Cholesky decomposition
You are encouraged to solve this task according to the task description, using any language you may know.
Every symmetric, positive definite matrix A can be decomposed into a product of a unique lower triangular matrix L and its transpose:
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
Ada
decomposition.ads: <lang Ada>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;</lang>
decomposition.adb: <lang Ada>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;</lang>
Example usage: <lang Ada>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;</lang>
Output:
Example 1: A: 25.000 15.000 -5.000 15.000 18.000 0.000 -5.000 0.000 11.000 L: 5.000 0.000 0.000 3.000 3.000 0.000 -1.000 1.000 3.000 Example 2: A: 18.000 22.000 54.000 42.000 22.000 70.000 86.000 62.000 54.000 86.000 174.000 134.000 42.000 62.000 134.000 106.000 L: 4.243 0.000 0.000 0.000 5.185 6.566 0.000 0.000 12.728 3.046 1.650 0.000 9.899 1.625 1.850 1.393
ALGOL 68
Note: This specimen retains the original C coding style. diff
<lang algol68>#!/usr/local/bin/a68g --script #
MODE FIELD=LONG REAL; PROC (FIELD)FIELD field sqrt = long sqrt; INT field prec = 5; FORMAT field fmt = $g(-(2+1+field prec),field prec)$;
MODE MAT = [0,0]FIELD;
PROC cholesky = (MAT a) MAT:(
[UPB a, 2 UPB a]FIELD l;
FOR i FROM LWB a TO UPB a DO
FOR j FROM 2 LWB a TO i DO
FIELD s := 0;
FOR k FROM 2 LWB a TO j-1 DO
s +:= l[i,k] * l[j,k]
OD;
l[i,j] := IF i = j
THEN field sqrt(a[i,i] - s)
ELSE 1.0 / l[j,j] * (a[i,j] - s) FI
OD;
FOR j FROM i+1 TO 2 UPB a DO
l[i,j]:=0 # Not required if matrix is declared as triangular #
OD
OD;
l
);
PROC print matrix v1 =(MAT a)VOID:(
FOR i FROM LWB a TO UPB a DO
FOR j FROM 2 LWB a TO 2 UPB a DO
printf(($g(-(2+1+field prec),field prec)$, a[i,j]))
OD;
printf($l$)
OD
);
PROC print matrix =(MAT a)VOID:(
FORMAT vector fmt = $"("f(field fmt)n(2 UPB a-2 LWB a)(", " f(field fmt))")"$;
FORMAT matrix fmt = $"("f(vector fmt)n( UPB a- LWB a)(","lxf(vector fmt))")"$;
printf((matrix fmt, a))
);
main: (
MAT m1 = ((25, 15, -5),
(15, 18, 0),
(-5, 0, 11));
MAT c1 = cholesky(m1);
print matrix(c1);
printf($l$);
MAT m2 = ((18, 22, 54, 42),
(22, 70, 86, 62),
(54, 86, 174, 134),
(42, 62, 134, 106));
MAT c2 = cholesky(m2);
print matrix(c2)
)</lang> Output:
(( 5.00000, 0.00000, 0.00000), ( 3.00000, 3.00000, 0.00000), (-1.00000, 1.00000, 3.00000)) (( 4.24264, 0.00000, 0.00000, 0.00000), ( 5.18545, 6.56591, 0.00000, 0.00000), (12.72792, 3.04604, 1.64974, 0.00000), ( 9.89949, 1.62455, 1.84971, 1.39262))
C
<lang 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;
}</lang> Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Common Lisp
<lang lisp>;; Calculates the Cholesky decomposition matrix L
- for a positive-definite, symmetric nxn matrix A.
(defun chol (A)
(let* ((n (car (array-dimensions A)))
(L (make-array `(,n ,n) :initial-element 0)))
(do ((k 0 (incf k))) ((> k (- n 1)) nil)
;; First, calculate diagonal elements L_kk.
(setf (aref L k k)
(sqrt (- (aref A k k)
(do* ((j 0 (incf j))
(sum (expt (aref L k j) 2)
(incf sum (expt (aref L k j) 2))))
((> j (- k 1)) sum)))))
;; Then, all elements below a diagonal element, L_ik, i=k+1..n.
(do ((i (+ k 1) (incf i)))
((> i (- n 1)) nil)
(setf (aref L i k)
(/ (- (aref A i k)
(do* ((j 0 (incf j))
(sum (* (aref L i j) (aref L k j))
(incf sum (* (aref L i j) (aref L k j)))))
((> j (- k 1)) sum)))
(aref L k k)))))
;; Return the calculated matrix L. L))</lang>
<lang lisp>;; Example 1: (setf A (make-array '(3 3) :initial-contents '((25 15 -5) (15 18 0) (-5 0 11)))) (chol A)
- 2A((5.0 0 0)
(3.0 3.0 0) (-1.0 1.0 3.0))</lang>
<lang lisp>;; Example 2: (setf B (make-array '(4 4) :initial-contents '((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106)))) (chol B)
- 2A((4.2426405 0 0 0)
(5.18545 6.565905 0 0) (12.727922 3.0460374 1.6497375 0) (9.899495 1.6245536 1.849715 1.3926151))</lang>
<lang lisp>;; case of matrix stored as a list of lists (inner lists are rows of matrix)
- as above, returns the Cholesky decomposition matrix of a square positive-definite, symmetric matrix
(defun cholesky (m)
(let ((l (list (list (sqrt (caar m))))) x (j 0) i)
(dolist (cm (cdr m) (mapcar #'(lambda (x) (nconc x (make-list (- (length m) (length x)) :initial-element 0))) l))
(setq x (list (/ (car cm) (caar l))) i 0)
(dolist (cl (cdr l))
(setf (cdr (last x)) (list (/ (- (elt cm (incf i)) (*v x cl)) (car (last cl))))))
(setf (cdr (last l)) (list (nconc x (list (sqrt (- (elt cm (incf j)) (*v x x))))))))))
- where *v is the scalar product defined as
(defun *v (v1 v2) (reduce #'+ (mapcar #'* v1 v2)))</lang>
<lang lisp>;; example 1 CL-USER> (setf a '((25 15 -5) (15 18 0) (-5 0 11))) ((25 15 -5) (15 18 0) (-5 0 11)) CL-USER> (cholesky a) ((5 0 0) (3 3 0) (-1 1 3)) CL-USER> (format t "~{~{~5d~}~%~}" (cholesky a))
5 0 0 3 3 0 -1 1 3
NIL</lang>
<lang lisp>;; example 2 CL-USER> (setf a '((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106))) ((18 22 54 42) (22 70 86 62) (54 86 174 134) (42 62 134 106)) CL-USER> (cholesky a) ((4.2426405 0 0 0) (5.18545 6.565905 0 0) (12.727922 3.0460374 1.6497375 0) (9.899495 1.6245536 1.849715 1.3926151)) CL-USER> (format t "~{~{~10,5f~}~%~}" (cholesky a))
4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89950 1.62455 1.84971 1.39262
NIL</lang>
D
<lang d>import std.stdio, std.math, std.numeric;
T[][] cholesky(T)(in T[][] A) {
auto L = new T[][](A.length, A.length);
foreach (r, row; L)
row[r+1 .. $] = 0;
foreach (i; 0 .. A.length)
foreach (j; 0 .. i+1) {
T 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() {
double[][] m1 = [[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]];
foreach (row; cholesky(m1))
writeln(row);
writeln();
double[][] m2 = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]];
foreach (row; cholesky(m2))
writeln(row);
}</lang> Output:
[5, 0, 0] [3, 3, 0] [-1, 1, 3] [4.24264, 0, 0, 0] [5.18545, 6.56591, 0, 0] [12.7279, 3.04604, 1.64974, 0] [9.89949, 1.62455, 1.84971, 1.39262]
Icon and Unicon
<lang Icon> procedure cholesky (array)
result := make_square_array (*array)
every (i := 1 to *array) do {
every (k := 1 to i) do {
sum := 0
every (j := 1 to (k-1)) do {
sum +:= result[i][j] * result[k][j]
}
if (i = k)
then result[i][k] := sqrt(array[i][i] - sum)
else result[i][k] := 1.0 / result[k][k] * (array[i][k] - sum)
}
}
return result
end
procedure make_square_array (n)
result := [] every (1 to n) do push (result, list(n, 0)) return result
end
procedure print_array (array)
every (row := !array) do {
every writes (!row || " ")
write ()
}
end
procedure do_cholesky (array)
write ("Input:")
print_array (array)
result := cholesky (array)
write ("Result:")
print_array (result)
end
procedure main ()
do_cholesky ([[25,15,-5],[15,18,0],[-5,0,11]]) do_cholesky ([[18,22,54,42],[22,70,86,62],[54,86,174,134],[42,62,134,106]])
end </lang>
Output:
Input: 25 15 -5 15 18 0 -5 0 11 Result: 5.0 0 0 3.0 3.0 0 -1.0 1.0 3.0 Input: 18 22 54 42 22 70 86 62 54 86 174 134 42 62 134 106 Result: 4.242640687 0 0 0 5.185449729 6.565905201 0 0 12.72792206 3.046038495 1.649742248 0 9.899494937 1.624553864 1.849711005 1.392621248
J
Solution: <lang j>mp=: +/ . * NB. matrix product h =: +@|: NB. conjugate transpose
cholesky=: 3 : 0
n=. #A=. y
if. 1>:n do.
assert. (A=|A)>0=A NB. check for positive definite
%:A
else.
p=. >.n%2 [ q=. <.n%2
X=. (p,p) {.A [ Y=. (p,-q){.A [ Z=. (-q,q){.A
L0=. cholesky X
L1=. cholesky Z-(T=.(h Y) mp %.X) mp Y
L0,(T mp L0),.L1
end.
)</lang> See Choleski Decomposition essay on the J Wiki.
Examples: <lang j> eg1=: 25 15 _5 , 15 18 0 ,: _5 0 11
eg2=: 18 22 54 42 , 22 70 86 62 , 54 86 174 134 ,: 42 62 134 106 cholesky eg1 5 0 0 3 3 0
_1 1 3
cholesky eg2
4.24264 0 0 0 5.18545 6.56591 0 0 12.7279 3.04604 1.64974 0 9.89949 1.62455 1.84971 1.39262</lang>
Java
<lang java5>import java.util.Arrays;
public class Cholesky { public static double[][] chol(double[][] a){ int m = a.length; double[][] l = new double[m][m]; //automatically initialzed to 0's for(int i = 0; i< m;i++){ for(int k = 0; k < (i+1); k++){ double sum = 0; for(int j = 0; j < k; j++){ sum += l[i][j] * l[k][j]; } l[i][k] = (i == k) ? Math.sqrt(a[i][i] - sum) : (1.0 / l[k][k] * (a[i][k] - sum)); } } return l; }
public static void main(String[] args){ double[][] test1 = {{25, 15, -5}, {15, 18, 0}, {-5, 0, 11}}; System.out.println(Arrays.deepToString(chol(test1))); double[][] test2 = {{18, 22, 54, 42}, {22, 70, 86, 62}, {54, 86, 174, 134}, {42, 62, 134, 106}}; System.out.println(Arrays.deepToString(chol(test2))); } }</lang> Output:
[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[4.242640687119285, 0.0, 0.0, 0.0], [5.185449728701349, 6.565905201197403, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.899494936611667, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]
PicoLisp
<lang 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 (* 1.0 S))
(*/ S 1.0 (get L J J)) ) ) ) ) )
(for R L
(for N R (prin (align 9 (round N 5))))
(prinl) ) ) )</lang>
Test: <lang PicoLisp>(cholesky
'((25.0 15.0 -5.0) (15.0 18.0 0) (-5.0 0 11.0)) )
(prinl)
(cholesky
(quote
(18.0 22.0 54.0 42.0)
(22.0 70.0 86.0 62.0)
(54.0 86.0 174.0 134.0)
(42.0 62.0 134.0 106.0) ) )</lang>
Output:
5.00000 0.00000 0.00000 3.00000 3.00000 0.00000 -1.00000 1.00000 3.00000 4.24264 0.00000 0.00000 0.00000 5.18545 6.56591 0.00000 0.00000 12.72792 3.04604 1.64974 0.00000 9.89949 1.62455 1.84971 1.39262
Python
<lang python>import math, pprint
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] = math.sqrt(A[i][i] - s) if (i == j) else \
(1.0 / L[j][j] * (A[i][j] - s))
return L
m1 = [[25, 15, -5],
[15, 18, 0],
[-5, 0, 11]]
pprint.pprint(cholesky(m1)) print
m2 = [[18, 22, 54, 42],
[22, 70, 86, 62],
[54, 86, 174, 134],
[42, 62, 134, 106]]
pprint.pprint(cholesky(m2))</lang> Output:
[[5.0, 0.0, 0.0], [3.0, 3.0, 0.0], [-1.0, 1.0, 3.0]] [[4.2426406871192848, 0.0, 0.0, 0.0], [5.1854497287013492, 6.5659052011974026, 0.0, 0.0], [12.727922061357857, 3.0460384954008553, 1.6497422479090704, 0.0], [9.8994949366116671, 1.624553864213788, 1.8497110052313648, 1.3926212476456026]]
Tcl
<lang tcl>proc cholesky a {
set m [llength $a]
set n [llength [lindex $a 0]]
set l [lrepeat $m [lrepeat $n 0.0]]
for {set i 0} {$i < $m} {incr i} {
for {set k 0} {$k < $i+1} {incr k} { set sum 0.0 for {set j 0} {$j < $k} {incr j} { set sum [expr {$sum + [lindex $l $i $j] * [lindex $l $k $j]}] } lset l $i $k [expr { $i == $k ? sqrt([lindex $a $i $i] - $sum) : (1.0 / [lindex $l $k $k] * ([lindex $a $i $k] - $sum)) }] }
} return $l
}</lang> Demonstration code: <lang tcl>set test1 {
{25 15 -5}
{15 18 0}
{-5 0 11}
} puts [cholesky $test1] set test2 {
{18 22 54 42}
{22 70 86 62}
{54 86 174 134}
{42 62 134 106}
} puts [cholesky $test2]</lang> Output:
{5.0 0.0 0.0} {3.0 3.0 0.0} {-1.0 1.0 3.0}
{4.242640687119285 0.0 0.0 0.0} {5.185449728701349 6.565905201197403 0.0 0.0} {12.727922061357857 3.0460384954008553 1.6497422479090704 0.0} {9.899494936611667 1.624553864213788 1.8497110052313648 1.3926212476456026}