← All parts of this equation

Equation 6 · Part 3 · Mathematics, Proof, and Scientific Computation in 2035: Scenarios, Signals, and Falsifiable Predictions

Symbol A^-1

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

What this part means

A−A^-1 is one of the signed contributions combined to compute the quantity on the left.

Its job in the formula

A−A^-1 is one of the signed contributions combined to compute the quantity on the left.

The passage around this formula

Numerical analysis has always run on an analogous verification principle, just usually informal: an error bound is only useful if 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 proving that an implementation’s actual rounding behavior…

Read this part in the article →

Learn the underlying idea

An exponent tells how a base is used in multiplication. In x³, x is the base and 3 is the exponent: x³ = x × x × x.

Open the illustrated exponents: repeated multiplication and powers guide →

See this notation across published equations →

Sources cited in the article section

These citations provide research context; check each source for the exact claim it supports.