Square-root Kalman Filters

• Square-root Kalman filters are factor-based methods that replace direct covariance propagation with the propagation of a matrix square root to preserve symmetry and positive definiteness.
• They employ techniques like QR, Cholesky, and SVD factorizations to mitigate roundoff errors and improve numerical stability in state estimation.
• Ensemble square-root filters deterministically transform ensemble anomalies so that the empirical analysis covariance exactly matches the Kalman update, benefiting high-dimensional applications.

Searching arXiv for recent and foundational square-root Kalman filter papers relevant to the requested encyclopedia entry. Square-root Kalman filters are Kalman filtering methods that propagate a matrix square root of the state-error covariance rather than the covariance itself. In the classical linear-Gaussian setting, the filter maintains a state estimate and covariance, but a square-root formulation replaces the covariance matrix $P$ by a factor such as an upper-triangular or lower-triangular matrix satisfying $P = S S^\top$ or $P = F^\top F$. This representation is used to maintain symmetry and positive definiteness, improve numerical stability, and reduce loss of numerical precision in finite arithmetic (Tracy, 2022). In contemporary usage, the term encompasses both classical covariance-factor filters and ensemble-based methods that deterministically transform ensemble anomalies so that the empirical analysis covariance satisfies the Kalman covariance update exactly (Lange et al., 2019, Kwiatkowski et al., 2014).

1. Definition and scope

In the standard discrete-time linear Gaussian state-space model, the state evolves as

$$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$

and the observation model is

$$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$

with $W$ and $V$ symmetric positive (semi)definite (Tracy, 2022). The conventional Kalman filter propagates the mean $\mu_{t|t}$ and covariance $\Sigma_{t|t}$, using prediction and measurement-update recursions involving $A \Sigma A^\top + W$, the innovation covariance, and the Kalman gain (Tracy, 2022).

A square-root Kalman filter stores instead a triangular square root of the covariance, for example

$P = S S^\top$0

with $P = S S^\top$1 upper triangular (Tracy, 2022), or equivalently $P = S S^\top$2 in other notational conventions (Lange et al., 2019). In the classical sense, it is “any numerically stable implementation that propagates a matrix square root $P = S S^\top$3 with $P = S S^\top$4, instead of propagating $P = S S^\top$5 directly, to maintain symmetry/positive definiteness and avoid loss of numerical precision” (Lange et al., 2019).

The term also has an ensemble meaning. Deterministic ensemble square-root filters remove stochastic perturbations of observations and instead update ensemble anomalies by a deterministic transform so that the empirical analysis covariance satisfies the Kalman covariance update exactly (Lange et al., 2019, Tong et al., 2015). This suggests that “square-root” denotes a broader factor-based viewpoint: either a matrix factor of a full covariance or an anomaly matrix whose sample covariance realizes the same update.

2. Numerical rationale and historical development

The central motivation for square-root filtering is numerical fragility in direct covariance propagation. When covariance matrices are updated explicitly, they can become ill-conditioned, especially in long-running filters or when the system has weakly observable or unobservable directions (Tracy, 2022). Finite-precision arithmetic can cause loss of symmetry and, more seriously, loss of positive definiteness, which in turn can make Cholesky factorization fail and cause matrix inversions or linear solves to become unstable (Tracy, 2022). These effects degrade the Kalman gain, produce inconsistency between the covariance and the actual estimation error, and can lead to divergence (Tracy, 2022).

Square-root methods address these issues by construction. Covariances represented as $P = S S^\top$6, $P = S S^\top$7, or analogous factors are symmetric positive semidefinite automatically (Tracy, 2022). The use of orthogonal transformations such as QR factorizations, Givens rotations, or Householder transformations improves conditioning because such transformations preserve norms and are themselves well-conditioned (Tracy, 2022).

Historically, early square-root filtering work, including the Apollo guidance context, reported that square-root filters can achieve accuracy comparable to a standard filter with roughly twice the word length, described as benefiting from “double the working precision” (Tracy, 2022). Later work retained the same fundamental justification: a factorized covariance representation reduces the dynamic range of intermediate quantities and avoids catastrophic cancellation in direct covariance updates (Tracy, 2022).

The literature referenced across these works spans classical square-root covariance filters, square-root information filters, UD factorization methods, and ensemble square-root filters such as ETKF, EAKF, and the Whitaker–Hamill filter (Tracy, 2022, Tong et al., 2015, Kwiatkowski et al., 2014). More recent developments extend square-root formulations to nonlinear cubature and unscented filters (Kulikova et al., 2023, Kulikova et al., 2023), robust MCC-based filters (Kulikova, 2023), tensor-network implementations for extremely high-dimensional regression (Menzen et al., 2024), and differentiable square-root filters for gradient-based learning in linear-Gaussian state-space models (Corenflos, 13 Mar 2026).

3. Algebraic structure of covariance-factor updates

A central algebraic device in modern square-root formulations is to obtain square roots of sums of positive semidefinite matrices by QR factorization. If

$P = S S^\top$8

then

$P = S S^\top$9

If the vertically stacked matrix is QR-factorized and the upper-triangular factor is denoted $P = F^\top F$0, then $P = F^\top F$1, so $P = F^\top F$2 is a square root of the sum (Tracy, 2022).

This pattern yields a uniform square-root treatment of the major covariance recursions in the Kalman filter. In the QR-only formulation of the linear-Gaussian filter, the predicted covariance square root is obtained as

$P = F^\top F$3

where $P = F^\top F$4 (Tracy, 2022). The innovation covariance square root is similarly

$P = F^\top F$5

with $P = F^\top F$6 (Tracy, 2022). Using the Joseph form

$P = F^\top F$7

the posterior covariance square root becomes

$P = F^\top F$8

(Tracy, 2022).

The same factor-based logic appears in nonlinear square-root cubature and unscented filters. In the square-root cubature Kalman filter, predicted and innovation covariances are obtained via triangularization of stacked matrices of propagated cubature-point deviations and noise square roots, rather than by explicit covariance formation (Zhao et al., 21 Nov 2025). In the SVD-based continuous-discrete extended-cubature filter, covariance propagation is written as a pre-array/post-array identity in which an SVD of a stacked matrix yields the singular vectors and singular values of the updated covariance factor directly (Kulikova et al., 2023). In Cholesky-based square-root MCC Kalman filtering, one-step condensed forms similarly obtain innovation and posterior covariance factors from a single orthogonal transformation of a pre-array (Kulikova, 2023).

A concise comparison of factorization primitives used in the cited literature is given below.

Setting	Factor form	Core factorization
Linear covariance SRKF	$P = F^\top F$9 or $$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$0	QR (Tracy, 2022)
Classical/continuous-discrete nonlinear filters	Cholesky or SVD covariance factors	QR, SVD, $$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$1-orthogonal transformations (Kulikova et al., 2023, Kulikova et al., 2023)
Ensemble square-root filters	Anomaly transform	deterministic anomaly transforms (Lange et al., 2019, Tong et al., 2015)
Tensor-network SRKF	$$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$2 in TTm form	thin SVD-based QR step (Menzen et al., 2024)

This structural commonality suggests that square-root filtering is best understood as a factor-propagation paradigm in which covariance addition and update are expressed through orthogonal or related stable matrix factorizations.

4. Main algorithmic variants

Classical covariance-factor filters maintain explicit square roots of the covariance matrix. In the QR-only linear filter, the full algorithm uses state prediction

$$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$3

innovation

$$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$4

square-root innovation covariance $$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$5, Kalman gain computed through triangular solves, and a QR-based posterior square-root update (Tracy, 2022). The formulation is notable for using only QR decompositions and triangular solves, rather than specialized update identities (Tracy, 2022).

Cholesky-based square-root filters remain standard in nonlinear filtering. The continuous-discrete unscented Kalman filtering framework based on MATLAB ODE solvers uses square-root moment differential equations or sigma-point differential equations, with measurement updates performed through $$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$6-orthogonal hyperbolic QR transformations to handle the indefinite weighted covariance structures induced by unscented weights (Kulikova et al., 2023). The continuous-discrete extended-cubature filter replaces direct covariance propagation by SVD-based square-root propagation within adaptive ODE integration, which is especially aimed at stiff systems and ill-conditioned covariances (Kulikova et al., 2023).

Robust variants integrate alternative objective functions into square-root filters. One-step condensed forms for maximum correntropy criterion Kalman filtering derive Cholesky-based square-root algorithms in an a priori form that permit easier propagation of the state estimate and do not require a back-substitution for computing the estimate (Kulikova, 2023). In nonlinear battery state-of-charge estimation, a generalized mixture minimum error entropy square-root cubature Kalman filter combines a square-root cubature base with robust entropy-based measurement updates under mixed non-Gaussian noise (Zhao et al., 21 Nov 2025).

SVD-based covariance filters replace Cholesky factors by singular vectors and singular values. An improved discrete-time SVD-based KF uses SVD of covariance, process noise covariance, and measurement noise covariance, and implements the covariance update via the Joseph form to avoid inversion of covariance singular values (Kulikova et al., 2016). This paper explicitly argues that the previous SVD-based implementation remained sensitive to roundoff and poorly treated ill-conditioned situations, whereas the new SVD-based method is algebraically equivalent to the conventional KF but more robust in such settings (Kulikova et al., 2016).

Square-root filtering also appears in learning-oriented formulations. In linear-Gaussian state-space models, square-root Kalman filters are a natural target for gradient-based parameter learning, but naive differentiation through the QR-based triangularization is problematic because the semi-orthogonal factor is non-unique when $$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$7 and the standard Jacobian formula diverges for rank-deficient inputs (Corenflos, 13 Mar 2026). The Gramian-differential approach resolves this by differentiating the identity $$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$8 directly, because the losses of interest depend on the triangularization only through the Gramian (Corenflos, 13 Mar 2026).

5. Ensemble square-root filters

In ensemble filtering, the forecast distribution is represented by an ensemble rather than a full covariance matrix. The ensemble mean and covariance are approximated by the sample mean and sample covariance of the ensemble members (Kwiatkowski et al., 2014). The stochastic EnKF perturbs observations so that the analysis covariance is matched only in expectation (Tong et al., 2015), whereas deterministic ensemble square-root filters transform the ensemble anomalies so that the sample covariance exactly equals the Kalman covariance update for the given forecast ensemble (Tong et al., 2015).

In the formulation studied by Lange and Stannat, the forecast anomaly matrix is

$$x_{t+1} = A x_t + B u_t + w_t, \qquad w_t \sim \mathcal{N}(0, W),$$9

with empirical covariance

$$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$0

(Lange et al., 2019). An ensemble square-root filter is characterized by a deterministic transform

$$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$1

such that the empirical analysis covariance satisfies exactly

$$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$2

(Lange et al., 2019). The paper studies EAKF, ETKF, and the unperturbed EnKF of Whitaker and Hamill as members of this class (Lange et al., 2019).

A distinct but complementary analysis establishes the large-ensemble limit of unbiased square-root ensemble filters. In that setting, the defining property is that the ensemble analysis mean and covariance are exactly

$$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$3

for each time step (Kwiatkowski et al., 2014). Under a linear model and an initial distribution with all moments finite, the ensemble mean and covariance converge in all $$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$4 to the corresponding Kalman quantities at the Monte Carlo rate $$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$5, with constants that do not depend on the model or data dimensions; the result also holds in separable Hilbert spaces (Kwiatkowski et al., 2014).

The continuous-time limit of deterministic ESRFs with deterministic model perturbations yields a universal ensemble Kalman–Bucy limit. In the linear case, under assumptions on the perturbations, the empirical mean and covariance converge to the Kalman–Bucy mean and Riccati covariance at order $$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$6, and the whole discrete ensemble converges in $$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$7, uniformly on compact time intervals, to the Ensemble Kalman–Bucy filtering equations (Lange et al., 2019). The same limiting equation arises simultaneously for EAKF, ETKF, and Whitaker–Hamill (Lange et al., 2019). This universality explains why many distinct discrete deterministic square-root transforms exhibit similar continuous-time behavior.

A further line of work studies the nonlinear long-time behavior of ESRFs. Under an observable energy criterion, ETKF and EAKF admit Lyapunov-type bounds that rule out catastrophic filter divergence; with additional nondegenerate system noise and controllability arguments, geometric ergodicity is obtained for the corresponding joint signal–ensemble Markov chain (Tong et al., 2015). This places finite-ensemble square-root filters within a rigorous stability framework for nonlinear systems.

6. Numerical properties, implementation, and applications

The principal numerical advantages of square-root filters stated across the cited works are preservation of symmetry and positive definiteness, improved conditioning, and resistance to roundoff-induced divergence (Tracy, 2022, Lange et al., 2019, Kulikova et al., 2016). In the QR-only linear formulation, covariances are always represented as triangular factors $$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$8 or $$y_t = C x_t + v_t, \qquad v_t \sim \mathcal{N}(0, V),$$9, and QR uses orthogonal matrices whose condition number is 1 (Tracy, 2022). In continuous-discrete nonlinear filtering, SVD-based square-root methods are emphasized precisely because they handle near-rank-deficient matrices and ill-conditioned measurement problems more safely than Cholesky-based covariance updates (Kulikova et al., 2023). In continuous-discrete unscented filtering, the paper argues that pseudo-square-root implementations based on rank-one Cholesky updates remain vulnerable, whereas $W$0-orthogonal square-root methods are more robust in ill-conditioned settings (Kulikova et al., 2023).

Implementation patterns recur. The filter assumes triangular or factorized square roots for the process and measurement covariances, typically obtained by Cholesky or SVD once and then reused unless the noise covariances are time-varying (Tracy, 2022, Kulikova et al., 2023). Each recursion forms stacked matrices, applies QR, SVD, or a related transformation, and retains the triangular or singular-value factor as the new covariance representation (Tracy, 2022, Kulikova et al., 2023). In modern software, these implementations are described as “library-call friendly” because they rely on robust BLAS/LAPACK routines (Tracy, 2022).

The range of applications is broad. In online Gaussian process regression with product kernels, a tensor-network square-root Kalman filter estimates $W$1 parameters on a standard laptop, and the square-root formulation prevents the loss of positive definiteness caused by TT rounding in the covariance-form tensor-network Kalman filter (Menzen et al., 2024). In battery state-of-charge estimation, the square-root cubature formulation is reported to avoid “asymmetric and non-positive definite covariance during iteration” and serves as the numerically stable basis for robust error-entropy updates under mixed non-Gaussian noise (Zhao et al., 21 Nov 2025). In intensive longitudinal data analysis, a square-root second-order extended Kalman filter is used to estimate smoothly time-varying parameters in a dynamic factor model, including treatment effects and time-varying latent dynamics (Fisher et al., 2020).

Some representative application directions from the cited literature are summarized below.

Application area	Square-root formulation	Reported purpose
Linear state estimation	QR-only SRKF (Tracy, 2022)	numerical robustness and implementation simplicity
Geoscientific data assimilation	ESRF / InFo-ESRF (Lange et al., 2019, Armstrong et al., 1 Mar 2025)	localized high-dimensional ensemble updates
Continuous-discrete nonlinear filtering	SVD or $W$2-orthogonal square-root filters (Kulikova et al., 2023, Kulikova et al., 2023)	stiff systems, discretization control, roundoff stability
Online GP regression	tensor-network square-root KF (Menzen et al., 2024)	high-dimensional recursive Bayesian regression
Robust nonlinear estimation	SRCKF with MCC or MEE-type criteria (Kulikova, 2023, Zhao et al., 21 Nov 2025)	non-Gaussian or impulsive noise
Time-varying latent-variable models	SR-SO-EKF (Fisher et al., 2020)	smoothly time-varying parameters

7. Extensions, limitations, and current directions

A recurring distinction in the literature is between classical square-root covariance filters and ensemble square-root filters. The former propagate a factor of the full covariance matrix, while the latter propagate or transform ensemble anomalies so that the empirical covariance follows the Kalman update (Lange et al., 2019, Kwiatkowski et al., 2014). These are conceptually related but algorithmically distinct. Confusing them is a common misconception; the ensemble variant is not simply a Cholesky factorization of a covariance matrix, but a deterministic transform in the ensemble subspace (Tong et al., 2015).

Another recurring theme is that square-root structure alone does not eliminate all numerical issues. The improved discrete-time SVD-KF demonstrates that an SVD-based strategy can still be sensitive to roundoff if it relies on an information-form covariance update and inversion of small singular values; the paper’s revised Joseph-form SVD implementation is designed specifically to avoid those weaknesses (Kulikova et al., 2016). Similarly, pseudo-square-root UKFs based on repeated rank-one Cholesky downdates remain vulnerable in ill-conditioned settings, motivating $W$3-orthogonal or SVD-based alternatives (Kulikova et al., 2023).

In the ensemble setting, the perturbation update remains a computational bottleneck under localization. The integral-form ESRF addresses this by discretizing an integral representation of the perturbation update equations and using a preconditioned Krylov method to update perturbations to a high degree of accuracy, thereby avoiding a direct evaluation of the matrix square root in the perturbation stage (Armstrong et al., 1 Mar 2025). This suggests that future square-root developments in data assimilation may increasingly be phrased in terms of operator equations rather than explicit factorizations.

Differentiable square-root filtering is another current direction. For gradient-based learning in state-space models, robust automatic differentiation of square-root filters requires respecting the fact that the filter outputs depend on triangularization only through the Gramian $W$4, not through the non-unique QR factors themselves (Corenflos, 13 Mar 2026). The Gramian-differential formulation resolves non-uniqueness and rank-deficiency issues and yields exact gradients for the Kalman log-marginal likelihood and filtered moments (Corenflos, 13 Mar 2026).

Across these diverse developments, the defining principle remains stable: square-root Kalman filters replace direct covariance recursion by propagation of a covariance factor, using algebraic operations that are better conditioned and more compatible with finite-precision computation. Whether implemented through QR, SVD, $W$5-orthogonal transformations, deterministic ensemble anomaly transforms, tensor-network factorizations, or Gramian-based differentiation rules, square-root filtering continues to provide the main factor-based framework for numerically robust Kalman estimation (Tracy, 2022, Lange et al., 2019, Menzen et al., 2024, Corenflos, 13 Mar 2026).