By H. Cabannes, M. Holt, V. V. Rusanov
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What is this shape and why does it happen? 31. The graphs of the 3n + 1 sequences starting at n = 108, 109, and 110 are very similar to each other. Why? 32. Let L(n) be the number of terms in the 3n + 1 sequence that starts with n. Write a Matlab function that computes L(n) without using any vectors or unpredictable amounts of storage. Plot L(n) for 1 ≤ n ≤ 1000. What is the maximum value of L(n) for n in this range, and for what value of n does it occur? Use threenplus1 to plot the sequence that starts with this particular value of n.
0000 Where did this come from? Well, the equations are singular, but consistent. 1 times the first. The computed x is one of infinitely many possible solutions. But the floating-point representation of the matrix A is not exactly singular because A(2,1) is not exactly 17/10. The solution process subtracts a multiple of the first equation from the second. 7/17, which turns out to be the floating-point number obtained by truncating, rather than rounding, the binary expansion of 1/10. The matrix A and the right-hand side b are modified by A(2,:) = A(2,:) - mu*A(1,:) b(2) = b(2) - mu*b(1) With exact computation, both A(2,2) and b(2) would become zero, but with floating-point arithmetic, they both become nonzero multiples of eps.
Floating-point operations on flints do not introduce any roundoff error, as long as the results are not too large. Addition, subtraction, and multiplication of flints produce the exact flint result, if it is not larger than 253 . Division and square root involving flints also produce a flint if the result is an integer. For example, sqrt(363/3) produces 11, with no roundoff. Two Matlab functions that take apart and put together floating-point numbers are log2 and pow2. ^E. Any zeros in X produce F = 0 and E = 0.
6th Int'l Conference on Numerical Methods in Fluid Dynamics by H. Cabannes, M. Holt, V. V. Rusanov