Customer Reviews

Computer Approximations

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When computers were slow and memory expensive, 2 score and 4 years ago, John Hart published "Computer Approximations" as a means of implementing complex numerical calculations for greater efficiency. Trigonometric functions expressed as a finite series can take up valuable resources in a real-time processing application. Hart shows you haw to implement some of the most widely used numerical functions in minimal memory and minimal time and still obtain high precision and accuracy in the results. The treatment of trigonometric and logarithmic functions alone is worth the price of the book.

As long as computers have finite precision and the world around them doesn't, there's going to be a need for approximations to exact functions. This book helps even modern practitioners work out the most effective and economical choices for common and not-so-common mathematical functions.

The book's most noteworthy feature appears at the end, where nearly half the book presents tables of coefficients for rational polynomial approximations, giving the reader a wide range of choices in both the functions being approximated and in the degree of the polynomials. I'd approach these carefully, though - the notation tends toward the opaque, and there's no clear statement of the range over which the approximation works best or the error in each approximation. Although they represent a useful starting point, serious practitioners will use the techniques elsewhere in this book with modern extended precision algebra-handling tools to work out values on their own.

Even though the bulk of this book dates back to 1968 - over 20 Moore's Law generations ago - most of the discussion remains salient. In fact, things like error analysis are fast becoming lost arts, so parts of the discussion remain especially valuable. The basics are timeless, though, and that includes things like piecewise approximations, basis functions, range reduction, and subtleties of different ways to evaluate polynomials - Horner's rule is hardly the last word, or even the best. Only discussions really show their age, all of them having to do with the details of number representation. This pre-IEEE-754 text doesn't give as much as modern texts do regarding the last bit of precision.

If you think things like sines, cosines, and logs have been done - well, given your computing needs, you're probably right. New computing fabrics continue to emerge, though, with new quirks and performance characteristics. The standard (and non-standard) functions need to be done again for each new platform. Despite its age, this book still offers some help to those of us with needs not met by the Fortran, C, or Java libraries.

-- wiredweird

It is a pity that this book had gone out of print for many years. The material in the book is still useful in today's work. If you enjoy implementing numerical solutions on computing hardware, hunt down a copy in the used book market.

This book is an excellent reference in the way it furnishes a comphrehensive list of algorithms for each function you want to approximate, trading numerical precisions for computational complexities. Taking the list and experiment with the implementations, perhaps extending it, and then verifying and seeing the trade-offs for yourself, is very useful as well as fun.

This book is actually difficult to find and is a bible for anyone writing trig functions or anything like this for a compiler or a particular processor. Most libraries these days are not optimized (even gcc's libm is coded wrong). picfloat (the floating lib for the pic is coded wrong).
This book allows you to choose the optimum algorithm based on your hardware (great for DSP's) and then choose your precision and look it up in a table and then choose your chebyshev polynomial (or combination of polynomials) that is optimized for your hardware and resolution. For example, this book will give you a 3 coefficient polynomial for sine for single precision calculation, whereas Taylor series (normally used) requires 6 coefficients for the same accuracy, so it's twice as fast on a regular microprocessor.
Every software engineer should have a copy of this book. I have yet to see a new book to replace it. It's actually out of print and these copies are specially printed or older prints. Get It....you'll need it some day.

I still use this book, which I bought many years ago; it should be in the library of every college and university math and computing department. Its strong point, from the perspective of the year 2007, is its excellent discussion of various problems in getting good numerical approximations, and ways of overcoming these difficulties. However, the tables of approximations themselves are mostly obsolete in view of all the work that's been done on this topic since this book was published. One cautionary note, in particular, is worth making. All the tables for the gamma function and its logarithm, most of the tables for the (complementary) error function, and all the tables for the Bessel functions and for complete elliptic integrals, were derived using an absolute accuracy criterion, not a relative accuracy criterion. This makes these coefficient values inappropriate for floating point routines, but because it's not trivial to find good approximations elsewhere in the literature for these functions, I have seen these tables used for constructing floating point routines, with predictably inaccurate results.
If you are faced with the task of writing routines for these particular functions, I suggest using the text of this book to provide guidance, but deriving coefficients either from routines known to be of high accuracy, or doing it yourself, using the Remez method to get coefficients for a polynomial or rational approximation.

It's a good book, but isn't a easy book to read and understand.
You need to read a lot of the book to understand all the tables that the books came and where to get the number to start to calculate something.