High Precision Calculation of the Digamma Function

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High Precision Calculation of the Digamma Function

• Mohammad I Rafiq
• School of Computer Science, Carnegie Mellon
  University
• Research Advisor Dr. Victor Adamchik

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Goal of the Project

• The main goal of this research project is to
  be able to compute the digamma and other related
  special functions, to an arbitrary precision.
  This has been implemented using NASA's ARPREC
  package. The other important aspects of this
  project are to deal with dynamically controlled
  precision, self correcting error codes and fixed
  precision calculations.

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Formulas for computation of the Digamma function
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Results
X -(110/213)-5 -110/213 110/213 110/213 5
Precision 1000 1000 1000 1000
Our time (sec) 3.69 3.31 2.15 3.32
Mathematica time (sec) 2.29 2.20 2.24 2.20
Accuracy 1000 1000 1000 1000
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Results(contd)
X -(110/213)-5 -110/213 110/213 110/213 5
Precision 2000 2000 2000 2000
Our time (sec) 32.36 33.23 32.01 31.93
Mathematica time (sec) 23.05 23.45 23.0 23.29
Accuracy 2000 2000 2000 2000
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• Problem
• Accuracy decreases as precision and x is
  increased e.g.
• 1) digamma(110/213 500) when calculated
  at a precision of 500 gives an accuracy of 362.
• 2) digamma(110/213 50) when calculated at a
  precision of 1000 gives an accuracy of 781.
• Solution
• To use an adaptive precision system rather
  than a fixed precision system. An adaptive
  precision system is a floating point system based
  on significance arithmetic which dynamically
  adjusts the number of digits used in
  computations. This is also known as self
  correcting error codes.

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Future Additions

• To add a two layer caching mechanism in order to
  improve the calculation times.

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• Splay Tree A stores the result corresponding to
  each input x for a particular precision. If the
  function is called multiple times for the same
  input x, but with a different precision then only
  the result of the higher precision is stored.
• Splay Tree B stores the coefficients Ci in the
  struct k_st, corresponding to each iteration i
  for each precision.
• Each time the function digamma(x,prec) is called,
  we first search through A for x. If we find x and
  the precision stored is equal or less than our
  prec then we truncate the stored result and
  output it. If we do not find x or our prec is
  greater than the stored precision, then we search
  through B for our prec. If we find our prec in
  the tree, we use the stored coefficients to
  calculate our result and also update Tree A with
  the result. If we do not find our prec in B, we
  calculate both the coefficients and the result
  and update both Trees A and B.