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Oxford Academic. Google Scholar. Christian Krattenthaler. Cite Citation. Abstract In this article, the authors collect the recent results concerning the representations of integers as sums of an even number of squares that are inspired by conjectures of Kac and Wakimoto. Issue Section:. You do not currently have access to this article.
Download all figures. What other numbers have just a single set of Fibonacci numbers that sum to them? Investigate the number of ones in the Zeckendorf representations. What patterns can you find?
If the sum of squares of two numbers is 97
Can you express your patterns as mathematical formulae? What about the size of the largest sets the number of ones in the maximal bit representations? Is there a formulae for this function from n to the size of the largest set? Things to do A few years ago, the speed limit in USA was 55 miles per hour mph. The speed limit on UK motorways is 70 mph. What is this in kilometres per hour? The speed limit in built up areas is 30 mph in the UK. What do you think the "30" signs would be replaced by if road signs went metric, i.
What is the equivalent speed in mph using the Fibonacci method? What is the equivalent speed in mph using the conversion factor of 1.
Check that if you halve the 65 column and double the 19 column the method still works. Try the Egyptian method on 32x Try it on 31x Try it the other way round, starting with 19 and stop when the Fibonacci number exceeds Try the same multiplications as above: 32x65 and 31x Column i the just the rabbit sequence with "0" replaced by F i 0s and "1" replaced by F i-1 1s followed by F i 0s.
N Fib rep of N 1 1 4 6 9 12 14 22 27 33 35 51 56 64 74 80 88 90 Is there a formula for the series 1,4,6,9,12,14, Fib i Because they all have a 0,1,2 representation and we can use these to find all the 1,0,T representations for a given number n. We take any 0,1,2 Fib 2 representation and subtract from it the number " We are only interested in 10T representations that begin with 1 or T is the value is negative.
Subtracting from gives 1TTT. Show Graphs:. Note the parentheses. This discrepancy is subtracted in the next run of the loop from the next element to be added, which unlike the sum is small enough for the subtraction to have an effect. The important thing to take away from this is that numerical errors are not noise introduced at random in a computation, but can be quantified and possibly compensated. See here for even more robust algorithms, a test case and link to a paper.
A similar algorithm exists for computing the mean and standard deviation. This time delta is local to the loop body, but the other variables have to be initialised to 0 as above:. This method is attributed to Welford and others. This paper contains a comparison of algorithms for computing the variance from a numeric point of view, among others the one that yields the above for the standard deviation equation 1.
The standard C header file float. Some of the constants are special representable numbers and can be marked on a simplified version of the scale above:. I have left the prefix out for the purpose of presentation.
Sums of Squares of Integers
The values of the most important constants for our 8-bit floating-point type are:. The information contained in these constants is partly redundant. The following relations hold:. Because all values involved are representable exactly, these equivalences are also true numerically, not just mathematically. These days float. I wrote a shell script to view the resulting constants. It supports passing the path to the C preprocessor in the CPP environment variable, which allows you to substitute the preprocessor from a crosscompiling environment to view the floating-point constants of the target architecture.
Dividing a number by EPSILON gives you an estimate of the scale at which the spacing of floating-point numbers gets larger than that number. The result is only approximate because the spacing changes in jumps at powers of two, not continuously.
This can indicate practical limits for certain calculations. Another standard constant helps to print out floating-point numbers in a way that allows to parse them again without loss of precision. Why would one want to do that? It can help inspect intermediate results, one could annotate them with comments, or even autogenerate a lab book.
In this situation, the roles of the decimal number and floating-point representation are reversed: The floating-point datum is assumed to be exact and to be reproduced without modification from the decimal textual representation, and the decimal number has to provide enough digits to allow that.
This is more feasible than the reverse problem because fractions with power-of-two denominators do not give rise to an infinite periodic string of digits in decimal. This representation is also exact, can be parsed by the strtod function and is accepted in C programs by the gcc compiler.
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On this topic, there is some reassuring news: Floating-point operations will not crash a C program. Supposedly comparing a NaN will raise an exception, and avoiding that requires using comparison macros intended for that purpose. But in my experiments on Linux I have not encountered any such exception, as you can verify by compiling and running this small program.
If you want to check for floating-point error conditions, the C99 standard offers a set of functions documented in this manual page. See also the more general manual page on mathematical functions' errors. Some dated texts recommend disabling compiler optimisations to avoid excessive inaccuracies. This does not seem to be necessary any more.
The -O flags of the Gnu C compiler do not enable any optimisations that are numerically unsafe. Those options that do have to be set explicitly. See the manual page for details. This page should give you food for thought, but do not despair.