Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES

Nicole Spillane; Pierre Matalon

REVIEW 1 major objections 3 minor 34 references

Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES

T0 review · 1 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper proves that any two nonincreasing convergence curves can be realized together: plain GMRES and weighted GMRES, or left- and right-preconditioned GMRES, can be made to follow different prescribed residual histories on one system…

desk verdict Solid new results on simultaneous convergence-curve prescription for weighted and preconditioned GMRES, with a small patchable gap in the converse of Theorem 15. read the letter →

arxiv 2506.17193 v2 pith:5MYYJPGX submitted 2025-06-20 math.NA cs.NA

Pierre Matalon , Nicole Spillane This is my paper · ORCID

classification math.NAcs.NA MSC 65F1065F0865F35

keywords GMRESweightedleftpreconditioningrightKrylovsubspacemethodsconvergencecurvesresidualdecreasevectorprescribedspectrum

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper extends the classical one-curve GMRES prescription result to pairs of curves. It proves that for any linear system and any prescribed convergence curve, there exists a weight matrix making weighted GMRES follow that curve, and it gives a full characterization of such weights. The central result is a necessary and sufficient condition under which ordinary GMRES and weighted GMRES can be made to realize two prescribed curves simultaneously on the same system, with the eigenvalues of the system matrix chosen arbitrarily. The same condition transfers to left versus right preconditioned GMRES, showing that any two curves are simultaneously possible when the preconditioner is not fixed. This matters because it demonstrates that changing the inner product or moving a preconditioner from one side to the other can produce arbitrarily different residual histories even when the spectrum is identical.

What carries the argument

The residual decrease vector $g$ and the nested residual basis $W$ carry the argument. A convergence curve is recorded as $g_j=\sqrt{\|r_{j-1}\|^2-\|r_j\|^2}$, so prescribing a curve is prescribing $g$. Two nested bases $W$ and $\widetilde W$ of the Krylov residual spaces are related by $\widetilde W = W T$ with $T$ block upper triangular of the form (5); enforcing the right-hand side $b=Wg=\widetilde W\tilde g$ gives $g=T\tilde g$, and requiring $\widetilde W$ to be $M$-orthonormal forces $TT^*=W^*M^{-1}W$, which turns the eigenvalues of $M$ into singular values of $T$. This chain of identities is what all the equivalence theorems hinge on.

What would settle it

Run the paper's constructive proof on a random admissible triple, for instance n=2 with M=diag(1,1/4), g=(2,1)^T, and tilde g=(1,1)^T, which satisfies the Theorem 15 condition, building the system via the prescribed-basis version of the classical construction and then computing the actual I-GMRES and M-GMRES residual norms. If either residual sequence deviates from the prescribed curve, the converse direction is false. A simpler check is to enumerate all unitary matrices W for a small n and ask whether the classical construction can realize W as the GMRES basis; the first failure would refute the theorem's converse.

Watch

Extended reading notes

Core claim

The paper's central discovery is a transfer principle for prescribed GMRES convergence. In the residual decrease vector formalism, where a convergence curve is encoded by $g$ with entries $\sqrt{\|r_{i-1}\|^2-\|r_i\|^2}$, two curves $g$ and $\tilde g$ can be realized simultaneously by I-GMRES and M-GMRES, for a fixed Hermitian positive definite weight $M$, if and only if there is a nonsingular block upper triangular matrix $T$ such that $g=T\tilde g$ and the singular values of $T$ are the inverse square roots of the eigenvalues of $M$. The analogous statement for a fixed preconditioner $H$ is that left and right preconditioned GMRES realize prescribed curves exactly when the same triangular condition holds with singular values of $T$ equal to the reciprocals of the singular values of $H$. In both settings the eigenvalues of the (preconditioned) system matrix can be prescribed arbitrarily, generalizing the classical one-curve theorem. When the weight or preconditioner is itself free, any two same-length nonincreasing curves are simultaneously possible.

Load-bearing premise

The proof that the algebraic condition is sufficient assumes you can build a linear system whose GMRES residual directions are exactly a pre-chosen set of orthonormal vectors; the classical theorem cited only guarantees some set of directions, and the paper does not pin down the strengthened version it relies on.

Editorial extensions

If this is right

For a fixed weight matrix, the possible weighted-GMRES curves compatible with a given unweighted curve are exactly parameterized by triangular matrices whose singular values are fixed by the eigenvalues of the weight; every such matrix yields a system realizing both curves.
When the weight is not fixed, any two same-length nonincreasing curves are simultaneously realizable, and the system matrix can carry any prescribed eigenvalues.
Left and right preconditioned GMRES are interchangeable in a strong sense: any two curves are simultaneously possible, and for a fixed preconditioner the realizability condition is the same singular-value condition on the curves.
For any system and preconditioner there is another system and preconditioner with the same preconditioned spectrum where the left/right convergence roles are exchanged, so neither side is inherently faster.
The Euclidean and weighted residual norms of the two methods can differ by a factor bounded by the square root of the condition number of the weight, and the paper's constructions show this bound can be approached, so the inner product choice alone can drastically change the observed convergence history.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: the singular-value condition in Theorem 15 is algorithmic in nature, so it could be used as a design tool to search for a weight matrix that pushes weighted GMRES toward a desired curve while preserving a given unweighted curve.
Beyond the paper: because weighted GMRES with weight $M=P^*P$ is equivalent to split preconditioning by $(P,P^{-1})$, the same flexibility should extend to split preconditioners, with the left factor controlling the norm and the right factor controlling the operator.
Beyond the paper: the full flexibility proved here likely does not survive restarting or blocking, since restart cycles couple the nested bases across cycles; testing the same pair of curves under restarted or block GMRES would isolate the mechanism that makes the unrestricted setting so permissive.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a real contribution to the Greenbaum-Ptak-Strakos line of work. It extends the GPS parametrization to weighted and preconditioned GMRES, gives a necessary and sufficient condition (Theorem 15) for simultaneous prescription in terms of a triangular T with prescribed singular values, and draws the left/right preconditioning consequences (Theorems 22 and 25). The proofs are mostly constructive, and the open MATLAB code that reproduces the experiments is a plus.

The main theorems are new and, as far as I can tell, correct. Theorem 10 is clean: for any fixed system and any prescribed convergence curve of matching length, it constructs an hpd weight realizing that curve. Theorem 15's condition g = T~g with singular values of T tied to the spectrum of M is exactly the right kind of statement, and the necessity direction is proved properly through the relationship between nested I- and M-orthonormal bases.

The soft spot is in the proof of the converse of Theorem 15, and it propagates to Theorem 25. After choosing a unitary W with TT* = W* M^{-1} W, the authors set b = Wg and invoke Theorem 5 to produce A realizing g. They then claim that ~W = WT inherits the wanted nested-basis property. That only follows if the first m columns of W are a nested basis of AK_m(A,b), and Theorem 5 as stated does not guarantee that—it guarantees existence of some A,b with prescribed curve and eigenvalues, not one whose residual basis is this particular W. If g has zero entries, the basis is not uniquely pinned down by b = Wg, so this matters.

The gap is easily patched: cite or prove the strengthened basis-prescribed version of the GPS theorem (the original construction gives it, or use a unitary similarity argument with A = W V* A0 V W* where V is the basis produced by a standard GPS construction). The authors should state this explicitly. As written, the converse direction is not fully justified, but the fix is short and the result almost certainly survives.

Minor point: Corollary 11 and Theorem 22 require the two curves to have the same length. That is worth stating more prominently, because 'any two' in the title could lead a reader to expect different-length curves too. The analysis in Section 2.3 is honest about the condition being non-obvious. No issues with the citation pattern; the DD29 self-citation is directly relevant.

Recommendation: send to peer review. A referee should ask for the basis-prescribed lemma, but the core ideas are valuable and the flaw is not load-bearing.

Referee Report

1 major / 3 minor

Summary. The paper studies prescribed convergence curves for GMRES in the Euclidean inner product (I-GMRES), in a weighted inner product (M-GMRES), and for left/right/split preconditioned GMRES. Building on the Greenbaum–Pták–Strakoš theorem (Theorem 5), it proves: (i) for any fixed (A,b) and any admissible decrease vector g̃, there exists an hpd weight M such that M-GMRES realizes g̃ (Theorem 10); (ii) any two curves of the same length can be realized simultaneously by I-GMRES and M-GMRES with prescribed spectrum (Corollary 11), together with a characterization of the weights that allow a fixed M to realize a given curve (Theorem 13); (iii) a necessary and sufficient condition, in terms of a block-triangular T with prescribed singular values and g = T g̃, for a fixed M to allow simultaneous realization (Theorem 15); and (iv) analogous results for left/right preconditioned GMRES (Theorems 21–25), including a reversal construction (Corollary 24). The paper also gives residual-norm bounds relating I- and M-GMRES (Theorem 16) and contains numerical illustrations. The proofs are largely constructive and are accompanied by open-source MATLAB code.

Significance. If the main theorems are established, the paper makes a genuine contribution to the inverse convergence-analysis literature. The results quantify how much the choice of inner product or the side of preconditioning can alter GMRES convergence independently of the spectrum, and the block-triangular singular-value condition in Theorem 15 is a new and checkable criterion. The paper is also honest about the fact that the characterization in Theorem 15 is not very transparent and analyzes it in Section 2.3. Strengths include the constructive proofs, the clean structural Lemmas 7 and 8, the explicit characterization theorems, and the openly available reproducible MATLAB code. The main weakness is a missing justification in the converse of Theorem 15, where a basis-prescribed version of the GPS construction is needed; as written that proof is incomplete, although the gap is repairable by a short unitary-similarity argument.

major comments (1)

[Section 2.2, proof of Theorem 15, after Eq. (17)] The converse direction applies Theorem 5 to the vector b := Wg and then asserts that the first m columns of ~W := WT form a nested basis of AK_m(A,b). Theorem 5 as stated guarantees only the existence of some system (A,b) realizing g; it does not guarantee that the prescribed unitary W is a nested residual basis for the constructed A. The same gap is inherited by Theorem 25, whose proof invokes Theorem 15. The gap is repairable: take (A0,b0) from Theorem 5 with a nested orthonormal basis V (so b0 = Vg and A0 K_i(A0,b0) = span{v1,...,vi}), and set A := W V* A0 V W*. Then A has the prescribed eigenvalues, b = Wg, and the first m columns of W form a nested basis of AK_m(A,b), so the subsequent verification of the M-orthonormality of WT goes through. Please insert this argument, or cite a basis-prescribed version explicitly such as the constructive proof in [8, Th. 3].

minor comments (3)

[Title and Abstract] The simultaneous-prescription theorems (Corollary 11 and Theorem 22) require the two curves to have the same length, i.e. the same GMRES breakdown iteration. The title's unqualified claim that any nonincreasing convergence curves are simultaneously possible is stronger than the theorems; please qualify the statement.
[Section 4.2, Experiment 5] The sentence 'the scaled vectors (sqrt(mu_i)/beta w_i)_i form an M-orthonormal nested basis' is not consistent with the definitions M = beta M0 and b = (1/sqrt(n)) Q1; with W = Q, the normalized vectors are w_i/sqrt(beta mu_i) and the corresponding squared decrease is beta mu_i/n. Please correct the scaling expression (the displayed formulas for ||~r_i||_M are consistent with the corrected normalization).
[Lemma 8] The proof of Lemma 8 is only a sketch ('justified by nestedness'). Since Lemma 8 underlies both Theorem 13 and Theorem 15, a short block-by-block derivation of the upper-triangular structure of T would improve readability and make the paper more self-contained.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the derivation is anchored in the external GPS theorem and explicit constructions, with a contextual self-citation and a non-circular proof gap in Theorem 15's converse.

full rationale

The derivation chain is anchored in the external Greenbaum-Ptak-Strakos theorem (Theorem 5, cited to [13]) and in self-contained structural lemmas (Lemmas 6-8). Theorems 10, 13, 15, 22, 23 and 25 construct the weight matrix M, the system (A,b), and the preconditioner H explicitly from the prescribed residual decrease vectors; no parameter is fitted to a subset of data and then renamed as a prediction. The only directly relevant self-citation is [30], used as context ('See also the literature review in [30]' and 'see also [30, Sec. 3.1]'); it is not load-bearing for any theorem. I therefore find no step where the claimed conclusion is assumed as an input. One correctness caveat should be separated from circularity: in the converse direction of Theorem 15 (Section 2.2, after equation (17)), the proof chooses a unitary W satisfying TT* = W* M^{-1} W, sets b := Wg, invokes Theorem 5, and then asserts that ~W := WT has its first m columns forming a nested basis of AK_m(A,b). The quoted Theorem 5 guarantees existence of some A realizing the curve g with this b, but it does not state that a prescribed unitary W can be made the nested residual basis. The needed basis-preserved version is true (by unitary similarity of the GPS construction), so this is an omitted justification, not a circular reduction. Theorem 25 inherits the same caveat, but this does not raise the circularity score.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central results rest on the Greenbaum-Ptak-Strakos existence theorem as an external input, standard linear algebra (spectral theorem, Cholesky, interlacing inequalities), and the paper's own lemmas on nested bases. No free parameters are fitted: the weight matrices and preconditioners are constructed explicitly from the prescribed curves. No new entities are postulated. The only load-bearing assumption that is not explicitly stated is a basis-prescribed strengthening of the GPS theorem used in the converse of Theorem 15.

assumptions (6)

domain assumption Greenbaum-Ptak-Strakos theorem (Theorem 5): for any residual decrease vector g of length m and any n nonzero complex numbers lambda_i, there exists a nonsingular A with eigenvalues lambda_i and a vector b such that GMRES(A,b) realizes g.
Invoked throughout (e.g., proofs of Corollary 11, Theorem 15, Theorem 22) as the external starting point; the paper relies on its constructive proof including, in the converse of Theorem 15, a basis-prescribed version not stated in Theorem 5.
domain assumption A strengthened, basis-prescribed version of the GPS theorem: given any unitary W whose first m columns are to be the nested residual basis, one can construct (A,b) with prescribed eigenvalues realizing a given curve g.
Used implicitly in the converse of Theorem 15 after equation (17) to ensure W serves as the nested residual basis. This is true from the constructive proof of [13] but is not stated or cited explicitly, which is the paper's main proof gap.
standard math The GMRES residual decrease vector formalism: for any nested orthonormal basis w_i of the residual Krylov space, |<b,w_i>| = sqrt(||r_{i-1}||^2 - ||r_i||^2) (Lemma 7 and equation (2)).
Proved in the paper from basic projection properties; standard.
standard math Spectral theorem and singular value decomposition: hpd matrices can be diagonalized by unitary similarity; TT* and M^{-1} sharing eigenvalues implies existence of unitary W with TT* = W*M^{-1}W.
Used in Theorem 15 converse and Theorem 16.
standard math Existence of Cholesky factorization M = P*P for any hpd M.
Used in Theorem 21 and 22 to convert weighted GMRES to split preconditioned GMRES.
domain assumption The Krylov breakdown index m = dim K_n(A,b) is independent of the inner product.
Used implicitly in Theorem 10 and Corollary 11 to require same-length curves; stated in Section 1.

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Pith. "Pith review of Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES." pith.science (2026). https://pith.science/paper/5MYYJPGX

@misc{pith2026250617193,
  author       = {Pith},
  title        = {Pith review of: Any nonincreasing convergence curves are simultaneously possible for GMRES and weighted GMRES, as well as for left and right preconditioned GMRES},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MYYJPGX}},
  note         = {Machine review of arXiv:2506.17193}
}

read the original abstract

The convergence of the GMRES linear solver is notoriously hard to predict. A particularly enlightening result by [Greenbaum, Pt\'ak, Strako\v{s}, 1996] is that, given any convergence curve, one can build a linear system for which GMRES realizes that convergence curve. What is even more extraordinary is that the eigenvalues of the problem matrix can be chosen arbitrarily. We build upon this idea to derive novel results about weighted GMRES. We prove that for any linear system and any prescribed convergence curve, there exists a weight matrix M for which weighted GMRES (i.e., GMRES in the inner product induced by M) realizes that convergence curve, and we characterize the form of M. Additionally, we exhibit a necessary and sufficient condition on M for the simultaneous prescription of two convergence curves, one realized by GMRES in the Euclidean inner product, and the other in the inner product induced by M. These results are then applied to infer some properties of preconditioned GMRES when the preconditioner is applied either on the left or on the right. For instance, we show that any two convergence curves are simultaneously possible for left and right preconditioned GMRES.

Figures

Figures reproduced from arXiv: 2506.17193 by the authors.

**Figure 1.** Experiment 1. Convergence curves of M-GMRES for a sample of random T −1 matrices. M has two distinct eigenvalues with large gap: 1 and 1012, with respective multiplicities 12 and 8. In the plots, the convergence curve of I-GMRES is represented in black with dot markers, while all other lines (colored, without markers) represent those of various instances of M-GMRES. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗

**Figure 2.** Experiment 2. Convergence curves of M-GMRES for a sample of random T −1 matrices. M has three distinct eigenvalues with large gap: 1, 1010 , 1020, with respective multiplicities 9, 7 and 4. In the plots, the convergence curve of I-GMRES is represented in black with dot markers, while all other lines (colored, without markers) represent those of various instances of M-GMRES. Experiment 2 (two eigenvalue jumps) The sp… view at source ↗

**Figure 3.** Experiment 3. Convergence curves of M-GMRES for a sample of random T −1 matrices. The spectrum of M is uniformly distributed on a logarithmic scale between 1 and 1012 . In the plots, the convergence curve of I-GMRES is represented in black with dot markers, while all other lines (colored, without markers) represent those of various instances of M-GMRES. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: Experiment 4. In the plots, the convergence curve of I-GMRES is represented in black with dot markers, while all other lines (colored, without markers) represent those of various instances of M-GMRES. The results are displayed in [PITH_FULL_IMAGE:figures/full_fig_p019…

**Figure 5.** Figure 5: Experiment 5 4.3 Left and right preconditioning Note that the results from the previous sections can also be interpreted as instances of left and right preconditioning strategies. In this section, we consider the matrix HB/mcfe4 from the University of Florida Sparse ma…

**Figure 6.** Figure 6: Preconditioners applied on the left or on the right. The norm that is minimized by the precondi [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: Residuals and errors yielded by left (L) and right (R) preconditioned GMRES. [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Arioli, V

M. Arioli, V. Pták, and Z. Strakoš. Krylov sequences of maximal length and convergence of GMRES.BIT Numerical Mathematics, 38(4):636–643, 1998.doi:10.1007/BF02510405

work page doi:10.1007/bf02510405 1998

[2] [2]

A comparative study of sparse approximate inverse preconditioners.Appl

Michele Benzi and Miroslav Tůma. A comparative study of sparse approximate inverse preconditioners.Appl. Numer. Math., 30(2-3):305–340, 1999.doi:10.1016/S0168-9274(98)00118-4

work page doi:10.1016/s0168-9274(98)00118-4 1999

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