sqrtsum

Sqrtsum is a term used to refer to the sum of square roots, typically written as S = sum sqrt(a_i) for a list of nonnegative numbers a_i. In mathematical contexts, sqrtsum describes the collective contribution of square-root terms and appears in problems ranging from analysis to number theory and geometry. When the a_i are integers, the quantity is often called the square-root sum of integers.

In mathematics, the square root function is nonnegative on nonnegative inputs, so each term sqrt(a_i) is a nonnegative real number. The overall sum is monotone in each argument: increasing any a_i cannot decrease S. Because square roots are irrational for most nonzero integers, the exact value of a sqrtsum is generally not expressible in a simple closed form, and numeric approximation is common in applications.

A notable computational problem associated with sqrtsum is the Square Root Sum problem (often denoted SQRT-SUM): given integers a_1, ..., a_n and a bound B, decide whether sum sqrt(a_i) > B. The exact complexity status of this decision problem is open; no polynomial-time algorithm is known, and it has been studied as a representative challenge in computational complexity, with connections to problems in computational geometry. Researchers investigate both exact approaches (involving algebraic number comparisons and high-precision arithmetic) and approximation methods (using controlled error bounds).

Applications of sqrtsum considerations arise in geometry, numerical analysis, and algorithm design, particularly where weights involve square roots and exact comparisons or bounds are required. The topic is often treated as a combination of analytic intuition and computational techniques. See also square root, sum, and algebraic number comparison.