# PageRank

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- | [[Image:Gleich.jpg|thumb|right|429px|CSUM() in Digital Signal Processing terms: | ||

- | z<sup>-1</sup> is a unit delay, Q is a floating-point quantizer to 64 bits, | ||

- | q<sub>i</sub> represents error due to quantization (additive by definition). <br>-Jon Dattorro]] | ||

- | <pre> | ||

- | function s_hat=csum(x) | ||

- | % CSUM Sum of elements using a compensated summation algorithm. | ||

- | % David Gleich, Stanford University, 2008 | ||

- | % | ||

- | % For large vectors, the native sum command in Matlab does | ||

- | % not appear to use a compensated summation algorithm which | ||

- | % can cause significant roundoff errors. | ||

- | % | ||

- | % This code implements a variant of Kahan's compensated | ||

- | % summation algorithm which often takes about twice as long, | ||

- | % but produces more accurate sums when the number of | ||

- | % elements is large. | ||

- | % | ||

- | % See also SUM | ||

- | % | ||

- | % Example: | ||

- | % v=rand(1e7,1); | ||

- | % sum1 = sum(v); | ||

- | % sum2 = csum(v); | ||

- | % fprintf('sum1 = %18.16e\nsum2 = %18.16e\n', sum1, sum2); | ||

- | s_hat=0; y=0; e=0; | ||

- | for i=1:numel(x) | ||

- | s_hat_old = s_hat; | ||

- | y = x(i) + e; | ||

- | s_hat = s_hat_old + y; | ||

- | e = (s_hat_old - s_hat) + y; %calculate difference first (Higham) | ||

- | end | ||

- | </pre> | ||

- | |||

- | === links === | ||

- | [http://www.google.com/books?id=FJyBjjtHREQC&dq=Accuracy+and+Stability+of+Numerical+Algorithms&printsec=frontcover&source=bn#PPA92,M1 Accuracy and Stability of Numerical Algorithms, Nicholas J. Higham, 1996] | ||

- | |||

- | For multiplier error feedback, see: | ||

- | |||

- | [http://www.stanford.edu/~dattorro/HiFi.pdf Implementation of Recursive Digital Filters for High-Fidelity Audio] | ||

- | |||

- | [http://www.stanford.edu/~dattorro/CorrectionsHiFi.pdf Comments on the above...] |