sqrtre2

Sqrtre2 is a numerical function designed to compute the principal square root of a nonnegative real input with enhanced convergence efficiency. It is named to reflect its use of a second-order refinement on top of a basic Newton iteration, yielding faster convergence toward sqrt(x) than a simple Newton step alone.

Definition and method: For x ≥ 0, sqrtre2(x) returns an approximation to sqrt(x). The algorithm uses an initial guess g0, typically g0 = x if x ≥ 1, otherwise g0 = 1. It then performs two refinement steps: first a Newton iteration g1 = (g0 + x/g0)/2, then a Halley refinement g2 = g1 * (g1^2 + 3x) / (3 g1^2 + x). The result g2 is the returned value (with potential additional iterations if higher precision is required).

Properties and behavior: The combination provides cubic convergence once near the root, improving accuracy rapidly. It is numerically stable for large and small x and avoids overflow in many inputs by using the scaling implicit in the iterations. The method yields the principal square root, i.e., the nonnegative solution; for negative inputs, the function is typically defined to return NaN or to raise an error in real-number contexts.

Implementation and use: Sqrtre2 is used in numeric libraries where a fast and accurate sqrt is needed without resorting to fixed hardware instructions. In practice, two iterations provide double-precision accuracy from most starting guesses. It is suitable for environments where a balance of speed and accuracy matters, such as graphics, simulations, and embedded systems.

See also: sqrt, Newton-Raphson method, Halley’s method, cubic convergence.