dim=1100;
A= RandomReal[{0,1},{dim,dim}];
A= A + Transpose[A] + DiagonalMatrix[RandomReal[dim{0.5,1.5}, dim]];
b=RandomReal[{0,1},dim];
Timing[
xSol=LinearSolve[A,b];
]

ListPlot[Eigenvalues[A]]

Tol = (10^(-12));
x = RandomReal[{0,1},dim];
rOld =b- A. x  ;
p = rOld;
n=300;
Do[
α= ((rOld.rOld)/(p.A.p));
x = x + α p;
rNew=rOld-α A.p;
If[ Norm[rNew]<Tol, Print[i];Break[]];
β = ((rNew.rNew)/(rOld.rOld));
p=rNew + β p;
rOld=rNew, 
{i,1,n}
]
Norm[x-xSol]