Create random symmetric matrix with given cond number to pass it to pcg

I want to generate random positive definitive symmetric matrix with given cond number, but the requirement is that PCG method should be able to solve it.

For now I use generation algorithm from this question. It does generate positive definitive symmetric random matrix with given cond number, but that is insufficient - pcg can't solve it even for smaller cond numbers. Pcg returns flag 4, which means that during algo it simply couldn't converge.

What am I doing wrong? Maybe there is better generating algorithm?

[Bruno Luong]

If you said "It does generate positive definitive symmetric random matrix with given cond number" then it meets what you ask for, why there is something supposingly "better" (and what sense)?

[Fedor]

The question is to generate matrix that COULD BE solved with pcg. I don't know the exact requirements of pcg honestly, but somehow it doesn't work at all with the generated matrix. So the question is, what can I tweak(or use different generation scheme) to make it work with pcg.

[Bruno Luong]

" I don't know the exact requirements of pcg honestly"

positive definitive symmetric matrix.

For now I use generation algorithm from this question. It does generate positive definitive symmetric

That's the problem, the algo in the question you quote: it does not generate positive matrix as you freely believe.

Answer 1

[Bruno Luong]

n=10; % size of the system
condtarget = 1000;

% generate ransom spd matrix
[Q,~]=qr(randn(n));
r = rand(n,1);
r = (r-min(r))/(max(r)-min(r));
d=exp(r*log(condtarget));
A=Q'*diag(d)*Q 

A = 10×10

61.2024 -52.3506 27.1572 10.1916 118.5038 -113.6603 51.9942 -15.6576 -36.1866 109.5894 -52.3506 54.0514 -19.7241 -14.7610 -97.4491 87.4631 -32.2037 3.9605 42.6244 -86.2163 27.1572 -19.7241 29.1843 -4.9656 72.9204 -77.2442 51.5620 -24.5594 -0.4552 73.7767 10.1916 -14.7610 -4.9656 21.8040 1.0998 8.1654 -16.8520 16.4248 -25.5273 -1.6470 118.5038 -97.4491 72.9204 1.0998 268.0601 -267.1190 146.9383 -59.9157 -45.3570 252.8113 -113.6603 87.4631 -77.2442 8.1654 -267.1190 279.6934 -160.8428 75.6400 28.7756 -258.5791 51.9942 -32.2037 51.5620 -16.8520 146.9383 -160.8428 113.5783 -59.3649 9.2172 151.0077 -15.6576 3.9605 -24.5594 16.4248 -59.9157 75.6400 -59.3649 46.1361 -20.6404 -68.2077 -36.1866 42.6244 -0.4552 -25.5273 -45.3570 28.7756 9.2172 -20.6404 52.0456 -32.5408 109.5894 -86.2163 73.7767 -1.6470 252.8113 -258.5791 151.0077 -68.2077 -32.5408 249.8764

cond(A)

ans = 1000.0000

b=rand(n,1) %  random rhs

b = 10×1

0.9430 0.1720 0.2828 0.8032 0.5086 0.7290 0.0177 0.6979 0.9675 0.4766

x=pcg(A,b)

pcg stopped at iteration 10 without converging to the desired tolerance 1e-06
because the maximum number of iterations was reached.
The iterate returned (number 10) has relative residual 0.00018.

x = 10×1

0.5857 -0.1797 0.0410 0.2153 0.1729 0.2925 0.0105 0.0112 0.5916 -0.1261

A*x % should close to b

ans = 10×1

0.9428 0.1718 0.2827 0.8033 0.5085 0.7288 0.0177 0.6980 0.9676 0.4766

[Fedor]

Yes, this works, thx!