Square root of 3 billion digits. The fraction (…) for the square root of three can be used as an approximation. Despite having a denominator of only 56, it differs from the correct value by less than (approximately ). The rounded value of 1.732 is correct to within 0.01% of the actual value. Archimedes reported , accurate to (six decimal places) and (four decimal places), respectively. It can be expressed as the continued fraction , expanded on the right. It can also be expressed by generalized continued fractions such as which is evaluated at every second term. The following nested square expressions converge to
Square root of a matrix but if the process converges, the matrix formula_26 converges quadratically to a square root , while formula_27 converges to its inverse, . Yet another iterative method is obtained by taking the well-known formula of the Babylonian method for computing the square root of a real number, and applying it to matrices. Let "X" = "I", where "I" is the identity matrix. The iteration is defined by Again, convergence is not guaranteed, but if the process converges, the matrix formula_29 converges quadratically to a square root "A". Compared to Denman–Beavers iteration, an advantage of the Babylonian method is that only one