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Since the additions, subtractions, and digit shifts (multiplications by powers of B) in Karatsuba's basic step take time proportional to n, their cost becomes negligible as n increases. More precisely, if T(n) denotes the total number of elementary operations that the algorithm performs when multiplying two n-digit numbers, then
It allows certain serial operations to be performed in parallel and the number of steps required for those operations to be reduced. A reduction operator stores the result of the partial tasks into a private copy of the variable. These private copies are then merged into a shared copy at the end. An operator is a reduction operator if:
On a single-step or immediate-execution calculator, the user presses a key for each operation, calculating all the intermediate results, before the final value is shown. [ 1 ] [ 2 ] [ 3 ] On an expression or formula calculator , one types in an expression and then presses a key, such as "=" or "Enter", to evaluate the expression.
The sum is so large that only the high-order digits of the input numbers are being accumulated. But on the next step, c, an approximation of the running error, counteracts the problem. y = 2.71828 - (-0.0415900) Most digits meet, since c is of a size similar to y. = 2.75987 The shortfall (low-order digits lost) of previous iteration ...
A prime modulus requires the computation of a double-width product and an explicit reduction step. If a modulus just less than a power of 2 is used (the Mersenne primes 2 31 − 1 and 2 61 − 1 are popular, as are 2 32 − 5 and 2 64 − 59), reduction modulo m = 2 e − d can be implemented more cheaply than a general double-width division ...
It uses a selection of full and half adders to sum the partial products in stages (the Dadda tree or Dadda reduction) until two numbers are left. The design is similar to the Wallace multiplier , but the different reduction tree reduces the required number of gates (for all but the smallest operand sizes) and makes it slightly faster (for all ...
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The considerations above concern the recursion depth only. Either way of iterating leads to the same number of reduction steps, involving the same rules (when the rules r11 and r12 are considered "the same"). As the example shows the reduction of (,) converges in 9 steps: 1 X r7, 3 X r8, 1 X r9, 2 X r10, 2 X r11/r12. The modus iterandi only ...