Numerical Linear Algebra
where the algebra stops being the arithmetic
Concept

Wrapping effect — where it appears

The compounding over-estimate an interval computation pays when a correlated quantity is forced back into an axis-aligned box at every step. It is measured as the enclosure's width divided by the width of the set it encloses, and on a rotation by 45° it grows by exactly √2 a step. The precision does not touch it: the loss is the shape of the representation rather than any rounding.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

10⁻⁹10⁻⁶10⁻³110³10⁶13212937κ · usignificand bitsκu = 1a bound was provedthe method refusedit never returns a wrong boundlargest κu with a proof0.45smallest κu without one0.89cases refused, of the grid13a refusal is not a wide bound — it is no bound at alland it is the only failure mode here

A bound that is proved

Every error statement on this site so far is a measurement of one run. Interval arithmetic makes a different kind of claim — the answer lies in this set, for this input, with no probability attached — and its failure mode is that it returns nothing at all. On a Hilbert system it proves a bound 23 times the error it bounds, and one size later it refuses.

arithmetic · Interval 036912151810¹10²10³rotationswidththe enclosurethe seta rotation is an isometrymeasured growth a step1.4√2, from the geometry1.4enclosure ÷ set after 201024no rounding error is responsible for any of thisa higher precision does not touch it

Nine steps of pessimism

A proved bound is 8 to 26 times the error it bounds, at every precision from 16 to 40 significand bits. A carried interval is (√2)ᵐ times too wide after m re-enclosures. The two cross between eight and nine, so the method everybody warns against is the tighter of the two for a short computation.

arithmetic · Interval

Named alongside it

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

Condition numberDirected roundingHilbert matrixInterval arithmeticUnit roundoffVerified computingCancellationForward errorResidual

All concepts