Concept

Scaled norm — where it appears

A norm applied after a diagonal rescaling, in which a badly scaled matrix's sensitivity is measured against the size of its own entries. It is what makes an error statement about a graded matrix mean anything, since an absolute norm measures every row against the largest entry in the matrix.

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

The units the matrix is measured in

One linear system, written twice. The rows of the second are the rows of the first in different units, the solution is identical to the last bit, and the condition number has moved by eight orders of magnitude. One of those two numbers is a fact about the problem and the other is a fact about the notation.

error · Scaling

A norm that overflows before it is a norm

The vector of sixteen thousands has a Euclidean norm of 4,000, which fp16 represents exactly. Written as the square root of the sum of squares it returns infinity, because squaring doubles the exponent — and the expression costs half the format's range on the one computation every iterative method performs at every step.

arithmetic · Overflow

Named alongside it

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

bfloat16 Catastrophic cancellation Componentwise condition number Condition number Dynamic range Equilibration Exact ground truth Forward error Half precision Hilbert matrix hypot Overflow

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