Matrix Inversion (and other geeky math things)

[mith]

Does anyone know anything about Cholesky decomposition? I'm wanting to write a program that involves inverting matricies, and I read somewhere that it is more stable that Gauss-Jordan elimination, but I haven't found anywhere detailing the method.

May be back with more random questions later, I'm terribly rusty with programming.

[Beartalon]

That's OK. I'm terribly rusty with this level of math.

[Chuck]

What's so hard? Do a screen print of the matrix, paste it into Paint, do a flip horizontal, and do a flip vertical.

[mole]

I found a "Cholesky's method" in my maths textbook, this looks like what you want:

code:

---

We have a matrix A





A  = a11 a12 a13


     a21 a22 a23


     a31 a32 a33





And we want two matrices B and BT


B  = b11  0   0


     b21 b22  0


     b31 b32 b33








BT = b11 b21 b31


     0  b22 b32


     0   0  b33 

---

First row/column: (b₁₁ to b_j1)
b₁₁ = sqrt(a₁₁)
b_j1 = a_j1/_b11

Diagonal: (b₁₁ to b_jj)
b_jj = sqrt(a_jj - sum [s = 1 to j - 1] (b_js²)

Everything else:
b_pj = 1/b_jj(a_pj - sum [s = 1 to j - 1] (b_ps b_sj)

I assume afterwards that you'd find B^-1B^-1T to get A^-1, like you do with the other decompositions.



[This message has been edited by mole (edited 11-30-2003 09:09 AM).]

[jesternl]

I thought decomposing was something Mozart did after he died

[math geek]

mole had it perfectly, with the sole comment that this algorithm applies only to symmetric and positively defined real matrices. And maybe to Hermitic matrices if working over the Complex plane.

[having read the thread again, i realised i mistook Cholesky for LU-decomposition - the will-reply-tomorow still applies though.]

actualy Cholesky decomposition works best with one of the L(ower) and U(pper) having 1's on their main diagonals. The algorhythm was taught to me at Numerical Computing(possibly incorrectly translated) and i have it at home. I think i can post more details tomorow.

And since my brain isn't in top shape right now, is inverting a triangular matrix so much easier that decomposing is feasible? EDIT: YES it is!

[This message has been edited by math geek (edited 12-20-2003 01:04 PM).]