double maxeig(double *xmat, mwSignedIndex n)
{
// xmat is symmetric n x n matrix
mwSignedIndex incx=1,indmax,maxloop=10000,k=0;
double alpha, beta, dmax, dmax_temp,dmax_tol;
double *bufveca=(double *) calloc(n,sizeof(double));
double *bufvecb=(double *) calloc(n,sizeof(double));
dmax_tol=.001;
// do power series to get approximation to max eigenvalue of A+X
alpha=0.0;cblas_dscal(n,alpha,bufveca,incx);bufveca[0]=1.0; // x_0 = [1,0,0,...] do something better later
beta=0.0;dmax=1.0;dmax_temp=0.0;
while ((dabsf(dmax-dmax_temp)>dmax_tol)&&(k<=maxloop)){
dmax_temp=dmax;
alpha=1.0;
cblas_dgemv(CblasColMajor,CblasNoTrans,n,n,alpha,xmat,n,bufveca,incx,beta,bufvecb,incx);
indmax=idxmax(bufvecb,n);dmax=bufvecb[indmax];
alpha=1.0/dmax;cblas_dscal(n,alpha,bufvecb,incx);
cblas_dcopy(n,bufvecb,incx,bufveca,incx);
k++;
}
alpha=1.0;
// compute Rayleigh Quotient to approximate max eigenvalue of A+X
cblas_dgemv(CblasColMajor,CblasNoTrans,n,n,alpha,xmat,n,bufvecb,incx,beta,bufveca,incx);
dmax=doubdot(bufvecb,bufveca,n);alpha=doubnorm2(bufvecb,n);dmax=dmax/alpha/alpha;
free(bufveca);free(bufvecb);
return dmax;
}
/*
Input parameters:
A : mxn matrix
m : number of rows of A
n : number of columns of A
f : Right-hand side nx1 column vector
s : nx1 start vector (the initial guess)
tol : relative residual error tolerance for break
condition
maxiter : Maximum number of iterations to perform
M : diag(M) is the inverse of the preconditioner where M is a vector
*/
void conjugate_gradient(double *A,ptrdiff_t m,ptrdiff_t n,double *D1,double *D2,double *f,double *s,double tol,int maxiter,double *M,double *xout,double *CGiters,double *flag)
{
double *u=(double *) calloc(n,sizeof(double)); // n x 1 solution vector
double *r=(double *) calloc(n,sizeof(double)); // n x 1 residual vector
double *p=(double *) calloc(n,sizeof(double)); // n x 1 direction vector
double *veca=(double *) calloc(m,sizeof(double)); // mx1 temp vector
double *vecb=(double *) calloc(n,sizeof(double)); // nx1 temp vector
double *vecc=(double *) calloc(n,sizeof(double)); // nx1 temp vector
double *vecd=(double *) calloc(m*n,sizeof(double)); // mxn temp matrix
double alpha,beta,rho,rho_new,normr,normf;
int niter=0,i,j;
ptrdiff_t incx = 1;
// initializations
for (i=0;i<n;i++)
for (j=0;j<m;j++)
vecd[j+i*m]=A[j+i*m]*D1[j]; // vecd stores the matrix D1*A
*flag=0;
cblas_dcopy(n,s,incx,u,incx);
alpha=1.0;beta=0.0;cblas_dgemv(CblasColMajor,CblasNoTrans,m,n,alpha,A,m,u,incx,beta,veca,incx); // veca contains A*s
alpha=1.0;beta=0.0;cblas_dgemv(CblasColMajor,CblasTrans,m,n,alpha,vecd,m,veca,incx,beta,vecb,incx); // vecb contains A'*(D1*(A*s))
cblas_dcopy(n,f,incx,r,incx);
alpha=-1.0;cblas_daxpy(n,alpha,vecb,incx,r,incx); // r contains f-A'*(D1*(A*s))
cblas_dcopy(n,s,incx,vecb,incx);
component_mult(vecb,D2,n); // vecb contains D2*s
alpha=-1.0;cblas_daxpy(n,alpha,vecb,incx,r,incx); // r contains f-A'*(D1*(A*s))-D2*s
cblas_dcopy(n,r,incx,p,incx);
component_mult(p,M,n); // p contains M*r
rho=doubdot(p,r,n); // rho is r'*M*r
normf=doubnorm2(f,n);
if (normf<.0001)
normf=1.0;
normr=doubnorm2(r,n);
while (normr/normf > tol) { // Test break condition
alpha=1.0;beta=0.0;cblas_dgemv(CblasColMajor,CblasNoTrans,m,n,alpha,A,m,p,incx,beta,veca,incx); // veca contains A*p
alpha=1.0;beta=0.0;cblas_dgemv(CblasColMajor,CblasTrans,m,n,alpha,vecd,m,veca,incx,beta,vecb,incx); // vecb contains A'*(D1*(A*p))
cblas_dcopy(n,p,incx,vecc,incx);
component_mult(vecc,D2,n); // vecc contains D2*p
alpha=1.0;cblas_daxpy(n,alpha,vecc,incx,vecb,incx); // vecb contains A'*(D1*(A*p))+D2*p
alpha=rho/doubdot(vecb,p,n);
cblas_daxpy(n,alpha,p,incx,u,incx); // u=u+alpha*p
alpha=-1.0*alpha;cblas_daxpy(n,alpha,vecb,incx,r,incx); // r=r-alpha*vecb
cblas_dcopy(n,r,incx,vecc,incx);
component_mult(vecc,M,n); // vecc contains M*r
rho_new=doubdot(vecc,r,n); // rho is r'*M*r
alpha=rho_new/rho;cblas_dscal(n,alpha,p,incx); //p=rho_new/rho * p;
alpha=1.0;cblas_daxpy(n,alpha,vecc,incx,p,incx); // p=(M*r)+p
rho=rho_new;
niter++;
if (niter == maxiter) { //if max. number of iterations is reached, break.
normr=doubnorm2(r,n);
*flag = 1;
break;
}
normr=doubnorm2(r,n);
}
cblas_dcopy(n,u,incx,xout,incx);
*CGiters=niter;
free(u);
free(r);
free(p);
free(veca);
free(vecb);
free(vecc);
free(vecd);
}
