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the mirror x, length 4.000 Hx = (-4.000, 0) v = x − αe₁ safe sign: α = −‖x‖, so v is formed from a sum and nothing cancels unsafe sign: α = +‖x‖ gives ‖v‖ only 35.1% of ‖x‖ + ‖x‖ — the digits go ‖HᵀH − I‖ 5·10⁻¹⁶ ‖Hx‖ − ‖x‖ 8.9·10⁻¹⁶ second component 2.2·10⁻¹⁶ built from a unit vector orthogonality is structural
Householder
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A reflection cannot stop being one
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One number that has to be right
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A triangle where the scalar was
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Eight blocks and sixty-four reflections
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The factor nobody forms
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7 essays · orthogonality
the same lattice, twice what the reduction may not change det, before 1 det, after 1 defect, before 7.1 defect, after 1 LLL steps 1 the determinant is the invariant and the defect is what is being reduced
Lattice reduction
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A basis that describes its lattice badly
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Rounding a coordinate in the wrong basis
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The knob and the rounding
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A relation among digits that were not there
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The room a relation has to stand out
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7 essays · exact
0 5 10 15 20 25 30 35 40 10⁻¹⁷ 10⁻¹⁴ 10⁻¹¹ 10⁻⁸ 10⁻⁵ 10⁻² 10¹ singular value, in order σ ⁄ σ₁ eight digits an independent draw per entry 1 ⁄ r, falling to the floor a cliff, and a control 1/r rank at 10⁻⁸ 5 log r rank at 10⁻⁸ 5 noise rank at 10⁻⁸ 96 σ₂ ⁄ σ₁ 0.024 σ₆ ⁄ σ₁ 2.8·10⁻⁹ the block has full rank and five useful columns
Off-diagonal rank
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A block nobody can call sparse
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A rank that is a number of digits
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The size the rank does not notice
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The kernel with nothing to compress
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The smaller cluster sets the rank
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7 essays · hierarchy
10⁻¹ 1 10¹ 10² -45 -35 -25 -15 -5 5 15 25 s, the smallest of the three interpolation points rightmost pole of the reduced model unstable above this line balanced truncation, order 3 exact, and unusable full system's pole -3.5 placements swept 19 unstable models 4 worst pole 24 their interpolation 1.3·10⁻¹⁴ balanced truncation -0.84 the conditions all hold and the model cannot be run
Reduced stability
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A model that cannot be run
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Half the conditions and a certificate
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A certificate written in coordinates
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The inner product the mesh already computed
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The smaller cell downstream
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7 essays · reduction
10² 10².³ 10².⁵⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁹⁶ 10⁻¹ 10⁻⁰.⁵ 1 rows in the sketch worst relative distortion dimension 64 dimension 256 5 seeds per point, band is best to worst the dimension does not appear
Sketching
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The dimension does not appear
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The sketch that is not the answer
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The sketch that is spent
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Built from products alone
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Sketching what is never unfolded
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7 essays · randomised
1 2 3 4 5 6 7 10⁻¹¹ 10⁻⁹ 10⁻⁷ 10⁻⁵ 10⁻³ 10⁻¹ order r kept error and bounds 2Σσ, and the error σᵣ₊₁ a bound that is an equality orders 7 bound ÷ error, worst 1 spread over the sweep 1 error ÷ σᵣ₊₁, worst 2.1 computed before the model and attained by it
Balanced truncation
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The bound that is known in advance
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The product nobody had to form
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The state that is removed is not a mode
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A mode that rings is counted twice
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Where a ringing mode stops being a pair
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6 essays · reduction
0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 x u exact central differences upwind the oscillation is exact ‖Ax − b‖/‖b‖ for the central answer 4.6·10⁻¹⁸ values outside [0, 1] 16 worst excursion 0.52 the dashed lines are 0 and 1, which the equation guarantees no solver was involved
Convection
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The stencil that is not symmetric
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The diffusion that makes the answer exact
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Exact along one axis
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The direction the diffusion does not go
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A different equation on every grid
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6 essays · iterative
2 1 -1 3 -3 -1 2 1 -2 1 2 -4 4 3 -1 2 as given rows in the order 1 2 3 4 4 3 -1 2 0 1.25 1.25 2.5 0 2.5 1.5 -3 0 -0.5 -0.5 2 after step 1 pivot 4 4 3 -1 2 0 2.5 1.5 -3 0 0 0.5 4 0 0 -0.2 1.4 after step 2 pivot 2.5 4 3 -1 2 0 2.5 1.5 -3 0 0 0.5 4 0 0 0 3 after step 3 pivot 0.5 ‖PA − LU‖/‖A‖ 0 largest multiplier 0.75 row order 4 3 2 1 the pivot is chosen
Elimination
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Elimination is a sequence of choices
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Which of the choices is doing the work
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The order the greedy rule cannot choose
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One step ahead is one step short
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A doubling that arrives two steps late
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6 essays · elimination
1 -2 -6 -7 -4 7 4 -2 6 -1 2 -5 -8 -5 2 4 -3 9 4 1 5 1 4 3 5 A, the matrix as given 1 -2 -6 -7 -4 0 18 40 55 27 0 0 112 217 207 0 0 394 301 171 0 0 172 79 153 after 2 fraction-free steps 3 × 3 minors 25 divisions, all exact the intermediates are minors which is why the divisions come out whole
Fraction-free
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Every intermediate is a minor
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A bound on every intermediate at once
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Three orders and one last entry
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The pivot is in every product
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What reading the next pivot buys
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6 essays · exact
a triangle and a square, joined K₂,₃ with a pendant edge 1 2 3 4 5 6 0 1 2 3 4 5 6 index eigenvalue the same, and not the same vertices each 6 edges each 7 spanning trees 12 spectra differ by 5.3·10⁻¹⁵ highest degree, left 4 highest degree, right 3 one has a triangle the other is bipartite
Graph invariant
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The spectrum is not the graph
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A finer invariant that hears less
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Each spectrum hears the other's pairs
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The group hears a switch only at two
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The set has to be even outward
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6 essays · graph
-14 -12 -10 -8 -6 -4 -2 0 10⁻¹⁷ 10⁻¹³ 10⁻⁹ 10⁻⁵ 10⁻¹ 10³ 10⁷ 10¹¹ 10¹⁵ log₁₀ μ — the barrier parameter condition number, and relative error κ₂, condensed κ₂, augmented error, condensed error, augmented against a BigInt answer κ₂ augmented, μ = 10⁻¹⁴ 3·10¹⁵ its relative error 10⁻¹⁵ κ₂ condensed 2.4·10¹⁶ its relative error 0.31 the same step, written two ways and only one of them is solvable
Interior-point conditioning
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A condition number sent to infinity
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The active set before the digits
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Two repairs for one symptom
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A test with no tolerance in it
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The multipliers were a set
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6 essays · constraint
the diagonal of the hat matrix, hᵢ = aᵢᵀ(AᵀA)⁻¹aᵢ · dashed: its average p/m = 0.200 p/m 1 0 the leave-one-out residual: eᵢ/(1 − hᵢ), and forty refits bars: closed form · dots: refitted without that point one number, two fields Σ hᵢ, exactly p 10 largest leverage 0.5 closed form against refits 8.9·10⁻¹³ 1 − h of the first row 0.5 y appears in the residual and nowhere in the leverage
Leverage
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Influence is decided before the data
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Two observations that hide each other
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One minus a leverage is a subtraction
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The factor a sparse code keeps anyway
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The weight the factor met first
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6 essays · leastsquares
how far is it from A to an orthogonal matrix? smaller is nearer · the polar factor minimises this in every unitarily invariant norm polar factor U 1.8554 QR, signs fixed 2.1265 QR as returned 3.8226 200 drawn at random κ = 10 polar factor 1.9 QR, signs fixed 2.1 QR as returned 3.8 best of 200 random 2.7 ‖A − QR‖ is the same either way and ‖A − Q‖ is not
Polar decomposition
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The nearest orthogonal matrix
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An iteration that only multiplies
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A test with no answer in it
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A rotation that comes back mirrored
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Five precise points are five points
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6 essays · orthogonality
0 2 4 6 8 10⁻¹⁶ 10⁻¹³ 10⁻¹⁰ 10⁻⁷ 10⁻⁴ 10⁻¹ log₁₀ γ, the change of units forward error as written after scaling one change of variable unscaled, worst 0.0013 scaled, worst 1.7·10⁻¹³ orders recovered 10 scaled coefficient spread 4.5 the answer was never the problem the units were
Polynomial scaling
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The scaling that buys ten orders
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An estimate that does not move
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Two groups need two reductions
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The middle group needs no reduction of its own
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An interior root needs its own scaling
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6 essays · polynomial
-18 -15 -12 -9 -6 -3 0 0 log₁₀ ‖PKPᵀ − LDLᵀ‖ / ‖K‖ share of orderings best worst existence and stability factorise 1 unregularised 0.69 worst growth 6.4·10⁵ growth × γ 0.64 the ordering is free to choose and not free of consequence
Quasi-definite
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The regularisation that legalises every order
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The perturbation that does the work
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A shift that certifies a saddle
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The residual turns before the error doubles
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The right-hand side that hides the saddle
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6 essays · constraint
-1 -0.582271 -0.164542 0.253186 0.670915 1.08864 1.50637 0 eigenvalue 4 negative 10 positive counted before it was formed positive 10 negative 4 at zero 0 innermost ratio 39 the zero block is a theorem and so is the count either side of it
Saddle-point systems
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The zero that is not a missing entry
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Two ways to remove a constraint
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A minimum the Hessian cannot see
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The shift that stops at the first right count
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A constraint the count stops seeing
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6 essays · constraint
nothing 8.3% the right-hand side 16.5% the matrix, slowly 85.9% everything 100.0% what can be reused: the answer what can be reused: the factorisation what can be reused: the factorisation, for a while what can be reused: nothing share of the cost of a sequence that shares nothing what 12 members cost in factorisations same: factorisations 1 rhs: factorisations 1 drift: factorisations 4 independent: factorisations 12 what changes between the members decides what may be carried
Sequence of solves
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The problem that arrives again
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The order a batch arrives in
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A warm start is degree zero
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A straight path has nothing for a parabola to fit
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The degree the history chooses
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6 essays · sequence
10² 10³ 10⁴ 10⁻¹⁷ 10⁻¹⁶ 10⁻¹⁵ 10⁻¹⁴ 10⁻¹³ 10⁻¹² steps taken ‖RᵀR − AᵀA‖ ⁄ ‖AᵀA‖ the bound, linear in the steps √k · u every step safe, the chain not drift after the run 3.9·10⁻¹⁴ the bound there 3.3·10⁻¹³ √k · u there 6.1·10⁻¹⁵ worst single amplification 2.7 worst leverage met 0.69 refreshes 1 backward stable once and three thousand times is a different claim
Sequence stability
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Stable once, and three thousand times
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The repair the drift did not need
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The answer the last window left
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The step the two rows owe
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A correction that reads every row
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6 essays · sequence
10⁻⁷ 10⁻⁵ 10⁻³ 10⁻¹⁰ 10⁻⁸ 10⁻⁶ magnitude spacing to the next number the smallest normal gradual flush to zero smallest normal 6.1·10⁻⁵ smallest subnormal 6·10⁻⁸ octaves of subnormals 10 pairs that lie under FTZ 10 the spacing stops halving and stays put which is what makes x − y = 0 mean x = y
Subnormals
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The numbers below the smallest one
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The well on the far side of the band
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The far well in a byte
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Room for the step, not the value
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An octave to land in
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6 essays · arithmetic
1 10¹ 10² 10³ 10⁻³ 10⁻² 10⁻¹ 1 10¹ 10² 10³ n ‖Aₙ − A‖ and the largest term's norm ‖Aₙ − A‖ the larger of its two terms an infimum that is not attained n 1024 ‖Aₙ − A‖ 0.0017 largest term 1024 their product 1.7 √3 1.7 the distance goes to zero and nothing reaches it
Tensor rank
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A nearest point that is not there
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A rank that is not a property of the tensor
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The rank that stops being typical
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A fit with no answer to find
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A stop that knows the distance
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6 essays · tensor
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