sqrtmur1
sqrtmur1 is a term used in theoretical discussions of numerical operations to describe a composite unary operator formed by applying a square root operation followed by a normalization step referred to as mur1. In this framework, sqrtmur1(x) is defined as mur1(sqrt(x)) for inputs x in the nonnegative real numbers. The mur1 component is a context-dependent normalization operation that may enforce constraints such as rounding, discretization, or domain reduction, and its exact behavior can vary between implementations or theoretical models.
- Range: sqrtmur1(x) lies in the range defined by the mur1 normalization
- If mur1 is fixed and continuous, sqrtmur1 inherits many basic properties from the composition of sqrt
- Monotonicity: If mur1 is monotone and sqrt is monotone, then sqrtmur1 is monotone on [0, ∞).
- Continuity: If mur1 and sqrt are continuous, sqrtmur1 is continuous on [0, ∞).
- Nonnegativity: For x ≥ 0, sqrtmur1(x) ≥ 0.
- Evaluation follows a two-step process: compute y = sqrt(x), then apply mur1 to y.
- Example with a hypothetical rounding rule: if mur1 rounds to the nearest tenth, sqrtmur1(2) ≈ round(√2, 0.1)
- sqrtmur1 is primarily a conceptual construct used to explore how a standard operation (square root) interacts
- It appears in discussions of algorithm design, numerical stability, and teaching examples illustrating the effects of