Biorthogonal Krylov Bases

• Biorthogonal Krylov bases are paired vector families spanning right and left Krylov subspaces, ensuring mutual biorthogonality.
• They enable efficient matrix function approximations and structured projections using short-term recurrences in nonsymmetric settings.
• These methods underpin advanced applications like time integration, inverse eigenvalue problems, and rational approximants in scientific computing.

A biorthogonal Krylov basis consists of two mutually biorthogonal families of vectors, each spanning a Krylov subspace generated by a matrix and its adjoint, respectively. Unlike standard orthogonal Krylov bases, which are typically constructed for Hermitian matrices using an inner-product-preserving process such as Arnoldi or Lanczos, biorthogonal Krylov approaches are tailored for general, often nonsymmetric matrices. They are constructed to enable efficient, structured, and compact representations of matrix functions, projections, and rational approximants, particularly in non-Hermitian or block-structured settings. These bases serve as the foundation for methods such as Biorthogonal Rosenbrock–Krylov (BOROK) integrators, rational Krylov algorithms, and inverse eigenvalue solvers in computational mathematics and scientific computing.

1. Mathematical Construction of Biorthogonal Krylov Bases

Given a general matrix $A \in \mathbb{C}^{N \times N}$, and two starting vectors $v, w \in \mathbb{C}^N$ (typically chosen such that $w^H v = 1$), the right Krylov subspace is defined as

$$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$

with the corresponding left Krylov subspace for the adjoint $A^H$,

$$\widetilde{\mathcal{K}}_m(A^H, w) = \operatorname{span}\{ w, A^H w, (A^H)^2 w, \dots, (A^H)^{m-1} w \}.$$

The goal is to construct two bases $V_m = [v_1, ..., v_m]$, $W_m = [w_1, ..., w_m]$ such that $W_m^H V_m = I_m$. The dominant algorithm for this task in the true (non-Hermitian) case is the biorthogonal Lanczos procedure, which produces a tridiagonal pencil that models $A$ in the projected subspace. In block settings and for rational Krylov spaces, this construction generalizes to the block and rational framework and further facilitates structured representations, such as tridiagonal × tridiagonal pencils (Buggenhout et al., 2018), block-tridiagonal recurrences (Faghih et al., 24 Jun 2025), and efficient short-term recurrences (Glandon et al., 2019).

2. The Biorthogonal Lanczos Algorithm

The biorthogonal Lanczos algorithm builds bases $v, w \in \mathbb{C}^N$0 and $v, w \in \mathbb{C}^N$1 via two-term recurrences. Initialization assigns $v, w \in \mathbb{C}^N$2 and selects $v, w \in \mathbb{C}^N$3 such that $v, w \in \mathbb{C}^N$4. At each iteration,

$v, w \in \mathbb{C}^N$5

with recurrences

$v, w \in \mathbb{C}^N$6

and normalization

$v, w \in \mathbb{C}^N$7

$v, w \in \mathbb{C}^N$8

The process gives rise to a tridiagonal projected matrix $v, w \in \mathbb{C}^N$9 defined as $w^H v = 1$0, $w^H v = 1$1, $w^H v = 1$2, with the biorthogonality relations

$w^H v = 1$3

and $w^H v = 1$4 (Glandon et al., 2019).

Block versions, required in multiple orthogonal polynomial recurrences and matrix function computations, rely on block-biorthogonal bases $w^H v = 1$5 with analogous recurrences and normalization structures (Faghih et al., 24 Jun 2025).

3. Rational and Extended Biorthogonal Krylov Methods

The framework generalizes to rational Krylov subspaces, defined by application of rational functions of $w^H v = 1$6 (with user-defined poles) to a starting vector or block,

$w^H v = 1$7

where the sequence of poles $w^H v = 1$8 controls the Krylov expansion (Buggenhout et al., 2018). Corresponding left subspaces are built for $w^H v = 1$9. Construction of mutually biorthogonal bases and projection leads, upon suitable normalization, to a structured matrix pencil $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$0, with $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$1 and $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$2 tridiagonal. Short recurrences (six-term in the general rational case) can be derived, with scaling and normalization imposed such that $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$3 at each step. Recurrence and normalization details allow the maintenance of the tridiagonal (or block tridiagonal) structure while avoiding full reorthogonalization.

4. Key Theoretical Properties and Residual Control

These constructions guarantee:

• Short-term recurrences: Owing to the biorthogonalized Lanczos process, the recursions for $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$4 require access only to the previous two vectors in each sequence, with the projected pencil remaining tridiagonal or block-tridiagonal (Glandon et al., 2019, Faghih et al., 24 Jun 2025).
• Biorthogonality: At every stage, $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$5 (or its block generalization), securing stability and structured projections.
• Projection identities: For the oblique (Petrov–Galerkin) projection, $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$6, and for powers and analytic functions $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$7, $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$8 up to the subspace (Glandon et al., 2019).
• Residual expressions: Exact control of the residual norm in projected linear solves, e.g., $$\mathcal{K}_m(A, v) = \operatorname{span}\{ v, Av, A^2 v, \dots, A^{m-1}v \},$$9 (Glandon et al., 2019), and for block methods explicit backward error estimates tied to the norm of block normalization matrices (Faghih et al., 24 Jun 2025).
• Structured storage and complexity: Only current and previous vector pairs plus tridiagonal or block-tridiagonal matrices need to be stored, yielding memory-optimal representation.

5. Algorithms: Adaptive, Rational, and Block Extensions

Biorthogonal Krylov basis construction has evolved:

• Adaptive Lanczos truncation: In BOROK methods, the basis expansion halts once the residual norm of the reduced system drops below a user-specified tolerance. The number of steps $A^H$0 is thus selected adaptively (Glandon et al., 2019).
• Basis extension: External vectors (e.g., arising in multi-stage integrators or block polynomial constructions) can be appended, preserving biorthogonality by small least-squares solves and updating the block structure of the tridiagonal or Hessenberg matrix (Glandon et al., 2019).
• Block biorthogonalization: For block starting data $A^H$1 of size $A^H$2, bases $A^H$3 extend in blocks of $A^H$4 per iteration, generating block-tridiagonal recurrences. Reorthogonalization may be necessary to prevent numerical degradation in finite precision; in ill-conditioned settings, failure to reorthogonalize yields rapidly worsening loss of biorthogonality (Faghih et al., 24 Jun 2025).

Pseudocode and algorithmic structure for rational, block, and classical cases are explicitly provided in the references and summarized above (Buggenhout et al., 2018, Faghih et al., 24 Jun 2025, Glandon et al., 2019).

6. Applications and Stability Considerations

Biorthogonal Krylov bases underpin several advanced algorithms:

• Time integration for ODEs/PDE discretizations: Biorthogonal Rosenbrock–Krylov (BOROK) methods achieve efficient, stable implicit time-stepping for stiff initial value problems by projecting the Jacobian onto the biorthogonal basis and replacing full $A^H$5-dimensional linear solves by reduced $A^H$6-dimensional ones, preserving high-order accuracy (Glandon et al., 2019).
• Inverse eigenvalue problems and multiple orthogonality: Block biorthogonal Lanczos methods solve for recurrence coefficients in the context of multiple orthogonal polynomials, with explicit stability and accuracy analysis. Full or partial reorthogonalization is essential to maintain backward stability in these generally ill-conditioned settings (Faghih et al., 24 Jun 2025).
• Matrix function evaluations and rational approximations: The rational (extended) biorthogonal Lanczos process is used to construct optimal rational approximants and in model reduction, relying on the efficient storage and structured recurrence relations (Buggenhout et al., 2018).
• Numerical stability: In exact arithmetic, breakdowns are rare except at convergence (which can be lucky). In floating point, (near-)zero denominators in normalization trigger breakdowns; look-ahead strategies or pivoting are deployed for recovery. Without reorthogonalization, exponential growth of $A^H$7 occurs, leading to loss of accuracy (Faghih et al., 24 Jun 2025).

7. Summary of Structural and Computational Features

Key structural, computational, and implementation aspects are summarized in the following table from the referenced algorithms:

Construct	Classical Biorthogonal Lanczos	Rational/Extended Biorthogonal	Block Biorthogonal
Subspace type	Powers of $A^H$8, $A^H$9	Rational Krylov with poles	Block powers/multi-vectors
Recurrence length	2-term	Up to 6-term (rational)	3-term (block-tridiagonal)
Basis size per iteration	1 vector	1 vector	$$\widetilde{\mathcal{K}}_m(A^H, w) = \operatorname{span}\{ w, A^H w, (A^H)^2 w, \dots, (A^H)^{m-1} w \}.$$0 vectors
Projected matrix structure	Tridiagonal	Tridiagonal pencil	Block-tridiagonal
Breakdown and remedy	Possible; lookahead/pivoting	As in classical	As in classical/blockified
Reorthogonalization needed	Sometimes (finite precision)	Sometimes	Often for stability

The universality of the biorthogonal Krylov approach arises from its generality for non-Hermitian and block-structured problems and its extensibility to rational, adaptive, and multi-stage regimes. Its theoretical and algorithmic foundation is essential for advanced time-stepping, eigenvalue computation, and rational approximation tasks in large-scale scientific computing (Glandon et al., 2019, Buggenhout et al., 2018, Faghih et al., 24 Jun 2025).