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the Cholesky decomposition
There are various methods for calculating the Cholesky decomposition. The computational complexity of commonly used algorithms is O(n3) in general.[citation needed] The algorithms described below all involve about n3/3 FLOPs, where n is the size of the matrix A. Hence, they are half the cost of the LU decomposition, which uses 2n3/3 FLOPs (see Trefethen and Bau 1997).
Which of the algorithms below is faster depends on the details of the implementation. Generally, the first algorithm will be slightly slower because it accesses the data in a more irregular manner.
The Cholesky algorithm
The Cholesky algorithm, used to calculate the decomposition matrix L, is a modified version of Gaussian elimination.
The recursive algorithm starts with i := 1 and
At step i, the matrix A(i) has the following form:
where Ii−1 denotes the identity matrix of dimension i − 1.
If we now define the matrix Li by
then we can write A(i) as
where
Note that bi bi* is an outer product, therefore this algorithm is called the outer product version in (Golub & Van Loan).
We repeat this for i from 1 to n. After n steps, we get A(n+1) = I. Hence, the lower triangular matrix L we are looking for is calculated as
[edit] The Cholesky–Banachiewicz and Cholesky–Crout algorithms
If we write out the equation A = LL*,
we obtain the following formula for the entries of L:
The expression under the square root is always positive if A is real and positive-definite.
For complex Hermitian matrix, the following formula applies:
So we can compute the (i, j) entry if we know the entries to the left and above. The computation is usually arranged in either of the following orders.
The Cholesky–Banachiewicz algorithm starts from the upper left corner of the matrix L and proceeds to calculate the matrix row by row.
The Cholesky–Crout algorithm starts from the upper left corner of the matrix L and proceeds to calculate the matrix column by column.
[edit] Stability of the computation
Suppose that we want to solve a well-conditioned system of linear equations. If the LU decomposition is used, then the algorithm is unstable unless we use some sort of pivoting strategy. In the latter case, the error depends on the so-called growth factor of the matrix, which is usually (but not always) small.
Now, suppose that the Cholesky decomposition is applicable. As mentioned above, the algorithm will be twice as fast. Furthermore, no pivoting is necessary and the error will always be small. Specifically, if we want to solve Ax = b, and y denotes the computed solution, then y solves the disturbed system (A + E)y = b where
Here, || ||2 is the matrix 2-norm, cn is a small constant depending on n, and ε denotes the unit round-off.
There is one small problem with the Cholesky decomposition. Note that we must compute square roots in order to find the Cholesky decomposition. If the matrix is real symmetric and positive definite, then the numbers under the square roots are always positive in exact arithmetic. Unfortunately, the numbers can become negative because of round-off errors, in which case the algorithm cannot continue. However, this can only happen if the matrix is very ill-conditioned.
[edit] Avoiding taking square roots
An alternative form is the factorization[2]
This form eliminates the need to take square roots. When A is positive definite the elements of the diagonal matrix D are all positive. However this factorization may be used for most invertible symmetric matrices; an example of an invertible matrix whose decomposition is undefined is one where the first entry is zero.
If A is real, the following recursive relations apply for the entries of D and L:
For complex Hermitian matrix A, the following formula applies:
The LDLT and LLT factorizations (note that L is different between the two) may be easily related:
The last expression is the product of a lower triangular matrix and its transpose, as is the LLT factorization.
[edit] Proof for positive semi-definite matrices
The above algorithms show that every positive definite matrix A has a Cholesky decomposition. This result can be extended to the positive semi-definite case by a limiting argument. The argument is not fully constructive, i.e., it gives no explicit numerical algorithms for computing Cholesky factors.
If A is an n-by-n positive semi-definite matrix, then the sequence {Ak} = {A + (1/k)In} consists of positive definite matrices. (This is an immediate consequence of, for example, the spectral mapping theorem for the polynomial functional calculus.) Also,
in operator norm. From the positive definite case, each Ak has Cholesky decomposition Ak = LkLk*. By property of the operator norm,
So {Lk} is a bounded set in the Banach space of operators, therefore relatively compact (because the underlying vector space is finite dimensional). Consequently it has a convergent subsequence, also denoted by {Lk}, with limit L. It can be easily checked that this L has the desired properties, i.e. A = LL* and L is lower triangular with non-negative diagonal entries: for all x and y,
Therefore A = LL*. Because the underlying vector space is finite dimensional, all topologies on the space of operators are equivalent. So Lk tends to L in norm means Lk tends to L entrywise. This in turn implies that, since each Lk is lower triangular with non-negative diagonal entries, L is also.
Generalization
The Cholesky factorization can be generalized to (not necessarily finite) matrices with operator entries. Let be a sequence of Hilbert spaces. Consider the operator matrix
acting on the direct sum
where each
is a bounded operator. If A is positive (semidefinite) in the sense that for all finite k and for any
we have , then there exists a lower triangular operator matrix L such that A = LL*. One can also take the diagonal entries of L to be positive.
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