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• A block size is a property of the machine
• Doing it twice
• Memory bought with messages
• The message and the word
• The order the products are taken in
• The same arithmetic at a different price
• The last digit is the cheapest
• The recursion that was never told the memory
• A reduction that changes the order
• Where the format starts paying
• The block size a recursion still has
• The plan that was right at rank four
• A ceiling is not a target
• The search that got worse as it widened
• A beam ranked on what remains
• Widen the beam where the ranking is right
• The leaf that sits on the edge

• A bound that is proved
• A coin flip that fixes the average
• A norm that overflows before it is a norm
• Eight bits, and a format that breaks the rules
• One exponent for thirty-two numbers
• Proving the answer is in the box
• The direction the error leans
• The numbers below the smallest one
• The other half of a format
• Where the box is cut
• Cancellation takes the answer, not a digit
• The order they are added in
• What a float can hold
• Nine steps of pessimism
• A bit buys an octave
• Buying the accuracy back
• The units that overflow before the answer does
• Three walks and one bound
• Where the hardware went

• A condition number for one eigenvalue
• An eigenvalue one vector cannot see
• An eigenvalue with no value
• How wide the block should be
• The algorithm the libraries actually run
• The best approximation there is
• The form a real matrix can reach
• The form that makes it affordable
• The gap decides the eigenvector
• The plane survives what its vectors do not
• The vector was what was wanted
• Two matrices and one problem
• Two shifts that are never formed
• An eigenvalue count that cannot be slightly wrong
• Symmetry is worth more than precision
• An eigenvalue that arrives twice
• The largest gap is inside the null space
• The error the method already knows
• The same budget, spent five ways
• A function of a matrix is not a function of its entries
• A good curve and a bad verdict
• A problem with no answer
• A spectral radius that grows first
• A threshold the matrix does not set
• Accurate is not a property of a method
• Keeping the vectors, and losing the bound
• Rank is a decision
• Restarting is a filter
• Small compared to what
• The cheap rank and what it cannot see
• The eigenvalues that are not there
• The series that has to be squared back

• A correction cheaper than the problem
• The observation that cannot be removed
• The projection and the right angle
• The road that squares the problem
• The valley with no bottom
• A constraint is a weight at infinity
• Influence is decided before the data
• A basis built from the points
• The two numbers a caller has
• When the matrix is wrong too
• Two observations that hide each other
• One minus a leverage is a subtraction
• The reference was a method
• The condition number that does not know
• Feasible and wrong
• The factor a sparse code keeps anyway
• The weight the factor met first

• A direction the smoother cannot see
• Aggregating what the matrix calls strong
• A hierarchy with no grid behind it
• An operator with no entries
• A rate that does not notice the size
• Exact along one axis
• One sequence and two recurrences
• Smoothing a whole line at once
• The coarse grid the matrix chooses
• The diffusion that makes the answer exact
• The formula that was already optimal
• The number that is re-derived
• The stencil that is not symmetric
• A Krylov space for a problem that is not linear
• Changing the condition number on purpose
• The coarse problem is a different problem
• A run that is over at step five
• A smoother that stops being one
• How much direction there was to lose
• A different equation on every grid
• A parameter that is also a price
• The switch does not know which side is better
• The answer that arrives when the space runs out
• A walk needs a length
• A proof that does not ask how large the matrix is
• A rate that is known in advance
• An iterate that must be made smaller
• An orthogonalisation nobody calls one
• Coarsening in one direction only
• The certificate that arrives soonest is worth least
• The direction the diffusion does not go
• The division that cannot be done
• The error smoothing cannot reach
• The rate the condition number predicts
• The residual the method reports
• The same problem on a coarser grid
• The same zero, and nothing was found
• The spectrum that predicts nothing
• The zero that means it is finished

• A limit the matrix never reaches
• An equation whose unknown is a matrix
• The matrix that is one row
• Two dimensions, and the cluster that thins
• A perturbation that keeps the symmetry
• The elimination the matrix does not need
• A speedup with a ceiling of its own
• The number that cannot rank them
• A nearby problem of the wrong kind
• A preconditioner that changes sign
• Four orders of conditioning, and four steps
• The circulant that cannot be indefinite
• The condition number of the model
• The circulant the problem did not contain
• Two near-zeros cost less than one
• The correction lost to its own two-by-two solve
• A zero no twist can step around

• An answer that is known
• The exact answer to a nearby problem
• The zero you are allowed to write
• Three errors and one number
• A small residual is not a small error
• The condition number is an amplifier
• The tail a sample never reaches
• A tolerance is priced by the problem
• A condition number scaling cannot move
• A tensor that cannot be decomposed
• An accuracy that is a backward error
• An estimate that can be fooled
• Deciding that a zero has arrived
• The fifth author
• The number that decides nothing
• The roots are not the coefficients
• The units the matrix is measured in
• Two condition numbers of one matrix

• A parameter chosen on a smaller problem
• Four knobs and one floor
• The part of a solver that may be rounded
• An answer that changes with the seed
• An expiry date the noise does not move
• A step that is not a unit of work
• The method that cannot use a smooth answer
• A preconditioner that arrives past the answer
• A parameter that counts steps
• The step that stops mattering
• What a cheap preconditioner has to leave alone
• A stopping rule that follows the run it is given
• The rule that is wrong in the right direction
• The parameter neither knob is
• A count that marks the edge and not the pace
• The shift had an edge, and the approximation moved it
• One arc, and what each filter pays to be on it

• A rank that is not a property of the tensor
• A compression of 10¹⁴ that still does not fit
• Five indices are cheaper than two
• A decomposition made only of SVDs
• A factorisation that is unique for once
• A nearest point that is not there
• A solve that is d decompositions
• An index that is a pair
• An iteration that walks out of the set
• The digit that costs more than the tensor
• The format that does not notice the dimension
• The orthogonality that cannot be diagonal
• The test that is a deadline
• The repair that costs exactly itself
• One term too many
• The rank a sweep can vouch for
• A fit that has an answer and cannot stop

• A reflection cannot stop being one
• Orthogonal is a number
• Two Gram–Schmidts
• A test with no answer in it
• The right-hand side as one more column
• A stable block is not a stable basis
• A rotation that comes back mirrored
• An iteration that only multiplies
• The nearest orthogonal matrix
• The tree the resistances choose
• Spread resistances make the loops easy
• The basis nobody chose on purpose
• One number that has to be right
• A triangle where the scalar was
• Five precise points are five points
• A mirror decided in the thin directions
• What the appended block inherits

• A rule that is correct and unusable
• Elimination is a sequence of choices
• The bound that is never attained
• The swap that is not optional
• The gap refinement can close
• A pivot that searches one row and one column
• A factorisation with nothing to pivot for
• The inverse that is never formed
• The pivot that reads the units
• When symmetry is not enough
• A worst case is as fragile as its margin
• The growth a boundary-value problem supplies
• Which of the choices is doing the work
• The order the greedy rule cannot choose
• Noise the growth amplifies
• A margin the factorisation records
• A threshold that holds the growth still

• A tolerance that reads its own residual
• Stable once, and three thousand times
• The accuracy that is thrown away
• The problem that arrives again
• What a rebuild is worth
• The penalty for keeping it is a ratio
• The repair the drift did not need
• A factorisation kept past its date
• What survives one step of the barrier
• Where the drift lands
• The order a batch arrives in
• A warm start is degree zero
• A guess worth two per cent
• One line that buys a quarter of the run
• A straight path has nothing for a parabola to fit
• The degree the history chooses
• The answer the last window left

• Structure and stability stop being separable
• The factor is not sparse
• The order decides the memory
• The order that was right last time
• Two ends of the same arrow
• What the symbolic phase can only bound
• A threshold between fill and growth
• The cliff behind the count
• An ordering that does not wait for the numbers
• The fill that is not independent
• The least fill there is
• An ordering that buys processors, not time
• Two minima that are one minimum
• The depth that is worse than both ends
• The column that was never fixed
• How few columns the search needs
• The freedom a symmetric factorisation does not have

• The basis decides what a filter is
• When the answer is a choice
• Where the answer stops being in the data
• The grid was the first filter
• Thirty-two coefficients instead of a noise level
• One draw in twenty
• The corner reads the norm it is drawn in
• A second blur, narrower than the first
• Choosing without knowing
• Noise that spares the answer and fools the rules
• Where the grid hands over to λ
• A better discretisation is a weaker filter
• A rule that has to be told how good its answer will be
• A second penalty is not a second parameter
• The grid on which the discretisation stops mattering
• A corner the penalty can afford
• The data count their dimensions, not the step's

• The kernel with nothing to compress
• The same matrix, numbered twice
• The size the rank does not notice
• The test that costs what it saves
• A knob calibrated in residuals
• The offset that moved the slope
• The count that is not the budget
• A block nobody can call sparse
• A rank that is a number of digits
• The accuracy worth paying for
• The knob that moved two things
• The rounding that was not the problem
• Which pairs are allowed to be small
• The partition that does not move
• Two knobs on one number
• A geometry setting that is a second accuracy
• A second objective that is the first one doubled
• A prediction that arrives a decade late

• The sketch that is not the answer
• The sketch that is spent
• The half of a problem a sketch may touch
• Counting what is inside a circle
• The dimension does not appear
• What a single draw cannot report
• A bound that holds with probability
• A rate that belongs to the matrix
• Built from products alone
• Counting what cannot be looked at
• Randomisation does not create structure
• Sketching what is never unfolded
• A rule that reads only its own probes
• The split nobody is in a position to choose
• The rank a certificate charges
• A sketch that finds the columns it can see
• The miss a normal table already priced

• The zero that is not a missing entry
• Two ways to remove a constraint
• Three eigenvalues, and two are the golden ratio
• A preconditioner that need not know the constraint
• A condition number sent to infinity
• A minimum the Hessian cannot see
• The active set before the digits
• One eigenvalue and two steps
• Where the augmentation puts the cost
• The regularisation that legalises every order
• A shift that certifies a saddle
• The perturbation that does the work
• Two repairs for one symptom
• A test with no tolerance in it
• The shift that stops at the first right count
• A constraint the count stops seeing
• A loop that asks the null space why

• A matrix that depends on its own eigenvalue
• The scaling that buys ten orders
• Six routes to one spectrum
• A spectrum that comes in reciprocal pairs
• A problem with infinitely many eigenvalues
• The problem the solver was actually given
• An error committed before the arithmetic
• The eigenvalues that are answers to nothing
• The points the algorithm chose
• A backward-stable answer to a problem nobody asked
• One mass removed, and one eigenvalue gone
• The number that moves when the problem does
• Where a contour's budget should go
• A ceiling with a knob on it
• A class a longer chain takes away
• A perturbation that moves every coefficient
• An estimate that does not move
• Every eigenvalue real, and a test that says so
• The conditioning that rises with the ceiling
• Two approximants and one matrix size

• A model that is a rational function
• The bound that is known in advance
• The product nobody had to form
• Exact at the points that were named
• A basis that is the same subspace and not the same thing
• Interpolating at the model’s own poles
• Where to put the poles of a rational function
• A model that cannot be run
• A model with no matrices behind it
• The state that is removed is not a mode
• The definition asks for more of what defeats it
• Bracketing an error nobody can measure
• A condition number that is not the model's
• Why a Gramian can be truncated at all
• Half the conditions and a certificate
• A certificate written in coordinates
• The inner product the mesh already computed

• The same program, twice
• A bound every answer satisfies
• Where the disagreement comes from
• The vector that hides it
• The sum that cannot be wrong
• What determinism costs
• Accuracy and agreement are different properties
• A matrix that is definite on one machine
• A square that evaluates negative
• A rank that depends on the thread count
• The length that changes the kernel
• What a regression test can ask for
• An inner product with no fixed sign
• The variation that comes with a seed
• Two machines, one certificate
• One multiply the compiler removed
• The licence is not the boundary
• The reading that never moves
• A stopping test is a race
• The tolerance that buys no agreement

• A matrix with no numbers in it
• The vertex nobody solves for
• A partition decided in the last digit
• A ranking that is an eigenvector
• The rate is the second eigenvalue
• A chain with no stationary vector
• An eigenvector that must not change sign
• A ranking whose order is not determined
• A distance computed by a solve
• A graph with a tenth of the edges
• Eliminating a vertex is a graph operation
• A count that comes out of a determinant
• The spectrum is not the graph
• A conductance the arcs do not measure
• Two Laplacians of one graph
• A bound with a square root in it
• A Laplacian that is not symmetric
• A preconditioner that is a tree
• The vector that has to be rounded

• An answer with no error in it
• Every intermediate is a minor
• How many primes the answer needs
• A prime that divides the answer
• A fraction recovered from one remainder
• The rank depends on the ring
• An exact answer to a measured problem
• The answer is longer than the question
• A basis that describes its lattice badly
• What a determinant does not determine
• Rounding a coordinate in the wrong basis
• The knob and the rounding
• A relation among digits that were not there
• A bound on every intermediate at once
• Three orders and one last entry
• The room a relation has to stand out
• Two precisions guard the other edge