Gaussian elimination is stable for the inverse of a diagonally dominant matrix
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Bibliographic record
Abstract
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B element-of upper M Subscript n Baseline left-parenthesis bold upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo> ∈ </mml:mo> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">B\in M_n({\mathbf {C}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a row diagonally dominant matrix, i.e., <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma Subscript i Baseline StartAbsoluteValue b Subscript i i Baseline EndAbsoluteValue equals sigma-summation Underscript StartLayout 1st Row j equals 1 j not-equals i EndLayout Overscript n Endscripts StartAbsoluteValue b Subscript i j Baseline EndAbsoluteValue comma i equals 1 comma ellipsis comma n comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi> σ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>i</mml:mi> </mml:mrow> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mstyle scriptlevel="1"> <mml:mtable rowspacing="0.1em" columnspacing="0em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mi>j</mml:mi> <mml:mo> ≠ </mml:mo> <mml:mi>i</mml:mi> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mstyle> </mml:mrow> <mml:mi>n</mml:mi> </mml:munderover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="1em"/> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma _i |b_{ii}| = \sum _{\substack {j=1 j\ne i }}^n |b_{ij}|, \quad i = 1,\ldots ,n,</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 less-than-or-equal-to sigma Subscript i Baseline greater-than 1 comma i equals 1 comma ellipsis comma n comma"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi> σ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo> … </mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">0 \le \sigma _i > 1,\ i= 1,\ldots ,n,</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma equals max Underscript 1 less-than-or-equal-to i less-than-or-equal-to n Endscripts sigma Subscript i Baseline period"> <mml:semantics> <mml:mrow> <mml:mi> σ </mml:mi> <mml:mo>=</mml:mo> <mml:munder> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo> ≤ </mml:mo> <mml:mi>i</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:munder> <mml:msub> <mml:mi> σ </mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\sigma = \max _{1\le i \le n} \sigma _i.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> We show that no pivoting is necessary when Gaussian elimination is applied to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper B Superscript negative 1 Baseline period"> <mml:semantics> <mml:mrow>
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it