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# Ponder This

Welcome to our monthly puzzles.
You are cordially invited to match wits with some of the best minds in IBM Research.

## May 2023 - Challenge

This problem was suggested by Lorenzo Gianferrari Pini and Radu-Alexandru Todor - thanks!

Let x\in\mathbb{R}^n be a length n vector (x_0, x_1, \ldots, x_{n-1}) and A\in\mathbb{R}^{n\times n} be an n\times n matrix. We seek to compute the quadratic form x^TAx.

Assume x is equally spaced between -1 and 1, e.g., for n=5, we have

x = (-1, -0.5, 0, 0.5, 1)

The matrix A is generated in the following manner:

Let k\in\mathbb{N} be a natural number and define a vector a\in\mathbb{R}^{2^k} that is equally spaced between 0 and 1, i.e., a_t =\frac{t}{2^k-1} for t=0,1,\ldots,2^k-1. The values of A will be taken from the vector a in the following manner:

We are given a sequence Q_0, Q_1,\ldots, Q_{n-1}, with each Q_i\in\mathbb{N}^k being a vector of k natural numbers. Given two such vectors, define Q_i==Q_j as the binary vector of length k that has 1s in the entries that are equal in Q_i, Q_j and 0s in the other entries. Let [Q_i==Q_j] be the natural number whose binary representation is Q_i==Q_j, where the least significant bit is on index 0. For example, if

Q_i = (1, 5, 7, 8)

Q_j = (2, 5, 6, 8)

Then

Q_i==Q_j = (0, 1, 0, 1)

And

[Q_i==Q_j] = 10 (since the vector (0, 1, 0, 1) stands for the binary representation 1010).

Now define A_{ij} = a_{[Q_i==Q_j]}.

So one can think of A_{ij} as taking on a value from a fixed list of 2^k values based on the "similarity" between Q_i and Q_j.

The values of the Q_i's are chosen pseudo-randomly using the formula

Q_i[t] = \left\lfloor2^k\cdot (\sin((i+1)\cdot (t+1))-\left\lfloor \sin((i+1)\cdot (t+1))\right\rfloor )\right\rfloor

For example, if k=5, then Q_{13}=(31, 8, 2, 15, 24)

Your goal: Find x^TAx (rounded to three decimals) for k=5, n=2^{20}.

A bonus "*" will be given for finding x^TAx (rounded to three decimals) for k=5, n=2^{30}.

We will post the names of those who submit a correct, original solution! If you don't want your name posted then please include such a statement in your submission!

We invite visitors to our website to submit an elegant solution. Send your submission to the ponder@il.ibm.com.

If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Challenge: 01/05/2023 @ 15:30 PM EST
Solution: 05/06/2023 @ 12:00 PM EST
List Updated: 20/06/2023 @ 14:35 PM EST

*Lazar Ilic(1/5/2023 4:26 PM IDT)
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