Morten was visiting Dyalog clients and forwarded a request: Can we have the Cholesky decomposition?

If A is a Hermitian, positive-definite matrix, its Cholesky decomposition [0] is a lower-triangular matrix L such that A ≡ L +.× +⍉L. The matrix L is a sort of “square root” of the matrix A.

For example:

   ⎕io←0 ⋄ ⎕rl←7*5 ⋄ ⎕pp←6

   A←t+.×⍉t←¯10+?5 5⍴20
   A
231   42  ¯63  16  26
 42  199 ¯127 ¯68  53
¯63 ¯127  245  66 ¯59
 16  ¯68   66 112 ¯75
 26   53  ¯59 ¯75  75

   L←Cholesky A
   L
15.1987   0        0        0       0
 2.7634  13.8334   0        0       0
¯4.1451  ¯8.35263 12.5719   0       0
 1.05272 ¯5.12592  2.1913   8.93392 0
 1.71067  3.48957 ¯1.81055 ¯6.15028 4.33502

   A ≡ L +.× +⍉L
1

For real matrices, “Hermitian” reduces to symmetric and the conjugate transpose +⍉ to transpose ⍉. The symmetry arises in solving least-squares problems.

Some writers asserted that an algorithm for the Cholesky decomposition “cannot be expressed without a loop” [1] and that “a Pascal program is a natural way of expressing the essentially iterative algorithm” [2]. You can judge for yourself whether the algorithm presented here belies these assertions.

The Algorithm [3]

A recursive solution for the Cholesky decomposition obtains by considering A as a 2-by-2 matrix of matrices. It is algorithmically interesting but not necessarily the best with respect to numerical stability.

Cholesky←{
 ⍝ Cholesky decomposition of a Hermitian positive-definite matrix
    1≥n←≢⍵:⍵*0.5
    p←⌈n÷2
    q←⌊n÷2
    X←(p,p)↑⍵ ⊣ Y←(p,-q)↑⍵ ⊣ Z←(-q,q)↑⍵
    L0←∇ X
    L1←∇ Z-T+.×Y ⊣ T←(+⍉Y)+.×⌹X
    ((p,n)↑L0)⍪(T+.×L0),L1
}

The recursive block matrix technique can be used for triangular matrix inversion [4], LU decomposition [5], and QR decomposition [6].

Proof of Correctness

The algorithm can be stated as a block matrix equation:

  ┌───┬───┐          ┌──────────────┬──────────────┐
  │ X │ Y │          │   L0 ← ∇ X   │       0      │
∇ ├───┼───┤  ←→  L ← ├──────────────┼──────────────┤ 
  │+⍉Y│ Z │          │    T+.×L0    │L1 ← ∇ Z-T+.×Y│
  └───┴───┘          └──────────────┴──────────────┘

where T←(+⍉Y)+.×⌹X. To verify that the result is correct, we need to show that A≡L+.×+⍉L and that L is lower triangular. For the first, we need to show:

┌───┬───┐     ┌──────┬───────┐     ┌────────┬────────┐
│ X │ Y │     │  L0  │   0   │     │  +⍉L0  │+⍉T+.×L0│
├───┼───┤  ≡  ├──────┼───────┤ +.× ├────────┼────────┤
│+⍉Y│ Z │     │T+.×L0│   L1  │     │    0   │  +⍉L1  │
└───┴───┘     └──────┴───────┘     └────────┴────────┘

that is:

(a)  X     ≡ L0 +.× +⍉L0
(b)  Y     ≡ L0 +.× +⍉ T+.×L0
(c)  (+⍉Y) ≡ (T+.×L0) +.× +⍉L0
(d)  Z     ≡ ((T+.×L0) +.× (+⍉T+.×L0)) + (L1+.×+⍉L1)

(a) holds because L0 is the Cholesky decomposition of X.

(b) is seen to be true as follows:
L0 +.× +⍉ T+.×L0
L0 +.× +⍉ ((+⍉Y)+.×⌹X)+.×L0 definition of T
L0 +.× (+⍉L0)+.×(+⍉⌹X)+.×Y +⍉A+.×B ←→ (+⍉B)+.×+⍉A and +⍉+⍉Y ←→ Y
(L0+.×+⍉L0)+.×(+⍉⌹X)+.×Y +.× is associative
X+.×(+⍉⌹X)+.×Y (a)
X+.×(⌹X)+.×Y X and hence ⌹X are Hermitian
I+.×Y associativity; matrix inverse
Y identity matrix

(c) follows from (b) by application of +⍉ to both sides of the equation.

(d) turns on that L1 is the Cholesky decomposition of Z-T+.×Y:

((T+.×L0)+.×(+⍉T+.×L0)) + (L1+.×+⍉L1)
((T+.×L0)+.×(+⍉T+.×L0)) + Z-T+.×Y
((T+.×L0)+.×(+⍉L0)+.×+⍉T) + Z-T+.×Y
(T+.×X+.×+⍉T) + Z-T+.×Y
(T+.×X+.×+⍉(+⍉Y)+.×⌹X) + Z-T+.×Y
(T+.×X+.×(+⍉⌹X)+.×Y) + Z-T+.×Y
(T+.×X+.×(⌹X)+.×Y) + Z-T+.×Y
(T+.×I+.×Y) + Z-T+.×Y
(T+.×Y) + Z-T+.×Y
Z

Finally, L is lower triangular if L0 and L1 are lower triangular, and they are by induction.

A Complex Example

   ⎕io←0 ⋄ ⎕rl←7*5

   A←t+.×+⍉t←(¯10+?5 5⍴20)+0j1ׯ10+?5 5⍴20
   A
382        17J131  ¯91J¯124 ¯43J0107  20J0035
 17J¯131  314     ¯107J0005 ¯60J¯154  26J¯137
¯91J0124 ¯107J¯05  379       49J0034  20J0137
¯43J¯107  ¯60J154   49J¯034 272       35J0103
 20J¯035   26J137   20J¯137  35J¯103 324

   L←Cholesky A

   A ≡ L +.× +⍉L
1
   0≠L
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1

A Personal Note

This way of computing the Cholesky decomposition was one of the topics of [7] and was the connection (through Professor Shlomo Moran) by which I acquired an Erdős number of 2.

References

1. Wikipedia, Cholesky decomposition, 2014-11-25.
2. Thomson, Norman, J-ottings 7, The Education Vector, Volume 12, Number 2, 1995, pp. 21-25.
3. Muller, Antje, Tineke van Woudenberg, and Alister Young, Two Numerical Algorithms in J, The Education Vector, Volume 12, Number 2, 1995, pp. 26-30.
4. Hui, Roger, Cholesky Decomposition, J Wiki Essay, 2005-10-14.
5. Hui, Roger, Triangular Matrix Inverse, J Wiki Essay, 2005-10-27.
6. Hui, Roger, LU Decomposition, J Wiki Essay, 2005-10-31.
7. Hui, Roger, QR Decomposition, J Wiki Essay, 2005-10-30.
8. Ibarra, Oscar, Shlomo Moran, and Roger Hui, A Generalization of the Fast LUP Matrix Decomposition Algorithm and Applications, Journal of Algorithms 3, 1982, pp. 45-56.

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