Problem Statement

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

L

{\displaystyle L}

is called the Cholesky factor of

A

{\displaystyle A}

, and can be interpreted as a generalized square root of

A

{\displaystyle A}

, 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 (

l

k k

{\displaystyle l_{kk}}

) of

L

{\displaystyle L}

there is a calculation pattern: or in general: For the elements below the diagonal (

l

i k

{\displaystyle l_{ik}}

, where

i

k

{\displaystyle i>k}

) 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

L

{\displaystyle L}

for every given symmetric, positive definite nxn matrix

A

{\displaystyle A}

. You should then test it on the following two examples and include your output. Example 1: Example 2:

Let's start with the solution:

(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))))


(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]]