Cholesky decomposition
Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose:
You are encouraged to solve this task according to the task description, using any language you may know.
is called the Cholesky factor of , and can be interpreted as a generalized square root of , as described in Cholesky decomposition.
In a 3x3 example, we have to solve the following system of equations:
We can see that for the diagonal elements () of there is a calculation pattern:
or in general:
For the elements below the diagonal (, where ) there is also a calculation pattern:
which can also be expressed in a general formula:
Task description
The task is to implement a routine which will return a lower Cholesky factor for every given symmetric, positive definite nxn matrix . You should then test it on the following two examples and include your output.
Example 1:
25 15 -5 5 0 0 15 18 0 --> 3 3 0 -5 0 11 -1 1 3
Example 2:
18 22 54 42 4.24264 0.00000 0.00000 0.00000 22 70 86 62 --> 5.18545 6.56591 0.00000 0.00000 54 86 174 134 12.72792 3.04604 1.64974 0.00000 42 62 134 106 9.89949 1.62455 1.84971 1.39262
- Note
- The Cholesky decomposition of a Pascal upper-triangle matrix is the Identity matrix of the same size.
- The Cholesky decomposition of a Pascal symmetric matrix is the Pascal lower-triangle matrix of the same size.
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))- Output:
[[5, 0, 0], [3, 3, 0], [-1, 1, 3]] [ [4.24264, 0, 0, 0] [5.18545, 6.56591, 0, 0] [12.7279, 3.04604, 1.64974, 0] [9.89949, 1.62455, 1.84971, 1.39262] ]
Ada
decomposition.ads:
with Ada.Numerics.Generic_Real_Arrays;
generic
with package Matrix is new Ada.Numerics.Generic_Real_Arrays (<>);
package Decomposition is
-- decompose a square matrix A by A = L * Transpose (L)
procedure Decompose (A : Matrix.Real_Matrix; L : out Matrix.Real_Matrix);
end Decomposition;
decomposition.adb:
with Ada.Numerics.Generic_Elementary_Functions;
package body Decomposition is
package Math is new Ada.Numerics.Generic_Elementary_Functions
(Matrix.Real);
procedure Decompose (A : Matrix.Real_Matrix; L : out Matrix.Real_Matrix) is
use type Matrix.Real_Matrix, Matrix.Real;
Order : constant Positive := A'Length (1);
S : Matrix.Real;
begin
L := (others => (others => 0.0));
for I in 0 .. Order - 1 loop
for K in 0 .. I loop
S := 0.0;
for J in 0 .. K - 1 loop
S := S +
L (L'First (1) + I, L'First (2) + J) *
L (L'First (1) + K, L'First (2) + J);
end loop;
-- diagonals
if K = I then
L (L'First (1) + K, L'First (2) + K) :=
Math.Sqrt (A (A'First (1) + K, A'First (2) + K) - S);
else
L (L'First (1) + I, L'First (2) + K) :=
1.0 / L (L'First (1) + K, L'First (2) + K) *
(A (A'First (1) + I, A'First (2) + K) - S);
end if;
end loop;
end loop;
end Decompose;
end Decomposition;
Example usage:
with Ada.Numerics.Real_Arrays;
with Ada.Text_IO;
with Decomposition;
procedure Decompose_Example is
package Real_Decomposition is new Decomposition
(Matrix => Ada.Numerics.Real_Arrays);
package Real_IO is new Ada.Text_IO.Float_IO (Float);
procedure Print (M : Ada.Numerics.Real_Arrays.Real_Matrix) is
begin
for Row in M'Range (1) loop
for Col in M'Range (2) loop
Real_IO.Put (M (Row, Col), 4, 3, 0);
end loop;
Ada.Text_IO.New_Line;
end loop;
end Print;
Example_1 : constant Ada.Numerics.Real_Arrays.Real_Matrix :=
((25.0, 15.0, -5.0),
(15.0, 18.0, 0.0),
(-5.0, 0.0, 11.0));
L_1 : Ada.Numerics.Real_Arrays.Real_Matrix (Example_1'Range (1),
Example_1'Range (2));
Example_2 : constant Ada.Numerics.Real_Arrays.Real_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));
L_2 : Ada.Numerics.Real_Arrays.Real_Matrix (Example_2'Range (1),
Example_2'Range (2));
begin
Real_Decomposition.Decompose (A => Example_1,
L => L_1);
Real_Decomposition.Decompose (A => Example_2,
L => L_2);
Ada.Text_IO.Put_Line ("Example 1:");
Ada.Text_IO.Put_Line ("A:"); Print (Example_1);
Ada.Text_IO.Put_Line ("L:"); Print (L_1);
Ada.Text_IO.New_Line;
Ada.Text_IO.Put_Line ("Example 2:");
Ada.Text_IO.Put_Line ("A:"); Print (Example_2);
Ada.Text_IO.Put_Line ("L:"); Print (L_2);
end Decompose_Example;
- Output:
Example 1: A: 25.000 15.000 -5.000 15.000 18.000 0.000 -5.000 0.000 11.000 L: 5.000 0.000 0.000 3.000 3.000 0.000 -1.000 1.000 3.000 Example 2: A: 18.000 22.000 54.000 42.000 22.000 70.000 86.000 62.000 54.000 86.000 174.000 134.000 42.000 62.000 134.000 106.000 L: 4.243 0.000 0.000 0.000 5.185 6.566 0.000 0.000 12.728 3.046 1.650 0.000 9.899 1.625 1.850 1.393
ALGOL 68
Note: This specimen retains the original C coding style. diff
#!/usr/local/bin/a68g --script #
MODE FIELD=LONG REAL;
PROC (FIELD)FIELD field sqrt = long sqrt;
INT field prec = 5;
FORMAT field fmt = $g(-(2+1+field prec),field prec)$;
MODE MAT = [0,0]FIELD;
PROC cholesky = (MAT a) MAT:(
[UPB a, 2 UPB a]FIELD l;
FOR i FROM LWB a TO UPB a DO
FOR j FROM 2 LWB a TO i DO
FIELD s := 0;
FOR k FROM 2 LWB a TO j-1 DO
s +:= l[i,k] * l[j,k]
OD;
l[i,j] := IF i = j
THEN field sqrt(a[i,i] - s)
ELSE 1.0 / l[j,j] * (a[i,j] - s) FI
OD;
FOR j FROM i+1 TO 2 UPB a DO
l[i,j]:=0 # Not required if matrix is declared as triangular #
OD
OD;
l
);
PROC print matrix v1 =(MAT a)VOID:(
FOR i FROM LWB a TO UPB a DO
FOR j FROM 2 LWB a TO 2 UPB a DO
printf(($g(-(2+1+field prec),field prec)$, a[i,j]))
OD;
printf($l$)
OD
);
PROC print matrix =(MAT a)VOID:(
FORMAT vector fmt = $"("f(field fmt)n(2 UPB a-2 LWB a)(", " f(field fmt))")"$;
FORMAT matrix fmt = $"("f(vector fmt)n( UPB a- LWB a)(","lxf(vector fmt))")"$;
printf((matrix fmt, a))
);
main: (
MAT m1 = ((25, 15, -5),
(15, 18, 0),
(-5, 0, 11));
MAT c1 = cholesky(m1);
print matrix(c1);
printf($l$);
MAT m2 = ((18, 22, 54, 42),
(22, 70, 86, 62),
(54, 86, 174, 134),
(42, 62, 134, 106));
MAT c2 = cholesky(m2);
print matrix(c2)
)- Output:
(( 5.00000, 0.00000, 0.00000), ( 3.00000, 3.00000, 0.00000), (-1.00000, 1.00000, 3.00000)) (( 4.24264, 0.00000, 0.00000, 0.00000), ( 5.18545, 6.56591, 0.00000, 0.00000), (12.72792, 3.04604, 1.64974, 0.00000), ( 9.89949, 1.62455, 1.84971, 1.39262))
Arturo
cholesky: function [m][
result: array.of: @[size m, size m] 0.0
loop 0..dec size m\0 'i [
loop 0..i 'j [
s: 0.0
loop 0..j 'k ->
s: s + result\[i]\[k] * result\[j]\[k]
result\[i]\[j]: (i = j)? -> sqrt m\[i]\[i] - s
-> (1.0 // result\[j]\[j]) * (m\[i]\[j] - s)
]
]
return result
]
printMatrix: function [a]->
loop a 'b ->
print to [:string] .format:"8.5f" b
m1: @[
@[25.0, 15.0, neg 5.0]
@[15.0, 18.0, 0.0]
@[neg 5.0, 0.0, 11.0]
]
printMatrix cholesky m1
print ""
m2: [
[18.0, 22.0, 54.0, 42.0]
[22.0, 70.0, 86.0, 62.0]
[54.0, 86.0, 174.0, 134.0]
[42.0, 62.0, 134.0, 106.0]
]
printMatrix cholesky m2
- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
ATS
%{^
#include <math.h>
#include <float.h>
%}
#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
(*------------------------------------------------------------------*)- Output:
$ patscc -std=gnu2x -g -O2 -DATS_MEMALLOC_GCBDW cholesky_decomposition_task.dats -lgc -lm && ./a.out
The Cholesky decomposition of
25 15 -5
15 18 0
-5 0 11
is
5 0 0
3 3 0
-1 1 3
The Cholesky decomposition of
18 22 54 42
22 70 86 62
54 86 174 134
42 62 134 106
is
4.24264 0 0 0
5.18545 6.56591 0 0
12.7279 3.04604 1.64974 0
9.89949 1.62455 1.84971 1.39262
AutoHotkey
Cholesky_Decomposition(A){
L := [], n := A.Count()
L[1,1] := Sqrt(A[1,1])
loop % n {
k := A_Index
loop % n-1 {
i := A_Index+1
Sigma := 0, j := 0
while (++j <= k-1)
Sigma += L[i, j] * L[k, j]
L[i, k] := (A[i, k] - Sigma) / L[k, k]
Sigma := 0, j := 0
while (++j <= k-1)
Sigma += (L[k, j])**2
L[k, k] := Sqrt(A[k, k] - Sigma)
}
}
loop % n{
k := A_Index
loop % n
L[k, A_Index] := L[k, A_Index] ? L[k, A_Index] : 0
}
return L
}
ShowMatrix(L){
for r, obj in L{
row := ""
for c, v in obj
row .= Format("{:.3f}", v) ", "
output .= "[" trim(row, ", ") "]`n,"
}
return "[" Trim(output, "`n,") "]"
}
Examples:
A := [[25, 15, -5]
, [15, 18, 0]
, [-5, 0 , 11]]
L1 := Cholesky_Decomposition(A)
A := [[18, 22, 54, 42]
, [22, 70, 86, 62]
, [54, 86, 174, 134]
, [42, 62, 134, 106]]
L2 := Cholesky_Decomposition(A)
MsgBox % Result := ShowMatrix(L1) "`n----`n" ShowMatrix(L2) "`n----"
return
- Output:
[[5.000, 0.000, 0.000] ,[3.000, 3.000, 0.000] ,[-1.000, 1.000, 3.000]] ---- [[4.243, 0.000, 0.000, 0.000] ,[5.185, 6.566, 0.000, 0.000] ,[12.728, 3.046, 1.650, 0.000] ,[9.899, 1.625, 1.850, 1.393]] ----
BBC BASIC
DIM m1(2,2)
m1() = 25, 15, -5, \
\ 15, 18, 0, \
\ -5, 0, 11
PROCcholesky(m1())
PROCprint(m1())
PRINT
@% = &2050A
DIM m2(3,3)
m2() = 18, 22, 54, 42, \
\ 22, 70, 86, 62, \
\ 54, 86, 174, 134, \
\ 42, 62, 134, 106
PROCcholesky(m2())
PROCprint(m2())
END
DEF PROCcholesky(a())
LOCAL i%, j%, k%, l(), s
DIM l(DIM(a(),1),DIM(a(),2))
FOR i% = 0 TO DIM(a(),1)
FOR j% = 0 TO i%
s = 0
FOR k% = 0 TO j%-1
s += l(i%,k%) * l(j%,k%)
NEXT
IF i% = j% THEN
l(i%,j%) = SQR(a(i%,i%) - s)
ELSE
l(i%,j%) = (a(i%,j%) - s) / l(j%,j%)
ENDIF
NEXT j%
NEXT i%
a() = l()
ENDPROC
DEF PROCprint(a())
LOCAL row%, col%
FOR row% = 0 TO DIM(a(),1)
FOR col% = 0 TO DIM(a(),2)
PRINT a(row%,col%);
NEXT
PRINT
NEXT row%
ENDPROC
Output:
5 0 0
3 3 0
-1 1 3
4.24264 0.00000 0.00000 0.00000
5.18545 6.56591 0.00000 0.00000
12.72792 3.04604 1.64974 0.00000
9.89949 1.62455 1.84971 1.39262
C
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
double *cholesky(double *A, int n) {
double *L = (double*)calloc(n * n, sizeof(double));
if (L == NULL)
exit(EXIT_FAILURE);
for (int i = 0; i < n; i++)
for (int j = 0; j < (i+1); j++) {
double s = 0;
for (int k = 0; k < j; k++)
s += L[i * n + k] * L[j * n + k];
L[i * n + j] = (i == j) ?
sqrt(A[i * n + i] - s) :
(1.0 / L[j * n + j] * (A[i * n + j] - s));
}
return L;
}
void show_matrix(double *A, int n) {
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
printf("%2.5f ", A[i * n + j]);
printf("\n");
}
}
int main() {
int n = 3;
double m1[] = {25, 15, -5,
15, 18, 0,
-5, 0, 11};
double *c1 = cholesky(m1, n);
show_matrix(c1, n);
printf("\n");
free(c1);
n = 4;
double m2[] = {18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106};
double *c2 = cholesky(m2, n);
show_matrix(c2, n);
free(c2);
return 0;
}
- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
C#
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace Cholesky
{
class Program
{
/// <summary>
/// This is example is written in C#, and compiles with .NET Framework 4.0
/// </summary>
/// <param name="args"></param>
static void Main(string[] args)
{
double[,] test1 = new double[,]
{
{25, 15, -5},
{15, 18, 0},
{-5, 0, 11},
};
double[,] test2 = new double[,]
{
{18, 22, 54, 42},
{22, 70, 86, 62},
{54, 86, 174, 134},
{42, 62, 134, 106},
};
double[,] chol1 = Cholesky(test1);
double[,] chol2 = Cholesky(test2);
Console.WriteLine("Test 1: ");
Print(test1);
Console.WriteLine("");
Console.WriteLine("Lower Cholesky 1: ");
Print(chol1);
Console.WriteLine("");
Console.WriteLine("Test 2: ");
Print(test2);
Console.WriteLine("");
Console.WriteLine("Lower Cholesky 2: ");
Print(chol2);
}
public static void Print(double[,] a)
{
int n = (int)Math.Sqrt(a.Length);
StringBuilder sb = new StringBuilder();
for (int r = 0; r < n; r++)
{
string s = "";
for (int c = 0; c < n; c++)
{
s += a[r, c].ToString("f5").PadLeft(9) + ",";
}
sb.AppendLine(s);
}
Console.WriteLine(sb.ToString());
}
/// <summary>
/// Returns the lower Cholesky Factor, L, of input matrix A.
/// Satisfies the equation: L*L^T = A.
/// </summary>
/// <param name="a">Input matrix must be square, symmetric,
/// and positive definite. This method does not check for these properties,
/// and may produce unexpected results of those properties are not met.</param>
/// <returns></returns>
public static double[,] Cholesky(double[,] a)
{
int n = (int)Math.Sqrt(a.Length);
double[,] ret = new double[n, n];
for (int r = 0; r < n; r++)
for (int c = 0; c <= r; c++)
{
if (c == r)
{
double sum = 0;
for (int j = 0; j < c; j++)
{
sum += ret[c, j] * ret[c, j];
}
ret[c, c] = Math.Sqrt(a[c, c] - sum);
}
else
{
double sum = 0;
for (int j = 0; j < c; j++)
sum += ret[r, j] * ret[c, j];
ret[r, c] = 1.0 / ret[c, c] * (a[r, c] - sum);
}
}
return ret;
}
}
}
- Output:
Test 1:
25.00000, 15.00000, -5.00000, 15.00000, 18.00000, 0.00000, -5.00000, 0.00000, 11.00000,
Lower Cholesky 1:
5.00000, 0.00000, 0.00000, 3.00000, 3.00000, 0.00000, -1.00000, 1.00000, 3.00000,
Test 2:
18.00000, 22.00000, 54.00000, 42.00000, 22.00000, 70.00000, 86.00000, 62.00000, 54.00000, 86.00000,174.00000,134.00000, 42.00000, 62.00000,134.00000,106.00000,
Lower Cholesky 2:
4.24264, 0.00000, 0.00000, 0.00000, 5.18545, 6.56591, 0.00000, 0.00000, 12.72792, 3.04604, 1.64974, 0.00000, 9.89949, 1.62455, 1.84971, 1.39262,
C++
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <vector>
template <typename scalar_type> class matrix {
public:
matrix(size_t rows, size_t columns)
: rows_(rows), columns_(columns), elements_(rows * columns) {}
matrix(size_t rows, size_t columns, scalar_type value)
: rows_(rows), columns_(columns), elements_(rows * columns, value) {}
matrix(size_t rows, size_t columns,
const std::initializer_list<std::initializer_list<scalar_type>>& values)
: rows_(rows), columns_(columns), elements_(rows * columns) {
assert(values.size() <= rows_);
size_t i = 0;
for (const auto& row : values) {
assert(row.size() <= columns_);
std::copy(begin(row), end(row), &elements_[i]);
i += columns_;
}
}
size_t rows() const { return rows_; }
size_t columns() const { return columns_; }
const scalar_type& operator()(size_t row, size_t column) const {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
scalar_type& operator()(size_t row, size_t column) {
assert(row < rows_);
assert(column < columns_);
return elements_[row * columns_ + column];
}
private:
size_t rows_;
size_t columns_;
std::vector<scalar_type> elements_;
};
template <typename scalar_type>
void print(std::ostream& out, const matrix<scalar_type>& a) {
size_t rows = a.rows(), columns = a.columns();
out << std::fixed << std::setprecision(5);
for (size_t row = 0; row < rows; ++row) {
for (size_t column = 0; column < columns; ++column) {
if (column > 0)
out << ' ';
out << std::setw(9) << a(row, column);
}
out << '\n';
}
}
template <typename scalar_type>
matrix<scalar_type> cholesky_factor(const matrix<scalar_type>& input) {
assert(input.rows() == input.columns());
size_t n = input.rows();
matrix<scalar_type> result(n, n);
for (size_t i = 0; i < n; ++i) {
for (size_t k = 0; k < i; ++k) {
scalar_type value = input(i, k);
for (size_t j = 0; j < k; ++j)
value -= result(i, j) * result(k, j);
result(i, k) = value/result(k, k);
}
scalar_type value = input(i, i);
for (size_t j = 0; j < i; ++j)
value -= result(i, j) * result(i, j);
result(i, i) = std::sqrt(value);
}
return result;
}
void print_cholesky_factor(const matrix<double>& matrix) {
std::cout << "Matrix:\n";
print(std::cout, matrix);
std::cout << "Cholesky factor:\n";
print(std::cout, cholesky_factor(matrix));
}
int main() {
matrix<double> matrix1(3, 3,
{{25, 15, -5},
{15, 18, 0},
{-5, 0, 11}});
print_cholesky_factor(matrix1);
matrix<double> matrix2(4, 4,
{{18, 22, 54, 42},
{22, 70, 86, 62},
{54, 86, 174, 134},
{42, 62, 134, 106}});
print_cholesky_factor(matrix2);
return 0;
}
- Output:
Matrix: 25.00000 15.00000 -5.00000 15.00000 18.00000 0.00000 -5.00000 0.00000 11.00000 Cholesky factor: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 Matrix: 18.00000 22.00000 54.00000 42.00000 22.00000 70.00000 86.00000 62.00000 54.00000 86.00000 174.00000 134.00000 42.00000 62.00000 134.00000 106.00000 Cholesky factor: 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Clojure
(defn cholesky
[matrix]
(let [n (count matrix)
A (to-array-2d matrix)
L (make-array Double/TYPE n n)]
(doseq [i (range n) j (range (inc i))]
(let [s (reduce + (for [k (range j)] (* (aget L i k) (aget L j k))))]
(aset L i j (if (= i j)
(Math/sqrt (- (aget A i i) s))
(* (/ 1.0 (aget L j j)) (- (aget A i j) s))))))
(vec (map vec L))))
Example:
(cholesky [[25 15 -5] [15 18 0] [-5 0 11]])
;=> [[ 5.0 0.0 0.0]
; [ 3.0 3.0 0.0]
; [-1.0 1.0 3.0]]
(cholesky [[18 22 54 42] [22 70 86 62] [54 86 174 134] [42 62 134 106]])
;=> [[ 4.242640687119285 0.0 0.0 0.0 ]
; [ 5.185449728701349 6.565905201197403 0.0 0.0 ]
; [12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 ]
; [ 9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026]]
Common Lisp
;; Calculates the Cholesky decomposition matrix L
;; for a positive-definite, symmetric nxn matrix A.
(defun chol (A)
(let* ((n (car (array-dimensions A)))
(L (make-array `(,n ,n) :initial-element 0)))
(do ((k 0 (incf k))) ((> k (- n 1)) nil)
;; First, calculate diagonal elements L_kk.
(setf (aref L k k)
(sqrt (- (aref A k k)
(do* ((j 0 (incf j))
(sum (expt (aref L k j) 2)
(incf sum (expt (aref L k j) 2))))
((> j (- k 1)) sum)))))
;; Then, all elements below a diagonal element, L_ik, i=k+1..n.
(do ((i (+ k 1) (incf i)))
((> i (- n 1)) nil)
(setf (aref L i k)
(/ (- (aref A i k)
(do* ((j 0 (incf j))
(sum (* (aref L i j) (aref L k j))
(incf sum (* (aref L i j) (aref L k j)))))
((> j (- k 1)) sum)))
(aref L k k)))))
;; Return the calculated matrix L.
L))
;; Example 1:
(setf A (make-array '(3 3) :initial-contents '((25 15 -5) (15 18 0) (-5 0 11))))
(chol A)
#2A((5.0 0 0)
(3.0 3.0 0)
(-1.0 1.0 3.0))
;; Example 2:
(setf B (make-array '(4 4) :initial-contents '((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106))))
(chol B)
#2A((4.2426405 0 0 0)
(5.18545 6.565905 0 0)
(12.727922 3.0460374 1.6497375 0)
(9.899495 1.6245536 1.849715 1.3926151))
;; case of matrix stored as a list of lists (inner lists are rows of matrix)
;; as above, returns the Cholesky decomposition matrix of a square positive-definite, symmetric matrix
(defun cholesky (m)
(let ((l (list (list (sqrt (caar m))))) x (j 0) i)
(dolist (cm (cdr m) (mapcar #'(lambda (x) (nconc x (make-list (- (length m) (length x)) :initial-element 0))) l))
(setq x (list (/ (car cm) (caar l))) i 0)
(dolist (cl (cdr l))
(setf (cdr (last x)) (list (/ (- (elt cm (incf i)) (*v x cl)) (car (last cl))))))
(setf (cdr (last l)) (list (nconc x (list (sqrt (- (elt cm (incf j)) (*v x x))))))))))
;; where *v is the scalar product defined as
(defun *v (v1 v2) (reduce #'+ (mapcar #'* v1 v2)))
;; example 1
CL-USER> (setf a '((25 15 -5) (15 18 0) (-5 0 11)))
((25 15 -5) (15 18 0) (-5 0 11))
CL-USER> (cholesky a)
((5 0 0) (3 3 0) (-1 1 3))
CL-USER> (format t "~{~{~5d~}~%~}" (cholesky a))
5 0 0
3 3 0
-1 1 3
NIL
;; example 2
CL-USER> (setf a '((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106)))
((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106))
CL-USER> (cholesky a)
((4.2426405 0 0 0) (5.18545 6.565905 0 0) (12.727922 3.0460374 1.6497375 0) (9.899495 1.6245536 1.849715 1.3926151))
CL-USER> (format t "~{~{~10,5f~}~%~}" (cholesky a))
4.24264 0.00000 0.00000 0.00000
5.18545 6.56591 0.00000 0.00000
12.72792 3.04604 1.64974 0.00000
9.89950 1.62455 1.84971 1.39262
NIL
D
import std.stdio, std.math, std.numeric;
T[][] cholesky(T)(in T[][] A) pure nothrow /*@safe*/ {
auto L = new T[][](A.length, A.length);
foreach (immutable r, row; L)
row[r + 1 .. $] = 0;
foreach (immutable i; 0 .. A.length)
foreach (immutable j; 0 .. i + 1) {
auto t = dotProduct(L[i][0 .. j], L[j][0 .. j]);
L[i][j] = (i == j) ? (A[i][i] - t) ^^ 0.5 :
(1.0 / L[j][j] * (A[i][j] - t));
}
return L;
}
void main() {
immutable double[][] m1 = [[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]];
writefln("%(%(%2.0f %)\n%)\n", m1.cholesky);
immutable double[][] m2 = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]];
writefln("%(%(%2.3f %)\n%)", m2.cholesky);
}
- Output:
5 0 0 3 3 0 -1 1 3 4.243 0.000 0.000 0.000 5.185 6.566 0.000 0.000 12.728 3.046 1.650 0.000 9.899 1.625 1.850 1.393
Delphi
See Pascal.
DWScript
function Cholesky(a : array of Float) : array of Float;
var
i, j, k, n : Integer;
s : Float;
begin
n:=Round(Sqrt(a.Length));
Result:=new Float[n*n];
for i:=0 to n-1 do begin
for j:=0 to i do begin
s:=0 ;
for k:=0 to j-1 do
s+=Result[i*n+k] * Result[j*n+k];
if i=j then
Result[i*n+j]:=Sqrt(a[i*n+i]-s)
else Result[i*n+j]:=1/Result[j*n+j]*(a[i*n+j]-s);
end;
end;
end;
procedure ShowMatrix(a : array of Float);
var
i, j, n : Integer;
begin
n:=Round(Sqrt(a.Length));
for i:=0 to n-1 do begin
for j:=0 to n-1 do
Print(Format('%2.5f ', [a[i*n+j]]));
PrintLn('');
end;
end;
var m1 := new Float[9];
m1 := [ 25.0, 15.0, -5.0,
15.0, 18.0, 0.0,
-5.0, 0.0, 11.0 ];
var c1 := Cholesky(m1);
ShowMatrix(c1);
PrintLn('');
var m2 : array of Float := [ 18.0, 22.0, 54.0, 42.0,
22.0, 70.0, 86.0, 62.0,
54.0, 86.0, 174.0, 134.0,
42.0, 62.0, 134.0, 106.0 ];
var c2 := Cholesky(m2);
ShowMatrix(c2);
F#
open Microsoft.FSharp.Collections
let cholesky a =
let calc (a: float[,]) (l: float[,]) i j =
let c1 j =
let sum = List.sumBy (fun k -> l.[j, k] ** 2.0) [0..j - 1]
sqrt (a.[j, j] - sum)
let c2 i j =
let sum = List.sumBy (fun k -> l.[i, k] * l.[j, k]) [0..j - 1]
(1.0 / l.[j, j]) * (a.[i, j] - sum)
if j > i then 0.0 else
if i = j
then c1 j
else c2 i j
let l = Array2D.zeroCreate (Array2D.length1 a) (Array2D.length2 a)
Array2D.iteri (fun i j _ -> l.[i, j] <- calc a l i j) l
l
let printMat a =
let arrow = (Array2D.length2 a |> float) / 2.0 |> int
let c = cholesky a
for row in 0..(Array2D.length1 a) - 1 do
for col in 0..(Array2D.length2 a) - 1 do
printf "%.5f,\t" a.[row, col]
printf (if arrow = row then "--> \t" else "\t\t")
for col in 0..(Array2D.length2 c) - 1 do
printf "%.5f,\t" c.[row, col]
printfn ""
let ex1 = array2D [
[25.0; 15.0; -5.0];
[15.0; 18.0; 0.0];
[-5.0; 0.0; 11.0]]
let ex2 = array2D [
[18.0; 22.0; 54.0; 42.0];
[22.0; 70.0; 86.0; 62.0];
[54.0; 86.0; 174.0; 134.0];
[42.0; 62.0; 134.0; 106.0]]
printfn "ex1:"
printMat ex1
printfn "ex2:"
printMat ex2
- Output:
ex1: 25.00000, 15.00000, -5.00000, 5.00000, 0.00000, 0.00000, 15.00000, 18.00000, 0.00000, --> 3.00000, 3.00000, 0.00000, -5.00000, 0.00000, 11.00000, -1.00000, 1.00000, 3.00000, ex2: 18.00000, 22.00000, 54.00000, 42.00000, 4.24264, 0.00000, 0.00000, 0.00000, 22.00000, 70.00000, 86.00000, 62.00000, 5.18545, 6.56591, 0.00000, 0.00000, 54.00000, 86.00000, 174.00000, 134.00000, --> 12.72792, 3.04604, 1.64974, 0.00000, 42.00000, 62.00000, 134.00000, 106.00000, 9.89949, 1.62455, 1.84971, 1.39262,
Fantom
**
** Cholesky decomposition
**
class Main
{
// create an array of Floats, initialised to 0.0
Float[][] makeArray (Int i, Int j)
{
Float[][] result := [,]
i.times { 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]])
}
}- Output:
[[25.0, 15.0, -5.0], [15.0, 18.0, 0.0], [-5.0, 0.0, 11.0]] [[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[18.0, 22.0, 54.0, 42.0], [22.0, 70.0, 86.0, 62.0], [54.0, 86.0, 174.0, 134.0], [42.0, 62.0, 134.0, 106.0]] [[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]
Fortran
Program Cholesky_decomp
! *************************************************!
! LBH @ ULPGC 06/03/2014
! Compute the Cholesky decomposition for a matrix A
! after the attached
! http://rosettacode.org/wiki/Cholesky_decomposition
! note that the matrix A is complex since there might
! be values, where the sqrt has complex solutions.
! Here, only the real values are taken into account
!*************************************************!
implicit none
INTEGER, PARAMETER :: m=3 !rows
INTEGER, PARAMETER :: n=3 !cols
COMPLEX, DIMENSION(m,n) :: A
REAL, DIMENSION(m,n) :: L
REAL :: sum1, sum2
INTEGER i,j,k
! Assign values to the matrix
A(1,:)=(/ 25, 15, -5 /)
A(2,:)=(/ 15, 18, 0 /)
A(3,:)=(/ -5, 0, 11 /)
! !!!!!!!!!!!another example!!!!!!!
! A(1,:) = (/ 18, 22, 54, 42 /)
! A(2,:) = (/ 22, 70, 86, 62 /)
! A(3,:) = (/ 54, 86, 174, 134 /)
! A(4,:) = (/ 42, 62, 134, 106 /)
! Initialize values
L(1,1)=real(sqrt(A(1,1)))
L(2,1)=A(2,1)/L(1,1)
L(2,2)=real(sqrt(A(2,2)-L(2,1)*L(2,1)))
L(3,1)=A(3,1)/L(1,1)
! for greater order than m,n=3 add initial row value
! for instance if m,n=4 then add the following line
! L(4,1)=A(4,1)/L(1,1)
do i=1,n
do k=1,i
sum1=0
sum2=0
do j=1,k-1
if (i==k) then
sum1=sum1+(L(k,j)*L(k,j))
L(k,k)=real(sqrt(A(k,k)-sum1))
elseif (i > k) then
sum2=sum2+(L(i,j)*L(k,j))
L(i,k)=(1/L(k,k))*(A(i,k)-sum2)
else
L(i,k)=0
end if
end do
end do
end do
! write output
do i=1,m
print "(3(1X,F6.1))",L(i,:)
end do
End program Cholesky_decomp
- Output:
5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0
FreeBASIC
' version 18-01-2017
' compile with: fbc -s console
Sub Cholesky_decomp(array() As Double)
Dim As Integer i, j, k
Dim As Double s, l(UBound(array), UBound(array, 2))
For i = 0 To UBound(array)
For j = 0 To i
s = 0
For k = 0 To j -1
s += l(i, k) * l(j, k)
Next
If i = j Then
l(i, j) = Sqr(array(i, i) - s)
Else
l(i, j) = (array(i, j) - s) / l(j, j)
End If
Next
Next
For i = 0 To UBound(array)
For j = 0 To UBound(array, 2)
Swap array(i, j), l(i, j)
Next
Next
End Sub
Sub Print_(array() As Double)
Dim As Integer i, j
For i = 0 To UBound(array)
For j = 0 To UBound(array, 2)
Print Using "###.#####";array(i,j);
Next
Print
Next
End Sub
' ------=< MAIN >=------
Dim m1(2,2) As Double => {{25, 15, -5}, _
{15, 18, 0}, _
{-5, 0, 11}}
Dim m2(3, 3) As Double => {{18, 22, 54, 42}, _
{22, 70, 86, 62}, _
{54, 86, 174, 134}, _
{42, 62, 134, 106}}
Cholesky_decomp(m1())
Print_(m1())
Print
Cholesky_decomp(m2())
Print_(m2())
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Frink
Frink's package Matrix.frink contains routines for Cholesky-Crout decomposition of a square Hermitian matrix (which can be real or complex.) This code is adapted from that more powerful class to work on raw 2-dimensional arrays. This also demonstrates Frink's layout routines.
Cholesky[array] :=
{
n = length[array]
L = new array[[n,n], 0]
for j = 0 to n-1
{
sum = 0
for k = 0 to j-1
sum = sum + (L@j@k)^2
L@j@j = sqrt[array@j@j - sum]
for i = j+1 to n-1
{
sum = 0
for k = 0 to j-1
sum = sum + L@i@k * L@j@k
L@i@j = (1 / L@j@j * (array@i@j -sum))
}
}
return L
}
A = [[ 25, 15, -5],
[ 15, 18, 0],
[ -5, 0, 11]]
println[formatTable[[[formatMatrix[A], "->", formatMatrix[Cholesky[A]]]]]]
B = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]]
println[formatTable[[[formatMatrix[B], "->", formatMatrix[formatFix[Cholesky[B], 1, 5]]]]]]- Output:
┌ ┐ ┌ ┐ │25 15 -5│ │ 5 0 0│ │ │ │ │ │15 18 0│ -> │ 3 3 0│ │ │ │ │ │-5 0 11│ │-1 1 3│ └ ┘ └ ┘ ┌ ┐ ┌ ┐ │18 22 54 42│ │ 4.24264 0.00000 0.00000 0.00000│ │ │ │ │ │22 70 86 62│ │ 5.18545 6.56591 0.00000 0.00000│ │ │ -> │ │ │54 86 174 134│ │12.72792 3.04604 1.64974 0.00000│ │ │ │ │ │42 62 134 106│ │ 9.89949 1.62455 1.84971 1.39262│ └ ┘ └ ┘
Go
Real
This version works with real matrices, like most other solutions on the page. The representation is packed, however, storing only the lower triange of the input symetric matrix and the output lower matrix. The decomposition algorithm computes rows in order from top to bottom but is a little different thatn Cholesky–Banachiewicz.
package main
import (
"fmt"
"math"
)
// symmetric and lower use a packed representation that stores only
// the lower triangle.
type symmetric struct {
order int
ele []float64
}
type lower struct {
order int
ele []float64
}
// symmetric.print prints a square matrix from the packed representation,
// printing the upper triange as a transpose of the lower.
func (s *symmetric) print() {
const eleFmt = "%10.5f "
row, diag := 1, 0
for i, e := range s.ele {
fmt.Printf(eleFmt, e)
if i == diag {
for j, col := diag+row, row; col < s.order; j += col {
fmt.Printf(eleFmt, s.ele[j])
col++
}
fmt.Println()
row++
diag += row
}
}
}
// lower.print prints a square matrix from the packed representation,
// printing the upper triangle as all zeros.
func (l *lower) print() {
const eleFmt = "%10.5f "
row, diag := 1, 0
for i, e := range l.ele {
fmt.Printf(eleFmt, e)
if i == diag {
for j := row; j < l.order; j++ {
fmt.Printf(eleFmt, 0.)
}
fmt.Println()
row++
diag += row
}
}
}
// choleskyLower returns the cholesky decomposition of a symmetric real
// matrix. The matrix must be positive definite but this is not checked.
func (a *symmetric) choleskyLower() *lower {
l := &lower{a.order, make([]float64, len(a.ele))}
row, col := 1, 1
dr := 0 // index of diagonal element at end of row
dc := 0 // index of diagonal element at top of column
for i, e := range a.ele {
if i < dr {
d := (e - l.ele[i]) / l.ele[dc]
l.ele[i] = d
ci, cx := col, dc
for j := i + 1; j <= dr; j++ {
cx += ci
ci++
l.ele[j] += d * l.ele[cx]
}
col++
dc += col
} else {
l.ele[i] = math.Sqrt(e - l.ele[i])
row++
dr += row
col = 1
dc = 0
}
}
return l
}
func main() {
demo(&symmetric{3, []float64{
25,
15, 18,
-5, 0, 11}})
demo(&symmetric{4, []float64{
18,
22, 70,
54, 86, 174,
42, 62, 134, 106}})
}
func demo(a *symmetric) {
fmt.Println("A:")
a.print()
fmt.Println("L:")
a.choleskyLower().print()
}
- Output:
A: 25.00000 15.00000 -5.00000 15.00000 18.00000 0.00000 -5.00000 0.00000 11.00000 L: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 A: 18.00000 22.00000 54.00000 42.00000 22.00000 70.00000 86.00000 62.00000 54.00000 86.00000 174.00000 134.00000 42.00000 62.00000 134.00000 106.00000 L: 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Hermitian
This version handles complex Hermitian matricies as described on the WP page. The matrix representation is flat, and storage is allocated for all elements, not just the lower triangles. The decomposition algorithm is Cholesky–Banachiewicz.
package main
import (
"fmt"
"math/cmplx"
)
type matrix struct {
stride int
ele []complex128
}
func like(a *matrix) *matrix {
return &matrix{a.stride, make([]complex128, len(a.ele))}
}
func (m *matrix) print(heading string) {
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:")
}
- Output:
A: [( 25.00 +0.00i) ( 15.00 +0.00i) ( -5.00 +0.00i)] [( 15.00 +0.00i) ( 18.00 +0.00i) ( 0.00 +0.00i)] [( -5.00 +0.00i) ( 0.00 +0.00i) ( 11.00 +0.00i)] Cholesky factor L: [( 5.00 +0.00i) ( 0.00 +0.00i) ( 0.00 +0.00i)] [( 3.00 +0.00i) ( 3.00 +0.00i) ( 0.00 +0.00i)] [( -1.00 +0.00i) ( 1.00 +0.00i) ( 3.00 +0.00i)] A: [( 18.00 +0.00i) ( 22.00 +0.00i) ( 54.00 +0.00i) ( 42.00 +0.00i)] [( 22.00 +0.00i) ( 70.00 +0.00i) ( 86.00 +0.00i) ( 62.00 +0.00i)] [( 54.00 +0.00i) ( 86.00 +0.00i) ( 174.00 +0.00i) ( 134.00 +0.00i)] [( 42.00 +0.00i) ( 62.00 +0.00i) ( 134.00 +0.00i) ( 106.00 +0.00i)] Cholesky factor L: [( 4.24 +0.00i) ( 0.00 +0.00i) ( 0.00 +0.00i) ( 0.00 +0.00i)] [( 5.19 +0.00i) ( 6.57 +0.00i) ( 0.00 +0.00i) ( 0.00 +0.00i)] [( 12.73 +0.00i) ( 3.05 +0.00i) ( 1.65 +0.00i) ( 0.00 +0.00i)] [( 9.90 +0.00i) ( 1.62 +0.00i) ( 1.85 +0.00i) ( 1.39 +0.00i)] A: [( 1.00 +0.00i) ( 0.00 -2.00i)] [( 0.00 +2.00i) ( 5.00 +0.00i)] Cholesky factor L: [( 1.00 +0.00i) ( 0.00 +0.00i)] [( 0.00 +2.00i) ( 1.00 +0.00i)]
Library gonum/mat
package main
import (
"fmt"
"gonum.org/v1/gonum/mat"
)
func cholesky(order int, elements []float64) fmt.Formatter {
var c mat.Cholesky
c.Factorize(mat.NewSymDense(order, elements))
return mat.Formatted(c.LTo(nil))
}
func main() {
fmt.Println(cholesky(3, []float64{
25, 15, -5,
15, 18, 0,
-5, 0, 11,
}))
fmt.Printf("\n%.5f\n", cholesky(4, []float64{
18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106,
}))
}
- Output:
⎡ 5 0 0⎤ ⎢ 3 3 0⎥ ⎣-1 1 3⎦ ⎡ 4.24264 0.00000 0.00000 0.00000⎤ ⎢ 5.18545 6.56591 0.00000 0.00000⎥ ⎢12.72792 3.04604 1.64974 0.00000⎥ ⎣ 9.89949 1.62455 1.84971 1.39262⎦
Library go.matrix
package main
import (
"fmt"
mat "github.com/skelterjohn/go.matrix"
)
func main() {
demo(mat.MakeDenseMatrix([]float64{
25, 15, -5,
15, 18, 0,
-5, 0, 11,
}, 3, 3))
demo(mat.MakeDenseMatrix([]float64{
18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106,
}, 4, 4))
}
func demo(m *mat.DenseMatrix) {
fmt.Println("A:")
fmt.Println(m)
l, err := m.Cholesky()
if err != nil {
fmt.Println(err)
return
}
fmt.Println("L:")
fmt.Println(l)
}
Output:
A:
{25, 15, -5,
15, 18, 0,
-5, 0, 11}
L:
{ 5, 0, 0,
3, 3, 0,
-1, 1, 3}
A:
{ 18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106}
L:
{ 4.242641, 0, 0, 0,
5.18545, 6.565905, 0, 0,
12.727922, 3.046038, 1.649742, 0,
9.899495, 1.624554, 1.849711, 1.392621}
Groovy
def decompose = { a ->
assert a.size > 0 && a[0].size == a.size
def m = a.size
def l = [].withEagerDefault { [].withEagerDefault { 0 } }
(0..<m).each { i ->
(0..i).each { k ->
Number s = (0..<k).sum { j -> l[i][j] * l[k][j] } ?: 0
l[i][k] = (i == k)
? Math.sqrt(a[i][i] - s)
: (1.0 / l[k][k] * (a[i][k] - s))
}
}
l
}
Test:
def test1 = [[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]]
def test2 = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]];
[test1,test2]. each { test ->
println()
decompose(test).each { println it[0..<(test.size)] }
}
- Output:
[5.0, 0, 0] [3.0, 3.0, 0] [-1.0, 1.0, 3.0] [4.242640687119285, 0, 0, 0] [5.185449728701349, 6.565905201197403, 0, 0] [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0] [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]
Haskell
We use the Cholesky–Banachiewicz algorithm described in the Wikipedia article.
For more serious numerical analysis there is a Cholesky decomposition function in the hmatrix package.
The Cholesky module:
module Cholesky (Arr, cholesky) where
import Data.Array.IArray
import Data.Array.MArray
import Data.Array.Unboxed
import Data.Array.ST
type Idx = (Int,Int)
type Arr = UArray Idx Double
-- Return the (i,j) element of the lower triangular matrix. (We assume the
-- lower array bound is (0,0).)
get :: Arr -> Arr -> Idx -> Double
get a l (i,j) | i == j = sqrt $ a!(j,j) - dot
| i > j = (a!(i,j) - dot) / l!(j,j)
| otherwise = 0
where dot = sum [l!(i,k) * l!(j,k) | k <- [0..j-1]]
-- Return the lower triangular matrix of a Cholesky decomposition. We assume
-- the input is a real, symmetric, positive-definite matrix, with lower array
-- bounds of (0,0).
cholesky :: Arr -> Arr
cholesky a = let n = maxBnd a
in runSTUArray $ do
l <- thaw a
mapM_ (update a l) [(i,j) | i <- [0..n], j <- [0..n]]
return l
where maxBnd = fst . snd . bounds
update a l i = unsafeFreeze l >>= \l' -> writeArray l i (get a l' i)
The main module:
import Data.Array.IArray
import Data.List
import Cholesky
fm _ [] = ""
fm _ [x] = fst x
fm width ((a,b):xs) = a ++ (take (width - b) $ cycle " ") ++ (fm width xs)
fmt width row (xs,[]) = fm width xs
fmt width row (xs,ys) = fm width xs ++ "\n" ++ fmt width row (splitAt row ys)
showMatrice row xs = ys where
vs = map (\s -> let sh = show s in (sh,length sh)) xs
width = (maximum $ snd $ unzip vs) + 1
ys = fmt width row (splitAt row vs)
ex1, ex2 :: Arr
ex1 = listArray ((0,0),(2,2)) [25, 15, -5,
15, 18, 0,
-5, 0, 11]
ex2 = listArray ((0,0),(3,3)) [18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106]
main :: IO ()
main = do
putStrLn $ showMatrice 3 $ elems $ cholesky ex1
putStrLn $ showMatrice 4 $ elems $ cholesky ex2
output:
5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0 4.242640687119285 0.0 0.0 0.0 5.185449728701349 6.565905201197403 0.0 0.0 12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 9.899494936611665 1.6245538642137891 1.849711005231382 1.3926212476455924
With Numeric.LinearAlgebra
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
- Output:
Herm (3><3) [ 25.0, 15.0, -5.0 , 15.0, 18.0, 0.0 , -5.0, 0.0, 11.0 ] (3><3) [ 5.0, 3.0, -1.0 , 0.0, 3.0, 1.0 , 0.0, 0.0, 3.0 ] Herm (4><4) [ 18.0, 22.0, 54.0, 42.0 , 22.0, 70.0, 86.0, 62.0 , 54.0, 86.0, 174.0, 134.0 , 42.0, 62.0, 134.0, 106.0 ] (4><4) [ 4.242640687119285, 5.185449728701349, 12.727922061357857, 9.899494936611665 , 0.0, 6.565905201197403, 3.0460384954008553, 1.6245538642137891 , 0.0, 0.0, 1.6497422479090704, 1.849711005231382 , 0.0, 0.0, 0.0, 1.3926212476455904 ] (4><4) [ 4.242640687119285, 0.0, 0.0, 0.0 , 5.185449728701349, 6.565905201197403, 0.0, 0.0 , 12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0 , 9.899494936611665, 1.6245538642137891, 1.849711005231382, 1.3926212476455904 ]
Icon and Unicon
- Output:
Input: 25 15 -5 15 18 0 -5 0 11 Result: 5.0 0 0 3.0 3.0 0 -1.0 1.0 3.0 Input: 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 Result: 4.242640687 0 0 0 5.185449729 6.565905201 0 0 12.72792206 3.046038495 1.649742248 0 9.899494937 1.624553864 1.849711005 1.392621248
Idris
works with Idris 0.10
Solution:
module Main
import Data.Vect
Matrix : Nat -> Nat -> Type -> Type
Matrix m n t = Vect m (Vect n t)
zeros : (m : Nat) -> (n : Nat) -> Matrix m n Double
zeros m n = replicate m (replicate n 0.0)
indexM : (Fin m, Fin n) -> Matrix m n t -> t
indexM (i, j) a = index j (index i a)
replaceAtM : (Fin m, Fin n) -> t -> Matrix m n t -> Matrix m n t
replaceAtM (i, j) e a = replaceAt i (replaceAt j e (index i a)) a
get : Matrix m m Double -> Matrix m m Double -> (Fin m, Fin m) -> Double
get a l (i, j) {m} = if i == j then sqrt $ indexM (j, j) a - dot
else if i > j then (indexM (i, j) a - dot) / indexM (j, j) l
else 0.0
where
-- Obtain indicies 0 to j -1
ks : List (Fin m)
ks = case (findIndices (\_ => True) a) of
[] => []
(x::xs) => init (x::xs)
dot : Double
dot = sum [(indexM (i, k) l) * (indexM (j, k) l) | k <- ks]
updateL : Matrix m m Double -> Matrix m m Double -> (Fin m, Fin m) -> Matrix m m Double
updateL a l idx = replaceAtM idx (get a l idx) l
cholesky : Matrix m m Double -> Matrix m m Double
cholesky a {m} =
foldl (\l',i =>
foldl (\l'',j => updateL a l'' (i, j)) l' (js i))
l is
where l = zeros m m
is : List (Fin m)
is = findIndices (\_ => True) a
js : Fin m -> List (Fin m)
js n = filter (<= n) is
ex1 : Matrix 3 3 Double
ex1 = cholesky [[25.0, 15.0, -5.0], [15.0, 18.0, 0.0], [-5.0, 0.0, 11.0]]
ex2 : Matrix 4 4 Double
ex2 = cholesky [[18.0, 22.0, 54.0, 42.0], [22.0, 70.0, 86.0, 62.0],
[54.0, 86.0, 174.0, 134.0], [42.0, 62.0, 134.0, 106.0]]
main : IO ()
main = do
print ex1
putStrLn "\n"
print ex2
putStrLn "\n"
- Output:
[[5, 0, 0], [3, 3, 0], [-1, 1, 3]] [[4.242640687119285, 0, 0, 0], [5.185449728701349, 6.565905201197403, 0, 0], [12.72792206135786, 3.046038495400855, 1.64974224790907, 0], [9.899494936611665, 1.624553864213789, 1.849711005231382, 1.392621247645587]]
J
Solution:
mp=: +/ . * NB. matrix product
h =: +@|: NB. conjugate transpose
cholesky=: 3 : 0
n=. #A=. y
if. 1>:n do.
assert. (A=|A)>0=A NB. check for positive definite
%:A
else.
'X Y t Z'=. , (;~n$(>.-:n){.1) <;.1 A
L0=. cholesky X
L1=. cholesky Z-(T=.(h Y) mp %.X) mp Y
L0,(T mp L0),.L1
end.
)
See Cholesky Decomposition essay on the J Wiki.
- Examples:
eg1=: 25 15 _5 , 15 18 0 ,: _5 0 11
eg2=: 18 22 54 42 , 22 70 86 62 , 54 86 174 134 ,: 42 62 134 106
cholesky eg1
5 0 0
3 3 0
_1 1 3
cholesky eg2
4.24264 0 0 0
5.18545 6.56591 0 0
12.7279 3.04604 1.64974 0
9.89949 1.62455 1.84971 1.39262
Using `math/lapack` addon
load 'math/lapack'
load 'math/lapack/potrf'
potrf_jlapack_ eg1
5 0 0
3 3 0
_1 1 3
potrf_jlapack_ eg2
4.24264 0 0 0
5.18545 6.56591 0 0
12.7279 3.04604 1.64974 0
9.89949 1.62455 1.84971 1.39262
Java
import java.util.Arrays;
public class Cholesky {
public static double[][] chol(double[][] a){
int m = a.length;
double[][] l = new double[m][m]; //automatically initialzed to 0's
for(int i = 0; i< m;i++){
for(int k = 0; k < (i+1); k++){
double sum = 0;
for(int j = 0; j < k; j++){
sum += l[i][j] * l[k][j];
}
l[i][k] = (i == k) ? Math.sqrt(a[i][i] - sum) :
(1.0 / l[k][k] * (a[i][k] - sum));
}
}
return l;
}
public static void main(String[] args){
double[][] test1 = {{25, 15, -5},
{15, 18, 0},
{-5, 0, 11}};
System.out.println(Arrays.deepToString(chol(test1)));
double[][] test2 = {{18, 22, 54, 42},
{22, 70, 86, 62},
{54, 86, 174, 134},
{42, 62, 134, 106}};
System.out.println(Arrays.deepToString(chol(test2)));
}
}
- Output:
[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]
JavaScript
const cholesky = function (array) {
const zeros = [...Array(array.length)].map( _ => Array(array.length).fill(0));
const L = zeros.map((row, r, xL) => row.map((v, c) => {
const sum = row.reduce((s, _, i) => i < c ? s + xL[r][i] * xL[c][i] : s, 0);
return xL[r][c] = c < r + 1 ? r === c ? Math.sqrt(array[r][r] - sum) : (array[r][c] - sum) / xL[c][c] : v;
}));
return L;
}
let arr3 = [[25, 15, -5], [15, 18, 0], [-5, 0, 11]];
console.log(cholesky(arr3));
let arr4 = [[18, 22, 54, 42], [22, 70, 86, 62], [54, 86, 174, 134], [42, 62, 134, 106]];
console.log(cholesky(arr4));
- Output:
0: (3) [5, 0, 0] 1: (3) [3, 3, 0] 2: (3) [-1, 1, 3] 0: (4) [4.242640687119285, 0, 0, 0] 1: (4) [5.185449728701349, 6.565905201197403, 0, 0] 2: (4) [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0] 3: (4) [9.899494936611665, 1.6245538642137891, 1.849711005231382, 1.3926212476455924]
jq
Infrastructure:
# Create an m x n matrix
def matrix(m; n; init):
if m == 0 then []
elif m == 1 then [range(0; n)] | map(init)
elif m > 0 then
matrix(1; n; init) as $row
| [range(0; m)] | map( $row )
else error("matrix\(m);_;_) invalid")
end ;
# Print a matrix neatly, each cell ideally occupying n spaces,
# but without truncation
def neatly(n):
def right: tostring | ( " " * (n-length) + .);
. as $in
| length as $length
| reduce range (0; $length) as $i
(""; . + reduce range(0; $length) as $j
(""; "\(.) \($in[$i][$j] | right )" ) + "\n" ) ;
def is_square:
type == "array" and (map(type == "array") | all) and
length == 0 or ( (.[0]|length) as $l | map(length == $l) | all) ;
# This implementation of is_symmetric/0 uses a helper function that circumvents
# limitations of jq 1.4:
def is_symmetric:
# [matrix, i,j, len]
def test:
if .[1] > .[3] then true
elif .[1] == .[2] then [ .[0], .[1] + 1, 0, .[3]] | test
elif .[0][.[1]][.[2]] == .[0][.[2]][.[1]]
then [ .[0], .[1], .[2]+1, .[3]] | test
else false
end;
if is_square|not then false
else [ ., 0, 0, length ] | test
end ;Cholesky Decomposition:
def cholesky_factor:
if is_symmetric then
length as $length
| . as $self
| reduce range(0; $length) as $k
( matrix(length; length; 0); # the matrix that will hold the answer
reduce range(0; $length) as $i
(.;
if $i == $k
then (. as $lower
| reduce range(0; $k) as $j
(0; . + ($lower[$k][$j] | .*.) )) as $sum
| .[$k][$k] = (($self[$k][$k] - $sum) | sqrt)
elif $i > $k
then (. as $lower
| reduce range(0; $k) as $j
(0; . + $lower[$i][$j] * $lower[$k][$j])) as $sum
| .[$i][$k] = (($self[$k][$i] - $sum) / .[$k][$k] )
else .
end ))
else error( "cholesky_factor: matrix is not symmetric" )
end ;Task 1:
[[25,15,-5],[15,18,0],[-5,0,11]] | cholesky_factor
- Output:
[[5,0,0],[3,3,0],[-1,1,3]]
Task 2:
[[18, 22, 54, 42], [22, 70, 86, 62], [54, 86, 174, 134], [42, 62, 134, 106]] | cholesky_factor | neatly(20)
- Output:
4.242640687119285 0 0 0
5.185449728701349 6.565905201197403 0 0
12.727922061357857 3.0460384954008553 1.6497422479090704 0
9.899494936611665 1.6245538642137891 1.849711005231382 1.3926212476455924Julia
Julia's strong linear algebra support includes Cholesky decomposition.
a = [25 15 5; 15 18 0; -5 0 11]
b = [18 22 54 22; 22 70 86 62; 54 86 174 134; 42 62 134 106]
println(a, "\n => \n", chol(a, :L))
println(b, "\n => \n", chol(b, :L))
- Output:
[25 15 5 15 18 0 -5 0 11] => [5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0] [18 22 54 22 22 70 86 62 54 86 174 134 42 62 134 106] => [4.242640687119285 0.0 0.0 0.0 5.185449728701349 6.565905201197403 0.0 0.0 12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026]
Kotlin
// version 1.0.6
fun cholesky(a: DoubleArray): DoubleArray {
val n = Math.sqrt(a.size.toDouble()).toInt()
val l = DoubleArray(a.size)
var s: Double
for (i in 0 until n)
for (j in 0 .. i) {
s = 0.0
for (k in 0 until j) s += l[i * n + k] * l[j * n + k]
l[i * n + j] = when {
(i == j) -> Math.sqrt(a[i * n + i] - s)
else -> 1.0 / l[j * n + j] * (a[i * n + j] - s)
}
}
return l
}
fun showMatrix(a: DoubleArray) {
val n = Math.sqrt(a.size.toDouble()).toInt()
for (i in 0 until n) {
for (j in 0 until n) print("%8.5f ".format(a[i * n + j]))
println()
}
}
fun main(args: Array<String>) {
val m1 = doubleArrayOf(25.0, 15.0, -5.0,
15.0, 18.0, 0.0,
-5.0, 0.0, 11.0)
val c1 = cholesky(m1)
showMatrix(c1)
println()
val m2 = doubleArrayOf(18.0, 22.0, 54.0, 42.0,
22.0, 70.0, 86.0, 62.0,
54.0, 86.0, 174.0, 134.0,
42.0, 62.0, 134.0, 106.0)
val c2 = cholesky(m2)
showMatrix(c2)
}
- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Lobster
Translated from the Go Real version: This version works with real matrices, like most other solutions on the page. The representation is packed, however, storing only the lower triange of the input symetric matrix and the output lower matrix. The decomposition algorithm computes rows in order from top to bottom but is a little different than Cholesky–Banachiewicz.
import std
// choleskyLower returns the cholesky decomposition of a symmetric real
// matrix. The matrix must be positive definite but this is not checked
def choleskyLower(order, a) -> [float]:
let l = map(a.length): 0.0
var row, col = 1, 1
var dr = 0 // index of diagonal element at end of row
var dc = 0 // index of diagonal element at top of column
for(a) e, i:
if i < dr:
let d = (e - l[i]) / l[dc]
l[i] = d
var ci, cx = col, dc
var j = i + 1
while j <= dr:
cx += ci
ci += 1
l[j] += d * l[cx]
j += 1
col += 1
dc += col
else:
l[i] = sqrt(e - l[i])
row += 1
dr += row
col = 1
dc = 0
return l
// symmetric.print prints a square matrix from the packed representation,
// printing the upper triange as a transpose of the lower
def print_symmetric(order, s):
//const eleFmt = "%10.5f "
var str = ""
var row, diag = 1, 0
for(s) e, i:
str += e + " " // format?
if i == diag:
var j, col = diag+row, row
while col < order:
str += s[j] + " " // format?
col++
j += col
print(str); str = ""
row += 1
diag += row
// lower.print prints a square matrix from the packed representation,
// printing the upper triangle as all zeros.
def print_lower(order, l):
//const eleFmt = "%10.5f "
var str = ""
var row, diag = 1, 0
for(l) e, i:
str += e + " " // format?
if i == diag:
var j = row
while j < order:
str += 0.0 + " " // format?
j += 1
print(str); str = ""
row += 1
diag += row
def demo(order, a):
print("A:")
print_symmetric(order, a)
print("L:")
print_lower(order, choleskyLower(order, a))
demo(3, [25.0,
15.0, 18.0,
-5.0, 0.0, 11.0])
demo(4, [18.0,
22.0, 70.0,
54.0, 86.0, 174.0,
42.0, 62.0, 134.0, 106.0])- Output:
A: 25.0 15.0 -5.0 15.0 18.0 0.0 -5.0 0.0 11.0 L: 5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0 A: 18.0 22.0 54.0 42.0 22.0 70.0 86.0 62.0 54.0 86.0 174.0 134.0 42.0 62.0 134.0 106.0 L: 4.242640687119 0.0 0.0 0.0 5.185449728701 6.565905201197 0.0 0.0 12.72792206135 3.046038495401 1.649742247909 0.0 9.899494936612 1.624553864214 1.849711005231 1.392621247646
Maple
The Cholesky decomposition is obtained by passing the `method = Cholesky' option to the LUDecomposition procedure in the LinearAlgebra pacakge. This is illustrated below for the two requested examples. The first is computed exactly; the second is also, but the subsequent application of `evalf' to the result produces a matrix with floating point entries which can be compared with the expected output in the problem statement.
> A := << 25, 15, -5; 15, 18, 0; -5, 0, 11 >>;
[25 15 -5]
[ ]
A := [15 18 0]
[ ]
[-5 0 11]
> B := << 18, 22, 54, 42; 22, 70, 86, 62; 54, 86, 174, 134; 42, 62, 134, 106>>;
[18 22 54 42]
[ ]
[22 70 86 62]
B := [ ]
[54 86 174 134]
[ ]
[42 62 134 106]
> use LinearAlgebra in
> LUDecomposition( A, method = Cholesky );
> LUDecomposition( B, method = Cholesky );
> evalf( % );
> end use;
[ 5 0 0]
[ ]
[ 3 3 0]
[ ]
[-1 1 3]
[ 1/2 ]
[3 2 0 0 0 ]
[ ]
[ 1/2 1/2 ]
[11 2 2 97 ]
[------- ------- 0 0 ]
[ 3 3 ]
[ ]
[ 1/2 1/2 ]
[ 1/2 30 97 2 6402 ]
[9 2 -------- --------- 0 ]
[ 97 97 ]
[ ]
[ 1/2 1/2 1/2]
[ 1/2 16 97 74 6402 8 33 ]
[7 2 -------- ---------- -------]
[ 97 3201 33 ]
[4.242640686 0. 0. 0. ]
[ ]
[5.185449728 6.565905202 0. 0. ]
[ ]
[12.72792206 3.046038495 1.649742248 0. ]
[ ]
[9.899494934 1.624553864 1.849711006 1.392621248]Mathematica / Wolfram Language
CholeskyDecomposition[{{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}]
Without the use of built-in functions, making use of memoization:
chol[A_] :=
Module[{L},
L[k_, k_] := L[k, k] = Sqrt[A[[k, k]] - Sum[L[k, j]^2, {j, 1, k-1}]];
L[i_, k_] := L[i, k] = L[k, k]^-1 (A[[i, k]] - Sum[L[i, j] L[k, j], {j, 1, k-1}]);
PadRight[Table[L[i, j], {i, Length[A]}, {j, i}]]
]
MATLAB / Octave
The cholesky decomposition chol() is an internal function
A = [
25 15 -5
15 18 0
-5 0 11 ];
B = [
18 22 54 42
22 70 86 62
54 86 174 134
42 62 134 106 ];
[L] = chol(A,'lower')
[L] = chol(B,'lower')
- Output:
> [L] = chol(A,'lower')
L =
5 0 0
3 3 0
-1 1 3
> [L] = chol(B,'lower')
L =
4.24264 0.00000 0.00000 0.00000
5.18545 6.56591 0.00000 0.00000
12.72792 3.04604 1.64974 0.00000
9.89949 1.62455 1.84971 1.39262
Maxima
/* Cholesky decomposition is built-in */
a: hilbert_matrix(4)$
b: cholesky(a);
/* matrix([1, 0, 0, 0 ],
[1/2, 1/(2*sqrt(3)), 0, 0 ],
[1/3, 1/(2*sqrt(3)), 1/(6*sqrt(5)), 0 ],
[1/4, 3^(3/2)/20, 1/(4*sqrt(5)), 1/(20*sqrt(7))]) */
b . transpose(b) - a;
matrix([0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 0, 0])
Nim
import math, strutils, strformat
type Matrix[N: static int, T: SomeFloat] = array[N, array[N, T]]
proc cholesky[Matrix](a: Matrix): Matrix =
for i in 0 ..< a[0].len:
for j in 0 .. i:
var s = 0.0
for k in 0 ..< j:
s += result[i][k] * result[j][k]
result[i][j] = if i == j: sqrt(a[i][i]-s)
else: 1.0 / result[j][j] * (a[i][j] - s)
proc `$`(a: Matrix): string =
result = ""
for b in a:
var line = ""
for c in b:
line.addSep(" ", 0)
line.add fmt"{c:8.5f}"
result.add line & '\n'
let m1 = [[25.0, 15.0, -5.0],
[15.0, 18.0, 0.0],
[-5.0, 0.0, 11.0]]
echo cholesky(m1)
let m2 = [[18.0, 22.0, 54.0, 42.0],
[22.0, 70.0, 86.0, 62.0],
[54.0, 86.0, 174.0, 134.0],
[42.0, 62.0, 134.0, 106.0]]
echo cholesky(m2)
- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Objeck
class Cholesky {
function : Main(args : String[]) ~ Nil {
n := 3;
m1 := [25.0, 15.0, -5.0, 15.0, 18.0, 0.0, -5.0, 0.0, 11.0];
c1 := Cholesky(m1, n);
ShowMatrix(c1, n);
IO.Console->PrintLine();
n := 4;
m2 := [18.0, 22.0, 54.0, 42.0, 22.0, 70.0, 86.0, 62.0,
54.0, 86.0, 174.0, 134.0, 42.0, 62.0, 134.0, 106.0];
c2 := Cholesky(m2, n);
ShowMatrix(c2, n);
}
function : ShowMatrix(A : Float[], n : Int) ~ Nil {
for (i := 0; i < n; i+=1;) {
for (j := 0; j < n; j+=1;) {
IO.Console->Print(A[i * n + j])->Print('\t');
};
IO.Console->PrintLine();
};
}
function : Cholesky(A : Float[], n : Int) ~ Float[] {
L := Float->New[n * n];
for (i := 0; i < n; i+=1;) {
for (j := 0; j < (i+1); j+=1;) {
s := 0.0;
for (k := 0; k < j; k+=1;) {
s += L[i * n + k] * L[j * n + k];
};
L[i * n + j] := (i = j) ?
(A[i * n + i] - s)->SquareRoot() :
(1.0 / L[j * n + j] * (A[i * n + j] - s));
};
};
return L;
}
}5 0 0 3 3 0 -1 1 3 4.24264069 0 0 0 5.18544973 6.5659052 0 0 12.7279221 3.0460385 1.64974225 0 9.89949494 1.62455386 1.84971101 1.39262125
OCaml
let cholesky inp =
let n = Array.length inp in
let res = Array.make_matrix n n 0.0 in
let factor i k =
let rec sum j =
if j = k then 0.0 else
res.(i).(j) *. res.(k).(j) +. sum (j+1) in
inp.(i).(k) -. sum 0 in
for col = 0 to n-1 do
res.(col).(col) <- sqrt (factor col col);
for row = col+1 to n-1 do
res.(row).(col) <- (factor row col) /. res.(col).(col)
done
done;
res
let pr_vec v = Array.iter (Printf.printf " %9.5f") v; print_newline()
let show = Array.iter pr_vec
let test a =
print_endline "\nin:"; show a;
print_endline "out:"; show (cholesky a)
let _ =
test [| [|25.0; 15.0; -5.0|];
[|15.0; 18.0; 0.0|];
[|-5.0; 0.0; 11.0|] |];
test [| [|18.0; 22.0; 54.0; 42.0|];
[|22.0; 70.0; 86.0; 62.0|];
[|54.0; 86.0; 174.0; 134.0|];
[|42.0; 62.0; 134.0; 106.0|] |];
- Output:
in: 25.00000 15.00000 -5.00000 15.00000 18.00000 0.00000 -5.00000 0.00000 11.00000 out: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 in: 18.00000 22.00000 54.00000 42.00000 22.00000 70.00000 86.00000 62.00000 54.00000 86.00000 174.00000 134.00000 42.00000 62.00000 134.00000 106.00000 out: 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
ooRexx
/*REXX program performs the Cholesky decomposition on a square matrix. */
niner = '25 15 -5' , /*define a 3x3 matrix. */
'15 18 0' ,
'-5 0 11'
call Cholesky niner
hexer = 18 22 54 42, /*define a 4x4 matrix. */
22 70 86 62,
54 86 174 134,
42 62 134 106
call Cholesky hexer
exit /*stick a fork in it, we're all done. */
/*----------------------------------------------------------------------------*/
Cholesky: procedure; parse arg mat; say; say; call tell 'input matrix',mat
do r=1 for ord
do c=1 for r; d=0; do i=1 for c-1; d=d+!.r.i*!.c.i; end /*i*/
if r=c then !.r.r=sqrt(!.r.r-d)
else !.r.c=1/!.c.c*(a.r.c-d)
end /*c*/
end /*r*/
call tell 'Cholesky factor',,!.,'-'
return
/*----------------------------------------------------------------------------*/
err: say; say; say '***error***!'; say; say arg(1); say; say; exit 13
/*----------------------------------------------------------------------------*/
tell: parse arg hdr,x,y,sep; n=0; if sep=='' then sep='-'
dPlaces= 5 /*n decimal places past the decimal point*/
width =10 /*width of field used to display elements*/
if y=='' then !.=0
else do row=1 for ord; do col=1 for ord; x=x !.row.col; end; end
w=words(x)
do ord=1 until ord**2>=w; end /*a fast way to find matrix's order*/
say
if ord**2\==w then call err "matrix elements don't form a square matrix."
say center(hdr, ((width+1)*w)%ord, sep)
say
do row=1 for ord; z=''
do col=1 for ord; n=n+1
a.row.col=word(x,n)
if col<=row then !.row.col=a.row.col
z=z right( format(a.row.col,, dPlaces) / 1, width)
end /*col*/
say z
end /*row*/
return
/*----------------------------------------------------------------------------*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); i=''; m.=9
numeric digits 9; numeric form; h=d+6; if x<0 then do; x=-x; i='i'; end
parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_%2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/
numeric digits d; return (g/1)i /*make complex if X < 0.*/
PARI/GP
cholesky(M) =
{
my (L = matrix(#M,#M));
for (i = 1, #M,
for (j = 1, i,
s = sum (k = 1, j-1, L[i,k] * L[j,k]);
L[i,j] = if (i == j, sqrt(M[i,i] - s), (M[i,j] - s) / L[j,j])
)
);
L
}Output: (set displayed digits with: \p 5)
gp > cholesky([25,15,-5;15,18,0;-5,0,11]) [ 5.0000 0 0] [ 3.0000 3.0000 0] [-1.0000 1.0000 3.0000] gp > cholesky([18,22,54,42;22,70,86,62;54,86,174,134;42,62,134,106]) [4.2426 0 0 0] [5.1854 6.5659 0 0] [12.728 3.0460 1.6497 0] [9.8995 1.6246 1.8497 1.3926]
Pascal
program CholeskyApp;
type
D2Array = array of array of double;
function cholesky(const A: D2Array): D2Array;
var
i, j, k: integer;
s: double;
begin
setlength(Result, length(A), length(A));
for i := low(Result) to high(Result) do
for j := 0 to i do
begin
s := 0;
for k := 0 to j - 1 do
s := s + Result[i][k] * Result[j][k];
if i = j then
Result[i][j] := sqrt(A[i][i] - s)
else
Result[i][j] := (A[i][j] - s) / Result[j][j]; // save one multiplication compared to the original
end;
end;
procedure printM(const A: D2Array);
var
i, j: integer;
begin
for i := low(A) to high(A) do
begin
for j := low(A) to high(A) do
write(A[i, j]: 8: 5);
writeln;
end;
end;
const
m1: array[0..2, 0..2] of double = ((25, 15, -5), (15, 18, 0), (-5, 0, 11));
m2: array[0..3, 0..3] of double = ((18, 22, 54, 42), (22, 70, 86, 62), (54, 86,
174, 134), (42, 62, 134, 106));
var
index, i: integer;
cIn, cOut: D2Array;
begin
setlength(cIn, length(m1), length(m1));
for index := low(m1) to high(m1) do
begin
SetLength(cIn[index], length(m1[index]));
for i := 0 to High(m1[Index]) do
cIn[index][i] := m1[index][i];
end;
cOut := cholesky(cIn);
printM(cOut);
writeln;
setlength(cIn, length(m2), length(m2));
for index := low(m2) to high(m2) do
begin
SetLength(cIn[index], length(m2[Index]));
for i := 0 to High(m2[Index]) do
cIn[index][i] := m2[index][i];
end;
cOut := cholesky(cIn);
printM(cOut);
end.
- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
PascalABC.NET
function Cholesky(a: array of real): array of real;
begin
var n := Round(Sqrt(a.Length));
Result := new real[n * n];
for var i := 0 to n - 1 do
begin
for var j := 0 to i do
begin
var s := 0.0;
for var 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 real);
begin
var n := Round(Sqrt(a.Length));
for var i := 0 to n - 1 do
begin
for var j := 0 to n - 1 do
write(a[i * n + j]:10:5);
PrintLn('');
end;
end;
begin
var m1 := new real[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(m1);
println('=>');
ShowMatrix(c1);
PrintLn('');
var m2: array of real :=
|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(m2);
println('=>');
ShowMatrix(c2);
end.
Perl
sub cholesky {
my $matrix = shift;
my $chol = [ map { [(0) x @$matrix ] } @$matrix ];
for my $row (0..@$matrix-1) {
for my $col (0..$row) {
my $x = $$matrix[$row][$col];
$x -= $$chol[$row][$_]*$$chol[$col][$_] for 0..$col;
$$chol[$row][$col] = $row == $col ? sqrt $x : $x/$$chol[$col][$col];
}
}
return $chol;
}
my $example1 = [ [ 25, 15, -5 ],
[ 15, 18, 0 ],
[ -5, 0, 11 ] ];
print "Example 1:\n";
print +(map { sprintf "%7.4f\t", $_ } @$_), "\n" for @{ cholesky $example1 };
my $example2 = [ [ 18, 22, 54, 42],
[ 22, 70, 86, 62],
[ 54, 86, 174, 134],
[ 42, 62, 134, 106] ];
print "\nExample 2:\n";
print +(map { sprintf "%7.4f\t", $_ } @$_), "\n" for @{ cholesky $example2 };
- Output:
Example 1: 5.0000 0.0000 0.0000 3.0000 3.0000 0.0000 -1.0000 1.0000 3.0000 Example 2: 4.2426 0.0000 0.0000 0.0000 5.1854 6.5659 0.0000 0.0000 12.7279 3.0460 1.6497 0.0000 9.8995 1.6246 1.8497 1.3926
Phix
with javascript_semantics function cholesky(sequence matrix) integer l = length(matrix) sequence chol = repeat(repeat(0,l),l) for row=1 to l do for col=1 to row do atom x = matrix[row][col] for i=1 to col do x -= chol[row][i] * chol[col][i] end for chol[row][col] = iff(row == col ? sqrt(x) : x/chol[col][col]) end for end for return chol end function ppOpt({pp_Nest,1}) pp(cholesky({{ 25, 15, -5 }, { 15, 18, 0 }, { -5, 0, 11 }})) pp(cholesky({{ 18, 22, 54, 42}, { 22, 70, 86, 62}, { 54, 86, 174, 134}, { 42, 62, 134, 106}}))
- Output:
{{5,0,0},
{3,3,0},
{-1,1,3}}
{{4.242640687,0,0,0},
{5.185449729,6.565905201,0,0},
{12.72792206,3.046038495,1.649742248,0},
{9.899494937,1.624553864,1.849711005,1.392621248}}
PicoLisp
(scl 9)
(load "@lib/math.l")
(de cholesky (A)
(let L (mapcar '(() (need (length A) 0)) A)
(for (I . R) A
(for J I
(let S (get R J)
(for K (inc J)
(dec 'S (*/ (get L I K) (get L J K) 1.0)) )
(set (nth L I J)
(if (= I J)
(sqrt S 1.0)
(*/ S 1.0 (get L J J)) ) ) ) ) )
(for R L
(for N R (prin (align 9 (round N 5))))
(prinl) ) ) )Test:
(cholesky
'((25.0 15.0 -5.0) (15.0 18.0 0) (-5.0 0 11.0)) )
(prinl)
(cholesky
(quote
(18.0 22.0 54.0 42.0)
(22.0 70.0 86.0 62.0)
(54.0 86.0 174.0 134.0)
(42.0 62.0 134.0 106.0) ) )- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
PL/I
(subscriptrange):
decompose: procedure options (main); /* 31 October 2013 */
declare a(*,*) float controlled;
allocate a(3,3) initial (25, 15, -5,
15, 18, 0,
-5, 0, 11);
put skip list ('Original matrix:');
put edit (a) (skip, 3 f(4));
call cholesky(a);
put skip list ('Decomposed matrix');
put edit (a) (skip, 3 f(4));
free a;
allocate a(4,4) initial (18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106);
put skip list ('Original matrix:');
put edit (a) (skip, (hbound(a,1)) f(12) );
call cholesky(a);
put skip list ('Decomposed matrix');
put edit (a) (skip, (hbound(a,1)) f(12,5) );
cholesky: procedure(a);
declare a(*,*) float;
declare L(hbound(a,1), hbound(a,2)) float;
declare s float;
declare (i, j, k) fixed binary;
L = 0;
do i = lbound(a,1) to hbound(a,1);
do j = lbound(a,2) to i;
s = 0;
do k = lbound(a,2) to j-1;
s = s + L(i,k) * L(j,k);
end;
if i = j then
L(i,j) = sqrt(a(i,i) - s);
else
L(i,j) = (a(i,j) - s) / L(j,j);
end;
end;
a = L;
end cholesky;
end decompose;ACTUAL RESULTS:-
Original matrix:
25 15 -5
15 18 0
-5 0 11
Decomposed matrix
5 0 0
3 3 0
-1 1 3
Original matrix:
18 22 54 42
22 70 86 62
54 86 174 134
42 62 134 106
Decomposed matrix
4.24264 0.00000 0.00000 0.00000
5.18545 6.56591 0.00000 0.00000
12.72792 3.04604 1.64974 0.00000
9.89950 1.62455 1.84971 1.39262
PowerShell
function cholesky ($a) {
$l = @()
if ($a) {
$n = $a.count
$end = $n - 1
$l = 1..$n | foreach {$row = @(0) * $n; ,$row}
foreach ($k in 0..$end) {
$m = $k - 1
$sum = 0
if(0 -lt $k) {
foreach ($j in 0..$m) {$sum += $l[$k][$j]*$l[$k][$j]}
}
$l[$k][$k] = [Math]::Sqrt($a[$k][$k] - $sum)
if ($k -lt $end) {
foreach ($i in ($k+1)..$end) {
$sum = 0
if (0 -lt $k) {
foreach ($j in 0..$m) {$sum += $l[$i][$j]*$l[$k][$j]}
}
$l[$i][$k] = ($a[$i][$k] - $sum)/$l[$k][$k]
}
}
}
}
$l
}
function show($a) {$a | foreach {"$_"}}
$a1 = @(
@(25, 15, -5),
@(15, 18, 0),
@(-5, 0, 11)
)
"a1 ="
show $a1
""
"l1 ="
show (cholesky $a1)
""
$a2 = @(
@(18, 22, 54, 42),
@(22, 70, 86, 62),
@(54, 86, 174, 134),
@(42, 62, 134, 106)
)
"a2 ="
show $a2
""
"l2 ="
show (cholesky $a2)
Output:
a1 = 25 15 -5 15 18 0 -5 0 11 l1 = 5 0 0 3 3 0 -1 1 3 a2 = 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 l2 = 4.24264068711928 0 0 0 5.18544972870135 6.5659052011974 0 0 12.7279220613579 3.04603849540086 1.64974224790907 0 9.89949493661167 1.62455386421379 1.84971100523138 1.39262124764559
Python
Python2.X version
from __future__ import print_function
from pprint import pprint
from math import sqrt
def cholesky(A):
L = [[0.0] * len(A) for _ in xrange(len(A))]
for i in xrange(len(A)):
for j in xrange(i+1):
s = sum(L[i][k] * L[j][k] for k in xrange(j))
L[i][j] = sqrt(A[i][i] - s) if (i == j) else \
(1.0 / L[j][j] * (A[i][j] - s))
return L
if __name__ == "__main__":
m1 = [[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]]
pprint(cholesky(m1))
print()
m2 = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]]
pprint(cholesky(m2), width=120)
- Output:
[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]
Python3.X version using extra Python idioms
Factors out accesses to A[i], L[i], and L[j] by creating Ai, Li and Lj respectively as well as using enumerate instead of range(len(some_array)).
def cholesky(A):
L = [[0.0] * len(A) for _ in range(len(A))]
for i, (Ai, Li) in enumerate(zip(A, L)):
for j, Lj in enumerate(L[:i+1]):
s = sum(Li[k] * Lj[k] for k in range(j))
Li[j] = sqrt(Ai[i] - s) if (i == j) else \
(1.0 / Lj[j] * (Ai[j] - s))
return L
- Output:
(As above)
q
solve:{[A;B] $[0h>type A;B%A;inv[A] mmu B]}
ak:{[m;k] (),/:m[;k]til k:k-1}
akk:{[m;k] m[k;k:k-1]}
transpose:{$[0h=type x;flip x;enlist each x]}
mult:{[A;B]$[0h=type A;A mmu B;A*B]}
cholesky:{[A]
{[A;L;n]
l_k:solve[L;ak[A;n]];
l_kk:first over sqrt[akk[A;n] - mult[transpose l_k;l_k]];
({$[0h<type x;enlist x;x]}L,'0f),enlist raze transpose[l_k],l_kk
}[A]/[sqrt A[0;0];1_1+til count first A]
}
show cholesky (25 15 -5f;15 18 0f;-5 0 11f)
-1"";
show cholesky (18 22 54 42f;22 70 86 62f;54 86 174 134f;42 62 134 106f)
- Output:
5 0 0 3 3 0 -1 1 3 4.242641 0 0 0 5.18545 6.565905 0 0 12.72792 3.046038 1.649742 0 9.899495 1.624554 1.849711 1.392621
R
t(chol(matrix(c(25, 15, -5, 15, 18, 0, -5, 0, 11), nrow=3, ncol=3)))
# [,1] [,2] [,3]
# [1,] 5 0 0
# [2,] 3 3 0
# [3,] -1 1 3
t(chol(matrix(c(18, 22, 54, 42, 22, 70, 86, 62, 54, 86, 174, 134, 42, 62, 134, 106), nrow=4, ncol=4)))
# [,1] [,2] [,3] [,4]
# [1,] 4.242641 0.000000 0.000000 0.000000
# [2,] 5.185450 6.565905 0.000000 0.000000
# [3,] 12.727922 3.046038 1.649742 0.000000
# [4,] 9.899495 1.624554 1.849711 1.392621
Racket
#lang racket
(require math)
(define (cholesky A)
(define mref matrix-ref)
(define n (matrix-num-rows A))
(define L (for/vector ([_ n]) (for/vector ([_ n]) 0)))
(define (set L i j x) (vector-set! (vector-ref L i) j x))
(define (ref L i j) (vector-ref (vector-ref L i) j))
(for* ([i n] [k n])
(set L i k
(cond
[(= i k)
(sqrt (- (mref A i i) (for/sum ([j k]) (sqr (ref L k j)))))]
[(> i k)
(/ (- (mref A i k) (for/sum ([j k]) (* (ref L i j) (ref L k j))))
(ref L k k))]
[else 0])))
L)
(cholesky (matrix [[25 15 -5]
[15 18 0]
[-5 0 11]]))
(cholesky (matrix [[18 22 54 42]
[22 70 86 62]
[54 86 174 134]
[42 62 134 106]]))
Output:
'#(#(5 0 0)
#(3 3 0)
#(-1 1 3))
'#(#(4.242640687119285 0 0 0)
#( 5.185449728701349 6.565905201197403 0 0)
#(12.727922061357857 3.0460384954008553 1.6497422479090704 0)
#( 9.899494936611665 1.6245538642137891 1.849711005231382 1.3926212476455924))
Raku
(formerly Perl 6)
sub cholesky(@A) {
my @L = @A »×» 0;
for ^@A -> \i {
for 0..i -> \j {
@L[i;j] = (i == j ?? &sqrt !! 1/@L[j;j] × * )\ # select function
(@A[i;j] - [+] (@L[i;*] Z× @L[j;*])[^j]) # provide value
}
}
@L
}
.fmt('%3d').say for cholesky [
[25],
[15, 18],
[-5, 0, 11],
];
say '';
.fmt('%6.3f').say for cholesky [
[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106],
];
- Output:
5 3 3 -1 1 3 4.243 0.000 0.000 0.000 5.185 6.566 0.000 0.000 12.728 3.046 1.650 0.000 9.899 1.625 1.850 1.393
REXX
If trailing zeros are wanted after the decimal point for values of zero (0), the / 1 (a division by unity performs
REXX number normalization) can be removed from the line (number 40) which contains the source statement:
- z=z right( format(@.row.col, , dPlaces) / 1, width)
/*REXX program performs the Cholesky decomposition on a square matrix & displays results*/
niner = '25 15 -5' , /*define a 3x3 matrix with elements. */
'15 18 0' ,
'-5 0 11'
call Cholesky niner
hexer = 18 22 54 42, /*define a 4x4 matrix with elements. */
22 70 86 62,
54 86 174 134,
42 62 134 106
call Cholesky hexer
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
Cholesky: procedure; parse arg mat; say; say; call tell 'input array',mat
do r=1 for ord
do c=1 for r; $=0; do i=1 for c-1; $= $ + !.r.i * !.c.i; end /*i*/
if r=c then !.r.r= sqrt(!.r.r - $)
else !.r.c= 1 / !.c.c * (@.r.c - $)
end /*c*/
end /*r*/
call tell 'Cholesky factor',,!.,'─'
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
err: say; say; say '***error***!'; say; say arg(1); say; say; exit 13
/*──────────────────────────────────────────────────────────────────────────────────────*/
tell: parse arg hdr,x,y,sep; #=0; if sep=='' then sep= '═'
dPlaces= 5 /*# dec. places past the decimal point.*/
width =10 /*field width used to display elements.*/
if y=='' then !.=0
else do row=1 for ord; do col=1 for ord; x=x !.row.col; end; end
w=words(x)
do ord=1 until ord**2>=w; end /*a fast way to find the matrix's order*/
say
if ord**2\==w then call err "matrix elements don't form a square matrix."
say center(hdr, ((width + 1) * w) % ord, sep)
say
do row=1 for ord; z=
do col=1 for ord; #= # + 1
@.row.col= word(x, #)
if col<=row then !.row.col= @.row.col
z=z right( format(@.row.col, , dPlaces) / 1, width)
end /*col*/ /* ↑↑↑ */
say z /* └┴┴──◄──normalization for zero*/
end /*row*/
return
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); numeric digits; h=d+6
numeric form; m.=9; parse value format(x,2,1,,0) 'E0' with g 'E' _ .; g=g*.5'e'_ %2
do j=0 while h>9; m.j=h; h=h%2+1; end /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g=(g+x/g)*.5; end /*k*/; return g/1
- output:
═══════════input matrix══════════
25 15 -5
15 18 0
-5 0 11
─────────Cholesky factor─────────
5 0 0
3 3 0
-1 1 3
════════════════input matrix════════════════
18 22 54 42
22 70 86 62
54 86 174 134
42 62 134 106
──────────────Cholesky factor───────────────
4.24264 0 0 0
5.18545 6.56591 0 0
12.72792 3.04604 1.64974 0
9.89949 1.62455 1.84971 1.39262
Ring
# Project : Cholesky decomposition
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
nextOutput:
5 0 0 3 3 0 -1 1 3 4.24264 0 0 0 5.18545 6.56591 0 0 12.72792 3.04604 1.64974 0 9.89949 1.62455 1.84971 1.39262
Ruby
require 'matrix'
class Matrix
def cholesky_factor
raise ArgumentError, "must provide symmetric matrix" unless symmetric?
l = Array.new(row_size) {Array.new(row_size, 0)}
(0 ... row_size).each do |k|
(0 ... row_size).each do |i|
if i == k
sum = (0 .. k-1).inject(0.0) {|sum, j| sum + l[k][j] ** 2}
val = Math.sqrt(self[k,k] - sum)
l[k][k] = val
elsif i > k
sum = (0 .. k-1).inject(0.0) {|sum, j| sum + l[i][j] * l[k][j]}
val = (self[k,i] - sum) / l[k][k]
l[i][k] = val
end
end
end
Matrix[*l]
end
end
puts Matrix[[25,15,-5],[15,18,0],[-5,0,11]].cholesky_factor
puts Matrix[[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]].cholesky_factor
- Output:
Matrix[[5.0, 0, 0], [3.0, 3.0, 0], [-1.0, 1.0, 3.0]] Matrix[[4.242640687119285, 0, 0, 0], [5.185449728701349, 6.565905201197403, 0, 0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0], [9.899494936611665, 1.6245538642137891, 1.849711005231382, 1.3926212476455924]]
Rust
fn cholesky(mat: Vec<f64>, n: usize) -> Vec<f64> {
let mut res = vec![0.0; mat.len()];
for i in 0..n {
for j in 0..(i+1){
let mut s = 0.0;
for k in 0..j {
s += res[i * n + k] * res[j * n + k];
}
res[i * n + j] = if i == j { (mat[i * n + i] - s).sqrt() } else { (1.0 / res[j * n + j] * (mat[i * n + j] - s)) };
}
}
res
}
fn show_matrix(matrix: Vec<f64>, n: usize){
for i in 0..n {
for j in 0..n {
print!("{:.4}\t", matrix[i * n + j]);
}
println!("");
}
println!("");
}
fn main(){
let dimension = 3 as usize;
let m1 = vec![25.0, 15.0, -5.0,
15.0, 18.0, 0.0,
-5.0, 0.0, 11.0];
let res1 = cholesky(m1, dimension);
show_matrix(res1, dimension);
let dimension = 4 as usize;
let m2 = vec![18.0, 22.0, 54.0, 42.0,
22.0, 70.0, 86.0, 62.0,
54.0, 86.0, 174.0, 134.0,
42.0, 62.0, 134.0, 106.0];
let res2 = cholesky(m2, dimension);
show_matrix(res2, dimension);
}
- Output:
5.0000 0.0000 0.0000 3.0000 3.0000 0.0000 -1.0000 1.0000 3.0000 4.2426 0.0000 0.0000 0.0000 5.1854 6.5659 0.0000 0.0000 12.7279 3.0460 1.6497 0.0000 9.8995 1.6246 1.8497 1.3926
Scala
case class Matrix( val matrix:Array[Array[Double]] ) {
// Assuming matrix is positive-definite, symmetric and not empty...
val rows,cols = matrix.size
def getOption( r:Int, c:Int ) : Option[Double] = Pair(r,c) match {
case (r,c) if r < rows && c < rows => Some(matrix(r)(c))
case _ => None
}
def isLowerTriangle( r:Int, c:Int ) : Boolean = { c <= r }
def isDiagonal( r:Int, c:Int ) : Boolean = { r == c}
override def toString = matrix.map(_.mkString(", ")).mkString("\n")
/**
* Perform Cholesky Decomposition of this matrix
*/
lazy val cholesky : Matrix = {
val l = Array.ofDim[Double](rows*cols)
for( i <- (0 until rows); j <- (0 until cols) ) yield {
val s = (for( k <- (0 until j) ) yield { l(i*rows+k) * l(j*rows+k) }).sum
l(i*rows+j) = (i,j) match {
case (r,c) if isDiagonal(r,c) => scala.math.sqrt(matrix(i)(i) - s)
case (r,c) if isLowerTriangle(r,c) => (1.0 / l(j*rows+j) * (matrix(i)(j) - s))
case _ => 0
}
}
val m = Array.ofDim[Double](rows,cols)
for( i <- (0 until rows); j <- (0 until cols) ) m(i)(j) = l(i*rows+j)
Matrix(m)
}
}
// A little test...
val a1 = Matrix(Array[Array[Double]](Array(25,15,-5),Array(15,18,0),Array(-5,0,11)))
val a2 = Matrix(Array[Array[Double]](Array(18,22,54,42), Array(22,70,86,62), Array(54,86,174,134), Array(42,62,134,106)))
val l1 = a1.cholesky
val l2 = a2.cholesky
// Given test results
val r1 = Array[Double](5,0,0,3,3,0,-1,1,3)
val r2 = Array[Double](4.24264,0.00000,0.00000,0.00000,5.18545,6.56591,0.00000,0.00000,
12.72792,3.04604,1.64974,0.00000,9.89949,1.62455,1.84971,1.39262)
// Verify assertions
(l1.matrix.flatten.zip(r1)).foreach{ case (result,test) =>
assert(math.round( result * 100000 ) * 0.00001 == math.round( test * 100000 ) * 0.00001)
}
(l2.matrix.flatten.zip(r2)).foreach{ case (result,test) =>
assert(math.round( result * 100000 ) * 0.00001 == math.round( test * 100000 ) * 0.00001)
}
Scilab
The Cholesky decomposition is builtin, and an upper triangular matrix is returned, such that $A=L^TL$.
a = [25 15 -5; 15 18 0; -5 0 11];
chol(a)
ans =
5. 3. -1.
0. 3. 1.
0. 0. 3.
a = [18 22 54 42; 22 70 86 62;
54 86 174 134; 42 62 134 106];
chol(a)
ans =
4.2426407 5.1854497 12.727922 9.8994949
0. 6.5659052 3.0460385 1.6245539
0. 0. 1.6497422 1.849711
0. 0. 0. 1.3926212
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;Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89950 1.62455 1.84971 1.39262
Sidef
func cholesky(matrix) {
var chol = matrix.len.of { matrix.len.of(0) }
for row in ^matrix {
for col in (0..row) {
var x = matrix[row][col]
for i in (0..col) {
x -= (chol[row][i] * chol[col][i])
}
chol[row][col] = (row == col ? x.sqrt : x/chol[col][col])
}
}
return chol
}
Examples:
var example1 = [ [ 25, 15, -5 ],
[ 15, 18, 0 ],
[ -5, 0, 11 ] ];
say "Example 1:";
cholesky(example1).each { |row|
say row.map {'%7.4f' % _}.join(' ');
}
var example2 = [ [ 18, 22, 54, 42],
[ 22, 70, 86, 62],
[ 54, 86, 174, 134],
[ 42, 62, 134, 106] ];
say "\nExample 2:";
cholesky(example2).each { |row|
say row.map {'%7.4f' % _}.join(' ');
}
- Output:
Example 1: 5.0000 0.0000 0.0000 3.0000 3.0000 0.0000 -1.0000 1.0000 3.0000 Example 2: 4.2426 0.0000 0.0000 0.0000 5.1854 6.5659 0.0000 0.0000 12.7279 3.0460 1.6497 0.0000 9.8995 1.6246 1.8497 1.3926
Smalltalk
FloatMatrix>>#cholesky
| l |
l := FloatMatrix zero: numRows.
1 to: numRows do: [:i |
1 to: i do: [:k | | rowSum lkk factor aki partialSum |
i = k
ifTrue: [
rowSum := (1 to: k - 1) sum: [:j | | lkj |
lkj := l at: j @ k.
lkj squared].
lkk := (self at: k @ k) - rowSum.
lkk := lkk sqrt.
l at: k @ k put: lkk]
ifFalse: [
factor := l at: k @ k.
aki := self at: k @ i.
partialSum := (1 to: k - 1) sum: [:j | | ljk lji |
lji := l at: j @ i.
ljk := l at: j @ k.
lji * ljk].
l at: k @ i put: aki - partialSum * factor reciprocal]]].
^l
Stata
See Cholesky square-root decomposition in Stata help.
mata
: a=25,15,-5\15,18,0\-5,0,11
: a
[symmetric]
1 2 3
+----------------+
1 | 25 |
2 | 15 18 |
3 | -5 0 11 |
+----------------+
: cholesky(a)
1 2 3
+----------------+
1 | 5 0 0 |
2 | 3 3 0 |
3 | -1 1 3 |
+----------------+
: a=18,22,54,42\22,70,86,62\54,86,174,134\42,62,134,106
: a
[symmetric]
1 2 3 4
+-------------------------+
1 | 18 |
2 | 22 70 |
3 | 54 86 174 |
4 | 42 62 134 106 |
+-------------------------+
: cholesky(a)
1 2 3 4
+---------------------------------------------------------+
1 | 4.242640687 0 0 0 |
2 | 5.185449729 6.565905201 0 0 |
3 | 12.72792206 3.046038495 1.649742248 0 |
4 | 9.899494937 1.624553864 1.849711005 1.392621248 |
+---------------------------------------------------------+
Swift
func cholesky(matrix: [Double], n: Int) -> [Double] {
var res = [Double](repeating: 0, count: matrix.count)
for i in 0..<n {
for j in 0..<i+1 {
var s = 0.0
for k in 0..<j {
s += res[i * n + k] * res[j * n + k]
}
if i == j {
res[i * n + j] = (matrix[i * n + i] - s).squareRoot()
} else {
res[i * n + j] = (1.0 / res[j * n + j] * (matrix[i * n + j] - s))
}
}
}
return res
}
func printMatrix(_ matrix: [Double], n: Int) {
for i in 0..<n {
for j in 0..<n {
print(matrix[i * n + j], terminator: " ")
}
print()
}
}
let res1 = cholesky(
matrix: [25.0, 15.0, -5.0,
15.0, 18.0, 0.0,
-5.0, 0.0, 11.0],
n: 3
)
let res2 = cholesky(
matrix: [18.0, 22.0, 54.0, 42.0,
22.0, 70.0, 86.0, 62.0,
54.0, 86.0, 174.0, 134.0,
42.0, 62.0, 134.0, 106.0],
n: 4
)
printMatrix(res1, n: 3)
print()
printMatrix(res2, n: 4)
- Output:
5.0 0.0 0.0 3.0 3.0 0.0 -1.0 1.0 3.0 4.242640687119285 0.0 0.0 0.0 5.185449728701349 6.565905201197403 0.0 0.0 12.727922061357857 3.0460384954008553 1.6497422479090704 0.0 9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026
Tcl
proc cholesky a {
set m [llength $a]
set n [llength [lindex $a 0]]
set l [lrepeat $m [lrepeat $n 0.0]]
for {set i 0} {$i < $m} {incr i} {
for {set k 0} {$k < $i+1} {incr k} {
set sum 0.0
for {set j 0} {$j < $k} {incr j} {
set sum [expr {$sum + [lindex $l $i $j] * [lindex $l $k $j]}]
}
lset l $i $k [expr {
$i == $k
? sqrt([lindex $a $i $i] - $sum)
: (1.0 / [lindex $l $k $k] * ([lindex $a $i $k] - $sum))
}]
}
}
return $l
}
Demonstration code:
set test1 {
{25 15 -5}
{15 18 0}
{-5 0 11}
}
puts [cholesky $test1]
set test2 {
{18 22 54 42}
{22 70 86 62}
{54 86 174 134}
{42 62 134 106}
}
puts [cholesky $test2]
- Output:
{5.0 0.0 0.0} {3.0 3.0 0.0} {-1.0 1.0 3.0}
{4.242640687119285 0.0 0.0 0.0} {5.185449728701349 6.565905201197403 0.0 0.0} {12.727922061357857 3.0460384954008553 1.6497422479090704 0.0} {9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026}
VBA
This function returns the lower Cholesky decomposition of a square matrix fed to it. It does not check for positive semi-definiteness, although it does check for squareness. It assumes that Option Base 0 is set, and thus the matrix entry indices need to be adjusted if Base is set to 1. It also assumes a matrix of size less than 256x256. To handle larger matrices, change all Byte-type variables to Long. It takes the square matrix range as an input, and can be implemented as an array function on the same sized square range of cells as output. For example, if the matrix is in cells A1:E5, highlighting cells A10:E14, typing "=Cholesky(A1:E5)" and htting Ctrl-Shift-Enter will populate the target cells with the lower Cholesky decomposition.
Function Cholesky(Mat As Range) As Variant
Dim A() As Double, L() As Double, sum As Double, sum2 As Double
Dim m As Byte, i As Byte, j As Byte, k As Byte
'Ensure matrix is square
If Mat.Rows.Count <> Mat.Columns.Count Then
MsgBox ("Correlation matrix is not square")
Exit Function
End If
m = Mat.Rows.Count
'Initialize and populate matrix A of values and matrix L which will be the lower Cholesky
ReDim A(0 To m - 1, 0 To m - 1)
ReDim L(0 To m - 1, 0 To m - 1)
For i = 0 To m - 1
For j = 0 To m - 1
A(i, j) = Mat(i + 1, j + 1).Value2
L(i, j) = 0
Next j
Next i
'Handle the simple cases explicitly to save time
Select Case m
Case Is = 1
L(0, 0) = Sqr(A(0, 0))
Case Is = 2
L(0, 0) = Sqr(A(0, 0))
L(1, 0) = A(1, 0) / L(0, 0)
L(1, 1) = Sqr(A(1, 1) - L(1, 0) * L(1, 0))
Case Else
L(0, 0) = Sqr(A(0, 0))
L(1, 0) = A(1, 0) / L(0, 0)
L(1, 1) = Sqr(A(1, 1) - L(1, 0) * L(1, 0))
For i = 2 To m - 1
sum2 = 0
For k = 0 To i - 1
sum = 0
For j = 0 To k
sum = sum + L(i, j) * L(k, j)
Next j
L(i, k) = (A(i, k) - sum) / L(k, k)
sum2 = sum2 + L(i, k) * L(i, k)
Next k
L(i, i) = Sqr(A(i, i) - sum2)
Next i
End Select
Cholesky = L
End Function
V (Vlang)
import math
// Symmetric and Lower use a packed representation that stores only
// the Lower triangle.
struct Symmetric {
order int
ele []f64
}
struct Lower {
mut:
order int
ele []f64
}
// Symmetric.print prints a square matrix from the packed representation,
// printing the upper triange as a transpose of the Lower.
fn (s Symmetric) print() {
mut row, mut diag := 1, 0
for i, e in s.ele {
print("${e:10.5f} ")
if i == diag {
for j, col := diag+row, row; col < s.order; j += col {
print("${s.ele[j]:10.5f} ")
col++
}
println('')
row++
diag += row
}
}
}
// Lower.print prints a square matrix from the packed representation,
// printing the upper triangle as all zeros.
fn (l Lower) print() {
mut row, mut diag := 1, 0
for i, e in l.ele {
print("${e:10.5f} ")
if i == diag {
for _ in row..l.order {
print("${0.0:10.5f} ")
}
println('')
row++
diag += row
}
}
}
// cholesky_lower returns the cholesky decomposition of a Symmetric real
// matrix. The matrix must be positive definite but this is not checked.
fn (a Symmetric) cholesky_lower() Lower {
mut l := Lower{a.order, []f64{len: a.ele.len}}
mut row, mut col := 1, 1
mut dr := 0 // index of diagonal element at end of row
mut dc := 0 // index of diagonal element at top of column
for i, e in a.ele {
if i < dr {
d := (e - l.ele[i]) / l.ele[dc]
l.ele[i] = d
mut ci, mut cx := col, dc
for j := i + 1; j <= dr; j++ {
cx += ci
ci++
l.ele[j] += d * l.ele[cx]
}
col++
dc += col
} else {
l.ele[i] = math.sqrt(e - l.ele[i])
row++
dr += row
col = 1
dc = 0
}
}
return l
}
fn main() {
demo(Symmetric{3, [
f64(25),
15, 18,
-5, 0, 11]})
demo(Symmetric{4, [
f64(18),
22, 70,
54, 86, 174,
42, 62, 134, 106]})
}
fn demo(a Symmetric) {
println("A:")
a.print()
println("L:")
a.cholesky_lower().print()
}- Output:
A: 25.00000 15.00000 -5.00000 15.00000 18.00000 0.00000 -5.00000 0.00000 11.00000 L: 5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 A: 18.00000 22.00000 54.00000 42.00000 22.00000 70.00000 86.00000 62.00000 54.00000 86.00000 174.00000 134.00000 42.00000 62.00000 134.00000 106.00000 L: 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.3926
Wren
import "./matrix" for Matrix
import "./fmt" for Fmt
var arrays = [
[ [25, 15, -5],
[15, 18, 0],
[-5, 0, 11] ],
[ [18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106] ]
]
for (array in arrays) {
System.print("Original:")
Fmt.mprint(array, 3, 0)
System.print("\nLower Cholesky factor:")
Fmt.mprint(Matrix.new(array).cholesky(), 8, 5)
System.print()
}
- Output:
Original: | 25 15 -5| | 15 18 0| | -5 0 11| Lower Cholesky factor: | 5.00000 0.00000 0.00000| | 3.00000 3.00000 0.00000| |-1.00000 1.00000 3.00000| Original: | 18 22 54 42| | 22 70 86 62| | 54 86 174 134| | 42 62 134 106| Lower Cholesky factor: | 4.24264 0.00000 0.00000 0.00000| | 5.18545 6.56591 0.00000 0.00000| |12.72792 3.04604 1.64974 0.00000| | 9.89949 1.62455 1.84971 1.39262|
XPL0
real L(4*4);
func real Cholesky(A, N);
real A; int N;
real S;
int I, J, K;
[for I:= 0 to N*N-1 do L(I):= 0.;
for I:= 0 to N-1 do
for J:= 0 to I do
[S:= 0.;
for K:= 0 to J-1 do
S:= S + L(I*N+K) * L(J*N+K);
L(I*N+J):= if I = J then sqrt(A(I*N+I) - S)
else (1.0 / L(J*N+J) * (A(I*N+J) - S));
];
return L;
];
proc ShowMatrix(A, N);
real A; int N;
int I, J;
[for I:= 0 to N-1 do
[for J:= 0 to N-1 do
RlOut(0, A(I*N+J));
CrLf(0);
];
];
int N;
real M1, C1, M2, C2;
[N:= 3;
M1:= [25., 15., -5.,
15., 18., 0.,
-5., 0., 11.];
C1:= Cholesky(M1, N);
ShowMatrix(C1, N);
CrLf(0);
N:= 4;
M2:= [18., 22., 54., 42.,
22., 70., 86., 62.,
54., 86., 174., 134.,
42., 62., 134., 106.];
C2:= Cholesky(M2, N);
ShowMatrix(C2, N);
]- Output:
5.00000 0.00000 0.00000
3.00000 3.00000 0.00000
-1.00000 1.00000 3.00000
4.24264 0.00000 0.00000 0.00000
5.18545 6.56591 0.00000 0.00000
12.72792 3.04604 1.64974 0.00000
9.89949 1.62455 1.84971 1.39262
zkl
Using the GNU Scientific Library:
var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library)
fcn lowerCholesky(m){ // trans: C
rows:=m.rows;
lcm:=GSL.Matrix(rows,rows); // zero filled
foreach i,j in (rows,i+1){
s:=(0).reduce(j,'wrap(s,k){ s + lcm[i,k]*lcm[j,k] },0.0);
lcm[i,j]=( if(i==j)(m[i,i] - s).sqrt()
else 1.0/lcm[j,j]*(m[i,j] - s) );
}
lcm
}- Output:
lowerCholesky(GSL.Matrix(3,3).set(25, 15, -5, // example 1 15, 18, 0, -5, 0, 11)) .format(6).println(); 5.00, 0.00, 0.00 3.00, 3.00, 0.00 -1.00, 1.00, 3.00
- Output:
lowerCholesky(GSL.Matrix(4,4).set( // example 2
18, 22, 54, 42,
22, 70, 86, 62,
54, 86, 174, 134,
42, 62, 134, 106) )
.format(8,4).println();
4.2426, 0.0000, 0.0000, 0.0000
5.1854, 6.5659, 0.0000, 0.0000
12.7279, 3.0460, 1.6497, 0.0000
9.8995, 1.6246, 1.8497, 1.3926
Or, using lists:
fcn cholesky(mat){
rows:=mat.len();
r:=(0).pump(rows,List().write, (0).pump(rows,List,0.0).copy); // matrix of zeros
foreach i,j in (rows,i+1){
s:=(0).reduce(j,'wrap(s,k){ s + r[i][k]*r[j][k] },0.0);
r[i][j]=( if(i==j)(mat[i][i] - s).sqrt()
else 1.0/r[j][j]*(mat[i][j] - s) );
}
r
}ex1:=L( L(25.0,15.0,-5.0), L(15.0,18.0,0.0), L(-5.0,0.0,11.0) );
printM(cholesky(ex1));
println("-----------------");
ex2:=L( L(18.0, 22.0, 54.0, 42.0,),
L(22.0, 70.0, 86.0, 62.0,),
L(54.0, 86.0, 174.0, 134.0,),
L(42.0, 62.0, 134.0, 106.0,) );
printM(cholesky(ex2));fcn printM(m){ m.pump(Console.println,rowFmt) }
fcn rowFmt(row){ ("%9.5f "*row.len()).fmt(row.xplode()) }- Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 ----------------- 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
ZX Spectrum Basic
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