I don’t recall having seen this matrix earlier than, however I cannot be stunned to be taught that anyone else already is aware of all about it, particularly if that individual’s identify is Nick.

### Contents

`Q`

I have been investigating the matrices generated by this elegant one-liner.

Q = @(n) (-n:n).^2 + (-n:n)'.^2;

The `Q` is for “quadratic”.

The center column accommodates the squares of the integers from `-n` to `n`. So does the center row. The apostrophe summons singleton enlargement. The ensuing matrix has order `2*n+1`. Right here is `Q(5)`.

Q5 = Q(5)

Q5 = 50 41 34 29 26 25 26 29 34 41 50 41 32 25 20 17 16 17 20 25 32 41 34 25 18 13 10 9 10 13 18 25 34 29 20 13 8 5 4 5 8 13 20 29 26 17 10 5 2 1 2 5 10 17 26 25 16 9 4 1 0 1 4 9 16 25 26 17 10 5 2 1 2 5 10 17 26 29 20 13 8 5 4 5 8 13 20 29 34 25 18 13 10 9 10 13 18 25 34 41 32 25 20 17 16 17 20 25 32 41 50 41 34 29 26 25 26 29 34 41 50

I just like the contour plot.

contourf(Q(100)) axis sq. colorbar title('Q(100)')

`D`

For an additional weblog put up beneath growth, I want a logical masks that carves a round area out of graphic. This disc does the job.

D = @(n) Q(n) <= n^2;

Right here is my carver.

```
spy(D(100))
title('D(100)')
```

Did you discover the digits within the depend of nonzeros in `D(100)`? It occurs every time `n` is an influence of 10.

fprintf('%15s %12sn','n','nnz(D(n))') for n = 10.^(0:4) fprintf('%15d %12dn',n, nnz(D(n))) finish

n nnz(D(n)) 1 5 10 317 100 31417 1000 3141549 10000 314159053

#### O.E.I.S.

A basic drawback, described within the On-line Encyclopedia of Integer Sequences, asks what number of factors with integer coordinates lie inside a disc of accelerating radius. Our nonzero depend gives the reply.

fprintf('%15s %8sn','n','a(n)') for n = [1:15 99:101 499:501 999:1001] if mod(n,100) == 99 fprintf('%15s %8sn','-','-') finish a(n) = nnz(D(n)); fprintf('%15d %8dn',n,a(n)) finish

n a(n) 1 5 2 13 3 29 4 49 5 81 6 113 7 149 8 197 9 253 10 317 11 377 12 441 13 529 14 613 15 709 - - 99 30757 100 31417 101 32017 - - 499 782197 500 785349 501 788509 - - 999 3135157 1000 3141549 1001 3147833

`R`

Taking the reciprocals of the matrix entries, and lowering the vary of the nameless index, produces a matrix that behaves a bit just like the Hilbert matrix, `hilb(n)`.

R = @(n) 1./((1:n).^2 + (1:n)'.^2);

Listed below are the 5-by-5’s.

```
format rat
R5 = R(5)
H5 = hilb(5)
```

R5 = 1/2 1/5 1/10 1/17 1/26 1/5 1/8 1/13 1/20 1/29 1/10 1/13 1/18 1/25 1/34 1/17 1/20 1/25 1/32 1/41 1/26 1/29 1/34 1/41 1/50 H5 = 1 1/2 1/3 1/4 1/5 1/2 1/3 1/4 1/5 1/6 1/3 1/4 1/5 1/6 1/7 1/4 1/5 1/6 1/7 1/8 1/5 1/6 1/7 1/8 1/9

#### Situation

Going away from the diagonal, the weather of `R(n)` decay extra quickly than these of `hilb(n)`, so `R(n)` is healthier conditioned than `hilb(n)`.

format quick e fprintf('%15s %12s %12sn','n','cond R','cond H') for n = 1:12 fprintf('%15d %12.2e %12.2en',n,cond(R(n)),cond(hilb(n))) finish

n cond R cond H 1 1.00e+00 1.00e+00 2 1.53e+01 1.93e+01 3 2.04e+02 5.24e+02 4 2.59e+03 1.55e+04 5 3.21e+04 4.77e+05 6 3.89e+05 1.50e+07 7 4.67e+06 4.75e+08 8 5.54e+07 1.53e+10 9 6.53e+08 4.93e+11 10 7.65e+09 1.60e+13 11 8.92e+10 5.22e+14 12 1.04e+12 1.62e+16

#### Further Credit score

What’s the rank of `Q(n)`? Why? See a paper in SIAM Assessment by Strang and Moler.

Why is the desk of values for `nnz(D(10^okay))` so quick? How may you prolong this desk?

Examine `R(n)`. Is it optimistic particular? What are its eigenvalues? What’s its inverse? What’s the signal sample of the weather of its inverse? For what values of `n` are you able to compute the inverse reliably utilizing floating level arithmetic? How does all this evaluate with `hilb(n)` and `invhilb(n)` ?

Feedback within the Feedback, or in e mail to me, are welcome.

Printed with MATLAB® R2022a