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Equation 6 · Part 1 · Mathematics, Proof, and Scientific Computation in 2035: Scenarios, Signals, and Falsifiable Predictions

Symbol kappa

κ(A)=∥A∥ ∥A−1∥\kappa(A) = \|A\| \, \|A^{-1}\|
κ\kappa

What this part means

kappa is part of the quantity the equation computes from the expression on the right.

Its job in the formula

kappa is part of the quantity the equation computes from the expression on the right.

The passage around this formula

…it is derived from and checkable against a stated model of arithmetic. The classical example is the sensitivity of a linear system to input error, captured by the condition number of a matrix A : κ(A)=∥A∥ ∥A−1∥\kappa(A) = \|A\| \, \|A^{-1}\|. If κ(A)\kappa(A) is large, small errors in the input data or in floating-point rounding can produce large errors in the computed solution, independent of which algorithm is used to solve the system — a property of the problem, not the solver. Bringing formal verification to numerical software means…

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Learn the underlying idea

A function assigns an output to each allowed input. The expression f(x) means “apply f to x”.

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Sources cited in the article section

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