is a powerful tool for decomposing symmetric positive definite matrices. It's faster than LU factorization and has wide-ranging applications in linear algebra, optimization, and statistics.
This method breaks down a matrix into the product of a and its transpose. It's computationally efficient, numerically stable, and useful for and .
Cholesky Factorization Properties
Decomposition and Characteristics
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Cholesky factorization decomposes a A into the product of a lower triangular matrix and its transpose [A = LL^T](https://www.fiveableKeyTerm:a_=_ll^t)
Symmetric positive definite matrices have and satisfy the property xTAx>0 for all non-zero vectors x
Cholesky factor L remains unique with positive diagonal entries ensuring in computations
Factorization only exists for positive definite matrices making it a useful test for this property
Applications and Efficiency
Applicable in various fields (linear algebra, optimization, statistical computations)
Useful for solving systems of linear equations, computing determinants, and generating random samples from multivariate normal distributions
Computationally efficient requiring approximately n3/3 floating-point operations for an n × n matrix
Twice as fast as LU factorization for symmetric positive definite matrices
Cholesky Factorization Implementation
Standard and Blocked Algorithms
computes lower triangular matrix L column by column using inner products and square roots
improve on modern computer architectures
updates the entire matrix at each step as an alternative implementation
Variants and Specialized Implementations
handles symmetric matrices not necessarily positive definite
Produces factorization of the form [LDLT](https://www.fiveableKeyTerm:ldlt), where D represents a
Ensures factorization exists even for indefinite matrices useful in optimization algorithms
and allow efficient dynamic updates to factorization as underlying matrix changes
take advantage of multi-core processors or distributed computing environments
algorithms exploit sparsity patterns to reduce computational complexity and storage requirements
Applications of Cholesky Factorization
Solving Linear Systems and Least-Squares Problems
Solves system Ax = b by first solving Ly = b for y, then solving [LT](https://www.fiveableKeyTerm:lt)x=y for x
Solution process involves followed by , each with [O(n2)](https://www.fiveableKeyTerm:o(n2)) complexity
Applies to least-squares problems of the form [min ||Ax - b||_2](https://www.fiveableKeyTerm:min_||ax_-_b||_2) using ATAx=ATb
Numerically stable for well-conditioned A in least-squares problems
Combines with iterative refinement to improve solution accuracy in presence of rounding errors
Optimization and Large-Scale Problems
Used in to solve linear systems arising in or
Sparse Cholesky factorization with appropriate ordering techniques reduces computational cost for large, sparse systems
Efficient for dynamic updates in optimization algorithms through rank-one updates and downdates
Efficiency vs Stability of Cholesky Factorization
Computational Complexity and Stability
Time complexity [O(n3/3)](https://www.fiveableKeyTerm:o(n3/3)) for dense matrices more efficient than LU factorization for symmetric positive definite matrices
O(n2) for storing lower triangular factor L
Numerically stable without pivoting due to positive definiteness of the matrix
in computed Cholesky factor L bounded by O(ε[κ(A)](https://www.fiveableKeyTerm:κ(a))), where ε represents and κ(A) condition number of A
Backward error analysis shows computed Cholesky factor as exact Cholesky factor of slightly perturbed matrix
Performance Considerations
Accuracy degrades for ill-conditioned matrices but remains more stable than methods like Gaussian elimination
Block algorithms improve cache efficiency and overall performance on modern computer architectures
Parallel implementations achieve near-linear speedup on shared-memory systems for sufficiently large matrices