Starting diary file diary-23-Apr-2018.txt

A = grid5(10);
spy(A)
A
A =
   (1,1)        4
   (2,1)       -1
  (11,1)       -1
   (1,2)       -1
   (2,2)        4
   (3,2)       -1
  ...
 (100,100)      4

n = length(A)
n =
   100

condest(A)
ans =
   69.8634

help eig
 ...

[V,D] = eig(full(A));

d = diag(D)
d =
    0.1620
    0.3985
    0.3985
    0.6350
    0.7713
    0.7713
    1.0078
    ...
    7.6015
    7.8380

d(n)/d(1)
ans =
   48.3742

size(A)
ans =
   100   100

b = rand(n,1);

size(b)
ans =
   100     1
xx = A \ b;

norm(A*xx - b)
ans =
   1.8217e-14

norm(A*xx - b) / norm(A,'fro')
ans =
   4.1149e-16

help pcg
 pcg   Preconditioned Conjugate Gradients Method.
    ...

[x,flag,relres,iter,resvec] = pcg(A,b,1e-8,2*n);

norm(A*x-b)
ans =
   4.3046e-08

iter
iter =
    32

flag
flag =
     0

resvec
resvec =
    6.5469
   10.5379
    9.7603
    7.2655
    5.7331
    ...
    0.0000

semilogy(resvec / resvec(1))

lambda = linspace(.1,1,100);
plot(lambda)
length(lambda)
ans =
   100
A = randspd(lambda);
size(A)
ans =
   100   100
condest(A)
ans =
   60.5807
cond(full(A))
ans =
   10.0000

[x,flag,relres,iter,resvec] = pcg(A,b,1e-8,2*n);
iter
iter =
    28

semilogy(resvec / resvec(1))

lambda = [.1 1  .5+randn(1,98)/100];
figure, plot(lambda)
figure, hist(lambda,100)

A = randspd(lambda);

[x,flag,relres,iter,resvec] = pcg(A,b,1e-8,2*n);
iter
iter =
     8
figure(1)
hold on
semilogy(resvec / resvec(1), 'r')

load bcsstk08
A = BCSSTK08;
n = length(A)
n =
        1074
b = rand(n,1);

hold off
[x,flag,relres,iter,resvec] = pcg(A,b,1e-8,2*n);
iter
iter =
        2019
figure
semilogy(resvec / resvec(1))
flag
flag =
     1

[x,flag,relres,iter,resvec] = pcg(A,b,1e-8,50000);
semilogy(resvec / resvec(1))

condest(A)
ans =
   4.7262e+07

norm(A*x-b)
ans =
   1.5332e-07

xx = A \ b;
norm(A*xx-b)
ans =
   1.0262e-11