Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Reorthogonalisation — where it appears

Projecting a new basis vector against earlier ones again, to restore the orthogonality a three-term recurrence loses in floating point. It costs a pass over the whole basis, which is what the short recurrence was for, so the choice is between storage and the correctness of the bound.

Named by 13 essays across 6 fields — each of them below, with the objects they name alongside it.

015304560759010512010⁻²10⁻¹110¹steprelative sizeleast error: 20discrepancy stop: 7errorresidualthe knob is an integerleast error, at step20error there0.14error at step 1206the residual falls at every stepthe error turns and keeps rising

A parameter that counts steps

The regularisation field's knob is a positive real number chosen by one of three rules. The iterative field's is an integer nobody called a knob — where to stop. On the same problem the best step is 20 and the best λ is 0.025, and they reach 0.1426 and 0.1406.

combination · Iterative regularisation for each previous column i, subtract the projection of column j onto qᵢclassicalr[i][j] = qᵢ · a[j] ↑ the ORIGINAL columnv = v − r[i][j] · qᵢthe three worst |qᵢ · qⱼ|:columns 7 and 8: 1columns 6 and 8: 0.13columns 6 and 7: 0.13modifiedr[i][j] = qᵢ · v ↑ what is LEFT of itv = v − r[i][j] · qᵢthe three worst |qᵢ · qⱼ|:columns 1 and 8: 4.4·10⁻⁷columns 2 and 8: 2.7·10⁻⁷columns 3 and 8: 2.4·10⁻⁸The two R factors agree to 1.2·10⁻⁶ relative. The two Q factors do not.the 8×8 Hilbert matrixone word, eight orders

Two Gram–Schmidts

One argument changes. Classical Gram–Schmidt projects the original column onto each previous direction; modified projects what is left of it. In exact arithmetic the coefficients are identical. In floating point they differ by eight orders of magnitude in the thing that matters.

orthogonality · Gram–Schmidt 10²10³10⁴10⁵10⁶10⁷10⁸10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²condition number‖QᵀQ − I‖one passsweeptwicetreewhat the second pass removesfitted slope, one pass2fitted slope, two passes0.96rounds, two passes6rounds, the sweep24one pass squares the condition numberand two do not

Doing it twice

Cholesky QR squares the condition number — a fitted slope of 1.95 in κ against the Householder sweep's 1.00. Run the identical routine a second time on the Q it returned and the slope is 0.93, the orthogonality is at or below the sweep's at every κ, and the price is one more all-reduce.

cost · Communication 2345678110²10⁴10⁶10⁸10¹⁰vectors in the basisκ₂ of the basisas derivedsolves spread outone subspace, two spanning setseight moments at one point7.7·10⁹eight points, spread1growth per vector661/u4.5·10¹⁵the same subspaceand only one of them usable

A basis that is the same subspace and not the same thing

The interpolation conditions are conditions on a subspace, so any basis of it will do. The one a derivation writes down reaches a condition number of 7.7·10⁹ by its eighth vector, and the rate at which it gets there is set by a number the user chose with no information.

reduction · Moment matching 1591317212529333710⁻¹110¹10²bidiagonalisation stepsrelative errorleast without: 20no penaltypenalty insidewhat stopping is worthbest without a penalty0.14and at step 40161best with one0.14and at step 400.14the same floor, reached twiceand only one run stays on it

The step that stops mattering

Regularise the problem the iteration has built rather than the problem it was given, and the error curve stops turning. The unregularised run ends 1,127 times above its own best; the same run with a penalty inside it ends 1.000000000003 times above.

combination · Iterative regularisation 26101418222630343810⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹stepsizeleast error: 20‖QᵀQ − I‖filter disagreementan identity with an expiry datedisagreement at step 1210⁻¹³disagreement at step 400.014least error at step20exact while the basis is orthogonaland false where the method is best

An expiry date the noise does not move

The polynomial description of conjugate gradients leaves the level of rounding at step 17 or 18 on this operator, at every noise level from 10% to 0.1%. The step worth stopping at moves from 3 to 44 across the same range. They coincide at about 1% noise, which is where the coincidence was first read, and it is a fact about the noise rather than about the method.

combination · Iterative regularisation 10¹10³10⁵10⁷10⁹10¹¹10¹³10⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹condition number κ(A)‖QᵀQ − I‖classical once, Householdermodified once, Householderclassical twice, Householderclassical twice, Cholesky QRκ²uκuat κ = 10⁸classical once, Householder0.0042modified once, Householder5.7·10⁻⁹classical twice, Householder3.1·10⁻¹⁵classical twice, Cholesky QR10⁻¹⁵64×16 in blocks of 4, three seedsHouseholder inside does not help between

A stable block is not a stable basis

Block Gram–Schmidt orthogonalises twice over — between blocks, and inside each one. Householder inside the blocks does not stop the classical between-block step losing orthogonality like κ², 4.2·10⁻³ at κ = 4.3·10⁷, and a second pass does not stop Cholesky QR inside the blocks breaking down at κ = 10⁸. Each level fails only on ill-conditioning placed at its own level, and one variant holds 3·10⁻¹⁵ on every placement.

orthogonality · Gram–Schmidt 08162432404810⁻¹10²10⁵10⁸10¹¹10¹⁴10¹⁷10²⁰degreeκ of the design matrix1/u: past this a double holds nothingmonomialsChebyshevArnoldi on the pointsκ at degree 48monomials1.5·10¹⁷Chebyshev1.6·10⁷Arnoldi on the points1200 two intervals with a gap between themthree bases for one space of polynomials

A basis built from the points

A polynomial fit computed in monomials and in an orthogonal basis gives the same curve on exact data, and the valley essay drew the two lying on top of each other. Add 0.1% noise and they separate — by 1.7·10⁻⁵ at degree 40 and 0.004 at degree 48 — because the fitted curve moves with the basis by its condition number times the rounding times the residual. Chebyshev polynomials keep that small only on points spread like their weight; on a sample with a hole in it they reach κ = 1.55·10⁷. A basis orthogonalised against the sample points themselves stays at 1 on every set.

leastsquares · Fitting 1357910⁻⁵10⁻⁴10⁻³10⁻²10⁻¹1eigenvalue λamplificationkeptdiscardeda filter on the starting vectormeasured against the polynomial3.9·10⁻¹¹worst kept direction1best discarded direction0.8the roots are the discarded Ritz valuesand the vertical lines are where they sit

Restarting is a filter

A restart throws away the Ritz values it does not want and begins again from a new starting vector. Written in the eigenbasis, that vector's components have been multiplied by a polynomial with its roots at the discarded values — measured component by component, and agreeing with the polynomial to rounding.

spectra · Lanczos 1234567891010⁻⁵⁵10⁻⁴⁹10⁻⁴³10⁻³⁷10⁻³¹10⁻²⁵10⁻¹⁹10⁻¹³10⁻⁷10⁻¹cyclesizethe residualrepairedthe reported boundwhat the stopping rule readsreported at the last cycle9.4·10⁻⁴¹the residual there5.7·10⁻⁵with the border recomputed3.8·10⁻¹⁴products, cheap and repaired9a bound with nothing under itand one product a cycle to fix it

Keeping the vectors, and losing the bound

Thick restarting keeps the Ritz vectors instead of filtering the starting vector — the same eigenvalues for a third of the products with A. Its residual bound reaches 9.4·10⁻⁴¹ while the residual it bounds sits at 5.7·10⁻⁵, and the eigenvalues are correct to 4.3·10⁻¹⁴ the whole time, so nothing reports it.

spectra · Lanczos an eigenvalue repeated 2 times, in a 200-dimensional problemblock of 1never returns them allblock of 232 products, 16 stepsblock of 351 products, 17 stepsblock of 460 products, 15 stepsblock of 570 products, 14 stepsblock of 684 products, 14 stepsthe narrowest that workscheapest width2its products with A32the widest drawn84narrower than the multiplicity is not slowit is absent

How wide the block should be

A block narrower than the multiplicity does not converge slowly — it never returns the missing copy at all. Above the multiplicity every extra column buys iterations at about ten products with A each. And the mechanism that is supposed to make the choice unimportant never fires from a random start.

spectra · Invariant subspace three seeds, medianblock MGS, Qᵀb, κ 10⁸0.23block MGS, b as a block1.1·10⁻⁹Householder2.8·10⁻⁹10²10⁴10⁶10⁸10¹⁰10¹²10⁻¹⁶10⁻¹²10⁻⁸10⁻⁴110⁴10⁸κ(A)relative error of xblock MGS, Qᵀbblock CGS, either routeblock MGS, b as a blockHouseholderthe same Q in both block MGS routesonly the order in which b meets it differs

What the appended block inherits

Modified Gram–Schmidt on [A b] solves least squares as well as Householder, although its Q is not orthogonal. A block code appends b as one more block. Block modified Gram–Schmidt inherits the rescue at every placement of the ill-conditioning: at κ = 10⁸ the appended block gives 6.9·10⁻¹⁰ where the same Q through Qᵀb gives 8.9·10⁻³. Block classical Gram–Schmidt gets the same wrong answer both ways, to the last bit. And the variant whose Q is orthogonal to 10⁻¹⁵ — two passes with Cholesky QR inside — is a hundred thousand times worse than Householder when the ill-conditioning is inside the blocks, because its R is wrong.

orthogonality · Gram–Schmidt λ = 105 timesλ = 9.55 timesλ = 95 timesλ = 8.55 timesλ = 2.952 timesλ = 2.92 timeseigenvalues that arrived more than once — the matrix has 40 distinct onesa spectrum with the wrong multiplicitiesextra copies, no reorthogonalisation25extra copies, full reorthogonalisation0worst relative error among the copies1.9·10⁻⁸steps taken of 80 asked for, full40no arithmetic error was madeevery one of these is right to eight digits

An eigenvalue that arrives twice

A matrix with forty distinct eigenvalues, handed to Lanczos for eighty steps, returns twenty-five extra copies of thirteen of them — the largest arriving five times. Every copy is accurate to 1.9·10⁻⁸ relative. No arithmetic error was made, nothing overflowed, and a caller counting eigenvalues gets the wrong multiplicity from a computation in which no individual number is wrong.

spectra · Lanczos

Named alongside it

The objects these essays reach for when they reach for this one.

Krylov subspaceRitz valuesCondition numberLoss of orthogonalityOrthogonalityStopping criterionBackward stabilityBlock methodsCholesky QRFilter factorsGram–SchmidtHouseholder reflection

All concepts