Numeric precision in Microsoft Excel: Difference between revisions

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When one of these roots is very large compared to the other, that is, when the square root is close to the value ''b'', the evaluation of the root corresponding to subtraction of the two terms becomes very inaccurate due to round-off.

It is possible to determine the round-off error by using the [[Taylor series]] formula for the square root:<ref name=Ryzhik>

<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=English |isbn=0-12-384933-0 |id=ISBN 978-0-12-384933-5 |lccn=2014010276 <!-- |url=http://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik |chapter=1.112. Power series |page=25}}</ref>

{{cite book |chapter=§1.112 Power series |title=Table of integrals, series and products |author=IS Gradshteyn & IM Ryzhik |edition =7th |publisher=Academic Press |year=2007 |isbn=0-12-373637-4 |page = 25 |url=http://books.google.com/books?id=aBgFYxKHUjsC&pg=PA25}}

</ref>

:<math>\sqrt{b^2-4ac} = b \ \sqrt{1-\frac{4ac}{b^2}} \approx b \left( 1 -\frac{2ac}{b^2} + \frac{2 a^2 c^2 }{b^4} + \cdots \right ). </math>