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Finite-Precision Symmetric Krylov Methods: Exact Rounding Examples, Block Paige Identities,and a Variable-Block Lanczos Model

Mohit Sinha

arXiv:2609.08066v1math.NAcs.NA

TL;DR

Short-recurrence Krylov methods that are equivalent in exact arithmetic can diverge under floating-point rounding. The paper analyzes rounding and convergence in scalar and block methods, and shows how computed recurrences and Ritz-value clusters relate to solution accuracy and independent directions.

  • Problem

    Finite-precision rounding can make equivalent CG and direct Lanczos–Galerkin methods behave differently, while stored residuals and block-vector rank may depart from their exact-arithmetic counterparts.

    Source: §1 · p. 2

  • Method

    The paper combines rounding constructions and convergence bounds with precision analysis, nearby symmetric models, and block Lanczos overlap and rank analysis.

    Source: §1 · p. 3, §7.2 · p. 32, §9 · p. 50

  • Results

    In the Hessian example, sixteen residual-qualified vectors span only five dimensions, while the rank-aware recurrence, ARPACK, PRIMME, and LOBPCG return sixteen independent directions.

    Source: §9 · p. 50

  • Takeaways & Limitations

    Residual accuracy and subspace dimension describe different parts of the outcome, distinguishing repeated eigenvalue approximations from repeated information in the original space.

    Source: §9 · p. 50

    The precision guarantee is tied to the specified dense evaluation order, and its exponent-range assumptions include intermediate scalar calculations.

    Source: §5.2 · p. 23, §9 · p. 50

Abstract

from arXiv · show

Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes--Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos--Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within $n$ updates for an $n\times n$ matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.

1 Introduction

Short-recurrence Krylov methods can produce different computed outcomes in finite precision despite exact-arithmetic equivalence. The paper develops rounding, convergence, and spectral analyses for scalar and block methods, including how computed Ritz-value clusters relate to independent directions.

1 Introduction

  • Finite-precision rounding affects residuals, Lanczos-vector orthogonality, and block numerical rank, so exact-arithmetic equivalence does not ensure matching computed behavior.A stored residual can drift from the true residual, and a residual block may lose columns.Source: §1 · p. 2
  • A steepest-descent construction cycles exactly, while a second positive definite family gives an exact CG solution after four updates but an inconsistent two-step Galerkin system.A common spectral enclosure gives comparison bounds under stated enlarged-model and transfer hypotheses.Source: §1 · p. 3
  • For a fixed dense evaluation order, the paper gives sufficient precision for a prescribed backward error within n updates and a computable stopping test.The results include an interpolation estimate and experiments examining the precision–iteration trade-off.Source: §1 · p. 3
  • Householder orthogonalization and singular-value truncation yield a variable-block recurrence with controlled errors and an exact block Lanczos relation for a nearby symmetric problem in a larger space.The analysis allows the next block to have fewer columns.Source: §1 · p. 3
  • A block Paige identity bounds overlap along a Ritz vector’s residual coordinate, while an inter-block recurrence tracks its evolution and supports a lower bound on nearby Ritz values.A physical-residual and numerical-rank estimate assesses how many associated Ritz vectors are independent in the original space.Source: §1 · pp. 3–4
  • In the Hessian example, sixteen residual-qualified vectors have numerical rank five, while rank-aware and established eigensolvers return sixteen independent directions.The analysis distinguishes repeated Ritz approximations from repeated information in the original space.Source: §1 · p. 4

2 Floating-point model and algorithms

The paper specifies floating-point assumptions and evaluation order because they determine computed recurrences. It also defines backward error and the residual gap used to assess computed solutions.

2 Floating-point model and algorithms

  • Each scalar assignment and vector component is stored before the next operation, so algebraically equivalent rearrangements can produce different iterates.For normal-range operations, the relative error model bounds each rounded arithmetic operation by unit roundoff.Source: §2 · p. 4
  • The normwise relative backward error measures the smallest relative normwise perturbation of A and b that makes the approximate solution an exact solution.The true residual is evaluated over the reals; its difference from the stored residual is the residual gap.Source: §2 · p. 4
  • The algorithms distinguish CG and Lanczos coefficients and specify left-to-right accumulation for matrix-vector products and inner products.The direct Galerkin approximation solves the projected tridiagonal system and forms the iterate from the basis vectors.Source: §2 · pp. 4–5

3 Steepest descent in finite precision

The section shows how rounding can create an exact steepest-descent cycle, while a sufficient smallness condition guarantees geometric decrease of the stored residual. It also establishes scale invariance for recursively updated CG residuals under representability and range assumptions.

3 Steepest descent in finite precision

  • For an exactly representable system, Algorithm 1’s stored and true residuals agree at every step, and the iteration enters a periodic orbit in which both residuals have constant nonzero norm.Source: §3 · pp. 5–6
  • The cycle’s residual gap is zero, and its first-component correction is too small to reach the next representable iterate.The construction uses κ2(A)u = 8, unlike the small-roundoff conditions used in convergence analyses.Source: §3 · p. 6, §3.1 · p. 7

3.1 Geometric decrease of the recursively updated residual

  • Under Theorem 2’s sufficient condition, the stored residual decreases geometrically while the relative-error model applies and no underflow or overflow occurs.In an unbounded-exponent model it falls below every positive threshold in finitely many steps.Source: §3.1 · p. 9
  • For diagonal matrices, the sufficient condition improves to one with linear dependence on κ2(A).The dense-matrix product-error bound introduces an additional condition-number factor.Source: §3.1 · p. 10
  • In the eleven-bit diagonal scan, four of ten starts cycle at κ2(A)u = 8, and all ten do so at sampled values at least 32.For dense binary64 examples with κ2(A) in {10^2, 10^3}, the stored residual continues decreasing after the true residual reaches its attainable-accuracy level; the squared norm eventually underflows while residual components remain nonzero.Source: §3.1 · p. 10, Figure 1 · p. 10

3.2 Scaling and recursively updated CG residuals

  • In binary round-to-nearest arithmetic, dyadically scaled CG residual and search-direction pairs retain the same subsequent coefficients and scale their subsequent vectors by the same power of two.This requires representable scalings, nonzero divisors, and no underflow or overflow; the result also holds for the steepest-descent residual and step size.Source: §3.2 · p. 11
  • Dyadic scaling preserves directional evolution under the stated range assumptions, showing that unit roundoff is a relative scale rather than an absolute residual-norm threshold.The recurrence depends on the current residual and search direction, the matrix, and the arithmetic.Source: §3.2 · p. 11

4 CG and Lanczos computations in finite precision

The paper constructs a finite-precision case where Hestenes–Stiefel CG stores the exact solution after four updates, but the two-step Lanczos–Galerkin system is inconsistent. It relates this divergence to local orthogonality errors and gives a common convergence bound for enlarged models under stated hypotheses.

4 CG and Lanczos computations in finite precision

  • CG stores the exact solution after four updates, while the two-step direct projected system is singular and inconsistent.The exact comparison is for the stated evaluation orders and applies at κ2(A) ≥ 16/u^2.Source: §4 · pp. 12–13

4.1 Paige’s local-orthogonality relation

  • The local orthogonality term can change the second diagonal coefficient by more than λmin(A), making the stored projected matrix singular although the exact Galerkin matrix is positive definite.The weighted adjacent-vector product is at most u/4, while the inner product has magnitude h, of order √u.Source: §4.1 · p. 14
  • A residual bound for computed Ritz values combines their distance from σ(A) with the Lanczos-relation error; sufficiently close positive-spectrum Ritz values make the projected matrix positive definite.Under the stated hypotheses, that positive definiteness gives a unique Galerkin solution.Source: §4.1 · pp. 14–15

4.2 A common spectral convergence bound

  • When both recurrences admit enlarged Lanczos models with spectra in a common enclosure, stated transfer and residual hypotheses yield comparable convergence bounds for CG and direct Lanczos.The bounds distinguish model residuals from physical residuals; the true CG residual also includes the residual gap.Source: §4.2 · pp. 15–16
  • On a dense Strakoš matrix with n = 48, binary64 residual histories are similar despite separating recovered tridiagonal coefficients, and computed methods need extra iterations to reach the reference residual level.The reference trajectory uses CG in 300-bit arithmetic.Source: §4.2 · p. 16

5 Backward error and precision bounds for CG

This section studies precision and backward-error requirements for dense CG, including condition- and dimension-dependent bounds, true-residual stopping, and evaluation-order effects that can cause breakdown on a two-dimensional SPD system. It also combines spectral-cluster interpolation sensitivity estimates with floating-point analysis to examine the trade-off between required precision and iteration budgets on tested matrices.

5.1 Evaluation order, scaling, and error constants

  • The analysis fixes left-to-right dense accumulation and uses exact dyadic scaling; a separately recomputed true-residual test certifies the backward-error target.The cross-multiplied test is equivalent to the matrix-only backward-error criterion for nonzero iterates and implies the normwise condition.Source: §5.1 · pp. 17–19

5.2 Sufficient precision and condition-number bounds

  • Under stated rounding, scaling, smallness, and exponent-range assumptions, p ≥ p_suff makes CG return a nonzero iterate with target backward error by i* ≤ n.For x0 = 0, the sufficient precision bound is O(n^3 log(2 + nκ2(A)/ε)); the corollary assumes unbounded exponent range and ω = 0.Source: §5.2 · p. 22
  • A two-dimensional SPD construction makes the first nonfused, left-to-right CG denominator zero, while a fused multiply-add avoids the failure, showing that the condition obstruction depends on evaluation order.The initial iterate has normwise backward error 1; the example motivates a condition-dependent lower bound for worst-case guarantees.Source: §5 · p. 16, §5.2 · pp. 22–23

5.3 Precision and the iteration budget

  • The iteration-budget subsection contrasts fixed-budget precision guarantees with allowing more steps, using polynomial estimates and computed CG trajectories.It notes that Chebyshev approximation for inversion on a positive spectral interval supplies an iteration bound, but the supplied passage does not include its order expression.Source: §5.3 · pp. 23–24
  • A lower bound for degree-at-most-n polynomials q with q(0)=1 quantifies uniform behavior over disjoint spectral intervals; for fixed nodes, Λ(δ) tends to 2 as δ tends to zero.For the CG application, actual cluster locations and enlarged-problem weights matter because CG minimizes a weighted error over those eigenvalues; Greenbaum’s spectral enclosure bounds possible spread.Source: §5.3 · pp. 24–25
  • For a 10^-10 backward-error target, the reported precision was 32–43 bits with 2n or 6n steps, versus 33–172 bits with n steps across 14 test matrices.The matrices had κ2(A) = 10^3; each reported precision met the target under an independently evaluated backward-error test.Source: §5.3 · p. 25, Figure 4 · p. 25
  • At n=12 and ρ=0.8, the twelve-step best backward errors were approximately 7.73×10^-11, 1.90×10^-11, 1.93×10^-10, and 4.16×10^-11 at p=54, 55, 56, and 57, respectively.The success criterion can therefore vary nonmonotonically with precision.Source: §5.3 · p. 25
  • A sufficient theorem controls the specified floating-point computation directly, complementing the cluster-model interpolation estimate and motivating both a stopping criterion and an iteration budget.Source: §5.3 · p. 25

6 Inexact matrix-vector products and preconditioning

The section reviews analyses of inexact matrix-vector products and preconditioning, then extends block Paige identities to computations with inexact products. For a fixed SPD preconditioner, the argument carries over in the M-inner product, with block-width-dependent error bounds.

6 Inexact matrix-vector products and preconditioning

  • Chen’s additive-error analysis obtains a perturbed three-term recurrence and Paige’s loss-of-orthogonality identity, extending the scalar Paige–Greenbaum framework to its stated inexact-product model.Related work also treats inexact preconditioning, relaxed product accuracy, and backward error.Source: §6 · p. 26

6.1 Block identities under inexact operations

  • The block Paige identities apply when local errors include inexact products, provided the collected recurrence retains the required dimensions, symmetry, and within-block orthogonality.The total perturbation combines rounding errors with the errors in the matrix-vector products.Source: §6.1 · pp. 26–27
  • For a fixed SPD preconditioner, the argument applies in the M-inner product because M^-1A is self-adjoint there.The transformed-coordinate block error is bounded by √ri ε0∥A∥2 when columnwise solve tolerances are used.Source: §6.1 · p. 27

7 Block Lanczos, rank deflation, and Ritz values

The section develops finite-precision models for variable-block Lanczos with Householder QR and compact SVD rank selection. It bounds local recurrence errors and shows that block-update order affects adjacent-block products and the constants used in later overlap bounds.

7 Block Lanczos, rank deflation, and Ritz values

  • The analysis distinguishes block width from the dimension added to the computed space, using a Gram-matrix argument to connect rank deflation with repeated approximations.The block Lanczos step uses an orthonormal basis for a rank-deficient residual block’s column space, reducing the next block’s size when needed.Source: §7 · pp. 27–28

7.1 Finite-precision arithmetic in a block step

  • Householder QR followed by compact-SVD truncation selects the next block, retaining singular values above an absolute threshold.The resulting local bounds remain finite as a retained singular value approaches the threshold.Source: §7.1 · p. 29

7.2 Rounding-error models for Householder QR and the compact SVD

  • The computed variable-block recurrence has a bounded local perturbation and controlled adjacent-block products under the stated arithmetic and orthogonalization assumptions.The theorem allows the next block to have fewer columns after singular directions are discarded.Source: §7.2 · p. 32
  • Symmetrized update variants accumulate larger adjacent products, while unsymmetrized and re-projected variants keep them small.Re-projection removes a term that can amplify an existing product when the coupling block is ill-conditioned; the products also enter the block Paige identity constants.Source: §7.2 · p. 33, Figure 5 · p. 34

7.3 Numerical rank and components of the residual

This section separates residual directions that enlarge the computed space from components already in it, then bounds how this distinction affects rank selection and coupling growth.

7.3 Numerical rank and components of the residual

  • The next-block rank is bounded by counts of residual singular values above thresholds adjusted by the local error.The lower and upper counts use τ_i + ϵ_i and τ_i − d_i, respectively.Source: §7.3 · pp. 34–35
  • Components of the residual in the existing space can prevent block-size reduction even when the new-direction component has lost numerical rank.The singular values of the new-direction component describe residual directions orthogonal to the computed space.Source: §7.3 · pp. 34–35
  • The coupling-matrix growth bound accumulates local errors through a product, giving polynomial growth under polynomial bounds on its factors.When the relevant quantities are bounded, the stated accumulation is linear in the number of steps.Source: §7.3 · p. 36

7.4 An orthonormal basis for an enlarged symmetric problem

The section turns the computed block recurrence into an exact block Lanczos relation for a nearby symmetric problem in a larger space. Its construction normalizes the blocks and uses their Gram matrix to control the enlargement.

7.4 An orthonormal basis for an enlarged symmetric problem

  • Normalizing the computed blocks and accounting for their Gram matrix yields the orthonormal basis used in the enlarged relation.The construction also includes a boundary block orthogonal to the computed block basis.Source: Figure 6 · p. 39, §7.4 · pp. 39–40
  • Theorem 6 constructs an exact block Lanczos relation for an enlarged symmetric matrix of the form I_k ⊗ A + Δ.The perturbation satisfies ∥Δ∥_2 ≤ 2∥R∥_2 under the theorem’s hypotheses.Source: §7.4 · pp. 39–40
  • When the coupling blocks have full row rank, the relation is an exact variable-block Lanczos recurrence through block k, up to within-block orthogonal changes of basis.The theorem’s enlargement preserves the starting block, so spectral information from T refers to that same initial block.Source: §7.4 · pp. 40–41

7.5 Exact-arithmetic block Ritz clusters and deflation

The section relates exact block Ritz clusters and next-block deflation through Gram matrices. It also cautions that block Lanczos can produce improper clusters away from the spectrum of A.

7.5 Exact-arithmetic block Ritz clusters and deflation

  • When C = 0, each eigenvalue of J_k occurs d times in the block Ritz problem, and the lifted Ritz vectors are independent.The rank of the next new block is rank(D − C^T S^-1 C).Source: §7.5 · pp. 41–42
  • The Gram-matrix Schur complement D − C^T S^-1 C measures the new directions available at the next step.Its rank gives the rank of the next new block.Source: §7.5 · pp. 41–42
  • Improper block Ritz clusters may lie away from every eigenvalue of A, so localization of stabilized values requires the computable boundary term.Source: §7.5 · p. 42

7.6 Block Paige identities

The block Paige identities relate Ritz-vector overlap to residual coordinates and lost orthogonality, extending scalar bounds while distinguishing directional overlap from the full block overlap.

7.6 Block Paige identities

  • The block Paige identity bounds a Ritz vector’s overlap along its residual coordinate, with the physical residual controlled by the boundary residual and local error.The identities also apply to the inexact products covered by Corollary 4.Source: §7.6 · pp. 42, 44
  • With a wider final block, the inner product g_i^T w_i controls only the component of g_i along w_i, not its full norm.In scalar Lanczos, the formulas recover Paige’s one-vector and two-vector identities.Source: §7.6 · p. 43
  • For scalar Lanczos, the identities determine the overlap quantities uniquely; for block width s ≥ 2, m(s − 1) overlap components remain unconstrained.The component along each residual coordinate is determined, and the off-diagonal Ritz Gram entries are determined by the overlaps.Source: §7.6 · p. 43
  • The inter-block dynamics govern components perpendicular to the residual coordinate, explaining how a block Ritz vector can overlap the next block despite a large boundary residual.When a residual coordinate is zero, all s components remain free in the diagonal equation.Source: §7.6 · p. 43
  • For width three, the full overlap can exceed the scalar bound even while directional inner products remain small; width one satisfies the scalar bound.The plotted reference constant includes the computed Ritz-pair correction.Source: §7.6 · pp. 43–44

7.7 Subspace overlap and physical Ritz vectors

The section tracks overlap between current Lanczos blocks and earlier physical Ritz subspaces, then uses spectral and physical-rank bounds to interpret additional nearby Ritz values.

7.7 Subspace overlap and physical Ritz vectors

  • Theorem 8 gives at least d + t eigenvalues of T, counted with multiplicity, in an interval around c from t selected overlap directions.The selected directions come from the component of GZ orthogonal to the existing subspace Z.Source: §7.7 · p. 45
  • Choosing fewer leading singular directions omits weaker overlap directions; for a fixed ordering, the bound radius is nondecreasing with the number retained.At a prescribed radius, the largest admissible t can be selected.Source: §7.7 · p. 46
  • Overlap with an earlier Ritz subspace accumulates through positive-semidefinite Gram increments, so each singular value of WZ,ℓ is nondecreasing while that subspace is fixed.Residuals of the earlier subspace and local errors enter the recurrence as inhomogeneous terms that can create and propagate overlap.Source: §7.7 · pp. 45–46
  • Under the stated physical-residual and spectral-separation conditions, at least q − r of q nearby Ritz vectors are redundant in the original space.For every η ≥ √q ρ/γ, the numerical rank of their lifted vectors is at most r.Source: §7.7 · pp. 46–47
  • The combined recurrence, eigenvalue count, and physical-rank test connect accumulated overlap to nearby Ritz values and the number of independent physical directions they represent.Components already in the existing space can keep singular values above the SVD threshold even when orthogonal residual components are small.Source: §7.7 · p. 47

7.8 A matrix-free softmax-Hessian calculation

A matrix-free softmax-Hessian experiment compares residual accuracy with the independence of returned Ritz directions. The residual-only recurrence returns sixteen residual-qualified vectors but only five numerical dimensions, unlike the comparison solvers and rank-aware calculation.

7.8 A matrix-free softmax-Hessian calculation

  • The experiment used a matrix-free damped Hessian for a 40,970-parameter output layer, requesting its sixteen largest eigenpairs across three fixed random seeds.Storing the dense binary64 Hessian would require 13.43 GB.Source: §7.8 · p. 47
  • 16 independent directions were returned by ARPACK, PRIMME, LOBPCG, and the rank-aware block calculation; the unprotected recurrence returned 5 despite 16 residual-qualified vectors.The rank-aware calculation used an SVD-truncated generalized Rayleigh–Ritz projection; all methods were audited at the same residual and independence tolerances.Source: §7.8 · pp. 47–48
  • In all three runs, measured overlap preceded numerical dependence of the leading Ritz set by eight block steps, while residual-only acceptance came more than forty steps after overlap crossed √u.The overlap tracks return of an earlier converged subspace; the final rank test identifies five independent directions among sixteen reported vectors.Source: §7.8 · p. 48

8 Formal certificates and reproducibility

The paper provides Lean 4 certificates for its principal rounding, recurrence, and spectral results, alongside reproducible arithmetic checks and numerical artifacts.

8 Formal certificates and reproducibility

  • Exact rational simulation and Lean 4 proofs check the steepest-descent two-cycle and CG/Lanczos separation under their stated rounding assumptions.The checks cover significand precisions 11, 24, 53, and 113, and the zero-denominator construction is checked from precision 2 through 256.Source: §8 · p. 48
  • Formalized results include scalar perturbation and accumulated-error bounds, transfer to backward error, and block Lanczos identities for variable-width blocks.The block proofs also cover the inter-block overlap recurrence and positive-semidefinite Gram increment.Source: §8 · p. 49
  • The formal spectral results establish the additional-eigenvalue count and, under residual and separation assumptions, a bound on numerical rank and redundant directions.The Lean sources, pinned dependencies, build instructions, plotted data, fixed matrix inputs, and regeneration scripts are available; all twelve modules build together.Source: §8 · p. 49

9 Conclusions

The conclusions show that rounding can separate algebraically equivalent Krylov methods and that block-rank changes connect local error, overlap, and spectral information. They also give precision and convergence bounds while identifying where sharper long-run estimates remain needed.

9 Conclusions

  • The steepest-descent example has a two-cycle where stored and true residuals agree, whereas a sufficient condition ensures geometric decrease of the stored residual.The cycle distinguishes a periodic orbit from stagnation caused by a residual gap; the diagonal case has a sharper bound.Source: §9 · pp. 49–50
  • The CG/Lanczos example shows different computed outcomes, with local orthogonality loss, projected-system singularity, and common-enclosure comparison bounds linking the analyses.Numerical examples show similar convergence histories even after the tridiagonal coefficients separate.Source: §9 · p. 50
  • For a fixed dense evaluation order, the precision theorem gives a sufficient bit count and computable backward-error stopping test within n updates, under exponent-range assumptions.Fixed-input experiments show the searched precision decreases as the iteration budget grows, so working precision and iteration count must be considered together.Source: §9 · p. 50
  • Block orthogonalization and rank truncation yield a variable-block recurrence with controlled errors and an exact block Lanczos relation for a nearby symmetric problem.The block Paige identity bounds overlap along a Ritz vector’s residual coordinate; an inter-block recurrence tracks remaining overlap components.Source: §9 · p. 50
  • In the Hessian calculation, sixteen residual-qualified vectors span only five dimensions, while the rank-aware recurrence, ARPACK, PRIMME, and LOBPCG return sixteen independent directions.Residual accuracy and subspace dimension describe different parts of the outcome.Source: §9 · p. 50
  • Gram identities connect overlap to additional nearby Ritz values, while a physical-residual and rank estimate bounds how many corresponding vectors are independent at the stated numerical-rank tolerance.The Hessian example distinguishes repeated eigenvalue approximations from repeated information.Source: §9 · p. 50
  • Sharper long-run estimates for the inter-block recurrence are still needed to quantify when overlap reaches the eigenvalue-count theorem’s required level.The interpretation connects local error analysis, overlap evolution, and information in a Ritz cluster.Source: §9 · p. 50
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