{"id":"a328fb2a-4e01-4cac-9f2b-fbc7255cc70a","arxiv_id":"2506.17193","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Weighted and preconditioned GMRES can realize any prescribed pair of convergence curves, with an exact singular-value condition on a triangular linking matrix governing the cases where the weight or preconditioner is fixed.","lead":"This paper proves that weighted GMRES and preconditioned GMRES can realize essentially any prescribed convergence behavior, even when the spectrum of the linear system is fixed. It gives exact conditions on the weight matrix or preconditioner that determine which pairs of convergence curves can occur together.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Converse of Theorem 15 implicitly assumes a basis-prescribed version of the Greenbaum–Pták–Strakoš theorem that is neither stated nor cited; the proof needs this strengthened result or a short unitary-similarity argument.","rationale":"The paper's main contributions are well-motivated and largely correct. Theorem 10 and Corollary 11 provide constructive, reproducible results for weighted GMRES, and Theorem 22 transfers them to preconditioning. The numerical experiments and open-source code support the claims. The only load-bearing weakness is the converse of Theorem 15 (and its echo in Theorem 25): it silently relies on a strengthened GPS theorem that allows prescribing the unitary nested residual basis W. The standard Theorem 5 does not state this, so the proof has a genuine gap. The gap is easily closed by a unitary similarity transformation, so the mathematical claim is very likely true; however, the manuscript should either cite the basis-prescribed version (e.g., the constructive proof in [13] or [8, Th. 3]) or add the short argument. This is exactly the reader's identified weakest assumption, and the conditional verdict is appropriate.","tokens_in":22586,"tokens_out":12942,"duration_ms":128370,"concrete_test":"Inspect the companion MATLAB code for the construction used in the converse of Theorem 15. If the routine accepts a prescribed unitary W and builds A by setting A = W H W* with an upper Hessenberg H whose subdiagonal is determined by g, then the basis-prescribed construction is present and the proof gap is purely a missing citation. Alternatively, add to the proof the explicit unitary-similarity argument: take (A0,b0) from Theorem 5 with nested basis V, set A = W V* A0 V W*, b = Wg, and verify that AK_i(A,b) = span{w_1,...,w_i}, so W is the required basis. If the code only calls a GPS routine that does not accept W, the gap remains.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The converse direction of Theorem 15 (Section 2.2, after Eq. (17)) picks a unitary W with TT* = W*M^{-1}W, sets b := Wg, and applies Theorem 5 to obtain A such that I-GMRES(A,b) realizes g. It then asserts that ~W := WT has its first m columns forming a nested basis of AK_m(A,b). This assertion is only justified if the first m columns of W themselves form such a nested basis, but Theorem 5 as stated guarantees only existence of some (A,b), not that a prescribed W is the nested residual basis. The same gap propagates into Theorem 25 via its use of Theorem 15. The missing statement is the basis-preservable GPS theorem: for any unitary W with first m columns forming a candidate nested basis and b = Wg, there exists A with prescribed eigenvalues such that W is the nested basis of AK_m(A,b). This is true—it follows from Theorem 5 by the unitary similarity A := W V* A0 V W*, where (A0,b0) realizes g with basis V—but the paper does not supply this argument. Without it, the converse of Theorem 15 is unproven as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22802,"tokens_out":15894,"duration_ms":158853,"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":[{"comment":"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].","section":"Section 2.2, proof of Theorem 15, after Eq. (17)"}],"minor_comments":[{"comment":"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":"Title and Abstract"},{"comment":"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).","section":"Section 4.2, Experiment 5"},{"comment":"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.","section":"Lemma 8"}],"recommendation":"major_revision","confidential_remarks":"The only substantive issue is the unproven step in the converse of Theorem 15. I am confident the paper should be reconsidered after the authors add the short unitary-similarity argument (or cite the basis-prescribed GPS version), and the same fix then covers Theorem 25. The numerical experiments are illustrative and do not overclaim. The self-citation [30] in the literature review is acceptable, but the editor may ask the authors to confirm that the overlap with that companion paper is limited to background; the novel results in this manuscript do not appear to depend on it. If the missing argument is supplied, I would support acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe 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.\n\nThe 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.\n\nMinor 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.\n\nRecommendation: 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.","headline":"Solid new results on simultaneous convergence-curve prescription for weighted and preconditioned GMRES, with a small patchable gap in the converse of Theorem 15.","tokens_in":23346,"tokens_out":5207,"would_cite":true,"duration_ms":51882,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F10","65F08","65F35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["GMRES","weighted GMRES","left preconditioning","right preconditioning","Krylov subspace methods","convergence curves","residual decrease vector","prescribed spectrum"],"falsifier":"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.","tokens_in":1898,"feed_emoji":"📉","tokens_out":6793,"duration_ms":114270,"temperature":0.7,"pith_summary":"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.","feed_headline":"Any two convergence curves are simultaneously possible for GMRES","feed_subtitle":"Choice of inner product or left/right preconditioning can completely reshape residual history, regardless of spectrum.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Provides the foundational theorem that any nonincreasing convergence curve and arbitrary spectrum are realizable by GMRES; the starting point for all simultaneous-prescription results.","marker":"[13]"},{"why":"Extends the construction to early termination and is used in the constructive proofs as the version of the prescription theorem that allows a prescribed nested basis.","marker":"[8]"},{"why":"Introduces weighted GMRES and supplies the early relation between nested bases in different inner products, which Lemma 8 generalizes.","marker":"[12]"},{"why":"Provides the nonstandard-inner-product framework and the comparison result that the paper's Theorem 16 is compared against.","marker":"[24]"},{"why":"Supplies the GMRES residual expressions used in Section 2.1 to compare I- and M-GMRES residuals.","marker":"[17]"},{"why":"Defines the GMRES algorithm, the minimal residual property, and the breakdown index that underpin the curve formalism.","marker":"[26]"}],"fun_headline_variants":["Any two nonincreasing GMRES curves are simultaneously possible","Weighted GMRES: any two curves, one matrix choice","Left and right preconditioned GMRES: any two curves simultaneously","Prescribe any two GMRES curves with a single linear system"],"cache_read_input_tokens":25472,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Any two nonincreasing GMRES curves are simultaneously possible","Weighted GMRES: any two curves, one matrix choice","Left and right preconditioned GMRES: any two curves simultaneously","Prescribe any two GMRES curves with a single linear system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000927,"raw_usage":{"total_tokens":4004,"prompt_tokens":1009,"completion_tokens":2995,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":625,"completion_tokens_details":{"reasoning_tokens":2921}},"tokens_in":625,"tokens_out":2995,"duration_ms":23570,"temperature":1.0,"reasoning_tokens":2921,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:10:46.206194+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Duintjer Tebbens and G","cited_arxiv_id":null,"evidence_quote":"Extends the construction to early termination and is used in the constructive proofs as the version of the prescription theorem that allows a prescribed nested basis."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces weighted GMRES and supplies the early relation between nested bases in different inner products, which Lemma 8 generalizes."},{"cited_title":"Pestana.Nonstandard Inner Products and Preconditioned Iterative Methods","cited_arxiv_id":null,"evidence_quote":"Provides the nonstandard-inner-product framework and the comparison result that the paper's Theorem 16 is compared against."},{"cited_title":"Saad.Iterative methods for sparse linear systems.Philadelphia, PA: SIAM Society for Industrial and Applied Mathematics, 2nd ed","cited_arxiv_id":null,"evidence_quote":"Defines the GMRES algorithm, the minimal residual property, and the breakdown index that underpin the curve formalism."}],"review_version":2}