G. Bohlender, Floating-point computation of functions with maximum accuracy, IEEE Trans. Comput., C-26 (1977), pp. 621--632.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of this paper is our complete analysis of Algorithm 1, and more generally our algorithms for exploiting any amount of extra precision F f . In contrast, prior work (see section 6) addresses just the case of at least double precision F 2f . A related accurate summation problem is distillation [17, 3], where the goal is to compute the exact floating point sum S represented as the sum of as few f bit quantities as possible: S = i=1 s i = i=1 t i where m is as small as possible. The idea is that each f bit quantity t i represents a subset of the bits in S. Early versions of this algorithm ....

....3, since N = 2 is the number of accumulators A j to sort and add at the end. Similarly, we can choose e to minimize the work in Algorithm 4. We consider these possibilities in the section comparing Algorithms 1 through 4. 4 Distillation in O(n) time We consider the problem of distillation [3, 18], or computing the exact sum S = i=1 s i of n f bit quantities represented as the sum S = i=1 d i of as few f bit quantities as possible. The goal is for the d i to contain pairwise disjoint subsets of all the bits of S, but in practice we settle for i 1 so that dm is within 1 ulp ....

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G. Bohlender. Floating point computation of functions with maximum accuracy. IEEE Trans. Comput., C-26:621--632, 1977.

.... accurate scientific computation are GMP (Gnu Multiple Precision) Brent s MP [3] and Bailey s package 1 [1] Expansions were introduced by Priest [12] based on some earlier compound operations defined to reduce the rounding error of a long expression [11, 10] or to compute an accurate rounding [2]. Actually, Dekker was the first in the past to propose this technique but he restricted himself to doubling the precision available to the user [8] Priest s expansions adapt to the working radix and the precision of the rounding attained by the floating point unit 2 [9] IEEE standard ....

G. Bohlender. Floating point computation of functions with maximum accuracy. IEEE Transactions on Computers, 26(7):621--632, 1977.

.... accurate scientific computation are GMP (Gnu Multiple Precision) Brent s MP [3] and Bailey s package 1 [1] Expansions were introduced by Priest [12] based on some earlier compound operations defined to reduce the rounding error of a long expression [11, 10] or to compute an accurate rounding [2]. Actually, Dekker was the first in the past to propose this technique but he restricted himself to doubling the precision available to the user [8] Priest s expansions adapt to the working radix and the precision of the rounding attained by the floating point unit 2 [9] IEEE standard ....

G. Bohlender. Floating point computation of functions with maximum accuracy. IEEE Transactions on Computers, 26(7):621--632, 1977.

....constructs floating point numbers x (k) 1 ; x (k) n such that P n i=1 x (k) i = P n i=1 x i , terminating when x (k) n approximates P n i=1 x i with relative error at most u. Kahan states that these algorithms appear to have average run times of order at least n log n. See [3], 19] 25] and [23] for further details and references. 4. Statistical Estimates of Accuracy. As we have noted, rounding error bounds can be very pessimistic, because they account for the worst case propagation of errors. An alternative way to compare summation methods is through statistical ....

Gerd Bohlender, Floating-point computation of functions with maximum accuracy, IEEE Trans. Comput., C-26 (1977), pp. 621--632.

.... Delta Delta ; x 0 n are remainders. Repeating the same process leads to values x (k) i ; under certain conditions x (k) 1 converges to a value s with an error of less than two units of the last place of the mantissa. This algorithm was modified and applied to compute dot products in [Boh 77, Boh 78] using an appropriate estimation of the sum of remainders P n i=2 x (k) i , the iteration can be terminated as soon as the required accuracy is reached. Secondly, it could be proved that (if one disregards zeros) the operands x (k) 1 Delta Delta Delta x (k) k 1 are ordered ....

Bohlender, G.: Floating-Point Computation of Functions with Maximum Accuracy. IEEE Transactions on Computers, vol. C-26, no. 7, July 1977.

.... computation are GMP (Gnu Multiple Precision) Brent s MP [7] and Bailey s package 1 [2, 3] Expansions where introduced by Priest [18] based on some earlier compound operations dened to reduce the rounding error of a long expression [17, 16] or to compute an error free IEEE standard rounding [6]. Dekker was actually the rst in the past to propose this technique but he restricted himself to doubling the precision of the arithmetic available for the user [11] Priest s expansions adapt to the dioeerent machines and they take advantages of both the working radix and the precision of the ....

Gerd Bohlender. Floating point computation of functions with maximum accuracy. IEEE Transactions on Computers, 26(7):621632, 1977.

....passes one obtains the first component of the final expansion, which is all he wished to find. Of course, if the summands are presorted, the doubly compensated summation method given above produces the first component at a cost comparable to two or three passes of Pichat s algorithm. Bohlender [7] later showed that Pichat s algorithm produces the entire expansion for the sum in at most n Gamma 1 passes, and he added a stopping criterion for the case where only a given number of leading components are desired. Many other algorithms which evaluate a sum or an inner product to working ....

Bohlender, G., Floating-Point Computation of Functions with Maximum Accuracy, IEEE Trans. Comput. C-26 (1977), 621--632.

....on computers and coprocessors. These two methods conceptually define the same function acc : a n2N F n Gamma F ffl The first method is close to the general purpose computer architecture. The dot product is obtained by issuing the following stream of operations. resetacc (X) acc (X, x[1]) acc (X, x[n] t = valueacc (X, n) Input: A vector of floating point values [x i ] n 1 Output: An approximation of oe = P n 1 x i A bound on the error ffi The level of cancellation Set the two registers X and X Gamma to 0:0 Repeat until all the vector has been read Read a ....

G. Bohlender, "Floating point computation of functions with maximum accuracy," IEEE Transaction on Computer, C26, 1977, pp. 621632.

....[13, 14] and several others [10, 12] but our hypotheses are slightly more general than theirs. Moreover, rather than simply extending the accuracy to approximately twice the working precision, as do most of the aforementioned references, our algorithms expand upon methods developed by Bohlender [1] and Kahan [9] which compute to arbitrarily high accuracy. Below, we give algorithms for exact addition and multiplication and arbitrarily accurate division of extended precision numbers using only fixed precision floating point arithmetic operations. We express the cost of these algorithms in ....

....x i in turn applying the basic sum err procedure. In fact, the first loop in our renormalization procedure above constitutes one pass of Pichat s algorithm. Pichat showed that this method converges; i.e. that after a sufficient number of passes, one obtains the first component of y. Bohlender [1] then showed that Pichat s algorithm can be used to obtain the entire expansion for y with at most n Gamma 1 passes, and he added a stopping criterion for the case where only a given number of leading components of y are desired. Many other algorithms which evaluate a sum or an inner product to ....

Bohlender, G., Floating-Point Computation of Functions with Maximum Accuracy, IEEE Trans. Comput. C-26 (1977), 621--632.

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Bohlender, G.: Floating-Point Computation of Functions with Maximum Accuracy. IEEE Transactions on Computers, Vol. C-26, no. 7, July 1977.

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G. Bohlender, Floating-point computation of functions with maximum accuracy, IEEE Trans. Comput., C-26 (1977), pp. 621--632.

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G. Bohlender. Floating point computation of functions with maximum accuracy. IEEE Trans. Comput., C-26:621-632, 1977.

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G. Bohlender. Floating point computation of functions with maximum accuracy. IEEE Trans. Comput., C-26:621-632, 1977.

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G. Bohlender, "Floating point computation of functions with maximum accuracy," IEEE Transaction on Computer, C26, 1977, pp. 621632.

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