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Randomized Complete Pivoting for Solving Symmetric Indefinite Linear Systems
Yuehua Feng, Jianwei Xiao, Ming Gu
TL;DR
Bunch-Kaufman and Aasen methods for symmetric indefinite systems can encounter numerical instability, motivating a pivoting strategy that balances efficiency and stability. The paper develops RCP using random projections and reports GECP-comparable element-growth bounds and numerical stability, with efficiency comparable to established methods.
Problem
Bunch-Kaufman and Aasen methods for symmetric indefinite systems can suffer occasional numerical instability from potentially exponential element growth or unbounded factorization entries.
Source: §1 · p. 2Method
Results
RCP’s element-growth upper bound is comparable to GECP’s, and experiments report stability comparable to GECP with only slightly slower runtime than Bunch-Kaufman and Aasen.
Source: §1 · p. 2, §6 · p. 31Takeaways & Limitations
RCP combines GECP-comparable stability with computational efficiency comparable to Bunch-Kaufman and Aasen in the reported experiments.
Source: §6 · p. 31The random matrix used in the updated projection may be correlated with the original projection, potentially making the new projection less effective.
Source: §3 · p. 8
Abstract
from arXiv · showhide
The Bunch-Kaufman algorithm and Aasen's algorithm are two of the most widely used methods for solving symmetric indefinite linear systems, yet they both are known to suffer from occasional numerical instability due to potentially exponential element growth or unbounded entries in the matrix factorization. In this work, we develop a randomized complete pivoting (RCP) algorithm for solving symmetric indefinite linear systems. RCP is comparable to the Bunch-Kaufman algorithm and Aasen's algorithm in computational efficiency, yet enjoys theoretical element growth and bounded entries in the factorization comparable to that of complete-pivoting, up to a theoretical failure probability that exponentially decays with an oversampling parameter. Our finite precision analysis shows that RCP is as numerically stable as Gaussian elimination with complete pivoting, and RCP has been observed to be numerically stable in our extensive numerical experiments.
1 Introduction
Symmetric indefinite systems arise in several scientific and optimization problems, but standard dense solvers face trade-offs between pivot-search cost and numerical stability. The paper introduces randomized complete pivoting (RCP), whose overhead diminishes with matrix dimensions and whose element-growth bound is comparable to complete pivoting.
1 Introduction
- Symmetric indefinite systems arise in acoustics, electromagnetism, structural physics, linear least squares, and saddle-point problems.Dense systems have four well-known approaches: Bunch-Parlett, Bunch-Kaufman, bounded Bunch-Kaufman, and Aasen’s algorithm.Source: §1 · p. 1
- Partial pivoting in block LDLT can be unstable: Bunch-Kaufman reduces comparisons from O(n^3) to O(n^2), but cannot control its multipliers.Complete pivoting offers strong backward error and element-growth bounds but requires searching the whole Schur complement; bounded Bunch-Kaufman can bound multipliers near one, at potentially high comparison cost.Source: §1 · p. 2
- RCP uses random projections to obtain an element-growth upper bound comparable to GECP and Bunch-Parlett, with overhead that diminishes as matrix dimensions increase.The paper also reports an error analysis demonstrating RCP’s numerical stability.Source: §1 · p. 2
- The paper presents RCP’s algorithm and operation-count analysis, then analyzes randomized-pivot reliability, element and column growth, and finite-precision behavior.Numerical experiments are reported to demonstrate effectiveness and reliability.Source: §1 · pp. 2–3
2 Preliminaries and Background
This background establishes the probabilistic projection tools, block LDLT framework, and Bunch-Kaufman pivoting strategy used to develop and analyze RCP.
2.1 Preliminaries
- Lemma 2.1 is a weakened Johnson-Lindenstrauss result, and its failure-probability dependence on the projection dimension is logarithmic, supporting high reliability of randomized algorithms.The ϵ-JL condition describes approximate preservation of a vector’s length under a random projection.Source: §2.1 · pp. 3–4
- The preliminaries also invoke the law of total probability and Hadamard’s inequality as tools for subsequent analysis.Hadamard’s inequality is identified as useful in the element-growth analysis.Source: §2.1 · pp. 3–4
2.2 The Block LDLT Factorization
- Block LDLT factors a symmetric indefinite matrix using a unit lower-triangular L and symmetric block-diagonal D with 1×1 or 2×2 blocks, applying permutations to successive Schur complements.Pivot choices affect both computational efficiency and numerical stability; element growth is a classical stability measure, with column-norm growth closely related.Source: §2.2 · pp. 4–5
2.3 The Bunch-Kaufman Algorithm
- Bunch-Kaufman partial pivoting (BKPP) is a computationally efficient pivoting strategy for computing LDLT factorization.Its parameter α is usually chosen so the element-growth bound over two consecutive 1×1 pivots equals the bound for one 2×2 pivot.Source: §2.3 · p. 5
- BKPP does not fully prevent ill-conditioning in 2×2 pivots, leaving potential numerical instability beyond large element growth.The strategy’s displayed pivot entries and index rules illustrate how its partial-pivot choices are made.Source: §2.3 · pp. 5–6, Figure 1 · p. 6, Figure 2 · p. 6
3 Randomized Complete Pivoting Algorithm
RCP selects a pivot using column norms in a random projection, then applies a simplified Bunch-Kaufman strategy to the symmetric matrix. Its blocked implementations trade off pivot batching and projection-update cost, while the operation analysis finds the added work is lower order for large matrices.
3 Randomized Complete Pivoting Algorithm
- RCP chooses the largest column 2-norm in a random projection, transfers that column pivot to A, then applies simplified Bunch-Kaufman pivoting.The special case b=q=1 is the unblocked RCP algorithm.Source: §3 · p. 7, §3.1 · p. 10
- Updating the projection from B costs O(pn), rather than directly recomputing it, but the updated random matrix may be correlated with Ω and less effective.The paper defers discussion of this potential loss of effectiveness to Section 4.Source: §3 · p. 8
3.1 Implementations of the RCP algorithm
- With block size b, q=1 accumulates pivot columns one at a time while q=b chooses b together; experiments favor q=1 for efficiency and element growth.Both variants delay the Schur-complement update until b elimination steps have been processed.Source: §3.1 · p. 10
3.2 Operation counts
- All three algorithms share a dominant 1/3 n^3 multiplication-and-addition cost; RCP adds 9.5n^2 flops when p=5, relatively insignificant for large n.The operation-count analysis is for BLAS-2 versions; the authors use BLAS-3 versions in practice for lower communication costs.Source: §3.2 · pp. 10–11
4 Analysis of randomized complete pivoting algorithm
This section analyzes RCP’s pivot reliability, factor entries, growth, and finite-precision effects, including potential instability in its random projections. It also describes a recomputation strategy for addressing projection instability.
4.1 Reliability of randomized column pivots
- RCP’s JL-based pivot selection chooses column norms within a factor of optimal with high probability, with the stated guarantee at least 1−δ for p≥4.Although the illustrative guarantee requires p≥538 for n=1000, ε=0.5, and δ=0.05, the authors report that p=5 often suffices in practice.Source: §4.1 · pp. 13, 15
4.2 Upper Bounds on entries of the L matrix
- With high probability, RCP bounds entries of L by O(√n), avoiding Bunch–Kaufman’s potential unbounded L entries, though symmetric complete pivoting gives O(1) bounds.The theorem’s bound depends on ε, δ, the oversampling parameter, and α in the SBKP strategy.Source: §4.2 · p. 15
4.3 Probability analysis on column growth factor
- RCP has a rigorous probabilistic column-norm growth bound; its element-growth upper bound is larger than GECP’s, but its dominant factor n ln(n) is far below 2^(n−1) for large n.The element-growth bound is given under Theorem 4.2’s assumptions and the SBKP strategy’s stated parameter choice.Source: §4.3 · pp. 17, 21
4.4 Finite precision analysis
- Theorem 4.3 gives a finite-precision error bound for RCP’s randomized projections across steps k=1,...,N−2.The analysis applies to the RCP method with the SBKP strategy.Source: §4.4 · p. 21
4.5 Potential for numerical instability in random projections
- Large element growth can cause rounding errors in updated projections, while near-rank-deficient inputs can yield low-quality pivots even when projection rounding errors are small.The passage notes a similar instability discussion for QR with column pivoting.Source: §4.5 · p. 23
- Algorithm 2 recomputes a random projection when rounding error becomes large and enforces a threshold on pivoted column norms to address rank deficiency.The algorithm permits at most r projection recomputations.Source: §4.5 · p. 23
- The implementation set r = 1 because experiments never required correction for instability caused by updating random projections.Algorithm 2 allows up to r recomputations.Source: §4.5 · p. 23
5 Numerical Experiments
Across the tested matrix types, RCP generally matched or improved on the comparison algorithms in stability measures while retaining competitive runtime. The results also show that its behavior was largely insensitive to the tested values of p on Type 6 matrices.
5 Numerical Experiments
- On Type 6 matrices, runtime ratios for RCP versus BK and Aasen decreased toward negligible levels as n increased, and tested p values had little impact on stability measures.The p comparison covered growth factor, backward error, ∥L∥1, and ∥L−1∥1; the same patterns were reported for other matrix types.Source: §5 · p. 26, Figure 7 · p. 27, Figure 8 · p. 28
- On Type 1 matrices, BK’s growth factor increased exponentially and broke down, while RCP was as efficient as bounded Bunch-Kaufman and Aasen’s algorithms and remained stable.The bounded Bunch-Kaufman runtime increased exponentially as matrix size grew; the other three algorithms’ runtimes remained small.Source: §5 · p. 27, Figure 9 · p. 29, Figure 10 · p. 29
- For Type 3–8 matrices, RCP outperformed the other algorithms in growth factor and backward error, and BK in both ∥L∥1 and ∥L−1∥1.Figures 11 and 12 report these comparisons.Source: §5 · p. 27, Figure 11 · p. 30, Figure 12 · p. 31
- For Type 9 matrices, RCP’s results were comparable to the other three algorithms, while its L and L−1 matrices had much smaller 1-norms.The comparison covers backward error, growth factor, ∥L∥1, and ∥L−1∥1.Source: §5 · p. 27
- For Type 10 rank-deficient matrices with rank below 55 and n ≥ 100, all four algorithms had growth factor 1; RCP’s backward error was comparable to the others.RCP’s ∥L∥1 and ∥L−1∥1 were as small as BBK and Aasen’s and much smaller than BK’s.Source: §5 · p. 27
6 Conclusions
The paper presents RCP for symmetric indefinite LDLT factorization and establishes a high-probability element-growth bound compatible with GECP. Its analysis and experiments indicate GECP-level numerical stability, with only a slight slowdown relative to Bunch-Kaufman and Aasen’s algorithms.
6 Conclusions
- RCP has a theoretical high-probability element-growth bound compatible with GECP, and experiments find it as stable as GECP while only slightly slower than Bunch-Kaufman and Aasen’s algorithms.The paper also reports an error analysis demonstrating RCP’s numerical stability.Source: §6 · p. 31