Chebyshev Inverse Decomposition (Cheb-inv)

• Chebyshev Decomposition of the Inverse (Cheb-inv) represents inverse functions or operators using Chebyshev polynomial approximations, offering both exact and numerical schemes.
• It is applied across diverse fields such as quantum inverse power iteration, inverse-transform sampling, and PDE integration, leveraging recurrence and orthogonality for efficiency.
• While Cheb-inv enables precise structural encoding of inverse information, its performance can be challenged by singularities and dense spectral conditions.

Searching arXiv for papers on Chebyshev decomposition of the inverse and closely related Chebyshev-based inverse representations. Search query: "Chebyshev decomposition of the inverse" Chebyshev Decomposition of the Inverse, often abbreviated “Cheb-inv,” denotes a family of constructions in which an inverse function, inverse map, or inverse operator is represented, approximated, or analyzed through Chebyshev polynomials or Chebyshev-based coefficient systems. In the most explicit contemporary usage, the term names the Chebyshev approximation of the shifted inverse Hamiltonian $(H-\omega I)^{-1}$ inside quantum inverse power iteration for excited-state preparation (Patil et al., 26 Jun 2026). In a broader technical sense, closely related work exhibits the same motif in exact derivative formulas for inverse trigonometric functions, inverse-transform sampling, explicit matrix inverses, and Fourier–Chebyshev inverse estimates for parabolic PDEs (Kronenburg, 2020).

1. Scope and main formulations

A broader reading of the literature places several distinct constructions under the same structural idea: the inverse object is not handled directly in its native form, but through Chebyshev polynomials, Chebyshev recurrences, or Chebyshev coefficient operators. The role of the Chebyshev basis varies by domain: exact closed form in some settings, approximation basis in others, and operator-theoretic tail control in yet others.

Setting	Inverse object	Chebyshev role
Quantum inverse power iteration	$(H-\omega I)^{-1}$	odd Chebyshev approximation implemented by QSVT (Patil et al., 26 Jun 2026)
Inverse trigonometric derivatives	$D^n\arctan x$, $D^n\operatorname{arctanh}x$	exact pullback through $U_{n-1}$ (Kronenburg, 2020)
Inverse transform sampling	$F^{-1}$	CDF represented by Chebyshev series and inverted numerically (Olver et al., 2013)
Almost-Toeplitz tridiagonal inverse	$Q^{-1}$	entries expressed through $U_n(c/2)$ (Tan, 2018)
Inverse localization length	$\kappa(E)$	Chebyshev expansion in energy from traces of $T_n(H)$ (Hatano et al., 2016)
Parabolic PDE validation	inverse of the linear time operator and approximate inverse of $(H-\omega I)^{-1}$0	Fourier–Chebyshev block inversion with explicit decay estimates (Cadiot et al., 28 Feb 2025)

This diversity matters because Cheb-inv is not a single algorithmic primitive. In some papers it is an exact representation theorem; in others it is a numerical inversion pipeline; in others it is an approximate inverse used inside a Newton-type or QSVT-based procedure. This suggests that the most stable encyclopedic definition is structural rather than domain-specific: Cheb-inv refers to inverse information encoded through Chebyshev objects.

2. Exact inverse-function representations

A canonical exact instance appears in the study of higher derivatives of inverse tangent and inverse hyperbolic tangent. For integers $(H-\omega I)^{-1}$1, the $(H-\omega I)^{-1}$2-th derivative of $(H-\omega I)^{-1}$3 admits the rational-polynomial form

$(H-\omega I)^{-1}$4

and the Chebyshev form

$(H-\omega I)^{-1}$5

The corresponding formula for $(H-\omega I)^{-1}$6 is

$(H-\omega I)^{-1}$7

These identities exhibit an exact Chebyshev decomposition of the derivative hierarchy of inverse functions: the nontrivial numerator structure is carried entirely by $(H-\omega I)^{-1}$8 after a simple change of variable (Kronenburg, 2020).

The relevant pullback map for $(H-\omega I)^{-1}$9,

$D^n\arctan x$0

maps $D^n\arctan x$1 into $D^n\arctan x$2, which is the standard Chebyshev domain. In this sense, the higher derivatives of $D^n\arctan x$3 are Chebyshev polynomials of the second kind evaluated on a transformed variable, multiplied by a simple weight $D^n\arctan x$4. The paper explicitly identifies this as a canonical Chebyshev decomposition of the $D^n\arctan x$5-th derivative of $D^n\arctan x$6.

The same work also derives explicit binomial expansions

$D^n\arctan x$7

and

$D^n\arctan x$8

from a comparison between two formulas for $D^n\arctan x$9. The logical direction is noteworthy: the Chebyshev identities are proved from inverse-function derivative formulas, not introduced independently. This makes the paper an exact, analytic prototype of Cheb-inv rather than an approximation scheme.

3. Numerical inversion of monotone maps and probability laws

In inverse-transform sampling, the inverse object is the quantile map $D^n\operatorname{arctanh}x$0. The one-dimensional algorithm in "Fast inverse transform sampling in one and two dimensions" does not directly fit a Chebyshev series to $D^n\operatorname{arctanh}x$1; instead, it approximates the density by

$D^n\operatorname{arctanh}x$2

integrates termwise to obtain a Chebyshev representation

$D^n\operatorname{arctanh}x$3

normalizes to $D^n\operatorname{arctanh}x$4, and then solves $D^n\operatorname{arctanh}x$5 for each $D^n\operatorname{arctanh}x$6 by bisection, evaluating the Chebyshev series with Clenshaw’s algorithm (Olver et al., 2013).

This is an implicit Cheb-inv construction. The forward map is decomposed in a Chebyshev basis, while the inverse is realized procedurally by rootfinding on that representation. The paper states that one could go one step further, sample $D^n\operatorname{arctanh}x$7 at Chebyshev nodes in $D^n\operatorname{arctanh}x$8-space, and fit a second Chebyshev series; that would produce an explicit Cheb-inv representation. The implemented method stops one step earlier because the implicit inverse already yields a fast and robust sampler.

Several computational details are central. Coefficients are computed from function values on Chebyshev grids by a fast cosine transform in $D^n\operatorname{arctanh}x$9. Integration of the Chebyshev series is performed in $U_{n-1}$0. Each inversion costs $U_{n-1}$1, and the paper notes that approximately 50 bisection iterations suffice for double-precision accuracy. In two dimensions, the density is first approximated by a low-rank sum

$U_{n-1}$2

after which the marginal and conditional densities are reduced to one-dimensional Chebyshev inversion steps. The resulting complexity is

$U_{n-1}$3

The numerical significance is that Cheb-inv here is an inversion architecture rather than a closed formula. The paper reports that in one dimension the method significantly outperforms MATLAB’s slicesample, and often also beats rejection sampling; in two dimensions it can be faster than rejection sampling by factors up to approximately $U_{n-1}$4 for smooth low-rank densities. A common misconception is therefore that Cheb-inv must mean a direct series for the inverse itself. In this line of work, the inverse is encoded indirectly through a Chebyshev decomposition of the forward operator.

4. Matrix, spectral, and PDE inverse operators

A concrete matrix-level Cheb-inv appears in the explicit inverse of the symmetric tridiagonal matrix

$U_{n-1}$5

If $U_{n-1}$6 exists, then for $U_{n-1}$7,

$U_{n-1}$8

where $U_{n-1}$9, $F^{-1}$0, and $F^{-1}$1. The recurrence is solved by the Chebyshev polynomials of the second kind: $F^{-1}$2 Accordingly, every inverse entry is a bilinear expression in $F^{-1}$3. The paper develops this for AR(1) precision matrices, CAR models on path graphs, and cubic spline systems, and derives explicit formulas for row sums, $F^{-1}$4, and $F^{-1}$5 (Tan, 2018).

A different inverse quantity appears in the Chebyshev expansion of the inverse localization length of Hermitian and non-Hermitian random chains. In the Hermitian case,

$F^{-1}$6

This expansion yields the energy-dependent inverse localization length from a single computation of the moments $F^{-1}$7, in contrast to transfer-matrix methods, which treat one energy per run. In the non-Hermitian case the paper combines Hermitization with a Chebyshev expansion in the auxiliary Hermitized spectrum and obtains an analogous formula for $F^{-1}$8 at fixed complex $F^{-1}$9 (Hatano et al., 2016).

In fully spectral validation of semilinear parabolic PDEs, the inverse object is the linear time operator in Fourier–Chebyshev coordinates and, on top of that, an approximate inverse of the full Fréchet derivative. After expanding in spatial Fourier modes and temporal Chebyshev modes, each Fourier block is governed by an operator $Q^{-1}$0 acting on the Chebyshev coefficients. The key estimate is

$Q^{-1}$1

with $Q^{-1}$2 determined by the linear Fourier eigenvalue. For $Q^{-1}$3, the paper gives explicit constants

$Q^{-1}$4

and uses them to construct a block-structured approximate inverse

$Q^{-1}$5

for the Fréchet derivative $Q^{-1}$6. The stated consequence is improved tail control and larger time steps in rigorous integration, including a global-existence result for a nontrivial 2D Navier–Stokes initial condition (Cadiot et al., 28 Feb 2025).

Across these examples, the inverse is not always a literal matrix inverse. It may be a Green kernel, an inverse localization length, or an inverse-like linear block inside a Newton–Kantorovich proof. What unifies them is that the inverse information is carried by Chebyshev recurrences, Chebyshev moments, or Chebyshev-mode resolvent bounds.

5. Cheb-inv in quantum inverse power iteration

The narrowest explicit use of the name “Cheb-inv” appears in quantum inverse power iteration for arbitrary excited states. Given a Hermitian Hamiltonian $Q^{-1}$7 and a shift $Q^{-1}$8, classical inverse power iteration would repeatedly apply

$Q^{-1}$9

to amplify the eigenvector whose eigenvalue is closest to $U_n(c/2)$0. The quantum difficulty is that $U_n(c/2)$1 is nonunitary. The Cheb-inv strategy is therefore to approximate the scalar inverse function $U_n(c/2)$2 on

$U_n(c/2)$3

by an odd Chebyshev series

$U_n(c/2)$4

where

$U_n(c/2)$5

and the maximal polynomial degree is $U_n(c/2)$6. After shifting and scaling,

$U_n(c/2)$7

the polynomial $U_n(c/2)$8 approximates $U_n(c/2)$9, hence $\kappa(E)$0 up to the scale factor $\kappa(E)$1. QSVT then implements this polynomial transformation on a block encoding of $\kappa(E)$2 (Patil et al., 26 Jun 2026).

The attraction of this construction is immediate: repeated application of an approximation to the shifted inverse reproduces the classical shift-and-invert mechanism in a QSVT-compatible form. The difficulty is equally explicit in the same paper. As $\kappa(E)$3 approaches the target eigenvalue, the condition number

$\kappa(E)$4

grows, so the polynomial approximation problem becomes harder rather than easier. The paper’s numerical analysis on $\kappa(E)$5 shows that for moderate degree the approximation $\kappa(E)$6 does not peak at the eigenvalue closest to zero; instead, its largest magnitude can occur farther from the origin. The observed consequences include slow convergence, amplification of off-target eigenstates, and “pseudo-convergence,” where fidelity initially increases and later drifts toward a different state.

The paper therefore contrasts Cheb-inv with eigenstate filtering (EF), a symmetric Chebyshev-based filter centered at the origin. EF-based QIPI is reported to be substantially more robust than Cheb-inv and other decomposition-based approaches because the symmetry of the filtering polynomial avoids divergence with respect to the choice of $\kappa(E)$7 and efficiently suppresses off-target eigenstates even in closely spaced spectra. Numerical simulations for molecular Hamiltonians of $\kappa(E)$8, LiH, and $\kappa(E)$9 show improved convergence and enhanced access to higher excited states relative to other quantum power methods. The authors conclude that Chebyshev-decomposition-based inverse schemes are unsuitable for targeting excited states in dense spectra, because the required polynomial degree increases rather than decreases with the quality of the energy guess.

This section is a useful corrective to an overly optimistic view of Cheb-inv. Chebyshev approximation is not automatically benign when the target function is singular. In QIPI, the singularity of $T_n(H)$0 near the shifted target is precisely what creates the instability.

6. Interpretation, limitations, and adjacent inverse decompositions

Several clarifications follow from the literature.

First, Cheb-inv does not require a literal Chebyshev series for the inverse map itself. In inverse-transform sampling, the implemented object is a Chebyshev series for the forward CDF plus numerical inversion. In rigorous PDE integration, the operative object is an inverse estimate for a block tridiagonal Fourier–Chebyshev operator together with an approximate inverse for the full Fréchet derivative. These are genuine inverse decompositions, but not of the same algebraic type as the exact $T_n(H)$1-based formulas for $T_n(H)$2 or the odd Chebyshev approximation to $T_n(H)$3 in QIPI.

Second, Cheb-inv is not tied to a single Chebyshev family. Exact inverse-trigonometric and tridiagonal-matrix formulas use the second-kind polynomials $T_n(H)$4, whereas inverse-transform sampling, inverse localization length, and QIPI are formulated with first-kind polynomials $T_n(H)$5 or $T_n(H)$6. The choice is dictated by the recurrence or orthogonality structure of the problem rather than by terminology.

Third, numerical stability is problem-dependent. In rigorous parabolic PDE integration, sharper inverse estimates of Fourier–Chebyshev blocks improve step-size bounds and support constructive Newton–Kantorovich arguments (Cadiot et al., 28 Feb 2025). In QIPI, the analogous attempt to approximate $T_n(H)$7 near zero is numerically unstable for excited-state targeting, and a different Chebyshev-based filter is preferable (Patil et al., 26 Jun 2026). A plausible implication is that Cheb-inv is most reliable when the inverse object is either exact and recurrence-driven, or approximated away from singular spectral geometry.

Finally, not every inverse decomposition is a Cheb-inv. "A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints" studies a different problem: for $T_n(H)$8 and a subspace $T_n(H)$9 satisfying

$(H-\omega I)^{-1}$00

there exists a unique pair

$(H-\omega I)^{-1}$01

such that

$(H-\omega I)^{-1}$02

The inverse component is characterized variationally as the unique maximizer of

$(H-\omega I)^{-1}$03

over $(H-\omega I)^{-1}$04 in $(H-\omega I)^{-1}$05, and the paper develops stability results and feasibility-preserving Newton-type algorithms (Dolinsky et al., 26 Jan 2026). This is an inverse decomposition in a rigorous optimization sense, but it does not introduce Chebyshev polynomials. Its relevance is chiefly contrastive: it isolates what is specific to Cheb-inv, namely the use of Chebyshev polynomial structure rather than only inversion plus linear constraints.

Taken together, these works support a precise encyclopedic interpretation. Cheb-inv is best understood as a structural pattern for encoding inverse information through Chebyshev objects. It ranges from exact formulas for inverse-function derivatives, to numerical inversion frameworks for monotone maps, to explicit matrix inverses and spectral inverse quantities, to QSVT-implementable polynomial surrogates for nonunitary inverses. Its strengths derive from recurrence structure, orthogonality, and coefficient control; its limitations emerge when the target inverse becomes too singular or too tightly embedded in dense spectra.