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π,e, ln(2), ln(10) arbitrary precision
 
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Permission to use, copy, and distribute this software and It’s documentation for any non commercial purpose is hereby granted without fee, provided: THE SOFTWARE IS PROVIDED "AS-IS" AND WITHOUT WARRANTY OF ANY KIND, EXPRESS, IMPLIED OR OTHERWISE, INCLUDING WITHOUT LIMITATION, ANY WARRANTY OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. IN NO EVENT SHALL Henrik Vestermark, BE LIABLE FOR ANY SPECIAL, INCIDENTAL, INDIRECT OR CONSEQUENTIAL DAMAGES OF ANY KIND, OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER OR NOT ADVISED OF THE POSSIBILITY OF DAMAGE, AND ON ANY THEORY OF LIABILITY, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.

Various Numerical Papers


Practical Implementation of Spigot Algorithms for transcendental constants
This paper examined the various modern version of spigot algorithm for calculating transcendental constant like p, e, ln(2) and ln(10) to unlimited precision. It layout the algorithm and the timing for the constants and compare it with a traditional implementationusing arbitrary precision arithmetic. It is found that the performance of spigot algorithm beats the traditional method using arbitrary precision with several factors and it is therefore recommend to be used instead, when performance is needed.
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January
2017

New Tool

Practical Implementation of π Algorithms
This paper examined the various modern version of algorithm for calculating p. That potential could be the based for using these algorithm when using arbitrary precision arithmetic that opens up for calculating p to Billions or Trillions of digits. Let it be noted that there is no engineering or pratical reason why you need to calculate p beyond the limitation in the IEEE754 standard for floating point precision as found in PC, computers etc. However these quest for more precision has let to a lot of new discovery of modern and faster algorithm that is presented in this paper.
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November
2016

New Tool

The Fundamental Financial Equation
The Fundamental Financial Equation link the 5 variables. Present value, Future value, period Payment, number of period and the interest rate together in this equation:
(PV+PMT(1+cIR)/IR)((1+IR)^NP-1)+PV+FV=0
That can be useful in a number of scenario including mortgage calculation. This paper highlight the equation and devise a method for how to calculate all the other variables when only 4 of the 5 variables are known, Including the Interest rate based on a Newton iteration.
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October
2016

New Tool

A Stopping criterias for polynomial root finders
Finding adequate stopping criteria for polynomial root finders is not always easy. To aggressive stopping criteria and you will never convergence to an acceptable root or too lax and you find roots with a lesser degree of accuracy than possible by the actually limitation of the machine precision. This paper discuss stopping criterias based on the round off errors when evaluation a polynomial at a given real or complex point.  
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March 2013

Polynomial deflation strategy when deflating polynominal roots
This paper outline polynominal deflationstrategy when using root finding methods that find one or two roots at a time that in turn is deflated into the polynomial and the process is repeated to find one or two more roots until all roots has been found. Now technically you can use either forward deflating or backward deflating or a hybrid composite deflation that combines the advantages of both the forward and backward deflating technique to preserve the accuracy of the deflated polynomial.
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March 2013

A Modified Newton and higher orders Iteration for multiple roots
In general Newton method for finding roots of polynomials is an effective and easy algorithm to both implement and use. However certain weakness is exposed when trying to find roots in a polynomial with multiple roots. This paper highlights the weakness and devised a modification to the general Newton algorithm that effectively can cope with the multiple roots issue. Furthermore we also address a solution for higher order methods as well which include Halley’s and Householders 3rd order method.
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March 2013
The Math behind arbitrary precision
We are all used to the fast microprocessors available nowadays and computational speed of basic arithmetic, trigonometric or logarithms functions is done at lighting fast speed. However when building an arbitrary integer and floating point packages that can handle decimal in the range from a few to several millions digits it  is s all back to the basic of math in order to build an arbitrary precision packages with reasonable speed.
This paper describes the underlying math behind this package.
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August 2013
Interval Arithmetic. A Practical Implementation
You can find example of interval arithmetic template class for the C++ programming language but none that is easy to understand and easy to implement and that is the reason why this paper was written. The paper highlights the implementation of a general interval template class supporting the float and double type of the C++ programming language.
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April
2015