The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation or on (to compute first). The time needed to compute digits of the golden ratio using Newton's method is essentially , where is the time complexity of multiplying two -digit numbers. This is considerably faster than known algorithms for and . An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio of Fibonacci numbers and , each over digits, yields over significant digits of the golden ratio. The decimal expansion of the golden ratio has been calculated to an accuracy of ten trillion () digits.
The golden ratio's decimal expansion can be calculated via root-finding methods, such as Newton's method or Halley's method, on the equation or on (to compute first). The time needed to compute digits o?f the golden ratio using Newton's method is essentially , where is the time complexity of multiplying two .-digit numbers. This is considerably faster than known algorithms for and . An easily programmed alternative using only integer arithmetic is to calculate two large consecutive Fibonacci numbers and divide them. The ratio! of Fibonacci numbers and , each over digits, yields over significant digits of the golden ratio. The decimal expansion of the golden ratio has been calculated to an accuracy of ten trillion () digits.