Cholesky decomposition: Difference between revisions
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Line 8:
In a 3x3 example, we have to solve the following system of equations:
:<math>\begin{align}
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 \\
Line 25:
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{align}
</math>
Line 77 ⟶ 75:
;Note:
# The Cholesky decomposition of a [[Pascal matrix generation|Pascal]] upper-triangle matrix is the [[wp:Identity matrix|Identity matrix]] of the same size.
# The Cholesky decomposition of a Pascal symmetric matrix is the Pascal lower-triangle matrix of the same size.
=={{header|11l}}==
{{trans|Python}}
<syntaxhighlight 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))</syntaxhighlight>
{{out}}
<pre>
[[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]
]
</pre>
=={{header|Ada}}==
{{works with|Ada 2005}}
decomposition.ads:
<
generic
with package Matrix is new Ada.Numerics.Generic_Real_Arrays (<>);
Line 88 ⟶ 129:
procedure Decompose (A : Matrix.Real_Matrix; L : out Matrix.Real_Matrix);
end Decomposition;</
decomposition.adb:
<
package body Decomposition is
Line 123 ⟶ 164:
end loop;
end Decompose;
end Decomposition;</
Example usage:
<
with Ada.Text_IO;
with Decomposition;
Line 170 ⟶ 211:
Ada.Text_IO.Put_Line ("A:"); Print (Example_2);
Ada.Text_IO.Put_Line ("L:"); Print (L_2);
end Decompose_Example;</
{{out}}
<pre>Example 1:
A:
Line 194 ⟶ 234:
12.728 3.046 1.650 0.000
9.899 1.625 1.850 1.393</pre>
=={{header|ALGOL 68}}==
{{trans|C}} Note: This specimen retains the original [[#C|C]] coding style. [http://rosettacode.org/mw/index.php?title=Cholesky_decomposition&action=historysubmit&diff=107753&oldid=107752 diff]
Line 200 ⟶ 239:
{{works with|ALGOL 68G|Any - tested with release [http://sourceforge.net/projects/algol68/files/algol68g/algol68g-1.18.0/algol68g-1.18.0-9h.tiny.el5.centos.fc11.i386.rpm/download 1.18.0-9h.tiny].}}
{{wont work with|ELLA ALGOL 68|Any (with appropriate job cards) - tested with release [http://sourceforge.net/projects/algol68/files/algol68toc/algol68toc-1.8.8d/algol68toc-1.8-8d.fc9.i386.rpm/download 1.8-8d] - due to extensive use of '''format'''[ted] ''transput''.}}
<
MODE FIELD=LONG REAL;
Line 258 ⟶ 297:
MAT c2 = cholesky(m2);
print matrix(c2)
)</
{{out}}
<pre>
(( 5.00000, 0.00000, 0.00000),
Line 269 ⟶ 308:
( 9.89949, 1.62455, 1.84971, 1.39262))
</pre>
=={{header|Arturo}}==
<syntaxhighlight lang="arturo">cholesky: function [m][
result: array.of: @[size m, size m] 0.0
loop 0..dec size m\0 'i [
loop 0..i 'j [
s: 0.0
loop 0..j 'k ->
s: s + result\[i]\[k] * result\[j]\[k]
result\[i]\[j]: (i = j)? -> sqrt m\[i]\[i] - s
-> (1.0 // result\[j]\[j]) * (m\[i]\[j] - s)
]
]
return result
]
printMatrix: function [a]->
loop a 'b ->
print to [:string] .format:"8.5f" b
m1: @[
@[25.0, 15.0, neg 5.0]
@[15.0, 18.0, 0.0]
@[neg 5.0, 0.0, 11.0]
]
printMatrix cholesky m1
print ""
m2: [
[18.0, 22.0, 54.0, 42.0]
[22.0, 70.0, 86.0, 62.0]
[54.0, 86.0, 174.0, 134.0]
[42.0, 62.0, 134.0, 106.0]
]
printMatrix cholesky m2</syntaxhighlight>
{{out}}
<pre> 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</pre>
=={{header|ATS}}==
<syntaxhighlight lang="ats">
%{^
#include <math.h>
#include <float.h>
%}
#include "share/atspre_staload.hats"
macdef NAN = g0f2f ($extval (float, "NAN"))
macdef Zero = g0i2f 0
macdef One = g0i2f 1
(* The sqrt(3) function made part of the ‘g0float’ typekind series.
(The ats2-xprelude package will do this for you, but it is easy
to do if you are not using a lot of math functions. *)
extern fn {tk : tkind} g0float_sqrt : g0float tk -<> g0float tk
overload sqrt with g0float_sqrt
implement g0float_sqrt<fltknd> x = $extfcall (float, "sqrtf", x)
implement g0float_sqrt<dblknd> x = $extfcall (double, "sqrt", x)
implement g0float_sqrt<ldblknd> x = $extfcall (ldouble, "sqrtl", x)
(*------------------------------------------------------------------*)
(* A "very little matrix library" *)
typedef Matrix_Index_Map (m1 : int, n1 : int, m0 : int, n0 : int) =
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(int i1, int j1) -<cloref0>
[i0, j0 : pos | i0 <= m0; j0 <= n0]
@(int i0, int j0)
datatype Real_Matrix (tk : tkind,
m1 : int, n1 : int,
m0 : int, n0 : int) =
| Real_Matrix of (matrixref (g0float tk, m0, n0),
int m1, int n1, int m0, int n0,
Matrix_Index_Map (m1, n1, m0, n0))
typedef Real_Matrix (tk : tkind, m1 : int, n1 : int) =
[m0, n0 : pos] Real_Matrix (tk, m1, n1, m0, n0)
typedef Real_Vector (tk : tkind, m1 : int, n1 : int) =
[m1 == 1 || n1 == 1] Real_Matrix (tk, m1, n1)
typedef Real_Row (tk : tkind, n1 : int) = Real_Vector (tk, 1, n1)
typedef Real_Column (tk : tkind, m1 : int) = Real_Vector (tk, m1, 1)
extern fn {tk : tkind}
Real_Matrix_make_elt :
{m0, n0 : pos}
(int m0, int n0, g0float tk) -< !wrt >
Real_Matrix (tk, m0, n0, m0, n0)
extern fn {tk : tkind}
Real_Matrix_copy :
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -< !refwrt > Real_Matrix (tk, m1, n1)
extern fn {tk : tkind}
Real_Matrix_copy_to :
{m1, n1 : pos}
(Real_Matrix (tk, m1, n1), (* destination *)
Real_Matrix (tk, m1, n1)) -< !refwrt >
void
extern fn {}
Real_Matrix_dimension :
{tk : tkind}
{m1, n1 : pos}
Real_Matrix (tk, m1, n1) -<> @(int m1, int n1)
extern fn {tk : tkind}
Real_Matrix_get_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1) -< !ref > g0float tk
extern fn {tk : tkind}
Real_Matrix_set_at :
{m1, n1 : pos}
{i1, j1 : pos | i1 <= m1; j1 <= n1}
(Real_Matrix (tk, m1, n1), int i1, int j1, g0float tk) -< !refwrt >
void
extern fn {}
Real_Matrix_reflect_lower_triangle :
(* This operation makes every It is a change in how INDEXING
works. All the storage is still in the lower triangle. *)
{tk : tkind}
{n1 : pos}
{m0, n0 : pos}
Real_Matrix (tk, n1, n1, m0, n0) -<>
Real_Matrix (tk, n1, n1, m0, n0)
extern fn {tk : tkind}
Real_Matrix_fprint :
{m, n : pos}
(FILEref, Real_Matrix (tk, m, n)) -<1> void
overload copy with Real_Matrix_copy
overload copy_to with Real_Matrix_copy_to
overload dimension with Real_Matrix_dimension
overload [] with Real_Matrix_get_at
overload [] with Real_Matrix_set_at
overload reflect_lower_triangle with
Real_Matrix_reflect_lower_triangle
(*------------------------------------------------------------------*)
(* Implementation of the "very little matrix library" *)
implement {tk}
Real_Matrix_make_elt (m0, n0, elt) =
Real_Matrix (matrixref_make_elt<g0float tk> (i2sz m0, i2sz n0, elt),
m0, n0, m0, n0, lam (i1, j1) => @(i1, j1))
implement {}
Real_Matrix_dimension A =
case+ A of Real_Matrix (_, m1, n1, _, _, _) => @(m1, n1)
implement {tk}
Real_Matrix_get_at (A, i1, j1) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_get_at<g0float tk> (storage, pred i0, n0, pred j0)
end
implement {tk}
Real_Matrix_set_at (A, i1, j1, x) =
let
val+ Real_Matrix (storage, _, _, _, n0, index_map) = A
val @(i0, j0) = index_map (i1, j1)
in
matrixref_set_at<g0float tk> (storage, pred i0, n0, pred j0, x)
end
implement {}
Real_Matrix_reflect_lower_triangle {..} {n1} A =
let
typedef t = intBtwe (1, n1)
val+ Real_Matrix (storage, n1, _, m0, n0, index_map) = A
in
Real_Matrix (storage, n1, n1, m0, n0,
lam (i, j) =>
index_map ((if j <= i then i else j) : t,
(if j <= i then j else i) : t))
end
implement {tk}
Real_Matrix_copy A =
let
val @(m1, n1) = dimension A
val C = Real_Matrix_make_elt<tk> (m1, n1, A[1, 1])
val () = copy_to<tk> (C, A)
in
C
end
implement {tk}
Real_Matrix_copy_to (Dst, Src) =
let
val @(m1, n1) = dimension Src
prval [m1 : int] EQINT () = eqint_make_gint m1
prval [n1 : int] EQINT () = eqint_make_gint n1
var i : intGte 1
in
for* {i : pos | i <= m1 + 1} .<(m1 + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m1; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n1 + 1} .<(n1 + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n1; j := succ j)
Dst[i, j] := Src[i, j]
end
end
implement {tk}
Real_Matrix_fprint {m, n} (outf, A) =
let
val @(m, n) = dimension A
var i : intGte 1
in
for* {i : pos | i <= m + 1} .<(m + 1) - i>.
(i : int i) =>
(i := 1; i <> succ m; i := succ i)
let
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
typedef FILEstar = $extype"FILE *"
extern castfn FILEref2star : FILEref -<> FILEstar
val _ = $extfcall (int, "fprintf", FILEref2star outf,
"%16.6g", A[i, j])
in
end;
fprintln! (outf)
end
end
(*------------------------------------------------------------------*)
(* Cholesky-Banachiewicz, in place. See
https://en.wikipedia.org/w/index.php?title=Cholesky_decomposition&oldid=1149960985#The_Cholesky%E2%80%93Banachiewicz_and_Cholesky%E2%80%93Crout_algorithms
I would use Cholesky-Crout if my matrices were stored in column
major order. But it makes little difference. *)
extern fn {tk : tkind}
Real_Matrix_cholesky_decomposition :
(* Only the lower triangle is considered. *)
{n : pos}
Real_Matrix (tk, n, n) -< !refwrt > void
overload cholesky_decomposition with
Real_Matrix_cholesky_decomposition
implement {tk}
Real_Matrix_cholesky_decomposition {n} A =
let
val @(n, _) = dimension A
(* I arrange the nested loops somewhat differently from how it is
done in the Wikipedia article's C snippet. *)
fun
repeat {i, j : pos | j <= i; i <= n + 1} (* <-- allowed values *)
.<(n + 1) - i, i - j>. (* <-- proof of termination *)
(i : int i, j : int j) :<!refwrt> void =
if i = n + 1 then
() (* All done. *)
else
let
fun
_sum {k : pos | k <= j} .<j - k>.
(x : g0float tk, k : int k) :<!refwrt> g0float tk =
if k = j then
x
else
_sum (x + (A[i, k] * A[j, k]), succ k)
val sum = _sum (Zero, 1)
in
if j = i then
begin
A[i, j] := sqrt (A[i, i] - sum);
repeat (succ i, 1)
end
else
begin
A[i, j] := (One / A[j, j]) * (A[i, j] - sum);
repeat (i, succ j)
end
end
in
repeat (1, 1)
end
(*------------------------------------------------------------------*)
fn {tk : tkind} (* We like Fortran, so COLUMN major here. *)
column_major_list_to_square_matrix
{n : pos}
(n : int n,
lst : list (g0float tk, n * n))
: Real_Matrix (tk, n, n) =
let
#define :: list_cons
prval () = mul_gte_gte_gte {n, n} ()
val A = Real_Matrix_make_elt (n, n, NAN)
val lstref : ref (List0 (g0float tk)) = ref lst
var j : intGte 1
in
for* {j : pos | j <= n + 1} .<(n + 1) - j>.
(j : int j) =>
(j := 1; j <> succ n; j := succ j)
let
var i : intGte 1
in
for* {i : pos | i <= n + 1} .<(n + 1) - i>.
(i : int i) =>
(i := 1; i <> succ n; i := succ i)
case- !lstref of
| hd :: tl =>
begin
A[i, j] := hd;
!lstref := tl
end
end;
A
end
implement
main0 () =
let
val _A =
column_major_list_to_square_matrix
(3, $list (25.0, 15.0, ~5.0,
0.0, 18.0, 0.0,
0.0, 0.0, 11.0))
val A = reflect_lower_triangle _A
and B = copy _A
val () =
begin
cholesky_decomposition B;
print! ("\nThe Cholesky decomposition of\n\n");
Real_Matrix_fprint (stdout_ref, A);
print! ("is\n");
Real_Matrix_fprint (stdout_ref, B)
end
val _A =
column_major_list_to_square_matrix
(4, $list (18.0, 22.0, 54.0, 42.0,
0.0, 70.0, 86.0, 62.0,
0.0, 0.0, 174.0, 134.0,
0.0, 0.0, 0.0, 106.0))
val A = reflect_lower_triangle _A
and B = copy _A
val () =
begin
cholesky_decomposition B;
print! ("\nThe Cholesky decomposition of\n\n");
Real_Matrix_fprint (stdout_ref, A);
print! ("is\n");
Real_Matrix_fprint (stdout_ref, B)
end
in
println! ()
end
(*------------------------------------------------------------------*)
</syntaxhighlight>
{{out}}
<pre>$ patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW cholesky_decomposition_task.dats -lgc -lm && ./a.out
The Cholesky decomposition of
25 15 -5
15 18 0
-5 0 11
is
5 0 0
3 3 0
-1 1 3
The Cholesky decomposition of
18 22 54 42
22 70 86 62
54 86 174 134
42 62 134 106
is
4.24264 0 0 0
5.18545 6.56591 0 0
12.7279 3.04604 1.64974 0
9.89949 1.62455 1.84971 1.39262
</pre>
=={{header|AutoHotkey}}==
<syntaxhighlight 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,") "]"
}</syntaxhighlight>
Examples:<syntaxhighlight 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</syntaxhighlight>
{{out}}
<pre>[[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]]
----</pre>
=={{header|BBC BASIC}}==
{{works with|BBC BASIC for Windows}}
<syntaxhighlight 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</syntaxhighlight>
'''Output:'''
<pre>
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
</pre>
=={{header|C}}==
<
#include <stdlib.h>
#include <math.h>
Line 321 ⟶ 891:
return 0;
}</
{{out}}
<pre>5.00000 0.00000 0.00000
3.00000 3.00000 0.00000
Line 331 ⟶ 901:
12.72792 3.04604 1.64974 0.00000
9.89949 1.62455 1.84971 1.39262</pre>
=={{header|C sharp|C#}}==
<syntaxhighlight 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;
}
}
}</syntaxhighlight>
{{out}}
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,
=={{header|C++}}==
<syntaxhighlight lang="cpp">#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <vector>
template <typename scalar_type> class matrix {
public:
matrix(size_t rows, size_t columns)
: rows_(rows), columns_(columns), elements_(rows * columns) {}
matrix(size_t rows, size_t columns, scalar_type value)
: rows_(rows), columns_(columns), elements_(rows * columns, value) {}
matrix(size_t rows, size_t columns,
const std::initializer_list<std::initializer_list<scalar_type>>& values)
: rows_(rows), columns_(columns), elements_(rows * columns) {
assert(values.size() <= rows_);
size_t i = 0;
for (const auto& row : values) {
assert(row.size() <= columns_);
std::copy(begin(row), end(row), &elements_[i]);
i += columns_;
}
}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
const scalar_type& operator()(size_t row, size_t column) const {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
scalar_type& operator()(size_t row, size_t column) {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
private:
size_t rows_;
size_t columns_;
std::vector<scalar_type> elements_;
};
template <typename scalar_type>
void print(std::ostream& out, const matrix<scalar_type>& a) {
size_t rows = a.rows(), columns = a.columns();
out << std::fixed << std::setprecision(5);
for (size_t row = 0; row < rows; ++row) {
for (size_t column = 0; column < columns; ++column) {
if (column > 0)
out << ' ';
out << std::setw(9) << a(row, column);
}
out << '\n';
}
}
template <typename scalar_type>
matrix<scalar_type> cholesky_factor(const matrix<scalar_type>& input) {
assert(input.rows() == input.columns());
size_t n = input.rows();
matrix<scalar_type> result(n, n);
for (size_t i = 0; i < n; ++i) {
for (size_t k = 0; k < i; ++k) {
scalar_type value = input(i, k);
for (size_t j = 0; j < k; ++j)
value -= result(i, j) * result(k, j);
result(i, k) = value/result(k, k);
}
scalar_type value = input(i, i);
for (size_t j = 0; j < i; ++j)
value -= result(i, j) * result(i, j);
result(i, i) = std::sqrt(value);
}
return result;
}
void print_cholesky_factor(const matrix<double>& matrix) {
std::cout << "Matrix:\n";
print(std::cout, matrix);
std::cout << "Cholesky factor:\n";
print(std::cout, cholesky_factor(matrix));
}
int main() {
matrix<double> matrix1(3, 3,
{{25, 15, -5},
{15, 18, 0},
{-5, 0, 11}});
print_cholesky_factor(matrix1);
matrix<double> matrix2(4, 4,
{{18, 22, 54, 42},
{22, 70, 86, 62},
{54, 86, 174, 134},
{42, 62, 134, 106}});
print_cholesky_factor(matrix2);
return 0;
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Clojure}}==
{{trans|Python}}
<syntaxhighlight 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))))</syntaxhighlight>
Example:
<syntaxhighlight 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]]</syntaxhighlight>
=={{header|Common Lisp}}==
<
;; for a positive-definite, symmetric nxn matrix A.
(defun chol (A)
Line 361 ⟶ 1,209:
;; Return the calculated matrix L.
L))</
<
(setf A (make-array '(3 3) :initial-contents '((25 15 -5) (15 18 0) (-5 0 11))))
(chol A)
#2A((5.0 0 0)
(3.0 3.0 0)
(-1.0 1.0 3.0))</
<
(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)
Line 376 ⟶ 1,224:
(5.18545 6.565905 0 0)
(12.727922 3.0460374 1.6497375 0)
(9.899495 1.6245536 1.849715 1.3926151))</
<
;; 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)
(setq x (list (/ (car cm) (caar l))) i 0)
(dolist (cl (cdr l))
;; where *v is the scalar product defined as
(defun *v (v1 v2) (reduce #'+ (mapcar #'* v1 v2)))</
<
CL-USER> (setf a '((25 15 -5) (15 18 0) (-5 0 11)))
((25 15 -5) (15 18 0) (-5 0 11))
Line 399 ⟶ 1,247:
3 3 0
-1 1 3
NIL</
<
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))
Line 411 ⟶ 1,259:
12.72792 3.04604 1.64974 0.00000
9.89950 1.62455 1.84971 1.39262
NIL</
=={{header|D}}==
<
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) {
L[i][j] = (i == j) ? (A[i][i] - t) ^^ 0.5 :
(1.0 / L[j][j] * (A[i][j] - t));
Line 430 ⟶ 1,277:
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);
}</syntaxhighlight>
{{out}}
<pre> 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</pre>
=={{header|Delphi}}==
See [[#Pascal|Pascal]].
=={{header|DWScript}}==
{{Trans|C}}
<syntaxhighlight 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);</syntaxhighlight>
=={{header|F_Sharp|F#}}==
<syntaxhighlight 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
</syntaxhighlight>
{{out}}
<pre>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,
</pre>
=={{header|Fantom}}==
<syntaxhighlight 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]])
}
}</syntaxhighlight>
{{out}}
<pre>
[[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]]
</pre>
=={{header|Fortran}}==
<syntaxhighlight 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</syntaxhighlight>
{{out}}
<pre>
5.0 0.0 0.0
3.0 3.0 0.0
-1.0 1.0 3.0
</pre>
=={{header|FreeBASIC}}==
{{trans|BBC BASIC}}
<syntaxhighlight 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</syntaxhighlight>
{{out}}
<pre> 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</pre>
=={{header|Frink}}==
Frink's package [https://frinklang.org/fsp/colorize.fsp?f=Matrix.frink Matrix.frink] contains routines for Cholesky-Crout decomposition of a square Hermitian matrix (which can be real or complex.) This code is adapted from that more powerful class to work on raw 2-dimensional arrays. This also demonstrates Frink's layout routines.
<syntaxhighlight lang="frink">Cholesky[array] :=
{
n = length[array]
L = new array[[n,n], 0]
for j = 0 to n-1
{
sum = 0
for k = 0 to j-1
sum = sum + (L@j@k)^2
L@j@j = sqrt[array@j@j - sum]
for i = j+1 to n-1
{
sum = 0
for k = 0 to j-1
sum = sum + L@i@k * L@j@k
L@i@j = (1 / L@j@j * (array@i@j -sum))
}
}
return L
}
A = [[ 25, 15, -5],
[ 15, 18, 0],
[ -5, 0, 11]]
println[formatTable[[[formatMatrix[A], "->", formatMatrix[Cholesky[A]]]]]]
B = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]]
println[formatTable[[[formatMatrix[B], "->", formatMatrix[formatFix[Cholesky[B], 1, 5]]]]]]</syntaxhighlight>
{{out}}
<pre>
┌ ┐ ┌ ┐
│25 15 -5│ │ 5 0 0│
│ │ │ │
│15 18 0│ -> │ 3 3 0│
│ │ │ │
│-5 0 11│ │-1 1 3│
└ ┘ └ ┘
┌ ┐ ┌ ┐
│18 22 54 42│ │ 4.24264 0.00000 0.00000 0.00000│
│ │ │ │
│22 70 86 62│ │ 5.18545 6.56591 0.00000 0.00000│
│ │ -> │ │
│54 86 174 134│ │12.72792 3.04604 1.64974 0.00000│
│ │ │ │
│42 62 134 106│ │ 9.89949 1.62455 1.84971 1.39262│
└ ┘ └ ┘
</pre>
=={{header|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.
<syntaxhighlight 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()
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
===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.
<syntaxhighlight 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:")
}</syntaxhighlight>
{{out}}
<pre>
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)]
</pre>
===Library gonum/mat===
<syntaxhighlight 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,
}))
}</syntaxhighlight>
{{out}}
<pre>
⎡ 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⎦
</pre>
===Library go.matrix===
<syntaxhighlight 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)
}</syntaxhighlight>
Output:
<pre>
A:
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}
</pre>
=={{header|Groovy}}==
{{Trans|Java}}
<syntaxhighlight 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
}</syntaxhighlight>
Test:
<syntaxhighlight lang="groovy">def test1 = [[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]]
def test2 = [
[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)] }
}</syntaxhighlight>
{{out}}
<pre>[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]</pre>
=={{header|Haskell}}==
We use the [http://en.wikipedia.org/wiki/Cholesky_decomposition#The_Cholesky.E2.80.93Banachiewicz_and_Cholesky.E2.80.93Crout_algorithms Cholesky–Banachiewicz algorithm] described in the Wikipedia article.
For more serious numerical analysis there is a Cholesky decomposition function in the [http://hackage.haskell.org/package/hmatrix hmatrix package].
The Cholesky module:
<syntaxhighlight 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)</syntaxhighlight>
The main module:
<syntaxhighlight 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</syntaxhighlight>
<b>output:</b>
<pre>
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
</pre>
===With Numeric.LinearAlgebra===
<syntaxhighlight lang="haskell">import 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
</syntaxhighlight>
{{out}}
<pre>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 ]</pre>
=={{header|Icon}} and {{header|Unicon}}==
<syntaxhighlight lang="icon">procedure cholesky (array)
result := make_square_array (*array)
every (i := 1 to *array) do {
Line 497 ⟶ 2,225:
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</syntaxhighlight>
{{out}}
<pre>
Input:
Line 521 ⟶ 2,247:
9.899494937 1.624553864 1.849711005 1.392621248
</pre>
=={{header|Idris}}==
'''works with Idris 0.10'''
'''Solution:'''
<syntaxhighlight 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"
</syntaxhighlight>
{{out}}
<pre>
[[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]]
</pre>
=={{header|J}}==
'''Solution:'''
<
h =: +@|: NB. conjugate transpose
Line 533 ⟶ 2,337:
%:A
else.
L0=. cholesky X
L1=. cholesky Z-(T=.(h Y) mp %.X) mp Y
L0,(T mp L0),.L1
end.
)</
See [[j:Essays/
{{out|Examples}}
<syntaxhighlight 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
Line 553 ⟶ 2,355:
5.18545 6.56591 0 0
12.7279 3.04604 1.64974 0
9.89949 1.62455 1.84971 1.39262</
'''Using `math/lapack` addon'''
<syntaxhighlight 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</syntaxhighlight>
=={{header|Java}}==
{{works with|Java|1.5+}}
<
public class Cholesky {
Line 587 ⟶ 2,400:
System.out.println(Arrays.deepToString(chol(test2)));
}
}</
{{out}}
<pre>[[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]]</pre>
=={{header|JavaScript}}==
<syntaxhighlight 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));
</syntaxhighlight>
{{output}}
<pre>
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]
</pre>
=={{header|jq}}==
{{Works with|jq|1.4}}
'''Infrastructure''':
<syntaxhighlight 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 ;
# Print a matrix neatly, each cell ideally occupying n spaces,
# but without truncation
def neatly(n):
def right: tostring | ( " " * (n-length) + .);
. as $in
| length as $length
| reduce range (0; $length) as $i
(""; . + reduce range(0; $length) as $j
(""; "\(.) \($in[$i][$j] | right )" ) + "\n" ) ;
def is_square:
type == "array" and (map(type == "array") | all) and
length == 0 or ( (.[0]|length) as $l | map(length == $l) | all) ;
# This implementation of is_symmetric/0 uses a helper function that circumvents
# limitations of jq 1.4:
def is_symmetric:
# [matrix, i,j, len]
def test:
if .[1] > .[3] then true
elif .[1] == .[2] then [ .[0], .[1] + 1, 0, .[3]] | test
elif .[0][.[1]][.[2]] == .[0][.[2]][.[1]]
then [ .[0], .[1], .[2]+1, .[3]] | test
else false
end;
if is_square|not then false
else [ ., 0, 0, length ] | test
end ;
</syntaxhighlight>'''Cholesky Decomposition''':<syntaxhighlight 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 ;</syntaxhighlight>
'''Task 1''':
[[25,15,-5],[15,18,0],[-5,0,11]] | cholesky_factor
{{Out}}
[[5,0,0],[3,3,0],[-1,1,3]]
'''Task 2''':
[[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]] | cholesky_factor | neatly(20)
{{Out}}
<syntaxhighlight 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</syntaxhighlight>
=={{header|Julia}}==
Julia's strong linear algebra support includes Cholesky decomposition.
<syntaxhighlight 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))
</syntaxhighlight>
{{out}}
<pre>
[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]
</pre>
=={{header|Kotlin}}==
{{trans|C}}
<syntaxhighlight 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)
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Lobster}}==
{{trans|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.
<syntaxhighlight 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])
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|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.
<syntaxhighlight 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]</syntaxhighlight>
=={{header|Mathematica}} / {{header|Wolfram Language}}==
<syntaxhighlight lang="mathematica">CholeskyDecomposition[{{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}]</syntaxhighlight>
Without the use of built-in functions, making use of memoization:
<syntaxhighlight lang="mathematica">chol[A_] :=
Module[{L},
L[k_, k_] := L[k, k] = Sqrt[A[[k, k]] - Sum[L[k, j]^2, {j, 1, k-1}]];
L[i_, k_] := L[i, k] = L[k, k]^-1 (A[[i, k]] - Sum[L[i, j] L[k, j], {j, 1, k-1}]);
PadRight[Table[L[i, j], {i, Length[A]}, {j, i}]]
]</syntaxhighlight>
=={{header|MATLAB}} / {{header|Octave}}==
The cholesky decomposition chol() is an internal function
<syntaxhighlight 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')
</syntaxhighlight>
{{out}}
<pre> > [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
</pre>
=={{header|Maxima}}==
<syntaxhighlight 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])</syntaxhighlight>
=={{header|Nim}}==
{{trans|C}}
<syntaxhighlight 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:
line.addSep(" ", 0)
line.add fmt"{c:8.5f}"
result.add line & '\n'
let m1 = [[25.0, 15.0, -5.0],
[15.0, 18.0, 0.0],
[-5.0, 0.0, 11.0]]
echo cholesky(m1)
let m2 = [[18.0, 22.0, 54.0, 42.0],
[22.0, 70.0, 86.0, 62.0],
[54.0, 86.0, 174.0, 134.0],
[42.0, 62.0, 134.0, 106.0]]
echo cholesky(m2)</syntaxhighlight>
{{out}}
<pre> 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</pre>
=={{header|Objeck}}==
{{trans|C}}
<syntaxhighlight 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;
}
}
</syntaxhighlight>
<pre>
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
</pre>
=={{header|OCaml}}==
<syntaxhighlight 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|] |];</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|ooRexx}}==
{{trans|REXX}}
<syntaxhighlight 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.*/</syntaxhighlight>
=={{header|PARI/GP}}==
<syntaxhighlight 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
}</syntaxhighlight>
Output: (set displayed digits with: \p 5)
<pre>
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]
</pre>
=={{header|Pascal}}==
<syntaxhighlight 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.</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Perl}}==
<syntaxhighlight 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 };
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Phix}}==
{{trans|Sidef}}
<!--<syntaxhighlight lang="phix">(phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">cholesky</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">matrix</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">integer</span> <span style="color: #000000;">l</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">matrix</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">sequence</span> <span style="color: #000000;">chol</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">l</span><span style="color: #0000FF;">),</span><span style="color: #000000;">l</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">row</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">l</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">col</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">row</span> <span style="color: #008080;">do</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">matrix</span><span style="color: #0000FF;">[</span><span style="color: #000000;">row</span><span style="color: #0000FF;">][</span><span style="color: #000000;">col</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">col</span> <span style="color: #008080;">do</span>
<span style="color: #000000;">x</span> <span style="color: #0000FF;">-=</span> <span style="color: #000000;">chol</span><span style="color: #0000FF;">[</span><span style="color: #000000;">row</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">chol</span><span style="color: #0000FF;">[</span><span style="color: #000000;">col</span><span style="color: #0000FF;">][</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #000000;">chol</span><span style="color: #0000FF;">[</span><span style="color: #000000;">row</span><span style="color: #0000FF;">][</span><span style="color: #000000;">col</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">row</span> <span style="color: #0000FF;">==</span> <span style="color: #000000;">col</span> <span style="color: #0000FF;">?</span> <span style="color: #7060A8;">sqrt</span><span style="color: #0000FF;">(</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">:</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">/</span><span style="color: #000000;">chol</span><span style="color: #0000FF;">[</span><span style="color: #000000;">col</span><span style="color: #0000FF;">][</span><span style="color: #000000;">col</span><span style="color: #0000FF;">])</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">return</span> <span style="color: #000000;">chol</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #7060A8;">ppOpt</span><span style="color: #0000FF;">({</span><span style="color: #004600;">pp_Nest</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cholesky</span><span style="color: #0000FF;">({{</span> <span style="color: #000000;">25</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">15</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">5</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">15</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">18</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span> <span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">5</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">11</span> <span style="color: #0000FF;">}}))</span>
<span style="color: #7060A8;">pp</span><span style="color: #0000FF;">(</span><span style="color: #000000;">cholesky</span><span style="color: #0000FF;">({{</span> <span style="color: #000000;">18</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">22</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">54</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">42</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">22</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">70</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">86</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">62</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">54</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">86</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">174</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">134</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span> <span style="color: #000000;">42</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">62</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">134</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">106</span><span style="color: #0000FF;">}}))</span>
<!--</syntaxhighlight>-->
{{out}}
<pre>
{{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}}
</pre>
=={{header|PicoLisp}}==
<
(load "@lib/math.l")
Line 605 ⟶ 3,216:
(set (nth L I J)
(if (= I J)
(sqrt
(*/ S 1.0 (get L J J)) ) ) ) ) )
(for R L
(for N R (prin (align 9 (round N 5))))
(prinl) ) ) )</
Test:
<
'((25.0 15.0 -5.0) (15.0 18.0 0) (-5.0 0 11.0)) )
Line 621 ⟶ 3,232:
(22.0 70.0 86.0 62.0)
(54.0 86.0 174.0 134.0)
(42.0 62.0 134.0 106.0) ) )</
{{out}}
<pre> 5.00000 0.00000 0.00000
3.00000 3.00000 0.00000
Line 631 ⟶ 3,242:
12.72792 3.04604 1.64974 0.00000
9.89949 1.62455 1.84971 1.39262</pre>
=={{header|PL/I}}==
<syntaxhighlight 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;</syntaxhighlight>
ACTUAL RESULTS:-
<pre>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
</pre>
=={{header|PowerShell}}==
<syntaxhighlight 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)
</syntaxhighlight>
<b>Output:</b>
<pre>
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
</pre>
=={{header|Python}}==
===Python2.X version===
<syntaxhighlight lang="python">from __future__ import print_function
from pprint import pprint
from math import sqrt
def cholesky(A):
Line 640 ⟶ 3,401:
for j in xrange(i+1):
s = sum(L[i][k] * L[j][k] for k in xrange(j))
L[i][j] =
(1.0 / L[j][j] * (A[i][j] - s))
return L
if __name__ == "__main__":
m1 = [
[
[-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)</syntaxhighlight>
{{out}}
<pre>[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]]
[[4.
[5.
[12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0],
[9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]
</pre>
===Python3.X version using extra Python idioms===
Factors out accesses to <code>A[i], L[i], and L[j]</code> by creating <code>Ai, Li and Lj</code> respectively as well as using <code>enumerate</code> instead of <code>range(len(some_array))</code>.
<syntaxhighlight 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</syntaxhighlight>
{{out}}
(As above)
=={{header|q}}==
<syntaxhighlight 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)</syntaxhighlight>
{{out}}
<pre>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
</pre>
=={{header|R}}==
<syntaxhighlight lang="r">t(chol(matrix(c(25, 15, -5, 15, 18, 0, -5, 0, 11), nrow=3, ncol=3)))
# [,1] [,2] [,3]
# [1,] 5 0 0
# [2,] 3 3 0
# [3,] -1 1 3
t(chol(matrix(c(18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106), nrow=4, ncol=4)))
# [,1] [,2] [,3] [,4]
# [1,] 4.242641 0.000000 0.000000 0.000000
# [2,] 5.185450 6.565905 0.000000 0.000000
# [3,] 12.727922 3.046038 1.649742 0.000000
# [4,] 9.899495 1.624554 1.849711 1.392621</syntaxhighlight>
=={{header|Racket}}==
<syntaxhighlight lang="racket">
#lang racket
(require math)
(define (cholesky A)
(define mref matrix-ref)
(define n (matrix-num-rows A))
(define L (for/vector ([_ n]) (for/vector ([_ n]) 0)))
(define (set L i j x) (vector-set! (vector-ref L i) j x))
(define (ref L i j) (vector-ref (vector-ref L i) j))
(for* ([i n] [k n])
(set L i k
(cond
[(= i k)
(sqrt (- (mref A i i) (for/sum ([j k]) (sqr (ref L k j)))))]
[(> i k)
(/ (- (mref A i k) (for/sum ([j k]) (* (ref L i j) (ref L k j))))
(ref L k k))]
[else 0])))
L)
(cholesky (matrix [[25 15 -5]
[15 18 0]
[-5 0 11]]))
(cholesky (matrix [[18 22 54 42]
[22 70 86 62]
[54 86 174 134]
[42 62 134 106]]))
</syntaxhighlight>
Output:
<syntaxhighlight 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))
</syntaxhighlight>
=={{header|Raku}}==
(formerly Perl 6)
<syntaxhighlight lang="raku" line>sub cholesky(@A) {
my @L = @A »×» 0;
for ^@A -> \i {
for 0..i -> \j {
@L[i;j] = (i == j ?? &sqrt !! 1/@L[j;j] × * )\ # select function
(@A[i;j] - [+] (@L[i;*] Z× @L[j;*])[^j]) # provide value
}
}
@L
}
.fmt('%3d').say for cholesky [
[25],
[15, 18],
[-5, 0, 11],
];
say '';
.fmt('%6.3f').say for cholesky [
[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106],
];</syntaxhighlight>
{{out}}
<pre> 5
3 3
-1 1 3
4.243 0.000 0.000 0.000
5.185 6.566 0.000 0.000
12.728 3.046 1.650 0.000
9.899 1.625 1.850 1.393</pre>
=={{header|REXX}}==
If trailing zeros are wanted after the decimal point for values of zero (0), the <big><big>'''/ 1'''</big></big> (a division by unity performs
<br>REXX number normalization) can be removed from the line (number 40) which contains the source statement:
::::: <b> z=z right( format(@.row.col, , dPlaces) / 1, width) </b>
<syntaxhighlight 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</syntaxhighlight>
{{out|output}}
<pre>
═══════════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
</pre>
=={{header|Ring}}==
<syntaxhighlight lang="ring">
# 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
</syntaxhighlight>
Output:
<pre>
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
</pre>
=={{header|Ruby}}==
<syntaxhighlight lang="ruby">require 'matrix'
class Matrix
def cholesky_factor
raise ArgumentError, "must provide symmetric matrix" unless symmetric?
l = Array.new(row_size) {Array.new(row_size, 0)}
(0 ... row_size).each do |k|
(0 ... row_size).each do |i|
if i == k
sum = (0 .. k-1).inject(0.0) {|sum, j| sum + l[k][j] ** 2}
val = Math.sqrt(self[k,k] - sum)
l[k][k] = val
elsif i > k
sum = (0 .. k-1).inject(0.0) {|sum, j| sum + l[i][j] * l[k][j]}
val = (self[k,i] - sum) / l[k][k]
l[i][k] = val
end
end
end
Matrix[*l]
end
end
puts Matrix[[25,15,-5],[15,18,0],[-5,0,11]].cholesky_factor
puts Matrix[[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]].cholesky_factor</syntaxhighlight>
{{out}}
<pre>
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]]
</pre>
=={{header|Rust}}==
{{trans|C}}
<syntaxhighlight 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);
}
</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Scala}}==
<syntaxhighlight 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)
}</syntaxhighlight>
=={{header|Scilab}}==
The Cholesky decomposition is builtin, and an upper triangular matrix is returned, such that $A=L^TL$.
<syntaxhighlight 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</syntaxhighlight>
=={{header|Seed7}}==
<syntaxhighlight 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
write(num digits 5 lpad 8);
end for;
writeln;
end for;
end func;
const matrix: m1 is [] (
[] (25.0, 15.0, -5.0),
[] (15.0, 18.0, 0.0),
[] (-5.0, 0.0, 11.0));
const matrix: m2 is [] (
[] (18.0, 22.0, 54.0, 42.0),
[] (22.0, 70.0, 86.0, 62.0),
[] (54.0, 86.0, 174.0, 134.0),
[] (42.0, 62.0, 134.0, 106.0));
const proc: main is func
begin
writeMat(cholesky(m1));
writeln;
writeMat(cholesky(m2));
end func;</syntaxhighlight>
Output:
<pre>
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
</pre>
=={{header|Sidef}}==
{{trans|Perl}}
<syntaxhighlight 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
}</syntaxhighlight>
Examples:
<syntaxhighlight 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(' ');
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|Smalltalk}}==
<syntaxhighlight 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
</syntaxhighlight>
=={{header|Stata}}==
See [http://www.stata.com/help.cgi?mf_cholesky Cholesky square-root decomposition] in Stata help.
<syntaxhighlight 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 |
+---------------------------------------------------------+</syntaxhighlight>
=={{header|Swift}}==
{{trans|Rust}}
<syntaxhighlight 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)</syntaxhighlight>
{{out}}
<pre>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</pre>
=={{header|Tcl}}==
{{trans|Java}}
<
set m [llength $a]
set n [llength [lindex $a 0]]
Line 686 ⟶ 4,150:
}
return $l
}</
Demonstration code:
<
{25 15 -5}
{15 18 0}
Line 700 ⟶ 4,164:
{42 62 134 106}
}
puts [cholesky $test2]</
{{out}}
<pre>
{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}
</pre>
=={{header|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 <code>Option Base 0</code> 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 <code>Byte</code>-type variables to <code>Long</code>. 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 "<code>=Cholesky(A1:E5)</code>" and htting <code>Ctrl-Shift-Enter</code> will populate the target cells with the lower Cholesky decomposition.
<syntaxhighlight 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
</syntaxhighlight>
=={{header|V (Vlang)}}==
{{trans|go}}
<syntaxhighlight lang="v (vlang)">import math
// Symmetric and Lower use a packed representation that stores only
// the Lower triangle.
struct Symmetric {
order int
ele []f64
}
struct Lower {
mut:
order int
ele []f64
}
// Symmetric.print prints a square matrix from the packed representation,
// printing the upper triange as a transpose of the Lower.
fn (s Symmetric) print() {
mut row, mut diag := 1, 0
for i, e in s.ele {
print("${e:10.5f} ")
if i == diag {
for j, col := diag+row, row; col < s.order; j += col {
print("${s.ele[j]:10.5f} ")
col++
}
println('')
row++
diag += row
}
}
}
// Lower.print prints a square matrix from the packed representation,
// printing the upper triangle as all zeros.
fn (l Lower) print() {
mut row, mut diag := 1, 0
for i, e in l.ele {
print("${e:10.5f} ")
if i == diag {
for _ in row..l.order {
print("${0.0:10.5f} ")
}
println('')
row++
diag += row
}
}
}
// cholesky_lower returns the cholesky decomposition of a Symmetric real
// matrix. The matrix must be positive definite but this is not checked.
fn (a Symmetric) cholesky_lower() Lower {
mut l := Lower{a.order, []f64{len: a.ele.len}}
mut row, mut col := 1, 1
mut dr := 0 // index of diagonal element at end of row
mut dc := 0 // index of diagonal element at top of column
for i, e in a.ele {
if i < dr {
d := (e - l.ele[i]) / l.ele[dc]
l.ele[i] = d
mut ci, mut cx := col, dc
for j := i + 1; j <= dr; j++ {
cx += ci
ci++
l.ele[j] += d * l.ele[cx]
}
col++
dc += col
} else {
l.ele[i] = math.sqrt(e - l.ele[i])
row++
dr += row
col = 1
dc = 0
}
}
return l
}
fn main() {
demo(Symmetric{3, [
f64(25),
15, 18,
-5, 0, 11]})
demo(Symmetric{4, [
f64(18),
22, 70,
54, 86, 174,
42, 62, 134, 106]})
}
fn demo(a Symmetric) {
println("A:")
a.print()
println("L:")
a.cholesky_lower().print()
}</syntaxhighlight>
{{out}}
<pre>
A:
25.00000 15.00000 -5.00000
15.00000 18.00000 0.00000
-5.00000 0.00000 11.00000
L:
5.00000 0.00000 0.00000
3.00000 3.00000 0.00000
-1.00000 1.00000 3.00000
A:
18.00000 22.00000 54.00000 42.00000
22.00000 70.00000 86.00000 62.00000
54.00000 86.00000 174.00000 134.00000
42.00000 62.00000 134.00000 106.00000
L:
4.24264 0.00000 0.00000 0.00000
5.18545 6.56591 0.00000 0.00000
12.72792 3.04604 1.64974 0.00000
9.89949 1.62455 1.84971 1.3926
</pre>
=={{header|Wren}}==
{{libheader|Wren-matrix}}
{{libheader|Wren-fmt}}
<syntaxhighlight lang="wren">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()
}</syntaxhighlight>
{{out}}
<pre>
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|
</pre>
=={{header|XPL0}}==
<syntaxhighlight lang "XPL0">real L(4*4);
func real Cholesky(A, N);
real A; int N;
real S;
int I, J, K;
[for I:= 0 to N*N-1 do L(I):= 0.;
for I:= 0 to N-1 do
for J:= 0 to I do
[S:= 0.;
for K:= 0 to J-1 do
S:= S + L(I*N+K) * L(J*N+K);
L(I*N+J):= if I = J then sqrt(A(I*N+I) - S)
else (1.0 / L(J*N+J) * (A(I*N+J) - S));
];
return L;
];
proc ShowMatrix(A, N);
real A; int N;
int I, J;
[for I:= 0 to N-1 do
[for J:= 0 to N-1 do
RlOut(0, A(I*N+J));
CrLf(0);
];
];
int N;
real M1, C1, M2, C2;
[N:= 3;
M1:= [25., 15., -5.,
15., 18., 0.,
-5., 0., 11.];
C1:= Cholesky(M1, N);
ShowMatrix(C1, N);
CrLf(0);
N:= 4;
M2:= [18., 22., 54., 42.,
22., 70., 86., 62.,
54., 86., 174., 134.,
42., 62., 134., 106.];
C2:= Cholesky(M2, N);
ShowMatrix(C2, N);
]</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|zkl}}==
Using the GNU Scientific Library:
<syntaxhighlight 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
}</syntaxhighlight>
{{out}}
<pre>
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
</pre>
{{out}}
<pre>
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
</pre>
Or, using lists:
{{trans|C}}
<syntaxhighlight 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
}</syntaxhighlight>
<syntaxhighlight 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));</syntaxhighlight>
<syntaxhighlight lang="zkl">fcn printM(m){ m.pump(Console.println,rowFmt) }
fcn rowFmt(row){ ("%9.5f "*row.len()).fmt(row.xplode()) }</syntaxhighlight>
{{out}}
<pre>
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
</pre>
=={{header|ZX Spectrum Basic}}==
{{trans|BBC_BASIC}}
<syntaxhighlight 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</syntaxhighlight>
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