Quick answer
Numerical Cholesky computes L such that A = L Lᵀ in about n³/3 flops for dense n×n SPD matrices, with good backward stability on well-conditioned SPD problems.
Formula
- A = L Lᵀ
- Dense cost ~ n³/3
- Prefer over LU when A is SPD
Introduction
Use the Cholesky Decomposition Calculator to see rounded factors on small matrices before you read about error growth on large ones.
Numerical analysis cares about operation counts, stability, and what happens when rounding makes a radicand slightly negative.
Pair this article with Cholesky vs LU and factorization calculator notes for a complete picture.
Why analysts prefer Cholesky on SPD problems
Operation count: Both Cholesky and LU are O(n³), but Cholesky constants are smaller on SPD matrices because symmetry halves the work in the classical algorithm.
Stability: For SPD matrices, Cholesky without pivoting is backward stable in standard analyses. LU may need pivoting even when Cholesky is safe.
Memory: Store only L and the lower triangle of A in optimized libraries; this site shows full grids for teaching clarity.
Near-singularity: When A is ill-conditioned, any direct method struggles. Cholesky failures often appear as tiny negative radicands due to rounding.
Sparse variants: Incomplete Cholesky preconditioners appear in iterative solvers for large sparse systems; the idea still begins with A = L Lᵀ.
Teaching simulators like this one use dense formulas so you can see each entry; production code calls optimized BLAS routines.
Numerical form of the updates
- Same formulas as exact Cholesky
- Watch radicands near zero
- Consider LDLᵀ or pivoting variants in production when SPD is doubtful
The mathematical formulas do not change; only the floating-point environment introduces edge cases.
Comparing reconstructed A from rounded L teaches backward error ideas without heavy notation.
When radicands go negative at size n, check conditioning before blaming the implementation.
Numerical workflow
- Confirm SPD theoretically. Numerical Cholesky should not run on indefinite matrices without modifications.
- Choose Cholesky over LU when valid. Save time and memory at scale.
- Monitor radicands. Tiny negatives suggest ill-conditioning or rounding.
- Use triangular solves after factorization. Forward solve with L, backward with Lᵀ.
- Validate on small n with the calculator. Build intuition before trusting black-box libraries.
Rounding illustration
Take a well-conditioned 3×3 SPD matrix from the examples article. Hand-compute L exactly, then compare to the calculator’s three-decimal display.
Reconstruct with rounded L and note the small drift in off-diagonals. That drift is the numerical story in miniature.
Repeat with a nearly singular SPD matrix if your course provides one; watch how early square roots shrink.