# Cholesky decomposition

Cholesky decomposition
You are encouraged to solve this task according to the task description, using any language you may know.

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

${\displaystyle A=LL^{T}}$

${\displaystyle L}$ 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

<lang 11l>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))</lang>

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 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;</lang>

package body Decomposition is

  package Math is new Ada.Numerics.Generic_Elementary_Functions
(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;</lang>

  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;

  Example_1 : constant Ada.Numerics.Real_Arrays.Real_Matrix :=
((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;</lang>

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.

<lang algol68>#!/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)


)</lang>

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))


## AutoHotkey

<lang 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,") "]"


}</lang> Examples:<lang AutoHotkey>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</lang>

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

<lang bbcbasic> 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</lang>


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

<lang c>#include <stdio.h>

1. include <stdlib.h>
2. 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;


}</lang>

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#

<lang csharp>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;
}
}


}</lang>

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++

<lang cpp>#include <cassert>

1. include <cmath>
2. include <iomanip>
3. include <iostream>
4. 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;


}</lang>

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

<lang clojure>(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))))</lang>


Example: <lang clojure>(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]]</lang>

## Common Lisp

<lang 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))</lang>


<lang lisp>;; Example 1: (setf A (make-array '(3 3) :initial-contents '((25 15 -5) (15 18 0) (-5 0 11)))) (chol A)

1. 2A((5.0 0 0)
   (3.0 3.0 0)
(-1.0 1.0 3.0))</lang>


<lang lisp>;; 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)

1. 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))</lang>


<lang lisp>;; 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)))</lang>

<lang lisp>;; 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</lang>

<lang lisp>;; 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</lang>

## D

<lang 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);  }</lang> 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 ## Delphi See Pascal. ## DWScript Translation of: C <lang delphi>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);</lang> ## F# <lang fsharp>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 </lang> 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 <lang fantom>** • Cholesky decomposition class Main {  // create an array of Floats, initialised to 0.0 Float[][] makeArray (Int i, Int j) { Float[][] result := [,] i.times { result.add ([,]) } i.times |Int x| { j.times { result[x].add(0f) } } 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]]) }  }</lang> 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 <lang 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</lang > Output:  5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0  ## FreeBASIC Translation of: BBC BASIC <lang freebasic>' 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</lang> 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 ## 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. <lang go>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()  }</lang> 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. <lang go>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) {  if heading > "" { fmt.Print("\n", heading, "\n") } 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.print(heading) a.choleskyDecomp().print("Cholesky factor L:")  }</lang> 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 <lang go>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, }))  }</lang> 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 <lang go>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)  }</lang> 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 <lang groovy>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  }</lang> Test: <lang groovy>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)] }  }</lang> 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] ## Haskell 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: <lang haskell>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 $do l <- thaw a mapM_ (update a l) [(i,j) | i <- [0..n], j <- [0..n]] return l where maxBnd = fst . snd . bounds update a l i = unsafeFreeze l >>= \l' -> writeArray l i (get a l' i)</lang>  The main module: <lang haskell>import Data.Array.IArray import Data.List import Cholesky fm _ [] = "" fm _ [x] = fst x fm width ((a,b):xs) = a ++ (take (width - b)$ 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 $cholesky ex1 putStrLn$ showMatrice 4 $elems$ cholesky ex2</lang>


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
9.899494936611665  1.6245538642137891 1.849711005231382  1.3926212476455924


### With Numeric.LinearAlgebra

a,b :: Matrix R a = (3><3)

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


b = (4><4)

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


main = do

 let sa = sym a
sb = sym b
print sa
print $chol sa print sb print$ chol sb
print $tr$ chol sb


</lang>

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

<lang Icon>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</lang>

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: <lang Idris>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 $indexM (j, j) a - dot  else if i > j then (indexM (i, j) a - dot) / indexM (j, j) l else 0.0 where -- Obtain indicies 0 to j -1 ks : List (Fin m) ks = case (findIndices (\_ => True) a) of [] => [] (x::xs) => init (x::xs) dot : Double dot = sum [(indexM (i, k) l) * (indexM (j, k) l) | k <- ks]  updateL : Matrix m m Double -> Matrix m m Double -> (Fin m, Fin m) -> Matrix m m Double updateL a l idx = replaceAtM idx (get a l idx) l cholesky : Matrix m m Double -> Matrix m m Double cholesky a {m} =  foldl (\l',i => foldl (\l,j => updateL a l (i, j)) l' (js i)) l is where l = zeros m m is : List (Fin m) is = findIndices (\_ => True) a   js : Fin m -> List (Fin m) js n = filter (<= n) is  ex1 : Matrix 3 3 Double ex1 = cholesky [[25.0, 15.0, -5.0], [15.0, 18.0, 0.0], [-5.0, 0.0, 11.0]] ex2 : Matrix 4 4 Double ex2 = cholesky [[18.0, 22.0, 54.0, 42.0], [22.0, 70.0, 86.0, 62.0],  [54.0, 86.0, 174.0, 134.0], [42.0, 62.0, 134.0, 106.0]]  main : IO () main = do  print ex1 putStrLn "\n" print ex2 putStrLn "\n"  </lang> Output: [[5, 0, 0], [3, 3, 0], [-1, 1, 3]] [[4.242640687119285, 0, 0, 0], [5.185449728701349, 6.565905201197403, 0, 0], [12.72792206135786, 3.046038495400855, 1.64974224790907, 0], [9.899494936611665, 1.624553864213789, 1.849711005231382, 1.392621247645587]]  ## J Solution: <lang j>mp=: +/ . * NB. matrix product h =: +@|: NB. conjugate transpose cholesky=: 3 : 0 n=. #A=. y if. 1>:n do. assert. (A=|A)>0=A NB. check for positive definite %:A else. 'X Y t Z'=. , (;~n$(>.-:n){.1) <;.1 A
L0=. cholesky X
L1=. cholesky Z-(T=.(h Y) mp %.X) mp Y
L0,(T mp L0),.L1
end.


)</lang> See Cholesky Decomposition essay on the J Wiki.

Examples:

<lang j> 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</lang> Using math/lapack addon <lang j> 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</lang>

## Java

Works with: Java version 1.5+

<lang java5>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))); } }</lang>

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

<lang 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)); </lang>

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: <lang jq># 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 ;

1. Print a matrix neatly, each cell ideally occupying n spaces,
2. 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 == $l) | all) ;  1. This implementation of is_symmetric/0 uses a helper function that circumvents 2. limitations of jq 1.4: def is_symmetric:  # [matrix, i,j, len] def test: if .[1] > .[3] then true elif .[1] == .[2] then [ .[0], .[1] + 1, 0, .[3]] | test elif .[0][.[1]][.[2]] == .[0][.[2]][.[1]] then [ .[0], .[1], .[2]+1, .[3]] | test else false end; if is_square|not then false else [ ., 0, 0, length ] | test end ;  </lang>Cholesky Decomposition:<lang jq>def cholesky_factor:  if is_symmetric then length as$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 ;</lang>


[[25,15,-5],[15,18,0],[-5,0,11]] | cholesky_factor

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


[[18, 22,  54,  42],
[22, 70,  86,  62],
[54, 86, 174, 134],
[42, 62, 134, 106]] | cholesky_factor | neatly(20)

Output:

<lang jq> 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</lang>


## Julia

Julia's strong linear algebra support includes Cholesky decomposition. <lang Julia> 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)) </lang>

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

<lang scala>// 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)


}</lang>

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. <lang Lobster>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])


</lang>

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. <lang Maple>> 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]</lang>


## Mathematica / Wolfram Language

<lang Mathematica>CholeskyDecomposition[{{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}]</lang> Without the use of built-in functions, making use of memoization: <lang Mathematica>chol[A_] :=

Module[{L},
L[k_, k_] := L[k, k] = Sqrt[Ak, k - Sum[L[k, j]^2, {j, 1, k-1}]];
L[i_, k_] := L[i, k] = L[k, k]^-1 (Ai, k - Sum[L[i, j] L[k, j], {j, 1, k-1}]);
PadRight[Table[L[i, j], {i, Length[A]}, {j, i}]]
]</lang>


## MATLAB / Octave

The cholesky decomposition chol() is an internal function <lang Matlab> 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')


</lang>

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

<lang 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])</lang> ## Nim Translation of: C <lang nim>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:


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)</lang>

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

<lang objeck> 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;
}


} </lang>

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

<lang 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|] |];</lang>

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

<lang oorexx>/*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.*/</lang>


## PARI/GP

<lang parigp>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


}</lang>

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

<lang 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.</lang>

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

<lang 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 }; </lang>

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

(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) ) ) )</lang>


Test: <lang PicoLisp>(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) ) )</lang>

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

<lang 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;</lang> 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

<lang 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) </lang> 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

<lang python>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)</lang>

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)).

<lang python>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</lang>

Output:

(As above)

## q

<lang 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)</lang>

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

<lang r>t(chol(matrix(c(25, 15, -5, 15, 18, 0, -5, 0, 11), nrow=3, ncol=3)))

1. [,1] [,2] [,3]
2. [1,] 5 0 0
3. [2,] 3 3 0
4. [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. [,1] [,2] [,3] [,4]
2. [1,] 4.242641 0.000000 0.000000 0.000000
3. [2,] 5.185450 6.565905 0.000000 0.000000
4. [3,] 12.727922 3.046038 1.649742 0.000000
5. [4,] 9.899495 1.624554 1.849711 1.392621</lang>

## Racket

<lang racket>

1. 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]]))


</lang> Output: <lang racket> '#(#(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))


</lang>

## Raku

(formerly Perl 6)

Works with: Rakudo version 2015.12

<lang perl6>sub cholesky(@A) {

   my @L = @A »*» 0;
for ^@A -> $i {  for 0..$i -> $j { @L[$i][$j] = ($i == $j ?? &sqrt !! 1/@L[$j][$j] * * )( @A[$i][$j] - [+] (@L[$i;*] Z* @L[$j;*])[^$j] ); }

   }
return @L;


} .say for cholesky [

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


];

.say for cholesky [

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


];</lang>

## 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)

<lang rexx>/*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</lang>  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 <lang ring> 1. Project : Cholesky decomposition load "stdlib.ring" 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  </lang> 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 <lang ruby>require 'matrix' class Matrix  def symmetric? return false if not square? (0 ... row_size).each do |i| (0 .. i).each do |j| return false if self[i,j] != self[j,i] end end true end   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</lang>  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 <lang rust>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);  } </lang> 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 <lang 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)  }</lang> ## Scilab The Cholesky decomposition is builtin, and an upper triangular matrix is returned, such that$A=L^TL$. <lang scilab>a = [25 15 -5; 15 18 0; -5 0 11]; chol(a) ans =   5. 3. -1. 0. 3. 1. 0. 0. 3.  a = [18 22 54 42; 22 70 86 62;  54 86 174 134; 42 62 134 106];  chol(a) ans =   4.2426407 5.1854497 12.727922 9.8994949 0. 6.5659052 3.0460385 1.6245539 0. 0. 1.6497422 1.849711 0. 0. 0. 1.3926212</lang>  ## Seed7 <lang seed7>$ 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
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;</lang>


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

<lang ruby>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


}</lang>

Examples: <lang ruby>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(' ');


}</lang>

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

<lang 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 </lang>

## Stata

See Cholesky square-root decomposition in Stata help. <lang stata>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  |
+---------------------------------------------------------+</lang>


## Swift

Translation of: Rust

<lang swift>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)</lang>

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

<lang tcl>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 $l  }</lang> Demonstration code: <lang tcl>set test1 {  {25 15 -5} {15 18 0} {-5 0 11}  } puts [cholesky$test1] set test2 {

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


} puts [cholesky \$test2]</lang>

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.

<lang vb>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 </lang>

## Wren

Library: Wren-matrix
Library: Wren-fmt

<lang ecmascript>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()


}</lang>

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|


## zkl

Using the GNU Scientific Library: <lang zkl>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


}</lang>

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

<lang zkl>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


}</lang> <lang zkl>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));</lang> <lang zkl>fcn printM(m){ m.pump(Console.println,rowFmt) } fcn rowFmt(row){ ("%9.5f "*row.len()).fmt(row.xplode()) }</lang>

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

<lang zxbasic>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 1010 READ a,b 1020 DIM m(a,b) 1040 FOR i=1 TO a 1050 FOR j=1 TO b 1060 READ m(i,j) 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</lang>