 for ord; n=n+1 a.row.col=word(x,n) if col<=row then !.row.col=a.row.col z=z right( format(a.row.col,, dPlaces) / 1, width) end /*col*/ say z end /*row*/ return /*----------------------------------------------------------------------------*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); i=''; m.=9 numeric digits 9; numeric form; h=d+6; if x<0 then do; x=-x; i='i'; end parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_%2 do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/ do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/ numeric digits d; return (g/1)i /*make complex if X < 0.*/ ## PARI/GP cholesky(M) = { my (L = matrix(#M,#M)); for (i = 1, #M, for (j = 1, i, s = sum (k = 1, j-1, L[i,k] * L[j,k]); L[i,j] = if (i == j, sqrt(M[i,i] - s), (M[i,j] - s) / L[j,j]) ) ); L } Output: (set displayed digits with: \p 5) gp > cholesky([25,15,-5;15,18,0;-5,0,11]) [ 5.0000 0 0] [ 3.0000 3.0000 0] [-1.0000 1.0000 3.0000] gp > cholesky([18,22,54,42;22,70,86,62;54,86,174,134;42,62,134,106]) [4.2426 0 0 0] [5.1854 6.5659 0 0] [12.728 3.0460 1.6497 0] [9.8995 1.6246 1.8497 1.3926]  ## Pascal program CholeskyApp; type D2Array = array of array of double; function cholesky(const A: D2Array): D2Array; var i, j, k: integer; s: double; begin setlength(Result, length(A), length(A)); for i := low(Result) to high(Result) do for j := 0 to i do begin s := 0; for k := 0 to j - 1 do s := s + Result[i][k] * Result[j][k]; if i = j then Result[i][j] := sqrt(A[i][i] - s) else Result[i][j] := (A[i][j] - s) / Result[j][j]; // save one multiplication compared to the original end; end; procedure printM(const A: D2Array); var i, j: integer; begin for i := low(A) to high(A) do begin for j := low(A) to high(A) do write(A[i, j]: 8: 5); writeln; end; end; const m1: array[0..2, 0..2] of double = ((25, 15, -5), (15, 18, 0), (-5, 0, 11)); m2: array[0..3, 0..3] of double = ((18, 22, 54, 42), (22, 70, 86, 62), (54, 86, 174, 134), (42, 62, 134, 106)); var index, i: integer; cIn, cOut: D2Array; begin setlength(cIn, length(m1), length(m1)); for index := low(m1) to high(m1) do begin SetLength(cIn[index], length(m1[index])); for i := 0 to High(m1[Index]) do cIn[index][i] := m1[index][i]; end; cOut := cholesky(cIn); printM(cOut); writeln; setlength(cIn, length(m2), length(m2)); for index := low(m2) to high(m2) do begin SetLength(cIn[index], length(m2[Index])); for i := 0 to High(m2[Index]) do cIn[index][i] := m2[index][i]; end; cOut := cholesky(cIn); printM(cOut); end.  Output:  5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262  ## Perl sub cholesky { my$matrix = shift;
my $chol = [ map { [(0) x @$matrix ] } @$matrix ]; for my$row (0..@$matrix-1) { for my$col (0..$row) { my$x = $$matrix[row][col]; x -=$$chol[$row][$_]*$$chol[col][_] for 0..col;$$chol[$row][$col] = $row ==$col ? sqrt $x :$x/chol[$col][$col];
}
}
return $chol; } my$example1 = [ [ 25, 15, -5 ],
[ 15, 18,  0 ],
[ -5,  0, 11 ] ];
print "Example 1:\n";
print +(map { sprintf "%7.4f\t", $_ } @$_), "\n" for @{ cholesky $example1 }; my$example2 = [ [ 18, 22,  54,  42],
[ 22, 70,  86,  62],
[ 54, 86, 174, 134],
[ 42, 62, 134, 106] ];
print "\nExample 2:\n";
print +(map { sprintf "%7.4f\t", $_ } @$_), "\n" for @{ cholesky example2 };  Output: Example 1: 5.0000 0.0000 0.0000 3.0000 3.0000 0.0000 -1.0000 1.0000 3.0000 Example 2: 4.2426 0.0000 0.0000 0.0000 5.1854 6.5659 0.0000 0.0000 12.7279 3.0460 1.6497 0.0000 9.8995 1.6246 1.8497 1.3926  ## Phix Translation of: Sidef with javascript_semantics function cholesky(sequence matrix) integer l = length(matrix) sequence chol = repeat(repeat(0,l),l) for row=1 to l do for col=1 to row do atom x = matrix[row][col] for i=1 to col do x -= chol[row][i] * chol[col][i] end for chol[row][col] = iff(row == col ? sqrt(x) : x/chol[col][col]) end for end for return chol end function ppOpt({pp_Nest,1}) pp(cholesky({{ 25, 15, -5 }, { 15, 18, 0 }, { -5, 0, 11 }})) pp(cholesky({{ 18, 22, 54, 42}, { 22, 70, 86, 62}, { 54, 86, 174, 134}, { 42, 62, 134, 106}}))  Output: {{5,0,0}, {3,3,0}, {-1,1,3}} {{4.242640687,0,0,0}, {5.185449729,6.565905201,0,0}, {12.72792206,3.046038495,1.649742248,0}, {9.899494937,1.624553864,1.849711005,1.392621248}}  ## PicoLisp (scl 9) (load "@lib/math.l") (de cholesky (A) (let L (mapcar '(() (need (length A) 0)) A) (for (I . R) A (for J I (let S (get R J) (for K (inc J) (dec 'S (*/ (get L I K) (get L J K) 1.0)) ) (set (nth L I J) (if (= I J) (sqrt S 1.0) (*/ S 1.0 (get L J J)) ) ) ) ) ) (for R L (for N R (prin (align 9 (round N 5)))) (prinl) ) ) ) Test: (cholesky '((25.0 15.0 -5.0) (15.0 18.0 0) (-5.0 0 11.0)) ) (prinl) (cholesky (quote (18.0 22.0 54.0 42.0) (22.0 70.0 86.0 62.0) (54.0 86.0 174.0 134.0) (42.0 62.0 134.0 106.0) ) ) Output:  5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262 ## PL/I (subscriptrange): decompose: procedure options (main); /* 31 October 2013 */ declare a(*,*) float controlled; allocate a(3,3) initial (25, 15, -5, 15, 18, 0, -5, 0, 11); put skip list ('Original matrix:'); put edit (a) (skip, 3 f(4)); call cholesky(a); put skip list ('Decomposed matrix'); put edit (a) (skip, 3 f(4)); free a; allocate a(4,4) initial (18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106); put skip list ('Original matrix:'); put edit (a) (skip, (hbound(a,1)) f(12) ); call cholesky(a); put skip list ('Decomposed matrix'); put edit (a) (skip, (hbound(a,1)) f(12,5) ); cholesky: procedure(a); declare a(*,*) float; declare L(hbound(a,1), hbound(a,2)) float; declare s float; declare (i, j, k) fixed binary; L = 0; do i = lbound(a,1) to hbound(a,1); do j = lbound(a,2) to i; s = 0; do k = lbound(a,2) to j-1; s = s + L(i,k) * L(j,k); end; if i = j then L(i,j) = sqrt(a(i,i) - s); else L(i,j) = (a(i,j) - s) / L(j,j); end; end; a = L; end cholesky; end decompose; ACTUAL RESULTS:- Original matrix: 25 15 -5 15 18 0 -5 0 11 Decomposed matrix 5 0 0 3 3 0 -1 1 3 Original matrix: 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 Decomposed matrix 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89950 1.62455 1.84971 1.39262  ## PowerShell function cholesky (a) {
$l = @() if ($a) {
$n =$a.count
$end =$n - 1
$l = 1..$n | foreach {$row = @(0) *$n; ,$row} foreach ($k in 0..$end) {$m = $k - 1$sum = 0
if(0 -lt $k) { foreach ($j in 0..$m) {$sum += $l[$k][$j]*$l[$k][$j]}
}
$l[$k][$k] = [Math]::Sqrt($a[$k][$k] - $sum) if ($k -lt $end) { foreach ($i in ($k+1)..$end) {
$sum = 0 if (0 -lt$k) {
foreach ($j in 0..$m) {$sum +=$l[$i][$j]*$l[$k][$j]} }$l[$i][$k] = ($a[$i][$k] -$sum)/$l[$k][$k] } } } }$l
}

function show($a) {$a | foreach {"$_"}}$a1 = @(
@(25, 15, -5),
@(15, 18, 0),
@(-5, 0, 11)
)
"a1 ="
show $a1 "" "l1 =" show (cholesky$a1)
""
$a2 = @( @(18, 22, 54, 42), @(22, 70, 86, 62), @(54, 86, 174, 134), @(42, 62, 134, 106) ) "a2 =" show$a2
""
"l2 ="
show (cholesky $a2)  Output: a1 = 25 15 -5 15 18 0 -5 0 11 l1 = 5 0 0 3 3 0 -1 1 3 a2 = 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 l2 = 4.24264068711928 0 0 0 5.18544972870135 6.5659052011974 0 0 12.7279220613579 3.04603849540086 1.64974224790907 0 9.89949493661167 1.62455386421379 1.84971100523138 1.39262124764559  ## Python ### Python2.X version from __future__ import print_function from pprint import pprint from math import sqrt def cholesky(A): L = [[0.0] * len(A) for _ in xrange(len(A))] for i in xrange(len(A)): for j in xrange(i+1): s = sum(L[i][k] * L[j][k] for k in xrange(j)) L[i][j] = sqrt(A[i][i] - s) if (i == j) else \ (1.0 / L[j][j] * (A[i][j] - s)) return L if __name__ == "__main__": m1 = [[25, 15, -5], [15, 18, 0], [-5, 0, 11]] pprint(cholesky(m1)) print() m2 = [[18, 22, 54, 42], [22, 70, 86, 62], [54, 86, 174, 134], [42, 62, 134, 106]] pprint(cholesky(m2), width=120)  Output: [[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]  ### Python3.X version using extra Python idioms Factors out accesses to A[i], L[i], and L[j] by creating Ai, Li and Lj respectively as well as using enumerate instead of range(len(some_array)). def cholesky(A): L = [[0.0] * len(A) for _ in range(len(A))] for i, (Ai, Li) in enumerate(zip(A, L)): for j, Lj in enumerate(L[:i+1]): s = sum(Li[k] * Lj[k] for k in range(j)) Li[j] = sqrt(Ai[i] - s) if (i == j) else \ (1.0 / Lj[j] * (Ai[j] - s)) return L  Output: (As above) ## q solve:{[A;B]$[0h>type A;B%A;inv[A] mmu B]}
ak:{[m;k] (),/:m[;k]til k:k-1}
akk:{[m;k] m[k;k:k-1]}
transpose:{$[0h=type x;flip x;enlist each x]} mult:{[A;B]$[0h=type A;A mmu B;A*B]}
cholesky:{[A]
{[A;L;n]
l_k:solve[L;ak[A;n]];
l_kk:first over sqrt[akk[A;n] - mult[transpose l_k;l_k]];
({$[0h<type x;enlist x;x]}L,'0f),enlist raze transpose[l_k],l_kk }[A]/[sqrt A[0;0];1_1+til count first A] } show cholesky (25 15 -5f;15 18 0f;-5 0 11f) -1""; show cholesky (18 22 54 42f;22 70 86 62f;54 86 174 134f;42 62 134 106f)  Output: 5 0 0 3 3 0 -1 1 3 4.242641 0 0 0 5.18545 6.565905 0 0 12.72792 3.046038 1.649742 0 9.899495 1.624554 1.849711 1.392621  ## R t(chol(matrix(c(25, 15, -5, 15, 18, 0, -5, 0, 11), nrow=3, ncol=3))) # [,1] [,2] [,3] # [1,] 5 0 0 # [2,] 3 3 0 # [3,] -1 1 3 t(chol(matrix(c(18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106), nrow=4, ncol=4))) # [,1] [,2] [,3] [,4] # [1,] 4.242641 0.000000 0.000000 0.000000 # [2,] 5.185450 6.565905 0.000000 0.000000 # [3,] 12.727922 3.046038 1.649742 0.000000 # [4,] 9.899495 1.624554 1.849711 1.392621  ## Racket #lang racket (require math) (define (cholesky A) (define mref matrix-ref) (define n (matrix-num-rows A)) (define L (for/vector ([_ n]) (for/vector ([_ n]) 0))) (define (set L i j x) (vector-set! (vector-ref L i) j x)) (define (ref L i j) (vector-ref (vector-ref L i) j)) (for* ([i n] [k n]) (set L i k (cond [(= i k) (sqrt (- (mref A i i) (for/sum ([j k]) (sqr (ref L k j)))))] [(> i k) (/ (- (mref A i k) (for/sum ([j k]) (* (ref L i j) (ref L k j)))) (ref L k k))] [else 0]))) L) (cholesky (matrix [[25 15 -5] [15 18 0] [-5 0 11]])) (cholesky (matrix [[18 22 54 42] [22 70 86 62] [54 86 174 134] [42 62 134 106]]))  Output: '#(#(5 0 0) #(3 3 0) #(-1 1 3)) '#(#(4.242640687119285 0 0 0) #( 5.185449728701349 6.565905201197403 0 0) #(12.727922061357857 3.0460384954008553 1.6497422479090704 0) #( 9.899494936611665 1.6245538642137891 1.849711005231382 1.3926212476455924))  ## Raku (formerly Perl 6) sub cholesky(@A) { my @L = @A »×» 0; for ^@A -> \i { for 0..i -> \j { @L[i;j] = (i == j ?? &sqrt !! 1/@L[j;j] × * )\ # select function (@A[i;j] - [+] (@L[i;*] Z× @L[j;*])[^j]) # provide value } } @L } .fmt('%3d').say for cholesky [ [25], [15, 18], [-5, 0, 11], ]; say ''; .fmt('%6.3f').say for cholesky [ [18, 22, 54, 42], [22, 70, 86, 62], [54, 86, 174, 134], [42, 62, 134, 106], ];  Output:  5 3 3 -1 1 3 4.243 0.000 0.000 0.000 5.185 6.566 0.000 0.000 12.728 3.046 1.650 0.000 9.899 1.625 1.850 1.393 ## REXX If trailing zeros are wanted after the decimal point for values of zero (0), the / 1 (a division by unity performs REXX number normalization) can be removed from the line (number 40) which contains the source statement: z=z right( format(@.row.col, , dPlaces) / 1, width) /*REXX program performs the Cholesky decomposition on a square matrix & displays results*/ niner = '25 15 -5' , /*define a 3x3 matrix with elements. */ '15 18 0' , '-5 0 11' call Cholesky niner hexer = 18 22 54 42, /*define a 4x4 matrix with elements. */ 22 70 86 62, 54 86 174 134, 42 62 134 106 call Cholesky hexer exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ Cholesky: procedure; parse arg mat; say; say; call tell 'input array',mat do r=1 for ord do c=1 for r;$=0;  do i=1  for c-1;  $=$  +  !.r.i * !.c.i;  end /*i*/
if r=c  then !.r.r= sqrt(!.r.r - $) else !.r.c= 1 / !.c.c * (@.r.c -$)
end   /*c*/
end      /*r*/
call tell  'Cholesky factor',,!.,'─'
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
err:   say;   say;   say '***error***!';      say;    say arg(1);   say;   say;    exit 13
/*──────────────────────────────────────────────────────────────────────────────────────*/
tell:  parse arg hdr,x,y,sep;   #=0;          if sep==''  then sep= '═'
dPlaces= 5                                /*# dec. places past the decimal point.*/
width  =10                                /*field width used to display elements.*/
if y==''  then !.=0
else do row=1  for ord;  do col=1  for ord;  x=x !.row.col;  end;   end
w=words(x)
do ord=1  until ord**2>=w;  end   /*a fast way to find the matrix's order*/
say
if ord**2\==w  then call err  "matrix elements don't form a square matrix."
say center(hdr, ((width + 1) * w) % ord,  sep)
say
do   row=1  for ord;         z=
do col=1  for ord;         #= # + 1
@.row.col= word(x, #)
if col<=row  then  !.row.col= @.row.col
z=z  right( format(@.row.col, , dPlaces) / 1,   width)
end   /*col*/                   /*       ↑↑↑                           */
say z                             /*       └┴┴──◄──normalization for zero*/
end        /*row*/
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt:  procedure; parse arg x;  if x=0  then return 0;  d=digits(); numeric digits;  h=d+6
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_ %2
do j=0  while h>9;      m.j=h;              h=h%2+1;       end  /*j*/
do k=j+5  to 0  by -1;  numeric digits m.k; g=(g+x/g)*.5;  end  /*k*/; return g/1

output:
═══════════input matrix══════════

25         15         -5
15         18          0
-5          0         11

─────────Cholesky factor─────────

5          0          0
3          3          0
-1          1          3

════════════════input matrix════════════════

18         22         54         42
22         70         86         62
54         86        174        134
42         62        134        106

──────────────Cholesky factor───────────────

4.24264          0          0          0
5.18545    6.56591          0          0
12.72792    3.04604    1.64974          0
9.89949    1.62455    1.84971    1.39262


## Ring

# Project : Cholesky decomposition

decimals(5)
m1 = [[25, 15, -5],
[15, 18,  0],
[-5,  0, 11]]
cholesky(m1)
printarray(m1)
see nl

m2 = [[18, 22,  54,  42],
[22, 70,  86,  62],
[54, 86, 174, 134],
[42, 62, 134, 106]]
cholesky(m2)
printarray(m2)

func cholesky(a)
l = newlist(len(a), len(a))
for i = 1 to len(a)
for j = 1 to i
s = 0
for k = 1 to j
s = s + l[i][k] * l[j][k]
next
if i = j
l[i][j] = sqrt(a[i][i] - s)
else
l[i][j] = (a[i][j] - s) / l[j][j]
ok
next
next
a = l

func printarray(a)
for row = 1 to len(a)
for col = 1 to len(a)
see "" + a[row][col] + "  "
next
see nl
next

Output:

5  0  0
3  3  0
-1  1  3

4.24264  0  0  0
5.18545  6.56591  0  0
12.72792  3.04604  1.64974  0
9.89949  1.62455  1.84971  1.39262


## Ruby

require 'matrix'

class Matrix
def cholesky_factor
raise ArgumentError, "must provide symmetric matrix" unless symmetric?
l = Array.new(row_size) {Array.new(row_size, 0)}
(0 ... row_size).each do |k|
(0 ... row_size).each do |i|
if i == k
sum = (0 .. k-1).inject(0.0) {|sum, j| sum + l[k][j] ** 2}
val = Math.sqrt(self[k,k] - sum)
l[k][k] = val
elsif i > k
sum = (0 .. k-1).inject(0.0) {|sum, j| sum + l[i][j] * l[k][j]}
val = (self[k,i] - sum) / l[k][k]
l[i][k] = val
end
end
end
Matrix[*l]
end
end

puts Matrix[[25,15,-5],[15,18,0],[-5,0,11]].cholesky_factor
puts Matrix[[18, 22,  54,  42],
[22, 70,  86,  62],
[54, 86, 174, 134],
[42, 62, 134, 106]].cholesky_factor

Output:
Matrix[[5.0, 0, 0], [3.0, 3.0, 0], [-1.0, 1.0, 3.0]]
Matrix[[4.242640687119285, 0, 0, 0],
[5.185449728701349, 6.565905201197403, 0, 0],
[12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0],
[9.899494936611665, 1.6245538642137891, 1.849711005231382, 1.3926212476455924]]


## Rust

Translation of: C
fn cholesky(mat: Vec<f64>, n: usize) -> Vec<f64> {
let mut res = vec![0.0; mat.len()];
for i in 0..n {
for j in 0..(i+1){
let mut s = 0.0;
for k in 0..j {
s += res[i * n + k] * res[j * n + k];
}
res[i * n + j] = if i == j { (mat[i * n + i] - s).sqrt() } else { (1.0 / res[j * n + j] * (mat[i * n + j] - s)) };
}
}
res
}

fn show_matrix(matrix: Vec<f64>, n: usize){
for i in 0..n {
for j in 0..n {
print!("{:.4}\t", matrix[i * n + j]);
}
println!("");
}
println!("");
}

fn main(){
let dimension = 3 as usize;
let m1 = vec![25.0, 15.0, -5.0,
15.0, 18.0,  0.0,
-5.0,  0.0, 11.0];
let res1 = cholesky(m1, dimension);
show_matrix(res1, dimension);

let dimension = 4 as usize;
let m2 = vec![18.0, 22.0,  54.0,  42.0,
22.0, 70.0,  86.0,  62.0,
54.0, 86.0, 174.0, 134.0,
42.0, 62.0, 134.0, 106.0];
let res2 = cholesky(m2, dimension);
show_matrix(res2, dimension);
}

Output:
5.0000	0.0000	0.0000
3.0000	3.0000	0.0000
-1.0000	1.0000	3.0000

4.2426	0.0000	0.0000	0.0000
5.1854	6.5659	0.0000	0.0000
12.7279	3.0460	1.6497	0.0000
9.8995	1.6246	1.8497	1.3926


## Scala

case class Matrix( val matrix:Array[Array[Double]] ) {

// Assuming matrix is positive-definite, symmetric and not empty...

val rows,cols = matrix.size

def getOption( r:Int, c:Int ) : Option[Double] = Pair(r,c) match {
case (r,c) if r < rows && c < rows => Some(matrix(r)(c))
case _ => None
}

def isLowerTriangle( r:Int, c:Int ) : Boolean = { c <= r }
def isDiagonal( r:Int, c:Int ) : Boolean = { r == c}

override def toString = matrix.map(_.mkString(", ")).mkString("\n")

/**
* Perform Cholesky Decomposition of this matrix
*/
lazy val cholesky : Matrix = {

val l = Array.ofDim[Double](rows*cols)

for( i <- (0 until rows); j <- (0 until cols) ) yield {

val s = (for( k <- (0 until j) ) yield { l(i*rows+k) * l(j*rows+k) }).sum

l(i*rows+j) = (i,j) match {
case (r,c) if isDiagonal(r,c) => scala.math.sqrt(matrix(i)(i) - s)
case (r,c) if isLowerTriangle(r,c) => (1.0 / l(j*rows+j) * (matrix(i)(j) - s))
case _ => 0
}
}

val m = Array.ofDim[Double](rows,cols)
for( i <- (0 until rows); j <- (0 until cols) ) m(i)(j) = l(i*rows+j)
Matrix(m)
}
}

// A little test...
val a1 = Matrix(Array[Array[Double]](Array(25,15,-5),Array(15,18,0),Array(-5,0,11)))
val a2 = Matrix(Array[Array[Double]](Array(18,22,54,42), Array(22,70,86,62), Array(54,86,174,134), Array(42,62,134,106)))

val l1 = a1.cholesky
val l2 = a2.cholesky

// Given test results
val r1 = Array[Double](5,0,0,3,3,0,-1,1,3)
val r2 = Array[Double](4.24264,0.00000,0.00000,0.00000,5.18545,6.56591,0.00000,0.00000,
12.72792,3.04604,1.64974,0.00000,9.89949,1.62455,1.84971,1.39262)

// Verify assertions
(l1.matrix.flatten.zip(r1)).foreach{ case (result,test) =>
assert(math.round( result * 100000 ) * 0.00001 == math.round( test * 100000 ) * 0.00001)
}

(l2.matrix.flatten.zip(r2)).foreach{ case (result,test) =>
assert(math.round( result * 100000 ) * 0.00001 == math.round( test * 100000 ) * 0.00001)
}


## Scilab

The Cholesky decomposition is builtin, and an upper triangular matrix is returned, such that $A=L^TL$.

a = [25 15 -5; 15 18 0; -5 0 11];
chol(a)
ans  =

5.   3.  -1.
0.   3.   1.
0.   0.   3.

a = [18 22 54 42; 22 70 86 62;
54 86 174 134; 42 62 134 106];

chol(a)
ans  =

4.2426407   5.1854497   12.727922   9.8994949
0.          6.5659052   3.0460385   1.6245539
0.          0.          1.6497422   1.849711
0.          0.          0.          1.3926212


$include "seed7_05.s7i"; include "float.s7i"; include "math.s7i"; const type: matrix is array array float; const func matrix: cholesky (in matrix: a) is func result var matrix: cholesky is 0 times 0 times 0.0; local var integer: i is 0; var integer: j is 0; var integer: k is 0; var float: sum is 0.0; begin cholesky := length(a) times length(a) times 0.0; for key i range cholesky do for j range 1 to i do sum := 0.0; for k range 1 to j do sum +:= cholesky[i][k] * cholesky[j][k]; end for; if i = j then cholesky[i][i] := sqrt(a[i][i] - sum) else cholesky[i][j] := (a[i][j] - sum) / cholesky[j][j]; end if; end for; end for; end func; const proc: writeMat (in matrix: a) is func local var integer: i is 0; var float: num is 0.0; begin for key i range a do for num range a[i] do write(num digits 5 lpad 8); end for; writeln; end for; end func; const matrix: m1 is [] ( [] (25.0, 15.0, -5.0), [] (15.0, 18.0, 0.0), [] (-5.0, 0.0, 11.0)); const matrix: m2 is [] ( [] (18.0, 22.0, 54.0, 42.0), [] (22.0, 70.0, 86.0, 62.0), [] (54.0, 86.0, 174.0, 134.0), [] (42.0, 62.0, 134.0, 106.0)); const proc: main is func begin writeMat(cholesky(m1)); writeln; writeMat(cholesky(m2)); end func; Output:  5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89950 1.62455 1.84971 1.39262  ## Sidef Translation of: Perl func cholesky(matrix) { var chol = matrix.len.of { matrix.len.of(0) } for row in ^matrix { for col in (0..row) { var x = matrix[row][col] for i in (0..col) { x -= (chol[row][i] * chol[col][i]) } chol[row][col] = (row == col ? x.sqrt : x/chol[col][col]) } } return chol }  Examples: var example1 = [ [ 25, 15, -5 ], [ 15, 18, 0 ], [ -5, 0, 11 ] ]; say "Example 1:"; cholesky(example1).each { |row| say row.map {'%7.4f' % _}.join(' '); } var example2 = [ [ 18, 22, 54, 42], [ 22, 70, 86, 62], [ 54, 86, 174, 134], [ 42, 62, 134, 106] ]; say "\nExample 2:"; cholesky(example2).each { |row| say row.map {'%7.4f' % _}.join(' '); }  Output: Example 1: 5.0000 0.0000 0.0000 3.0000 3.0000 0.0000 -1.0000 1.0000 3.0000 Example 2: 4.2426 0.0000 0.0000 0.0000 5.1854 6.5659 0.0000 0.0000 12.7279 3.0460 1.6497 0.0000 9.8995 1.6246 1.8497 1.3926  ## Smalltalk  FloatMatrix>>#cholesky | l | l := FloatMatrix zero: numRows. 1 to: numRows do: [:i | 1 to: i do: [:k | | rowSum lkk factor aki partialSum | i = k ifTrue: [ rowSum := (1 to: k - 1) sum: [:j | | lkj | lkj := l at: j @ k. lkj squared]. lkk := (self at: k @ k) - rowSum. lkk := lkk sqrt. l at: k @ k put: lkk] ifFalse: [ factor := l at: k @ k. aki := self at: k @ i. partialSum := (1 to: k - 1) sum: [:j | | ljk lji | lji := l at: j @ i. ljk := l at: j @ k. lji * ljk]. l at: k @ i put: aki - partialSum * factor reciprocal]]]. ^l  ## Stata See Cholesky square-root decomposition in Stata help. mata : a=25,15,-5\15,18,0\-5,0,11 : a [symmetric] 1 2 3 +----------------+ 1 | 25 | 2 | 15 18 | 3 | -5 0 11 | +----------------+ : cholesky(a) 1 2 3 +----------------+ 1 | 5 0 0 | 2 | 3 3 0 | 3 | -1 1 3 | +----------------+ : a=18,22,54,42\22,70,86,62\54,86,174,134\42,62,134,106 : a [symmetric] 1 2 3 4 +-------------------------+ 1 | 18 | 2 | 22 70 | 3 | 54 86 174 | 4 | 42 62 134 106 | +-------------------------+ : cholesky(a) 1 2 3 4 +---------------------------------------------------------+ 1 | 4.242640687 0 0 0 | 2 | 5.185449729 6.565905201 0 0 | 3 | 12.72792206 3.046038495 1.649742248 0 | 4 | 9.899494937 1.624553864 1.849711005 1.392621248 | +---------------------------------------------------------+  ## Swift Translation of: Rust func cholesky(matrix: [Double], n: Int) -> [Double] { var res = [Double](repeating: 0, count: matrix.count) for i in 0..<n { for j in 0..<i+1 { var s = 0.0 for k in 0..<j { s += res[i * n + k] * res[j * n + k] } if i == j { res[i * n + j] = (matrix[i * n + i] - s).squareRoot() } else { res[i * n + j] = (1.0 / res[j * n + j] * (matrix[i * n + j] - s)) } } } return res } func printMatrix(_ matrix: [Double], n: Int) { for i in 0..<n { for j in 0..<n { print(matrix[i * n + j], terminator: " ") } print() } } let res1 = cholesky( matrix: [25.0, 15.0, -5.0, 15.0, 18.0, 0.0, -5.0, 0.0, 11.0], n: 3 ) let res2 = cholesky( matrix: [18.0, 22.0, 54.0, 42.0, 22.0, 70.0, 86.0, 62.0, 54.0, 86.0, 174.0, 134.0, 42.0, 62.0, 134.0, 106.0], n: 4 ) printMatrix(res1, n: 3) print() printMatrix(res2, n: 4)  Output: 5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0 4.242640687119285 0.0 0.0 0.0 5.185449728701349 6.565905201197403 0.0 0.0 12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026 ## Tcl Translation of: Java proc cholesky a { set m [llength$a]
set n [llength [lindex $a 0]] set l [lrepeat$m [lrepeat $n 0.0]] for {set i 0} {$i < $m} {incr i} { for {set k 0} {$k < $i+1} {incr k} { set sum 0.0 for {set j 0} {$j < $k} {incr j} { set sum [expr {$sum + [lindex $l$i $j] * [lindex$l $k$j]}]
}
lset l $i$k [expr {
$i ==$k
? sqrt([lindex $a$i $i] -$sum)
: (1.0 / [lindex $l$k $k] * ([lindex$a $i$k] - $sum)) }] } } return$l
}


Demonstration code:

set test1 {
{25 15 -5}
{15 18  0}
{-5  0 11}
}
puts [cholesky $test1] set test2 { {18 22 54 42} {22 70 86 62} {54 86 174 134} {42 62 134 106} } puts [cholesky$test2]

Output:
{5.0 0.0 0.0} {3.0 3.0 0.0} {-1.0 1.0 3.0}
{4.242640687119285 0.0 0.0 0.0} {5.185449728701349 6.565905201197403 0.0 0.0} {12.727922061357857 3.0460384954008553 1.6497422479090704 0.0} {9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026}


## VBA

This function returns the lower Cholesky decomposition of a square matrix fed to it. It does not check for positive semi-definiteness, although it does check for squareness. It assumes that Option Base 0 is set, and thus the matrix entry indices need to be adjusted if Base is set to 1. It also assumes a matrix of size less than 256x256. To handle larger matrices, change all Byte-type variables to Long. It takes the square matrix range as an input, and can be implemented as an array function on the same sized square range of cells as output. For example, if the matrix is in cells A1:E5, highlighting cells A10:E14, typing "=Cholesky(A1:E5)" and htting Ctrl-Shift-Enter will populate the target cells with the lower Cholesky decomposition.

Function Cholesky(Mat As Range) As Variant

Dim A() As Double, L() As Double, sum As Double, sum2 As Double
Dim m As Byte, i As Byte, j As Byte, k As Byte

'Ensure matrix is square
If Mat.Rows.Count <> Mat.Columns.Count Then
MsgBox ("Correlation matrix is not square")
Exit Function
End If

m = Mat.Rows.Count

'Initialize and populate matrix A of values and matrix L which will be the lower Cholesky
ReDim A(0 To m - 1, 0 To m - 1)
ReDim L(0 To m - 1, 0 To m - 1)
For i = 0 To m - 1
For j = 0 To m - 1
A(i, j) = Mat(i + 1, j + 1).Value2
L(i, j) = 0
Next j
Next i

'Handle the simple cases explicitly to save time
Select Case m
Case Is = 1
L(0, 0) = Sqr(A(0, 0))

Case Is = 2
L(0, 0) = Sqr(A(0, 0))
L(1, 0) = A(1, 0) / L(0, 0)
L(1, 1) = Sqr(A(1, 1) - L(1, 0) * L(1, 0))

Case Else
L(0, 0) = Sqr(A(0, 0))
L(1, 0) = A(1, 0) / L(0, 0)
L(1, 1) = Sqr(A(1, 1) - L(1, 0) * L(1, 0))
For i = 2 To m - 1
sum2 = 0
For k = 0 To i - 1
sum = 0
For j = 0 To k
sum = sum + L(i, j) * L(k, j)
Next j
L(i, k) = (A(i, k) - sum) / L(k, k)
sum2 = sum2 + L(i, k) * L(i, k)
Next k
L(i, i) = Sqr(A(i, i) - sum2)
Next i
End Select
Cholesky = L
End Function

## V (Vlang)

Translation of: go
import math

// Symmetric and Lower use a packed representation that stores only
// the Lower triangle.

struct Symmetric {
order int
ele   []f64
}

struct Lower  {
mut:
order int
ele   []f64
}

// Symmetric.print prints a square matrix from the packed representation,
// printing the upper triange as a transpose of the Lower.
fn (s Symmetric) print() {
mut row, mut diag := 1, 0
for i, e in s.ele {
print("${e:10.5f} ") if i == diag { for j, col := diag+row, row; col < s.order; j += col { print("${s.ele[j]:10.5f} ")
col++
}
println('')
row++
diag += row
}
}
}

// Lower.print prints a square matrix from the packed representation,
// printing the upper triangle as all zeros.
fn (l Lower) print() {
mut row, mut diag := 1, 0
for i, e in l.ele {
print("${e:10.5f} ") if i == diag { for _ in row..l.order { print("${0.0:10.5f} ")
}
println('')
row++
diag += row
}
}
}

// cholesky_lower returns the cholesky decomposition of a Symmetric real
// matrix.  The matrix must be positive definite but this is not checked.
fn (a Symmetric) cholesky_lower() Lower {
mut l := Lower{a.order, []f64{len: a.ele.len}}
mut row, mut col := 1, 1
mut dr := 0 // index of diagonal element at end of row
mut dc := 0 // index of diagonal element at top of column
for i, e in a.ele {
if i < dr {
d := (e - l.ele[i]) / l.ele[dc]
l.ele[i] = d
mut ci, mut cx := col, dc
for j := i + 1; j <= dr; j++ {
cx += ci
ci++
l.ele[j] += d * l.ele[cx]
}
col++
dc += col
} else {
l.ele[i] = math.sqrt(e - l.ele[i])
row++
dr += row
col = 1
dc = 0
}
}
return l
}

fn main() {
demo(Symmetric{3, [
f64(25),
15, 18,
-5, 0, 11]})
demo(Symmetric{4, [
f64(18),
22, 70,
54, 86, 174,
42, 62, 134, 106]})
}

fn demo(a Symmetric) {
println("A:")
a.print()
println("L:")
a.cholesky_lower().print()
}
Output:
A:
25.00000   15.00000   -5.00000
15.00000   18.00000    0.00000
-5.00000    0.00000   11.00000
L:
5.00000    0.00000    0.00000
3.00000    3.00000    0.00000
-1.00000    1.00000    3.00000
A:
18.00000   22.00000   54.00000   42.00000
22.00000   70.00000   86.00000   62.00000
54.00000   86.00000  174.00000  134.00000
42.00000   62.00000  134.00000  106.00000
L:
4.24264    0.00000    0.00000    0.00000
5.18545    6.56591    0.00000    0.00000
12.72792    3.04604    1.64974    0.00000
9.89949    1.62455    1.84971    1.3926


## Wren

Library: Wren-matrix
Library: Wren-fmt
import "./matrix" for Matrix
import "./fmt" for Fmt

var arrays = [
[ [25, 15, -5],
[15, 18,  0],
[-5,  0, 11] ],

[ [18, 22,  54,  42],
[22, 70,  86,  62],
[54, 86, 174, 134],
[42, 62, 134, 106] ]
]

for (array in arrays) {
System.print("Original:")
Fmt.mprint(array, 3, 0)
System.print("\nLower Cholesky factor:")
Fmt.mprint(Matrix.new(array).cholesky(), 8, 5)
System.print()
}

Output:
Original:
| 25  15  -5|
| 15  18   0|
| -5   0  11|

Lower Cholesky factor:
| 5.00000  0.00000  0.00000|
| 3.00000  3.00000  0.00000|
|-1.00000  1.00000  3.00000|

Original:
| 18  22  54  42|
| 22  70  86  62|
| 54  86 174 134|
| 42  62 134 106|

Lower Cholesky factor:
| 4.24264  0.00000  0.00000  0.00000|
| 5.18545  6.56591  0.00000  0.00000|
|12.72792  3.04604  1.64974  0.00000|
| 9.89949  1.62455  1.84971  1.39262|


## XPL0

real L(4*4);

func real Cholesky(A, N);
real A;  int N;
real S;
int  I, J, K;
[for I:= 0 to N*N-1 do L(I):= 0.;
for I:= 0 to N-1 do
for J:= 0 to I do
[S:= 0.;
for K:= 0 to J-1 do
S:= S + L(I*N+K) * L(J*N+K);
L(I*N+J):= if I = J then sqrt(A(I*N+I) - S)
else (1.0 / L(J*N+J) * (A(I*N+J) - S));
];
return L;
];

proc ShowMatrix(A, N);
real A;  int N;
int  I, J;
[for I:= 0 to N-1 do
[for J:= 0 to N-1 do
RlOut(0, A(I*N+J));
CrLf(0);
];
];

int  N;
real M1, C1, M2, C2;
[N:= 3;
M1:=   [25., 15., -5.,
15., 18.,  0.,
-5.,  0., 11.];
C1:= Cholesky(M1, N);
ShowMatrix(C1, N);
CrLf(0);

N:= 4;
M2:=   [18., 22.,  54.,  42.,
22., 70.,  86.,  62.,
54., 86., 174., 134.,
42., 62., 134., 106.];
C2:= Cholesky(M2, N);
ShowMatrix(C2, N);
]
Output:
    5.00000    0.00000    0.00000
3.00000    3.00000    0.00000
-1.00000    1.00000    3.00000

4.24264    0.00000    0.00000    0.00000
5.18545    6.56591    0.00000    0.00000
12.72792    3.04604    1.64974    0.00000
9.89949    1.62455    1.84971    1.39262


## zkl

Using the GNU Scientific Library:

var [const] GSL=Import("zklGSL");	// libGSL (GNU Scientific Library)
fcn lowerCholesky(m){  // trans: C
rows:=m.rows;
lcm:=GSL.Matrix(rows,rows);	// zero filled
foreach i,j in (rows,i+1){
s:=(0).reduce(j,'wrap(s,k){ s + lcm[i,k]*lcm[j,k] },0.0);
lcm[i,j]=( if(i==j)(m[i,i] - s).sqrt()
else     1.0/lcm[j,j]*(m[i,j] - s) );
}
lcm
}
Output:
lowerCholesky(GSL.Matrix(3,3).set(25, 15, -5, 	// example 1
15, 18,  0,
-5,  0, 11))
.format(6).println();
5.00,  0.00,  0.00
3.00,  3.00,  0.00
-1.00,  1.00,  3.00

Output:
lowerCholesky(GSL.Matrix(4,4).set(	// example 2
18, 22,  54,  42,
22, 70,  86,  62,
54, 86, 174, 134,
42, 62, 134, 106) )
.format(8,4).println();
4.2426,  0.0000,  0.0000,  0.0000
5.1854,  6.5659,  0.0000,  0.0000
12.7279,  3.0460,  1.6497,  0.0000
9.8995,  1.6246,  1.8497,  1.3926


Or, using lists:

Translation of: C
fcn cholesky(mat){
rows:=mat.len();
r:=(0).pump(rows,List().write, (0).pump(rows,List,0.0).copy); // matrix of zeros
foreach i,j in (rows,i+1){
s:=(0).reduce(j,'wrap(s,k){ s + r[i][k]*r[j][k] },0.0);
r[i][j]=( if(i==j)(mat[i][i] - s).sqrt()
else    1.0/r[j][j]*(mat[i][j] - s) );
}
r
}
ex1:=L( L(25.0,15.0,-5.0), L(15.0,18.0,0.0), L(-5.0,0.0,11.0) );
printM(cholesky(ex1));
println("-----------------");
ex2:=L( L(18.0, 22.0,  54.0,  42.0,),
L(22.0, 70.0,  86.0,  62.0,),
L(54.0, 86.0, 174.0, 134.0,),
L(42.0, 62.0, 134.0, 106.0,) );
printM(cholesky(ex2));
fcn printM(m){ m.pump(Console.println,rowFmt) }
fcn rowFmt(row){ ("%9.5f "*row.len()).fmt(row.xplode()) }
Output:
  5.00000   0.00000   0.00000
3.00000   3.00000   0.00000
-1.00000   1.00000   3.00000
-----------------
4.24264   0.00000   0.00000   0.00000
5.18545   6.56591   0.00000   0.00000
12.72792   3.04604   1.64974   0.00000
9.89949   1.62455   1.84971   1.39262


## ZX Spectrum Basic

Translation of: BBC_BASIC
10 LET d=2000: GO SUB 1000: GO SUB 4000: GO SUB 5000
20 LET d=3000: GO SUB 1000: GO SUB 4000: GO SUB 5000
30 STOP
1000 RESTORE d
1020 DIM m(a,b)
1040 FOR i=1 TO a
1050 FOR j=1 TO b
1070 NEXT j
1080 NEXT i
1090 RETURN
2000 DATA 3,3,25,15,-5,15,18,0,-5,0,11
3000 DATA 4,4,18,22,54,42,22,70,86,62,54,86,174,134,42,62,134,106
4000 REM Cholesky decomposition
4005 DIM l(a,b)
4010 FOR i=1 TO a
4020 FOR j=1 TO i
4030 LET s=0
4050 FOR k=1 TO j-1
4060 LET s=s+l(i,k)*l(j,k)
4070 NEXT k
4080 IF i=j THEN LET l(i,j)=SQR (m(i,i)-s): GO TO 4100
4090 LET l(i,j)=(m(i,j)-s)/l(j,j)
4100 NEXT j
4110 NEXT i
4120 RETURN
5000 REM Print
5010 FOR r=1 TO a
5020 FOR c=1 TO b
5030 PRINT l(r,c);" ";
5040 NEXT c
5050 PRINT
5060 NEXT r
5070 RETURN