Numerical Linear Algebra
where the algebra stops being the arithmetic
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The thread: Where the algebra stops being the arithmetic — page 3

Essays 49 to 54 of the 54 on this theme, in the same order.
027548110813510⁻⁶10⁻⁴10⁻²110²10⁴10⁶power‖Aᵏ‖Kreiss constant 6760e · n · K‖Aᵏ‖ρᵏtwo routes to one peakspectral radius0.8peak of ‖Aᵏ‖2·10⁴Kreiss constant6757e · n · K1.1·10⁵everything here decays in the endand one of these curves says how much first Eigenvalues, singular values, rank

A spectral radius that grows first

ρ(A) below one guarantees that the powers of A go to zero and says nothing about what they do on the way. Here they rise by a factor of twenty thousand before turning over, and the peak is bracketed above and below by a constant computed from the resolvent norms outside the unit circle — two routes to one number, one through the plane and one through the powers.

 75 aggregates over 225 unknownsthe matrix chose thismean extent along y3mean extent along x1points adopted by pass two15no coordinate enters the methodand the shape follows the coupling Iterating, instead of factorising

Aggregating what the matrix calls strong

The depth phase measured every method it had on the 45°-rotated anisotropic operator — 0.784, 0.883, 0.844 — and diagnosed the failure as being in the discretisation rather than in the hierarchy. Smoothed aggregation is the standard answer to anisotropy. It returns 0.789.

 10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²10⁻¹⁴10⁻¹10¹²10²⁵10³⁸10⁵¹10⁶⁴δ, the gap between consecutive eigenvaluesrelative error in e^Aan answer with no correct digitsV f(Λ) V⁻¹scaling and squaring‖A_δ − A₀‖exact eigenvalues throughoutκ(V) at the smallest δ3.3·10⁸²eigen route2.9·10⁶⁵scaling and squaring4.1·10⁻¹²distance to the limit7·10⁻¹²the eigenvalues are the diagonaland they are exact at every stop Eigenvalues, singular values, rank

A function of a matrix is not a function of its entries

Everybody learns that f(A) means diagonalise, apply f to the eigenvalues, undiagonalise. That is a definition, not a method. On a matrix seven picometres from a defective one — with exact eigenvalues and eigenvectors from a closed form — the definition returns an answer wrong by sixty-five orders of magnitude, and a method that never mentions an eigenvalue returns the right one.

 0246810121410⁻¹⁷10⁻¹⁶10⁻¹⁵10⁻¹⁴10⁻¹³10⁻¹²10⁻¹¹10⁻¹⁰10⁻⁹10⁻⁸squarings, srelative error against the closed formbest: s = 2the approximant is out of rangethe squarings amplifyboth sides costerror at s = 06·10⁻¹²best, at s = 26.9·10⁻¹⁶error at s = 153.2·10⁻¹²‖A‖ / 2^s at the best1.1the squarings are exact in the algebraand each one doubles the rounding Eigenvalues, singular values, rank

The series that has to be squared back

The Taylor series for the matrix exponential is not wrong — every term is computed correctly — and on Moler and Van Loan's two-by-two its largest term is 5.4 million times the answer it sums to. The method that replaces it scales the matrix down and squares the result back, and both halves of that sentence cost: too few squarings and the approximant is out of range, too many and each one doubles the rounding.

 0481216202410⁻¹⁷10⁻¹⁴10⁻¹¹10⁻⁸10⁻⁵10⁻²10¹matrix–vector products, mrelative error in e^Abforming e^A: 3·10⁻¹⁶crosses at m = 18the vector, not the matrixsteps to the dense answer18dimension100Krylov megaflops0.36dense megaflops2the exponential that is computedis 18×18 Eigenvalues, singular values, rank

The vector was what was wanted

Nobody who computes a matrix exponential wants the matrix. They want e^{At}b — one vector, the state of a system at a later time. Twenty matrix–vector products get it to sixteen digits on a hundred-by-hundred problem, without ever forming a hundred-by-hundred exponential, and the exponential that does get computed is twenty by twenty.

 10⁻¹⁷10⁻¹⁵10⁻¹³10⁻¹¹10⁻⁹10⁻⁷10⁻⁵10⁻³10⁻¹10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²1εrelative error in J(x)vforwardcentralcancellationtruncationagainst a derivative that is exactforward floor1.3·10⁻¹⁰central floor1.1·10⁻¹²truncation slope, forward1truncation slope, central2no ε reaches the roundoffand the analytic derivative is free of the choice Iterating, instead of factorising

An operator with no entries

At the sizes where linear algebra is expensive the matrix does not exist. What exists is a subroutine that returns Av. Every Krylov method survives that unchanged; every algorithm that reads an entry disappears. And the derivative such a code computes is accurate to ten digits instead of sixteen, which turns out to cost nothing at all.

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