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45.1 Definitions for cholesky |
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Compute Cholesky decomposition of A,
a lower-triangular matrix L such that L . transpose(L) = A
.
Some examples follow.
Example 1:
(%i1) load("cholesky")$ (%i2) A : matrix ([a, b, c], [d, e, f], [g, h, i]); [ a b c ] [ ] (%o2) [ d e f ] [ ] [ g h i ] (%i3) A2 : transpose (A) . A; [ 2 2 2 ] [ g + d + a g h + d e + a b g i + d f + a c ] [ ] (%o3) [ 2 2 2 ] [ g h + d e + a b h + e + b h i + e f + b c ] [ ] [ 2 2 2 ] [ g i + d f + a c h i + e f + b c i + f + c ] (%i4) B : cholesky (A2)$ (%i5) B . transpose (B) - A2; [ 0 0 0 ] [ ] (%o5) [ 0 0 0 ] [ ] [ 0 0 0 ] |
Example 2:
(%i6) A : matrix ([2, 3, 4], [-2, 2,- 3], [11, -2, 3]); [ 2 3 4 ] [ ] (%o6) [ - 2 2 - 3 ] [ ] [ 11 - 2 3 ] (%i7) A2 : transpose (A) . A; [ 129 - 20 47 ] [ ] (%o7) [ - 20 17 0 ] [ ] [ 47 0 34 ] (%i8) B : cholesky (A2); [ sqrt(129) 0 0 ] [ ] [ 20 sqrt(1793) ] [ - --------- ---------- 0 ] (%o8) [ sqrt(129) sqrt(129) ] [ ] [ 47 940 sqrt(129) 153 ] [ --------- -------------- ---------- ] [ sqrt(129) 129 sqrt(1793) sqrt(1793) ] (%i9) B . transpose (B) - A2; [ 0 0 0 ] [ ] (%o9) [ 0 0 0 ] [ ] [ 0 0 0 ] |
To use this function write first load("cholesky")
.
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