---
title: Multistage Mixed Precision IR
url: https://www.emergentmind.com/topics/multistage-mixed-precision-ir-msir
type: topic
---

# Multistage Mixed Precision IR

A multistage mixed precision iterative refinement (MSIR) framework refers to an algorithmic paradigm that exploits several floating-point precision levels at distinct stages of a numerical linear algebra workflow, optimizing the balance between computational cost and attainable numerical accuracy. MSIR methods strategically start with low-precision computations for the most expensive operations and escalate to higher precisions in subsequent refinement phases to reach working-precision accuracy. This approach is particularly effective when deployed on modern hardware with native support for low-precision arithmetic, where the performance, energy, and memory savings are substantial. MSIR principles now underpin state-of-the-art algorithms for the solution of linear systems, matrix equations (e.g., Sylvester and Lyapunov), least-squares problems, and matrix decompositions, and have been rigorously analyzed in recent research [2503.03456][2307.03914][2501.04229][2107.06200][2410.06319][2012.00706][2401.03755][2405.18363][2202.10204].

## 1. MSIR Algorithmic Structure and Precision Hierarchy

MSIR is characterized by a deliberate, stage-wise allocation of computations to multiple precisions:
- **Low precision:** Used for factorization, matrix reductions (e.g., Schur or QR), sketching, or preconditioner construction, leveraging fast but inexact formats such as FP16 or FP32.
- **Intermediate/working precision:** Cast as the precision where iterates, corrections, solution updates, and possibly basis vector orthogonalizations are performed, often FP64 or double precision.
- **High (residual) precision:** Applied to critical scalar functionals (e.g., norms, residuals) and convergence monitors, sometimes requiring even quad precision to ensure correct termination, especially in extreme ill-conditioning regimes.

The division into two, three, or up to five arithmetic layers (as in state-of-the-art GMRES-based IR with adaptive sparse approximate inverse preconditioning [2307.03914][2202.10204]) is central to the flexibility and performance of MSIR. Precision assignments are tailored per stage, for example:

| Stage                 | Typical Operation                     | Precision      |
|-----------------------|---------------------------------------|---------------|
| Factorization/Sketch  | QR, Schur, SPAI, etc.                 | u_f (low)     |
| Correction Solves     | Triangular, GMRES, LSQR, etc.         | u, u_g, u_p   |
| Residual Computation  | Norms, backward errors                | u_r (high)    |
| Solution Updates      | Iterates, corrections                 | u             |
| Preconditioner Apply  | Bucketed or split preconditioner      | bucketed/u_s  |

This hierarchy supports a dynamic workflow that transitions from highly efficient, low-precision phases (dominant in flop count) to higher-precision refinement steps, with the goal of attaining error levels commensurate with the working precision.

## 2. MSIR in Direct and Iterative Linear Solvers

Canonical MSIR solvers for linear systems (Ax = b) and associated preconditioned iterative methods can be grouped as follows:

- **Standard IR (SIR):** Uses low-precision factorizations and corrective solves, limited to condition numbers κ(A) < u_f^{-1}.
- **GMRES-IR:** GMRES corrections are computed in higher precision, handling larger κ(A) but at increased computational cost.
- **Adaptive Multistage Schedulers:** Detect stalling via precision-insensitive convergence monitors (e.g., normwise contraction ratios, stagnation in corrections) and escalate the solver or factorization precision only as needed [2107.06200].

The iterative refinement process relies on an initial low-precision solution, high-precision residual computation, and iterative correction steps, with the possibility of substituting preconditioners (e.g., LU, SPAI, or bucketed adaptive-precision SPAI [2307.03914]) to further optimize the trade-off between memory/storage and solver convergence.

## 3. MSIR for Matrix Equations: The Sylvester Equation Paradigm

The multistage mixed-precision IR algorithm for the Sylvester equation (A X + X B = C) [2503.03456] exemplifies the MSIR approach:

1. **Low-precision Schur Reductions:** Compute A ≈ Q_A T_A Q_A^*, B ≈ Q_B T_B Q_B^* in low precision (u_low)—O(m^3+n^3) flops.
2. **Initial Low-precision Triangular Solve:** Transform the right-hand side and solve T_A Y_0 + Y_0 T_B = D to O(u_low) accuracy.
3. **Iterative Refinement in Working Precision:** Refine Y_i by stationary iteration in high precision with convergence guaranteed if ∥Δ_A∥ + ∥Δ_B∥ < sep_F(T_A,−T_B).
4. **Recovery Phase:** Since Q_A and Q_B are only unitary to O(u_low), two remedies are proposed: re-orthonormalization in working precision or explicit high-precision inversion.
5. **Cost Model:** The break-even point for MSIR superiority is derived explicitly. For k being the number of IR steps and ρ = cost_low/cost_high, the scheme is favored when k ≤ k^*(ρ).

This structure is typical of MSIR for general matrix equations and underscores the crucial role of both low-precision reductions and high-precision refinement.

## 4. MSIR in Least-Squares, Sketching, and Decomposition

Recent MSIR schemes extend beyond square systems to overdetermined (least-squares) problems and low-rank matrix decompositions:

- **Mixed Precision Sketching and Preconditioning:** Stage 1 performs randomized sketching of A in low precision, Stage 2 computes the QR of the sketch in potentially higher (yet still not “working”) precision, and Stages 3–4 achieve working-precision solution via adjusted LSQR and iterative refinement with the right-preconditioner [2410.06319].
- **Multistage IR for LS and WLS:** For least-squares and weighted least-squares, multistage schemes use varying precisions for QR (u_f), preconditioner solves (u_s), and solution updates/residuals (u) [2401.03755]. Multiple IR strategies (normal equations, semi-normal, augmented/multi-LS) exhibit complementary convergence properties depending on the problem's condition number and residual magnitude [2405.18363].
- **Interpolative Decompositions (ID):** The MSIR approach in model reduction utilizes low-precision pivoted QR (FP16/FP32) for column selection, with final assembly in double precision [2012.00706]. The error analysis shows double-precision-accurate ID is achievable provided σ_{k+1}/σ_1 ≫ u_L.

## 5. Error Analysis, Convergence, and Switching Logic

MSIR algorithms are designed around tight finite-precision analyses and robust switching logic:

- **Convergence Bounds:** For each IR variant, precise condition number and unit roundoff thresholds determine whether O(u) backward or forward errors are attainable. For example, in standard SIR, κ_∞(A) < u_f^{-1} suffices, while higher-order GMRES-based MSIR variants relax this to κ_∞(A) < u^{-1/3}u_f^{-2/3} or better [2107.06200].
- **Residual and Correction Monitors:** Cheaply computable estimates, such as z_i = ∥d_{i+1}∥_∞/∥x_i∥_∞ and contraction factors v_i, drive adaptive switches between algorithmic variants or upgrades in precision.
- **Preconditioner Adaptivity:** In sparse systems, bucketed SPAI approaches partition the preconditioner by magnitude and assign adaptive precision, reducing memory but increasing GMRES iterations proportionally to the relaxation tolerance [2307.03914].
- **Automatic Escalation:** The recursive MSIR philosophy starts at the lowest cost and climbs the solver strength/precision ladder only upon observed stagnation or contraction failures.

## 6. Performance Trade-offs and Practical Impact

The practical benefits of MSIR are quantifiable and confirmed by extensive numerical experiments:

- **Wall-clock Speedup:** In representative sparse large-scale problems, three-precision MSIR (e.g., FP16/FP32/FP64) yields 1.5–3× speedup over mono-precision, and up to 7–8× when the proportion of FP16 operations is maximized [2501.04229].
- **Memory and Bandwidth Savings:** Storing preconditioners and factorizations in low precision results in substantial reductions (e.g., up to 60% for adaptive SPAI [2307.03914]).
- **Energy Efficiency:** On hardware with fast FP16/FP32 TensorCores or similar accelerators, lower-precision work translates nearly proportionally into energy and time savings.
- **Convergence Robustness:** The adaptive, multistage nature of MSIR ensures convergence even in highly ill-conditioned regimes, invoking more robust (but costlier) kernels only as last resort.

## 7. Theoretical and Practical Limits

All MSIR variants are subject to theoretically derived bounds that guide both algorithmic deployment and parameter tuning:

- **Condition Number Windows:** Each combination of precisions and solver variants admits provable constraints, e.g., κ(A) < u^{-1/3}u_f^{-2/3} in five-precision GMRES-IR.
- **Breakdown Points:** Stalling in low precision due to overflows, underflows, or insufficient contraction triggers escalation in the MSIR stack.
- **Final Accuracy:** Provided singular value decays and/or preconditioner construction tolerances are set relative to the input problem and target working precision, MSIR approaches achieve results indistinguishable from all-double-precision workflows while incurring only a fraction of their cost.

MSIR is therefore a foundational methodology for fast, accurate, and scalable numerical linear algebra on heterogeneous precision platforms, and its descendants represent the current best practice for exploiting both algorithmic and hardware diversity [2503.03456][2307.03914][2107.06200][2410.06319][2012.00706][2501.04229][2401.03755][2405.18363][2202.10204].

Source: https://www.emergentmind.com/topics/multistage-mixed-precision-ir-msir