Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Exact cost — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 3456789101112110¹10²10³10⁴10⁵10⁶nwidest intermediate, in bitsrationals, not reducedfraction-free · rationals reduced · the answerone answer, three widthsn12answer40Hadamard bound52fraction-free40reduced rationals39unreduced1.4·10⁶the error is zero on every curvethe cost is the length of the numbers

    An answer with no error in it

    An integer matrix eliminated over the rationals rounds nothing, so the forward error is zero, the residual is the zero vector, and the identity this site is built on has no terms left. The cost does not vanish with the error. It moves into the length of the numbers, where three correct routes differ by four orders of magnitude.

    rung 1 · exact
  2. 3456789101110²10³nbitsthe answerthe question · det Abits in, bits outn11question484answer729det A33widest part37the answer is the floorand no route writes it down more cheaply

    The answer is longer than the question

    An exact solution of an integer system is a vector of fractions, each of them a ratio of two determinants. So the output carries 2n long integers where the input carried n² short ones, and no algorithm can write it down more cheaply — the length of the answer is a floor under every exact solver rather than a property of one.

    rung 2 · exact
  3. 10⁻¹⁶10⁻¹⁴10⁻¹²10⁻¹⁰10⁻⁸10⁻⁶10⁻⁴10⁻²110⁻⁷10⁻⁵10⁻³10⁻¹10¹relative error in the datarelative error in the answerthe answer's errorthe data's errornothing here is the arithmetic'sdata error10⁻¹²answer error10·10⁻⁴amplification10·10⁸rounding committed0residual0the residual is exactly zeroand the answer is wrong anyway

    An exact answer to a measured problem

    The residual is the zero vector, nothing was rounded at any step, and the answer is wrong in its first digit. Data accurate to fourteen places, an exact solve of the system it defines, and an error of 10⁻⁵ — because conditioning was never a statement about arithmetic and removing the arithmetic error removes none of it.

    rung 3 · exact

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