Kahan Summation Algorithm

In numerical analysis, the Kahan summation algorithm, also known as compensated summation,^[1] significantly reduces the numerical error in the total obtained by adding a sequence of finite-precision floating-point numbers, compared to the obvious approach. This is done by keeping a separate running compensation (a variable to accumulate small errors).

In particular, simply summing n numbers in sequence has a worst-case error that grows proportional to n, and a root mean square error that grows as $n\{\backslash displaystyle\; \{\backslash sqrt\; \{n\}\}\}$ for random inputs (the roundoff errors form a random walk).^[2] With compensated summation, the worst-case error bound is independent of n, so a large number of values can be summed with an error that only depends on the floating-point precision.^[2]

The algorithm is attributed to William Kahan.^[3] Similar, earlier techniques are, for example, Bresenham's line algorithm, keeping track of the accumulated error in integer operations (although first documented around the same time^[4]) and the delta-sigma modulation^[5] (integrating, not just summing the error).

The algorithm

In pseudocode, the algorithm is:

function KahanSum(input) var sum = 0.0 // Prepare the accumulator. var c = 0.0 // A running compensation for lost low-order bits. for i = 1 to input.length do // The array input has elements indexed input[1] to input[input.length]. var y = input[i] - c // c is zero the first time around. var t = sum + y // Alas, sum is big, y small, so low-order digits of y are lost. c = (t - sum) - y // (t - sum) cancels the high-order part of y; subtracting y recovers negative (low part of y) sum = t // Algebraically, c should always be zero. Beware overly-aggressive optimizing compilers! next i // Next time around, the lost low part will be added to y in a fresh attempt. return sum 

Worked example

This example will be given in decimal. Computers typically use binary arithmetic, but the principle being illustrated is the same. Suppose we are using six-digit decimal floating-point arithmetic, sum has attained the value 10000.0, and the next two values of input[i] are 3.14159 and 2.71828. The exact result is 10005.85987, which rounds to 10005.9. With a plain summation, each incoming value would be aligned with sum, and many low-order digits would be lost (by truncation or rounding). The first result, after rounding, would be 10003.1. The second result would be 10005.81828 before rounding and 10005.8 after rounding. This is not correct.

However, with compensated summation, we get the correct rounded result of 10005.9.

Assume that c has the initial value zero.

 y = 3.14159 - 0.00000 y = input[i] - c t = 10000.0 + 3.14159 = 10003.14159 But only six digits are retained. = 10003.1 Many digits have been lost! c = (10003.1 - 10000.0) - 3.14159 This must be evaluated as written! = 3.10000 - 3.14159 The assimilated part of y recovered, vs. the original full y. = -0.0415900 Trailing zeros shown because this is six-digit arithmetic. sum = 10003.1 Thus, few digits from input(i) met those of sum. 

The sum is so large that only the high-order digits of the input numbers are being accumulated. But on the next step, c gives the error.

 y = 2.71828 - (-0.0415900) The shortfall from the previous stage gets included. = 2.75987 It is of a size similar to y: most digits meet. t = 10003.1 + 2.75987 But few meet the digits of sum. = 10005.85987 And the result is rounded = 10005.9 To six digits. c = (10005.9 - 10003.1) - 2.75987 This extracts whatever went in. = 2.80000 - 2.75987 In this case, too much. = 0.040130 But no matter, the excess would be subtracted off next time. sum = 10005.9 Exact result is 10005.85987, this is correctly rounded to 6 digits. 

So the summation is performed with two accumulators: sum holds the sum, and c accumulates the parts not assimilated into sum, to nudge the low-order part of sum the next time around. Thus the summation proceeds with "guard digits" in c, which is better than not having any, but is not as good as performing the calculations with double the precision of the input. However, simply increasing the precision of the calculations is not practical in general; if input is already in double precision, few systems supply quadruple precision, and if they did, input could then be in quadruple precision.

Accuracy

A careful analysis of the errors in compensated summation is needed to appreciate its accuracy characteristics. While it is more accurate than naive summation, it can still give large relative errors for ill-conditioned sums.

Suppose that one is summing n values x_i, for i = 1, ... ,n. The exact sum is

$Sn=\sum i=1nxi\{\backslash displaystyle\; S\_\{n\}=\backslash sum\; \_\{i=1\}^\{n\}x\_\{i\}\}$ (computed with infinite precision).

With compensated summation, one instead obtains $Sn+En\{\backslash displaystyle\; S\_\{n\}+E\_\{n\}\}$, where the error $En\{\backslash displaystyle\; E\_\{n\}\}$ is bounded by^[2]

$|En|\le [2\epsilon +O(n\epsilon 2)]\sum i=1n|xi|,\{\backslash displaystyle\; |E\_\{n\}|\backslash leq\; \{\backslash big\; [\}2\backslash varepsilon\; +O(n\backslash varepsilon\; ^\{2\})\{\backslash big\; ]\}\backslash sum\; \_\{i=1\}^\{n\}|x\_\{i\}|,\}$

where ε is the machine precision of the arithmetic being employed (e.g. ε ≈ 10⁻¹⁶ for IEEE standard double-precision floating point). Usually, the quantity of interest is the relative error $|En|/|Sn|\{\backslash displaystyle\; |E\_\{n\}|/|S\_\{n\}|\}$, which is therefore bounded above by

$|En||Sn|\le [2\epsilon +O(n\epsilon 2)]\sum i=1n|xi||\sum i=1nxi|.\{\backslash displaystyle\; \{\backslash frac\; \{|E\_\{n\}|\}\{|S\_\{n\}|\}\}\backslash leq\; \{\backslash big\; [\}2\backslash varepsilon\; +O(n\backslash varepsilon\; ^\{2\})\{\backslash big\; ]\}\{\backslash frac\; \{\backslash sum\; \backslash limits\; \_\{i=1\}^\{n\}|x\_\{i\}|\}\{\backslash left|\backslash sum\; \backslash limits\; \_\{i=1\}^\{n\}x\_\{i\}\backslash right|\}\}.\}$

In the expression for the relative error bound, the fraction Σ|x_i|/|Σx_i| is the condition number of the summation problem. Essentially, the condition number represents the intrinsic sensitivity of the summation problem to errors, regardless of how it is computed.^[6] The relative error bound of every (backwards stable) summation method by a fixed algorithm in fixed precision (i.e. not those that use arbitrary-precision arithmetic, nor algorithms whose memory and time requirements change based on the data), is proportional to this condition number.^[2] An ill-conditioned summation problem is one in which this ratio is large, and in this case even compensated summation can have a large relative error. For example, if the summands x_i are uncorrelated random numbers with zero mean, the sum is a random walk, and the condition number will grow proportional to $n\{\backslash displaystyle\; \{\backslash sqrt\; \{n\}\}\}$. On the other hand, for random inputs with nonzero mean the condition number asymptotes to a finite constant as $n\to \infty \{\backslash displaystyle\; n\backslash to\; \backslash infty\; \}$. If the inputs are all non-negative, then the condition number is 1.

Given a condition number, the relative error of compensated summation is effectively independent of n. In principle, there is the O(nε²) that grows linearly with n, but in practice this term is effectively zero: since the final result is rounded to a precision ε, the nε² term rounds to zero, unless n is roughly 1/ε or larger.^[2] In double precision, this corresponds to an n of roughly 10¹⁶, much larger than most sums. So, for a fixed condition number, the errors of compensated summation are effectively O(ε), independent of n.

In comparison, the relative error bound for naive summation (simply adding the numbers in sequence, rounding at each step) grows as $O(\epsilon n)\{\backslash displaystyle\; O(\backslash varepsilon\; n)\}$ multiplied by the condition number.^[2] This worst-case error is rarely observed in practice, however, because it only occurs if the rounding errors are all in the same direction. In practice, it is much more likely that the rounding errors have a random sign, with zero mean, so that they form a random walk; in this case, naive summation has a root mean square relative error that grows as $O(\epsilon n)\{\backslash displaystyle\; O\backslash left(\backslash varepsilon\; \{\backslash sqrt\; \{n\}\}\backslash right)\}$ multiplied by the condition number.^[7] This is still much worse than compensated summation, however. However, if the sum can be performed in twice the precision, then ε is replaced by ε², and naive summation has a worst-case error comparable to the O(nε²) term in compensated summation at the original precision.

By the same token, the Σ|x_i| that appears in $En\{\backslash displaystyle\; E\_\{n\}\}$ above is a worst-case bound that occurs only if all the rounding errors have the same sign (and are of maximal possible magnitude).^[2] In practice, it is more likely that the errors have random sign, in which case terms in Σ|x_i| are replaced by a random walk, in which case, even for random inputs with zero mean, the error $En\{\backslash displaystyle\; E\_\{n\}\}$ grows only as $O(\epsilon n)\{\backslash displaystyle\; O\backslash left(\backslash varepsilon\; \{\backslash sqrt\; \{n\}\}\backslash right)\}$ (ignoring the nε² term), the same rate the sum $Sn\{\backslash displaystyle\; S\_\{n\}\}$ grows, canceling the $n\{\backslash displaystyle\; \{\backslash sqrt\; \{n\}\}\}$ factors when the relative error is computed. So, even for asymptotically ill-conditioned sums, the relative error for compensated summation can often be much smaller than a worst-case analysis might suggest.

Further enhancements

Neumaier^[8] introduced an improved version of Kahan algorithm, which he calls an "improved Kahan–Babuška algorithm", which also covers the case when the next term to be added is larger in absolute value than the running sum, effectively swapping the role of what is large and what is small. In pseudocode, the algorithm is:

function NeumaierSum(input) var sum = 0.0 var c = 0.0 // A running compensation for lost low-order bits. for i = 1 to input.length do var t = sum + input[i] if |sum| >= |input[i]| then c += (sum - t) + input[i] // If sum is bigger, low-order digits of input[i] are lost. else c += (input[i] - t) + sum // Else low-order digits of sum are lost. endif sum = t next i return sum + c // Correction only applied once in the very end. 

For many sequences of numbers, both algorithms agree, but a simple example due to Peters^[9] shows how they can differ. For summing $[1.0,+10100,1.0,-10100]\{\backslash displaystyle\; [1.0,+10^\{100\},1.0,-10^\{100\}]\}$ in double precision, Kahan's algorithm yields 0.0, whereas Neumaier's algorithm yields the correct value 2.0.

Higher-order modifications of better accuracy are also possible. For example a variant suggested by Klein^[10], which he called a second-order "iterative Kahan–Babuška algorithm". In pseudocode, the algorithm is:

function KleinSum(input) var s = 0.0 var cs = 0.0 var ccs = 0.0 for i = 1 to input.length do var t = s + input[i] if |s| >= |input[i]| then c = (s - t) + input[i] else c = (input[i] - t) + s endif s = t t = cs + c if |cs| >= |c| then cc = (cs - t) + c else cc = (c - t) + cs endif cs = t ccs = ccs + cc next i return s + cs + ccs 

Alternatives

Although Kahan's algorithm achieves $O(1)\{\backslash displaystyle\; O(1)\}$ error growth for summing n numbers, only slightly worse $O(logn)\{\backslash displaystyle\; O(\backslash log\; n)\}$ growth can be achieved by pairwise summation: one recursively divides the set of numbers into two halves, sums each half, and then adds the two sums.^[2] This has the advantage of requiring the same number of arithmetic operations as the naive summation (unlike Kahan's algorithm, which requires four times the arithmetic and has a latency of four times a simple summation) and can be calculated in parallel. The base case of the recursion could in principle be the sum of only one (or zero) numbers, but to amortize the overhead of recursion, one would normally use a larger base case. The equivalent of pairwise summation is used in many fast Fourier transform (FFT) algorithms and is responsible for the logarithmic growth of roundoff errors in those FFTs.^[11] In practice, with roundoff errors of random signs, the root mean square errors of pairwise summation actually grow as $O(logn)\{\backslash displaystyle\; O\backslash left(\{\backslash sqrt\; \{\backslash log\; n\}\}\backslash right)\}$.^[7]

Another alternative is to use arbitrary-precision arithmetic, which in principle need no rounding at all with a cost of much greater computational effort. A way of performing exactly rounded sums using arbitrary precision is to extend adaptively using multiple floating-point components. This will minimize computational cost in common cases where high precision is not needed.^[12][9] Another method that uses only integer arithmetic, but a large accumulator, was described by Kirchner and Kulisch;^[13] a hardware implementation was described by Müller, Rüb and Rülling.^[14]

Possible invalidation by compiler optimization

In principle, a sufficiently aggressive optimizing compiler could destroy the effectiveness of Kahan summation: for example, if the compiler simplified expressions according to the associativity rules of real arithmetic, it might "simplify" the second step in the sequence

t = sum + y;

c = (t - sum) - y;

to

c = ((sum + y) - sum) - y;

and then to

c = 0;

thus eliminating the error compensation.^[15] In practice, many compilers do not use associativity rules (which are only approximate in floating-point arithmetic) in simplifications, unless explicitly directed to do so by compiler options enabling "unsafe" optimizations,^{[16][17][18][19]} although the Intel C++ Compiler is one example that allows associativity-based transformations by default.^[20] The original K&R C version of the C programming language allowed the compiler to re-order floating-point expressions according to real-arithmetic associativity rules, but the subsequent ANSI C standard prohibited re-ordering in order to make C better suited for numerical applications (and more similar to Fortran, which also prohibits re-ordering),^[21] although in practice compiler options can re-enable re-ordering, as mentioned above.

Support by libraries

In general, built-in "sum" functions in computer languages typically provide no guarantees that a particular summation algorithm will be employed, much less Kahan summation.^{[citation needed]} The BLAS standard for linear algebra subroutines explicitly avoids mandating any particular computational order of operations for performance reasons,^[22] and BLAS implementations typically do not use Kahan summation.

The standard library of the Python computer language specifies an fsum function for exactly rounded summation, using the Shewchuk algorithm^[9] to track multiple partial sums.

In the Julia language, the default implementation of the sum function does pairwise summation for high accuracy with good performance,^[23] but an external library provides an implementation of Neumaier's variant named sum_kbn for the cases when higher accuracy is needed.^[24]

In the C♯ language, HPCsharp nuget package implements the Neumaier variant and pairwise summation: both as scalar, data-parallel using SIMD processor instructions, and parallel multi-core.^[25]

See also

References

1. ^ Strictly, there exist other variants of compensated summation as well: see Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (2 ed). SIAM. pp. 110–123. ISBN 978-0-89871-521-7.
2. ^ ^{a b c d e f g h} Higham, Nicholas J. (1993), "The accuracy of floating point summation" (PDF), SIAM Journal on Scientific Computing, 14 (4): 783–799, CiteSeerX 10.1.1.43.3535, doi:10.1137/0914050.
3. ^ Kahan, William (January 1965), "Further remarks on reducing truncation errors" (PDF), Communications of the ACM, 8 (1): 40, doi:10.1145/363707.363723, archived from the original (PDF) on 9 February 2018.
4. ^ Bresenham, Jack E. (January 1965). "Algorithm for computer control of a digital plotter" (PDF). IBM Systems Journal. 4 (1): 25–30. doi:10.1147/sj.41.0025.
5. ^ Inose, H.; Yasuda, Y.; Murakami, J. (September 1962). "A Telemetering System by Code Manipulation – ΔΣ Modulation". IRE Transactions on Space Electronics and Telemetry: 204–209.
6. ^ Trefethen, Lloyd N.; Bau, David (1997). Numerical Linear Algebra. Philadelphia: SIAM. ISBN 978-0-89871-361-9.
7. ^ ^{a b} Manfred Tasche and Hansmartin Zeuner, Handbook of Analytic-Computational Methods in Applied Mathematics, Boca Raton, FL: CRC Press, 2000.
8. ^ Neumaier, A. (1974). "Rundungsfehleranalyse einiger Verfahren zur Summation endlicher Summen" [Rounding Error Analysis of Some Methods for Summing Finite Sums] (PDF). Zeitschrift für Angewandte Mathematik und Mechanik (in German). 54 (1): 39–51. doi:10.1002/zamm.19740540106.
9. ^ ^{a b c} Raymond Hettinger, Recipe 393090: Binary floating point summation accurate to full precision, Python implementation of algorithm from Shewchuk (1997) article (28 March 2005).
10. ^ A., Klein (2006). "A generalized Kahan–Babuška-Summation-Algorithm". Computing. Springer-Verlag. 76: 279–293. doi:10.1007/s00607-005-0139-x.
11. ^ S. G. Johnson and M. Frigo, "Implementing FFTs in practice, in Fast Fourier Transforms, edited by C. Sidney Burrus(2008).
12. ^ Jonathan R. Shewchuk, Adaptive Precision Floating-Point Arithmetic and Fast Robust Geometric Predicates, Discrete and Computational Geometry, vol. 18, pp. 305–363 (October 1997).
13. ^ R. Kirchner, U. W. Kulisch, Accurate arithmetic for vector processors, Journal of Parallel and Distributed Computing 5 (1988) 250–270.
14. ^ M. Muller, C. Rub, W. Rulling [1], Exact accumulation of floating-point numbers, Proceedings 10th IEEE Symposium on Computer Arithmetic (Jun 1991), doi:10.1109/ARITH.1991.145535.
15. ^ Goldberg, David (March 1991), "What every computer scientist should know about floating-point arithmetic" (PDF), ACM Computing Surveys, 23 (1): 5–48, doi:10.1145/103162.103163.
16. ^ GNU Compiler Collection manual, version 4.4.3: 3.10 Options That Control Optimization, -fassociative-math (Jan. 21, 2010).
17. ^ Compaq Fortran User Manual for Tru64 UNIX and Linux Alpha Systems, section 5.9.7 Arithmetic Reordering Optimizations (retrieved March 2010).
18. ^ Börje Lindh, Application Performance Optimization, Sun BluePrints OnLine (March 2002).
19. ^ Eric Fleegal, "Microsoft Visual C++ Floating-Point Optimization", Microsoft Visual Studio Technical Articles (June 2004).
20. ^ Martyn J. Corden, "Consistency of floating-point results using the Intel compiler", Intel technical report (Sep. 18, 2009).
21. ^ MacDonald, Tom (1991). "C for Numerical Computing". Journal of Supercomputing. 5 (1): 31–48. doi:10.1007/BF00155856.
22. ^ BLAS Technical Forum, section 2.7 (August 21, 2001), Archived on Wayback Machine.
23. ^ RFC: use pairwise summation for sum, cumsum, and cumprod, github.com/JuliaLang/julia pull request #4039 (August 2013).
24. ^ KahanSummation library in Julia.
25. ^ HPCsharp nuget package of high performance algorithms.

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