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nr.cpp
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nr.cpp
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#include"nr.h"
#define TINY 1.0e-10
#define NMAX 5000
#define GET_PSUM \
for (j=1;j<=ndim;j++) {\
for (sum=0.0,i=1;i<=mpts;i++) sum += p[i][j];\
psum[j]=sum;}
#define SWAP(a,b) {swap=(a);(a)=(b);(b)=swap;}
void amoeba(double **p, double y[], int ndim, double ftol,
double (*funk)(double []), int *nfunk)
{
double amotry(double **p, double y[], double psum[], int ndim,
double (*funk)(double []), int ihi, double fac);
int i,ihi,ilo,inhi,j,mpts=ndim+1;
double rtol,sum,swap,ysave,ytry,*psum;
psum=vector(1,ndim);
*nfunk=0;
GET_PSUM
for (;;) {
ilo=1;
ihi = y[1]>y[2] ? (inhi=2,1) : (inhi=1,2);
for (i=1;i<=mpts;i++) {
if (y[i] <= y[ilo]) ilo=i;
if (y[i] > y[ihi]) {
inhi=ihi;
ihi=i;
} else if (y[i] > y[inhi] && i != ihi) inhi=i;
}
rtol=2.0*fabs(y[ihi]-y[ilo])/(fabs(y[ihi])+fabs(y[ilo])+TINY);
if (rtol < ftol) {
SWAP(y[1],y[ilo])
for (i=1;i<=ndim;i++) SWAP(p[1][i],p[ilo][i])
break;
}
if (*nfunk >= NMAX){ nrerror("NMAX exceeded");return;}
*nfunk += 2;
ytry=amotry(p,y,psum,ndim,funk,ihi,-1.0);
if (ytry <= y[ilo])
ytry=amotry(p,y,psum,ndim,funk,ihi,2.0);
else if (ytry >= y[inhi]) {
ysave=y[ihi];
ytry=amotry(p,y,psum,ndim,funk,ihi,0.5);
if (ytry >= ysave) {
for (i=1;i<=mpts;i++) {
if (i != ilo) {
for (j=1;j<=ndim;j++)
p[i][j]=psum[j]=0.5*(p[i][j]+p[ilo][j]);
y[i]=(*funk)(psum);
}
}
*nfunk += ndim;
GET_PSUM
}
} else --(*nfunk);
}
free_vector(psum,1,ndim);
}
double amotry(double **p, double y[], double psum[], int ndim,
double (*funk)(double []), int ihi, double fac)
{
int j;
double fac1,fac2,ytry,*ptry;
ptry=vector(1,ndim);
fac1=(1.0-fac)/ndim;
fac2=fac1-fac;
for (j=1;j<=ndim;j++) ptry[j]=psum[j]*fac1-p[ihi][j]*fac2;
ytry=(*funk)(ptry);
if (ytry < y[ihi]) {
y[ihi]=ytry;
for (j=1;j<=ndim;j++) {
psum[j] += ptry[j]-p[ihi][j];
p[ihi][j]=ptry[j];
}
}
free_vector(ptry,1,ndim);
return ytry;
}
void lubksb(double **a, int n, int *indx, double b[])
{
int i,ii=0,ip,j;
double sum;
for (i=1;i<=n;i++) {
ip=indx[i];
sum=b[ip];
b[ip]=b[i];
if (ii)
for (j=ii;j<=i-1;j++) sum -= a[i][j]*b[j];
else if (sum) ii=i;
b[i]=sum;
}
for (i=n;i>=1;i--) {
sum=b[i];
for (j=i+1;j<=n;j++) sum -= a[i][j]*b[j];
b[i]=sum/a[i][i];
}
}
void tqli(double d[], double e[], int n, double **z)
{
double pythag(double a, double b);
int m,l,iter,i,k;
double s,r,p,g,f,dd,c,b;
for (i=2;i<=n;i++) e[i-1]=e[i];
e[n]=0.0;
for (l=1;l<=n;l++) {
iter=0;
do {
for (m=l;m<=n-1;m++) {
dd=fabs(d[m])+fabs(d[m+1]);
if ((double)(fabs(e[m])+dd) == dd) break;
}
if (m != l) {
if (iter++ == 30) nrerror("Too many iterations in tqli");
g=(d[l+1]-d[l])/(2.0*e[l]);
r=pythag(g,1.0);
g=d[m]-d[l]+e[l]/(g+SIGN(r,g));
s=c=1.0;
p=0.0;
for (i=m-1;i>=l;i--) {
f=s*e[i];
b=c*e[i];
e[i+1]=(r=pythag(f,g));
if (r == 0.0) {
d[i+1] -= p;
e[m]=0.0;
break;
}
s=f/r;
c=g/r;
g=d[i+1]-p;
r=(d[i]-g)*s+2.0*c*b;
d[i+1]=g+(p=s*r);
g=c*r-b;
if (z!=NULL)
{
for (k=1;k<=n;k++) {
f=z[k][i+1];
z[k][i+1]=s*z[k][i]+c*f;
z[k][i]=c*z[k][i]-s*f;
}
}
}
if (r == 0.0 && i >= l) continue;
d[l] -= p;
e[l]=g;
e[m]=0.0;
}
} while (m != l);
}
}
void mnewt(int ntrial, double x[], int n, double tolx, double tolf,void (*usrfun)(double [],int n,double *fvec,double **fjac))
{
#define FREERETURN {free_matrix(fjac,1,n,1,n);free_vector(fvec,1,n);free_ivector(indx,1,n);free_vector(p,1,n);return;}
void lubksb(double **a, int n, int *indx, double b[]);
void ludcmp(double **a, int n, int *indx, double *d);
int k,i,*indx;
double errx,errf,d,*fvec,**fjac,*p;
indx=ivector(1,n);
p=vector(1,n);
fvec=vector(1,n);
fjac=matrix(1,n,1,n);
for (k=1;k<=ntrial;k++) {
usrfun(x,n,fvec,fjac);
errf=0.0;
for (i=1;i<=n;i++) errf += fabs(fvec[i]);
if (errf <= tolf){FREERETURN;}
for (i=1;i<=n;i++) p[i] = -fvec[i];
ludcmp(fjac,n,indx,&d);
lubksb(fjac,n,indx,p);
errx=0.0;
for (i=1;i<=n;i++) {
errx += fabs(p[i]);
x[i] += p[i];
}
if (errx <= tolx){FREERETURN;}
}
FREERETURN;
}
#define TINY 1.0e-10
void ludcmp(double **a, int n, int *indx, double *d)
{
int i,imax,j,k;
double big,dum,sum,temp;
double *vv;
vv=vector(1,n);
*d=1.0;
for (i=1;i<=n;i++) {
big=0.0;
for (j=1;j<=n;j++)
if ((temp=fabs(a[i][j])) > big) big=temp;
if (big == 0.0) nrerror("Singular matrix in routine ludcmp");
vv[i]=1.0/big;
}
for (j=1;j<=n;j++) {
for (i=1;i<j;i++) {
sum=a[i][j];
for (k=1;k<i;k++) sum -= a[i][k]*a[k][j];
a[i][j]=sum;
}
big=0.0;
for (i=j;i<=n;i++) {
sum=a[i][j];
for (k=1;k<j;k++)
sum -= a[i][k]*a[k][j];
a[i][j]=sum;
if ( (dum=vv[i]*fabs(sum)) >= big) {
big=dum;
imax=i;
}
}
if (j != imax) {
for (k=1;k<=n;k++) {
dum=a[imax][k];
a[imax][k]=a[j][k];
a[j][k]=dum;
}
*d = -(*d);
vv[imax]=vv[j];
}
indx[j]=imax;
if (a[j][j] == 0.0)a[j][j]=TINY;//Pivot is zero...
if (j != n) {
dum=1.0/(a[j][j]);
for (i=j+1;i<=n;i++) a[i][j] *= dum;
}
}
free_vector(vv,1,n);
}
#define SWAP(g,h) {y=(g);(g)=(h);(h)=y;}
void elmhes(double **a, int n)
{
int m,j,i;
double y,x;
for (m=2;m<n;m++) {
x=0.0;
i=m;
for (j=m;j<=n;j++) {
if (fabs(a[j][m-1]) > fabs(x)) {
x=a[j][m-1];
i=j;
}
}
if (i != m) {
for (j=m-1;j<=n;j++) SWAP(a[i][j],a[m][j])
for (j=1;j<=n;j++) SWAP(a[j][i],a[j][m])
}
if (x) {
for (i=m+1;i<=n;i++) {
if ((y=a[i][m-1]) != 0.0) {
y /= x;
a[i][m-1]=y;
for (j=m;j<=n;j++)
a[i][j] -= y*a[m][j];
for (j=1;j<=n;j++)
a[j][m] += y*a[j][i];
}
}
}
}
}
#undef SWAP
#define NRANSI
void hqr(double **a, int n, double wr[], double wi[])
{
int nn,m,l,k,j,its,i,mmin;
double z,y,x,w,v,u,t,s,r,q,p,anorm;
anorm=0.0;
for (i=1;i<=n;i++)
for (j=IMAX(i-1,1);j<=n;j++)
anorm += fabs(a[i][j]);
nn=n;
t=0.0;
while (nn >= 1) {
its=0;
do {
for (l=nn;l>=2;l--) {
s=fabs(a[l-1][l-1])+fabs(a[l][l]);
if (s == 0.0) s=anorm;
if ((double)(fabs(a[l][l-1]) + s) == s) {
a[l][l-1]=0.0;
break;
}
}
x=a[nn][nn];
if (l == nn) {
wr[nn]=x+t;
wi[nn--]=0.0;
} else {
y=a[nn-1][nn-1];
w=a[nn][nn-1]*a[nn-1][nn];
if (l == (nn-1)) {
p=0.5*(y-x);
q=p*p+w;
z=sqrt(fabs(q));
x += t;
if (q >= 0.0) {
z=p+SIGN(z,p);
wr[nn-1]=wr[nn]=x+z;
if (z) wr[nn]=x-w/z;
wi[nn-1]=wi[nn]=0.0;
} else {
wr[nn-1]=wr[nn]=x+p;
wi[nn-1]= -(wi[nn]=z);
}
nn -= 2;
} else {
if (its == 30) nrerror("Too many iterations in hqr");
if (its == 10 || its == 20) {
t += x;
for (i=1;i<=nn;i++) a[i][i] -= x;
s=fabs(a[nn][nn-1])+fabs(a[nn-1][nn-2]);
y=x=0.75*s;
w = -0.4375*s*s;
}
++its;
for (m=(nn-2);m>=l;m--) {
z=a[m][m];
r=x-z;
s=y-z;
p=(r*s-w)/a[m+1][m]+a[m][m+1];
q=a[m+1][m+1]-z-r-s;
r=a[m+2][m+1];
s=fabs(p)+fabs(q)+fabs(r);
p /= s;
q /= s;
r /= s;
if (m == l) break;
u=fabs(a[m][m-1])*(fabs(q)+fabs(r));
v=fabs(p)*(fabs(a[m-1][m-1])+fabs(z)+fabs(a[m+1][m+1]));
if ((double)(u+v) == v) break;
}
for (i=m+2;i<=nn;i++) {
a[i][i-2]=0.0;
if (i != (m+2)) a[i][i-3]=0.0;
}
for (k=m;k<=nn-1;k++) {
if (k != m) {
p=a[k][k-1];
q=a[k+1][k-1];
r=0.0;
if (k != (nn-1)) r=a[k+2][k-1];
if ((x=fabs(p)+fabs(q)+fabs(r)) != 0.0) {
p /= x;
q /= x;
r /= x;
}
}
if ((s=SIGN(sqrt(p*p+q*q+r*r),p)) != 0.0) {
if (k == m) {
if (l != m)
a[k][k-1] = -a[k][k-1];
} else
a[k][k-1] = -s*x;
p += s;
x=p/s;
y=q/s;
z=r/s;
q /= p;
r /= p;
for (j=k;j<=nn;j++) {
p=a[k][j]+q*a[k+1][j];
if (k != (nn-1)) {
p += r*a[k+2][j];
a[k+2][j] -= p*z;
}
a[k+1][j] -= p*y;
a[k][j] -= p*x;
}
mmin = nn<k+3 ? nn : k+3;
for (i=l;i<=mmin;i++) {
p=x*a[i][k]+y*a[i][k+1];
if (k != (nn-1)) {
p += z*a[i][k+2];
a[i][k+2] -= p*r;
}
a[i][k+1] -= p*q;
a[i][k] -= p;
}
}
}
}
}
} while (l < nn-1);
}
}
#undef NRANSI
#define ROTATE(a,i,j,k,l) g=a[i][j];h=a[k][l];a[i][j]=g-s*(h+g*tau);\
a[k][l]=h+s*(g-h*tau);
void jacobi(double **a, int n, double d[], double **v, int *nrot)
{
int j,iq,ip,i;
double tresh,theta,tau,t,sm,s,h,g,c,*b,*z;
b=vector(1,n);
z=vector(1,n);
for (ip=1;ip<=n;ip++) {
for (iq=1;iq<=n;iq++) v[ip][iq]=0.0;
v[ip][ip]=1.0;
}
for (ip=1;ip<=n;ip++) {
b[ip]=d[ip]=a[ip][ip];
z[ip]=0.0;
}
*nrot=0;
for (i=1;i<=50;i++) {
sm=0.0;
for (ip=1;ip<=n-1;ip++) {
for (iq=ip+1;iq<=n;iq++)
sm += fabs(a[ip][iq]);
}
if (sm == 0.0) {
free_vector(z,1,n);
free_vector(b,1,n);
return;
}
if (i < 4)
tresh=0.2*sm/(n*n);
else
tresh=0.0;
for (ip=1;ip<=n-1;ip++) {
for (iq=ip+1;iq<=n;iq++) {
g=100.0*fabs(a[ip][iq]);
if (i > 4 && (double)(fabs(d[ip])+g) == (double)fabs(d[ip])
&& (double)(fabs(d[iq])+g) == (double)fabs(d[iq]))
a[ip][iq]=0.0;
else if (fabs(a[ip][iq]) > tresh) {
h=d[iq]-d[ip];
if ((double)(fabs(h)+g) == (double)fabs(h))
t=(a[ip][iq])/h;
else {
theta=0.5*h/(a[ip][iq]);
t=1.0/(fabs(theta)+sqrt(1.0+theta*theta));
if (theta < 0.0) t = -t;
}
c=1.0/sqrt(1+t*t);
s=t*c;
tau=s/(1.0+c);
h=t*a[ip][iq];
z[ip] -= h;
z[iq] += h;
d[ip] -= h;
d[iq] += h;
a[ip][iq]=0.0;
for (j=1;j<=ip-1;j++) {
ROTATE(a,j,ip,j,iq)
}
for (j=ip+1;j<=iq-1;j++) {
ROTATE(a,ip,j,j,iq)
}
for (j=iq+1;j<=n;j++) {
ROTATE(a,ip,j,iq,j)
}
for (j=1;j<=n;j++) {
ROTATE(v,j,ip,j,iq)
}
++(*nrot);
}
}
}
for (ip=1;ip<=n;ip++) {
b[ip] += z[ip];
d[ip]=b[ip];
z[ip]=0.0;
}
}
nrerror("Too many iterations in routine jacobi");
}
#undef ROTATE
#undef NRANSI
#define RADIX 2.0
void balanc(double **a, int n)
{
int last,j,i;
double s,r,g,f,c,sqrdx;
sqrdx=RADIX*RADIX;
last=0;
while (last == 0) {
last=1;
for (i=1;i<=n;i++) {
r=c=0.0;
for (j=1;j<=n;j++)
if (j != i) {
c += fabs(a[j][i]);
r += fabs(a[i][j]);
}
if (c && r) {
g=r/RADIX;
f=1.0;
s=c+r;
while (c<g) {
f *= RADIX;
c *= sqrdx;
}
g=r*RADIX;
while (c>g) {
f /= RADIX;
c /= sqrdx;
}
if ((c+r)/f < 0.95*s) {
last=0;
g=1.0/f;
for (j=1;j<=n;j++) a[i][j] *= g;
for (j=1;j<=n;j++) a[j][i] *= f;
}
}
}
}
}
#undef RADIX
double pythag(double a, double b)
{
double absa,absb;
absa=fabs(a);
absb=fabs(b);
if (absa > absb) return absa*sqrt(1.0+SQR(absb/absa));
else return (absb == 0.0 ? 0.0 : absb*sqrt(1.0+SQR(absa/absb)));
}
void tred2(double **a, int n, double d[], double e[])
{
int l,k,j,i;
double scale,hh,h,g,f;
for (i=n;i>=2;i--) {
l=i-1;
h=scale=0.0;
if (l > 1) {
for (k=1;k<=l;k++)
scale += fabs(a[i][k]);
if (scale == 0.0)
e[i]=a[i][l];
else {
for (k=1;k<=l;k++) {
a[i][k] /= scale;
h += a[i][k]*a[i][k];
}
f=a[i][l];
g=(f >= 0.0 ? -sqrt(h) : sqrt(h));
e[i]=scale*g;
h -= f*g;
a[i][l]=f-g;
f=0.0;
for (j=1;j<=l;j++) {
a[j][i]=a[i][j]/h;
g=0.0;
for (k=1;k<=j;k++)
g += a[j][k]*a[i][k];
for (k=j+1;k<=l;k++)
g += a[k][j]*a[i][k];
e[j]=g/h;
f += e[j]*a[i][j];
}
hh=f/(h+h);
for (j=1;j<=l;j++) {
f=a[i][j];
e[j]=g=e[j]-hh*f;
for (k=1;k<=j;k++)
a[j][k] -= (f*e[k]+g*a[i][k]);
}
}
} else
e[i]=a[i][l];
d[i]=h;
}
d[1]=0.0;
e[1]=0.0;
/* Contents of this loop can be omitted if eigenvectors not
wanted except for statement d[i]=a[i][i]; */
for (i=1;i<=n;i++) {
l=i-1;
if (d[i]) {
for (j=1;j<=l;j++) {
g=0.0;
for (k=1;k<=l;k++)
g += a[i][k]*a[k][j];
for (k=1;k<=l;k++)
a[k][j] -= g*a[k][i];
}
}
d[i]=a[i][i];
a[i][i]=1.0;
for (j=1;j<=l;j++) a[j][i]=a[i][j]=0.0;
}
}
#define SWAP(a,b) tempr=(a);(a)=(b);(b)=tempr
void four1(double data[], unsigned long nn, int isign)
{
unsigned long n,mmax,m,j,istep,i;
double wtemp,wr,wpr,wpi,wi,theta;
double tempr,tempi;
n=nn << 1;
j=1;
for (i=1;i<n;i+=2) {
if (j > i) {
SWAP(data[j],data[i]);
SWAP(data[j+1],data[i+1]);
}
m=nn;
while (m >= 2 && j > m) {
j -= m;
m >>= 1;
}
j += m;
}
mmax=2;
while (n > mmax) {
istep=mmax << 1;
theta=isign*(6.28318530717959/mmax);
wtemp=sin(0.5*theta);
wpr = -2.0*wtemp*wtemp;
wpi=sin(theta);
wr=1.0;
wi=0.0;
for (m=1;m<mmax;m+=2) {
for (i=m;i<=n;i+=istep) {
j=i+mmax;
tempr=wr*data[j]-wi*data[j+1];
tempi=wr*data[j+1]+wi*data[j];
data[j]=data[i]-tempr;
data[j+1]=data[i+1]-tempi;
data[i] += tempr;
data[i+1] += tempi;
}
wr=(wtemp=wr)*wpr-wi*wpi+wr;
wi=wi*wpr+wtemp*wpi+wi;
}
mmax=istep;
}
}
#undef SWAP
void realft(double data[], unsigned long n, int isign)
{
void four1(double data[], unsigned long nn, int isign);
unsigned long i,i1,i2,i3,i4,np3;
double c1=0.5,c2,h1r,h1i,h2r,h2i;
double wr,wi,wpr,wpi,wtemp,theta;
theta=3.141592653589793/(double) (n>>1);
if (isign == 1) {
c2 = -0.5;
four1(data,n>>1,1);
} else {
c2=0.5;
theta = -theta;
}
wtemp=sin(0.5*theta);
wpr = -2.0*wtemp*wtemp;
wpi=sin(theta);
wr=1.0+wpr;
wi=wpi;
np3=n+3;
for (i=2;i<=(n>>2);i++) {
i4=1+(i3=np3-(i2=1+(i1=i+i-1)));
h1r=c1*(data[i1]+data[i3]);
h1i=c1*(data[i2]-data[i4]);
h2r = -c2*(data[i2]+data[i4]);
h2i=c2*(data[i1]-data[i3]);
data[i1]=h1r+wr*h2r-wi*h2i;
data[i2]=h1i+wr*h2i+wi*h2r;
data[i3]=h1r-wr*h2r+wi*h2i;
data[i4] = -h1i+wr*h2i+wi*h2r;
wr=(wtemp=wr)*wpr-wi*wpi+wr;
wi=wi*wpr+wtemp*wpi+wi;
}
if (isign == 1) {
data[1] = (h1r=data[1])+data[2];
data[2] = h1r-data[2];
} else {
data[1]=c1*((h1r=data[1])+data[2]);
data[2]=c1*(h1r-data[2]);
four1(data,n>>1,-1);
}
}
void asolve(unsigned long n, double b[], double x[], int itrnsp,double sa[],double *inv_vec,unsigned long *inv_ind)
{
dsprsax(inv_vec,inv_ind,b,x,n);
}
double snrm(unsigned long n, double sx[], int itol)
{
unsigned long i,isamax;
double ans;
if (itol <= 3) {
ans = 0.0;
for (i=1;i<=n;i++) ans += sx[i]*sx[i];
return sqrt(ans);
} else {
isamax=1;
for (i=1;i<=n;i++) {
if (fabs(sx[i]) > fabs(sx[isamax])) isamax=i;
}
return fabs(sx[isamax]);
}
}
void dsprsax(double sa[], unsigned long ija[], double x[], double b[], unsigned long n)
{
unsigned long i,k;
// if (ija[1] != n+2) nrerror("dsprsax: mismatched vector and matrix");
for (i=1;i<=n;i++) {
b[i]=sa[i]*x[i];
for (k=ija[i];k<=ija[i+1]-1;k++) b[i] += sa[k]*x[ija[k]];
}
}
void dsprstx(double sa[], unsigned long ija[], double x[], double b[], unsigned long n)
{
unsigned long i,j,k;
if (ija[1] != n+2) nrerror("mismatched vector and matrix in dsprstx");
for (i=1;i<=n;i++) b[i]=sa[i]*x[i];
for (i=1;i<=n;i++) {
for (k=ija[i];k<=ija[i+1]-1;k++) {
j=ija[k];
b[j] += sa[k]*x[i];
}
}
}
void atimes(unsigned long n, double x[], double r[], int itrnsp,double sa[],unsigned long ija[])
{
if (itrnsp) dsprstx(sa,ija,x,r,n);
else dsprsax(sa,ija,x,r,n);
}
#define NRANSI
#define SWAP(a,b) {temp=(a);(a)=(b);(b)=temp;}
void gaussj(double **a, int n, double **b, int m)
{
int *indxc,*indxr,*ipiv;
int i,icol,irow,j,k,l,ll;
double big,dum,pivinv,temp;
indxc=ivector(1,n);
indxr=ivector(1,n);
ipiv=ivector(1,n);
for (j=1;j<=n;j++) ipiv[j]=0;
for (i=1;i<=n;i++) {
big=0.0;
for (j=1;j<=n;j++)
if (ipiv[j] != 1)
for (k=1;k<=n;k++) {
if (ipiv[k] == 0) {
if (fabs(a[j][k]) >= big) {
big=fabs(a[j][k]);
irow=j;
icol=k;
}
}
}
++(ipiv[icol]);
if (irow != icol) {
for (l=1;l<=n;l++) SWAP(a[irow][l],a[icol][l])
for (l=1;l<=m;l++) SWAP(b[irow][l],b[icol][l])
}
indxr[i]=irow;
indxc[i]=icol;
if (a[icol][icol] == 0.0) nrerror("gaussj: Singular Matrix");
pivinv=1.0/a[icol][icol];
a[icol][icol]=1.0;
for (l=1;l<=n;l++) a[icol][l] *= pivinv;
for (l=1;l<=m;l++) b[icol][l] *= pivinv;
for (ll=1;ll<=n;ll++)
if (ll != icol) {
dum=a[ll][icol];
a[ll][icol]=0.0;
for (l=1;l<=n;l++) a[ll][l] -= a[icol][l]*dum;
for (l=1;l<=m;l++) b[ll][l] -= b[icol][l]*dum;
}
}
for (l=n;l>=1;l--) {
if (indxr[l] != indxc[l])
for (k=1;k<=n;k++)
SWAP(a[k][indxr[l]],a[k][indxc[l]]);
}
free_ivector(ipiv,1,n);
free_ivector(indxr,1,n);
free_ivector(indxc,1,n);
}
#undef SWAP
#undef NRANSI
#define EPS 1.0e-14
//This reduces a matrix to Gaussian normal form, or as close to this as possible (if A is singular).
void gauss_elimination(double **a,int n)
{
int i,k,l,m;
double *row;
double max;
int swaps;
double mult=0.0;
int n_max=0;
//Three steps, carried out for each row:
//1: We find the pivot element in each row. We swap the columns, such that the pivot element is at the right place.
//2: We divide the row by the pivot element.
//3: We subtract a multiple of the row from all other rows, such that all elements in the pivot columns become zero (besides the pivot element itself)
swaps=0;
for(i=1;i<=n;i++)
{
START:
max=fabs(a[i][i]);
n_max=i;
for (k=i+1;k<=n;k++)
{
if (fabs(a[k][i])>max)
{
max=fabs(a[k][i]);
n_max=k;
}
}
// cout << "column:" << i << "\n";
//pivot element is at a[i][n_max]
if (a[n_max][i]==0.0)
{
// cout << "No non-zero elements found. Returning...\n";
i++;
if (i<=n)
goto START;
else
return;
}
if (n_max!=i)
{
// cout << "Exchaning " << i << " for " << n_max << "\n";
//We exchange k & i.
row=a[i];
a[i]=a[n_max];
a[n_max]=row;
}
// cout << "Pivot element:" << a[i][i] << "\n";
//If the row contains only zeros, we swap it with one of the last rows and start again.
//pivoting is done. Now we divide the pivot row by the pivot element and subtract this row from the other ones.
// cout << "Multiplying...\n";
double mult=a[i][i];
for(k=i;k<=n;k++)
a[i][k]/=mult;
for(l=1;l<=n;l++)
if (l!=i)
{
mult=a[l][i];
for(m=i;m<=n;m++)
a[l][m]-=a[i][m]*mult;
// a[l][m]=a[l][m]-a[i][m]*a[l][i];
}
//Now all elements but the pivot element in the pivot column are zero. We can go on with the next column, until we reach the end of the matrix.
}
}
//This reduces a matrix to Gaussian normal form, or as close to this as possible (if A is singular).
void full_gauss_elimination(double **a,double **b,int n)
{
int i,k,l,m;
double max;
int swaps;
double temp;
double mult=0.0;
int n_max=0;
//Three steps, carried out for each row:
//1: We find the pivot element in each row. We swap the columns, such that the pivot element is at the right place.
//2: We divide the row by the pivot element.
//3: We subtract a multiple of the row from all other rows, such that all elements in the pivot columns become zero (besides the pivot element itself)
swaps=0;
for(i=1;i<=n;i++)
{
max=fabs(a[i][i]);
n_max=i;
for (k=i+1;k<=n;k++)
{
if (fabs(a[i][k])>max)
{
max=fabs(a[i][k]);
n_max=k;
}
}
// cout << "column:" << i << "\n";
//pivot element is at a[i][n_max]
if (a[i][n_max]==0.0)
{
cout << "Err: Singular matrix!\n";
return;
}
if (n_max!=i)
{
// cout << "Exchaning " << i << " for " << n_max << "\n";
//We exchange k & i.
for(k=i;k<=n;k++)
{
temp=a[i][k];
a[i][k]=a[i][n_max];
a[i][n_max]=temp;
}
}
// cout << "Pivot element:" << a[i][i] << "\n";
//If the row contains only zeros, we swap it with one of the last rows and start again.
//pivoting is done. Now we divide the pivot row by the pivot element and subtract this row from the other ones.
// cout << "Multiplying...\n";
double mult=a[i][i];
for(k=i;k<=n;k++)
a[i][k]/=mult;
for(l=1;l<=n;l++)
if (l!=i)
{
mult=a[l][i];
for(m=i;m<=n;m++)
a[l][m]-=a[i][m]*mult;
// a[l][m]=a[l][m]-a[i][m]*a[l][i];
}
//Now all elements but the pivot element in the pivot column are zero. We can go on with the next column, until we reach the end of the matrix.
}
}
void linbcg_single(unsigned long n, double b[], double x[], int itol, double tol,
int itmax, int *iter, double *err,double sa[],unsigned long ija[],double *inv_vec,unsigned long *inv_ind)
{
unsigned long j;
static double *p=dvector(1,n);
static double *r=dvector(1,n);
static double *z=dvector(1,n);
static double *za=dvector(1,n);
double my_err=1.0;
double lambda=0.0;
double zx,zz;
//dsprsax is the main bottleneck!!! make matrix multiplication faster, i.e.: make A and thus A^{-1} more sparse or accelerate matrix multiplication!
int cnt=0;
while(my_err>1e-4 && cnt<100)
{
dsprsax(sa,ija,x,r,n);
for (j=1;j<=n;j++) {
r[j]=b[j]-r[j];
}
// for(j=1;j<=n;j++)
// z[j]=r[j]/(sa[j]!=0.0?sa[j]:1e-10);
dsprsax(inv_vec,inv_ind,r,z,n);
zx=0.0;
zz=0.0;
my_err=0.0;
dsprsax(sa,ija,z,za,n);
for(j=1;j<=n;j++)
{
zx+=za[j]*r[j];
zz+=za[j]*za[j];
my_err+=fabs(r[j]);
}
my_err/=(double)n;
lambda=zx/zz;