void main ()
{
void impex(int, real_t, real_t, real_t [],
void (*)(real_t, real_t [], real_t [], int),
int (*)(real_t, real_t [], real_t **, int),
real_t, real_t, int, real_t, real_t [],
void (*)(real_t [], real_t [], int), int *fail,
void (*)(real_t *, real_t, real_t, real_t, real_t **,
real_t [], int, real_t));
void dupvec(int, int, int, real_t [], real_t []);
real_t vecvec(int, int, int, real_t [], real_t []);
void elmvec(int, int, int, real_t [], real_t [], real_t);
int n,fail,i,it;
real_t t,tend,eps,hmax,l,h2,y[4],sw[4],f1[4],f2[4],z[4],x[4],
n1,n2;
printf("The results with IMPEX are:\n\n");
n=3;
nje=nfe=0;
t=0.0;
tend=400.0;
eps=1.0e-5;
hmax=400.0;
y[1]=y[2]=y[3]=0.0;
sw[1]=sw[2]=sw[3]=1.0;
print[1]=0.1; print[2]=1.0; print[3]=10.0; print[4]=100.0;
print[5]=400.0;
dupvec(1,n,0,z,y);
for (i=1; i<=n; i++)
x[i] = (y[i] == 0.0) ? eps : (1.0+eps)*y[i];
n1=sqrt(vecvec(1,n,0,x,x))*eps;
f(t,x,f1,n);
for (it=1; it<=5; it++) {
f(t,z,f2,n);
elmvec(1,n,0,f2,f1,-1.0);
n2=n1/sqrt(vecvec(1,n,0,f2,f2));
dupvec(1,n,0,z,x);
elmvec(1,n,0,z,f2,n2);
}
f(t,z,f2,n);
elmvec(1,n,0,f2,f1,-1.0);
l=sqrt(vecvec(1,n,0,f2,f2))/n1;
h2=pow(eps*320.0,1.0/5.0)/(4.0*l);
printf("EPS = %e\nInterval of integration = (%1.0f,%3.0f)\n"
"Maximally allowed stepsize = %e\n\nLipschconst = %e\n"
"Starting stepsize = %e\nFunctional eval = %2d\n\n"
" X ERROR Y[1] Y[2] NFE NJE\n",
eps,t,tend,hmax,l,h2,nfe);
impex(n,t,tend,y,f,available,h2,hmax,0,eps,sw,update,&fail,
control);
printf("\nNumber of functional evaluations =%4d\n"
"Number of Jacobian evaluations = %3d\n",nfe,nje);
}
int DoTest(int N, TYPE *alpha0, int incX, int incY)
{
int iret;
const int npad=Mmax(4*Mabs(incY), 16);
const TYPE padval=(-2271.0);
TYPE *Yg, *Yt, *X, *x, *y;
#ifdef TREAL
TYPE alpha = *alpha0;
#else
TYPE *alpha = alpha0;
#endif
Yg = getvec(npad, padval, N, incY);
Yt = dupvec(npad, N, Yg, incY);
X = getvec(0, padval, N, incX); /* no padding for read-only X */
x = X;
y = Yg + (npad SHIFT);
if (incX < 1) x -= ((N-1)SHIFT) * incX;
if (incY < 1) y -= ((N-1)SHIFT) * incY;
good_axpy(N, alpha, x, incX, y, incY);
y = Yt + (npad SHIFT);
if (incY < 1) y -= ((N-1)SHIFT) * incY;
TEST_AXPY(N, alpha, x, incX, y, incY);
iret = CheckY(npad, padval, N, Yg, incY, Yt, incY);
free(X);
free(Yg);
free(Yt);
return(iret);
}
int DoTest(int N, TYPE *alpha0, int incY)
{
int iret;
const int npad=Mmax(4*Mabs(incY), 16);
const TYPE padval=(-2271.0);
TYPE *Yg, *Yt, *y;
#ifdef TREAL
TYPE alpha = *alpha0;
#else
TYPE *alpha = alpha0;
#endif
Yg = getvec(npad, padval, N, incY);
Yt = dupvec(npad, N, Yg, incY);
y = Yg + (npad SHIFT);
if (incY < 1) y -= ((N-1)SHIFT) * incY;
good_set(N, alpha, y, incY);
y = Yt + (npad SHIFT);
if (incY < 1) y -= ((N-1)SHIFT) * incY;
TEST_SET(N, alpha, Yt+(npad SHIFT), incY);
iret = CheckY(npad, padval, N, Yg, incY, Yt, incY);
free(Yg);
free(Yt);
return(iret);
}
void gssnewton(int m, int n, real_t par[], real_t rv[], real_t **jjinv,
int (*funct)(int, int, real_t[], real_t[]),
void (*jacobian)(int, int, real_t[], real_t[], real_t **),
real_t in[], real_t out[])
{
int *allocate_integer_vector(int, int);
real_t *allocate_real_vector(int, int);
real_t **allocate_real_matrix(int, int, int, int);
void free_integer_vector(int *, int);
void free_real_vector(real_t *, int);
void free_real_matrix(real_t **, int, int, int);
real_t vecvec(int, int, int, real_t [], real_t []);
void dupvec(int, int, int, real_t [], real_t []);
void elmvec(int, int, int, real_t [], real_t [], real_t);
void lsqortdec(real_t **, int, int, real_t [], real_t [], int []);
void lsqsol(real_t **, int, int, real_t [], int [], real_t []);
void lsqinv(real_t **, int, real_t [], int []);
int i,j,inr,mit,text,it,itmax,inrmax,tim,feval,fevalmax,conv,
testthf,dampingon,*ci,fail;
real_t rho,res1,res2,rn,reltolpar,abstolpar,abstolres,stap,normx,
**jac,*pr,*aid,*sol,*fu2,aux[6];
ci=allocate_integer_vector(1,n);
pr=allocate_real_vector(1,n);
aid=allocate_real_vector(1,n);
sol=allocate_real_vector(1,n);
fu2=allocate_real_vector(1,m);
jac=allocate_real_matrix(1,m+1,1,n);
itmax=fevalmax=in[5];
aux[2]=n*in[0];
tim=in[7];
reltolpar=in[1]*in[1];
abstolpar=in[2]*in[2];
abstolres=in[4]*in[4];
inrmax=in[6];
dupvec(1,n,0,pr,par);
if (m < n)
for (i=1; i<=n; i++) jac[m+1][i]=0.0;
text=4;
mit=0;
testthf=1;
res2=stap=out[5]=out[6]=out[7]=0.0;
(*funct)(m,n,par,fu2);
rn=vecvec(1,m,0,fu2,fu2);
out[3]=sqrt(rn);
feval=1;
dampingon=0;
fail=0;
it=1;
do {
out[5]=it;
(*jacobian)(m,n,par,fu2,jac);
if (!testthf) {
text=7;
fail=1;
break;
}
lsqortdec(jac,m,n,aux,aid,ci);
if (aux[3] != n) {
text=5;
fail=1;
break;
}
lsqsol(jac,m,n,aid,ci,fu2);
dupvec(1,n,0,sol,fu2);
stap=vecvec(1,n,0,sol,sol);
rho=2.0;
normx=vecvec(1,n,0,par,par);
if (stap > reltolpar*normx+abstolpar || it == 1 && stap > 0.0) {
inr=0;
do {
rho /= 2.0;
if (inr > 0) {
res1=res2;
dupvec(1,m,0,rv,fu2);
dampingon = inr > 1;
}
for (i=1; i<=n; i++) pr[i]=par[i]-sol[i]*rho;
feval++;
if (!(*funct)(m,n,pr,fu2)) {
text=6;
fail=1;
break;
}
res2=vecvec(1,m,0,fu2,fu2);
conv = inr >= inrmax;
inr++;
} while ((inr == 1) ? (dampingon || res2 >= rn) :
(!conv && (rn <= res1 || res2 < res1)));
if (fail) break;
if (conv) {
mit++;
if (mit < tim) conv=0;
} else
mit=0;
if (inr > 1) {
rho *= 2.0;
elmvec(1,n,0,par,sol,-rho);
rn=res1;
if (inr > 2) out[7]=it;
} else {
dupvec(1,n,0,par,pr);
rn=res2;
dupvec(1,m,0,rv,fu2);
}
if (rn <= abstolres) {
text=1;
itmax=it;
} else
if (conv && inrmax > 0) {
text=3;
itmax=it;
} else
dupvec(1,m,0,fu2,rv);
} else {
text=2;
rho=1.0;
itmax=it;
}
it++;
} while (it <= itmax && feval < fevalmax);
if (!fail) {
lsqinv(jac,n,aid,ci);
for (i=1; i<=n; i++) {
jjinv[i][i]=jac[i][i];
for (j=i+1; j<=n; j++) jjinv[i][j]=jjinv[j][i]=jac[i][j];
}
}
out[6]=sqrt(stap)*rho;
out[2]=sqrt(rn);
out[4]=feval;
out[1]=text;
out[8]=aux[3];
out[9]=aux[5];
free_integer_vector(ci,1);
free_real_vector(pr,1);
free_real_vector(aid,1);
free_real_vector(sol,1);
free_real_vector(fu2,1);
free_real_matrix(jac,1,m+1,1);
}
void praxis( int n, double *x, int *data, double (*funct)(double *, void *data), double *in, double *out) {
int illc,i,j,k,k2,nl,maxf,nf,kl,kt,ktm,emergency;
double s,sl,dn,dmin,fx,f1,lds,ldt,sf,df,qf1,qd0,qd1,qa,qb,qc,m2,m4,
small,vsmall,large,vlarge,scbd,ldfac,t2,macheps,reltol,
abstol,h,**v,*d,*y,*z,*q0,*q1,**a,em[8],l;
/*
* Seed random number generator
*/
#ifdef MSWIN
srand(34084320);
#else
srand48(34084320);
#endif
// for (i=0; i<8; ++i) x[i+1] = (double)data->x[i];
d=allocate_real_vector(1,n);
y=allocate_real_vector(1,n);
z=allocate_real_vector(1,n);
q0=allocate_real_vector(1,n);
q1=allocate_real_vector(1,n);
v=allocate_real_matrix(1,n,1,n);
a=allocate_real_matrix(1,n,1,n);
// heuristic numbers:
//
// If the axes may be badly scaled (which is to be avoided if
// possible), then set scbd = 10. otherwise set scbd=1.
//
// If the problem is known to be ill-conditioned, set ILLC = true.
//
// KTM is the number of iterations without improvement before the
// algorithm terminates. KTM = 4 is very cautious; usually KTM = 1
// is satisfactory.
//
macheps=in[0];
reltol=in[1];
abstol=in[2];
maxf=in[5];
h=in[6];
scbd=in[7];
ktm=in[8];
illc = in[9] < 0.0;
small=macheps*macheps;
vsmall=small*small;
large=1.0/small;
vlarge=1.0/vsmall;
m2=reltol;
m4=sqrt(m2);
srand(1);
ldfac = (illc ? 0.1 : 0.01);
kt=nl=0;
nf=1;
out[3]=qf1=fx=(*funct)(x, data);
abstol=t2=small+fabs(abstol);
dmin=small;
if (h < abstol*100.0) h=abstol*100;
ldt=h;
inimat(1,n,1,n,v,0.0);
for (i=1; i<=n; i++) v[i][i]=1.0;
d[1]=qd0=qd1=0.0;
dupvec(1,n,0,q1,x);
inivec(1,n,q0,0.0);
emergency=0;
while (1) {
sf=d[1];
d[1]=s=0.0;
praxismin(1,2,&(d[1]),&s,&fx,0,
n,x,v,&qa,&qb,&qc,qd0,qd1,q0,q1,&nf,
&nl,&fx,m2,m4,dmin,ldt,reltol,abstol,small,h,funct, data);
if (s <= 0.0) mulcol(1,n,1,1,v,v,-1.0);
if (sf <= 0.9*d[1] || 0.9*sf >= d[1]) inivec(2,n,d,0.0);
for (k=2; k<=n; k++) {
dupvec(1,n,0,y,x);
sf=fx;
illc = (illc || kt > 0);
while (1) {
kl=k;
df=0.0;
if (illc) {
/* random stop to get off resulting valley */
for (i=1; i<=n; i++) {
s=z[i]=(0.1*ldt+t2*pow(10.0,kt))*
#ifdef MSWIN
((double)(rand())/RAND_MAX-0.5);
#else
(drand48()-0.5);
#endif
elmveccol(1,n,i,x,v,s);
}
fx=(*funct)(x, data);
nf++;
}
for (k2=k; k2<=n; k2++) {
sl=fx;
s=0.0;
praxismin(k2,2,&(d[k2]),&s,&fx,0,
n,x,v,&qa,&qb,&qc,qd0,qd1,q0,q1,&nf,
&nl,&fx,m2,m4,dmin,ldt,reltol,abstol,small,h,funct, data);
s = illc ? d[k2]*(s+z[k2])*(s+z[k2]) : sl-fx;
if (df < s) {
df=s;
kl=k2;
}
}
if (!illc && df < fabs(100.0*macheps*fx))
illc=1;
else
break;
}
for (k2=1; k2<=k-1; k2++) {
s=0.0;
praxismin(k2,2,&(d[k2]),&s,&fx,0,
n,x,v,&qa,&qb,&qc,qd0,qd1,q0,q1,&nf,
&nl,&fx,m2,m4,dmin,ldt,reltol,abstol,small,h,funct, data);
}
f1=fx;
fx=sf;
lds=0.0;
for (i=1; i<=n; i++) {
sl=x[i];
x[i]=y[i];
y[i] = sl -= y[i];
lds += sl*sl;
}
lds=sqrt(lds);
if (lds > small) {
for (i=kl-1; i>=k; i--) {
for (j=1; j<=n; j++) v[j][i+1]=v[j][i];
d[i+1]=d[i];
}
d[k]=0.0;
dupcolvec(1,n,k,v,y);
mulcol(1,n,k,k,v,v,1.0/lds);
praxismin(k,4,&(d[k]),&lds,&f1,1,
n,x,v,&qa,&qb,&qc,qd0,qd1,q0,q1,&nf,
&nl,&fx,m2,m4,dmin,ldt,reltol,abstol,small,h,funct, data);
if (lds <= 0.0) {
lds = -lds;
mulcol(1,n,k,k,v,v,-1.0);
}
}
ldt *= ldfac;
if (ldt < lds) ldt=lds;
t2=m2*sqrt(vecvec(1,n,0,x,x))+abstol;
kt = (ldt > 0.5*t2) ? 0 : kt+1;
if (kt > ktm) {
out[1]=0.0;
emergency=1;
}
}
if (emergency) break;
/* quad */
s=fx;
fx=qf1;
qf1=s;
qd1=0.0;
for (i=1; i<=n; i++) {
s=x[i];
x[i]=l=q1[i];
q1[i]=s;
qd1 += (s-l)*(s-l);
}
l=qd1=sqrt(qd1);
s=0.0;
if ((qd0*qd1 > DBL_MIN) && (nl >=3*n*n)) {
praxismin(0,2,&s,&l,&qf1,1,
n,x,v,&qa,&qb,&qc,qd0,qd1,q0,q1,&nf,
&nl,&fx,m2,m4,dmin,ldt,reltol,abstol,small,h,funct, data);
qa=l*(l-qd1)/(qd0*(qd0+qd1));
qb=(l+qd0)*(qd1-l)/(qd0*qd1);
qc=l*(l+qd0)/(qd1*(qd0+qd1));
} else {
fx=qf1;
qa=qb=0.0;
qc=1.0;
}
qd0=qd1;
for (i=1; i<=n; i++) {
s=q0[i];
q0[i]=x[i];
x[i]=qa*s+qb*x[i]+qc*q1[i];
}
/* end of quad */
dn=0.0;
for (i=1; i<=n; i++) {
d[i]=1.0/sqrt(d[i]);
if (dn < d[i]) dn=d[i];
}
for (j=1; j<=n; j++) {
s=d[j]/dn;
mulcol(1,n,j,j,v,v,s);
}
if (scbd > 1.0) {
s=vlarge;
for (i=1; i<=n; i++) {
sl=z[i]=sqrt(mattam(1,n,i,i,v,v));
if (sl < m4) z[i]=m4;
if (s > sl) s=sl;
}
for (i=1; i<=n; i++) {
sl=s/z[i];
z[i]=1.0/sl;
if (z[i] > scbd) {
sl=1.0/scbd;
z[i]=scbd;
}
mulrow(1,n,i,i,v,v,sl);
}
}
for (i=1; i<=n; i++) ichrowcol(i+1,n,i,i,v);
em[0]=em[2]=macheps;
em[4]=10*n;
em[6]=vsmall;
dupmat(1,n,1,n,a,v);
if (qrisngvaldec(a,n,n,d,v,em) != 0) {
out[1]=2.0;
emergency=1;
}
if (emergency) break;
if (scbd > 1.0) {
for (i=1; i<=n; i++) mulrow(1,n,i,i,v,v,z[i]);
for (i=1; i<=n; i++) {
s=sqrt(tammat(1,n,i,i,v,v));
d[i] *= s;
s=1.0/s;
mulcol(1,n,i,i,v,v,s);
}
}
for (i=1; i<=n; i++) {
s=dn*d[i];
d[i] = (s > large) ? vsmall :
((s < small) ? vlarge : 1.0/(s*s));
}
/* sort */
for (i=1; i<=n-1; i++) {
k=i;
s=d[i];
for (j=i+1; j<=n; j++)
if (d[j] > s) {
k=j;
s=d[j];
}
if (k > i) {
d[k]=d[i];
d[i]=s;
for (j=1; j<=n; j++) {
s=v[j][i];
v[j][i]=v[j][k];
v[j][k]=s;
}
}
}
/* end of sort */
dmin=d[n];
if (dmin < small) dmin=small;
illc = (m2*d[1]) > dmin;
if (nf >= maxf) {
out[1]=1.0;
break;
}
}
out[2]=fx;
out[4]=nf;
out[5]=nl;
out[6]=ldt;
free_real_vector(d,1);
free_real_vector(y,1);
free_real_vector(z,1);
free_real_vector(q0,1);
free_real_vector(q1,1);
free_real_matrix(v,1,n,1);
free_real_matrix(a,1,n,1);
// for (i=0; i<40; ++i) data->x[i] = (double)x[i+1];
}
int peidefunct(int nrow, int ncol, real_t par[], real_t res[],
int n, int m, int nobs, int *nbp, int first, int *sec,
int *max, int *nis, real_t eps1, int weight, int bp[],
real_t save[], real_t ymax[], real_t y[], real_t **yp,
real_t **fy, real_t **fp, int cobs[], real_t tobs[],
real_t obs[], real_t in[], real_t aux[], int clean,
int (*deriv)(int,int,real_t [],real_t [],real_t,real_t []),
int (*jacdfdy)(int,int,real_t [],real_t [],real_t,real_t **),
int (*jacdfdp)(int,int,real_t [],real_t [],real_t,real_t **),
void (*callystart)(int,int,real_t [],real_t [],real_t[]),
void (*monitor)(int,int,int,real_t [],real_t [],int,int))
{
/* this function is internally used by PEIDE */
void peidereset(int, int, real_t, real_t, real_t, real_t, real_t [],
real_t [], real_t *, real_t *, real_t *, int *);
void peideorder(int, int, real_t, real_t [], real_t [],
real_t *, real_t *, real_t *, real_t *, real_t *, int *);
void peidestep(int, int, int, real_t, real_t, real_t, real_t,
real_t [], real_t [], real_t [], real_t [], int *, real_t *);
real_t peideinterpol(int, int, int, real_t, real_t []);
int l,k,knew,fails,same,kpold,n6,nnpar,j5n,cobsii,*p,evaluate,
evaluated,decompose,conv,extra,npar,i,j,jj,ii;
real_t xold,hold,a0,tolup,tol,toldwn,tolconv,h,ch,chnew,error,
dfi,tobsdif,a[6],*delta,*lastdelta,*df,*y0,**jacob,xend,
hmax,hmin,eps,s,aa,x,t,c;
p=allocate_integer_vector(1,n);
delta=allocate_real_vector(1,n);
lastdelta=allocate_real_vector(1,n);
df=allocate_real_vector(1,n);
y0=allocate_real_vector(1,n);
jacob=allocate_real_matrix(1,n,1,n);
if (*sec) {
*sec=0;
goto Finish;
}
xend=tobs[nobs];
eps=in[2];
npar=m;
extra=(*nis)=0;
ii=1;
jj = (*nbp == 0) ? 0 : 1;
n6=n*6;
inivec(-3,-1,save,0.0);
inivec(n6+1,(6+m)*n,y,0.0);
inimat(1,nobs+(*nbp),1,m+(*nbp),yp,0.0);
t=tobs[1];
x=tobs[0];
(*callystart)(n,m,par,y,ymax);
hmax=tobs[1]-tobs[0];
hmin=hmax*in[1];
/* evaluate jacobian */
evaluate=0;
decompose=evaluated=1;
if (!(*jacdfdy)(n,m,par,y,x,fy)) {
save[-3]=4.0;
goto Finish;
}
nnpar=n*npar;
Newstart:
k=1;
kpold=0;
same=2;
peideorder(n,k,eps,a,save,&tol,&tolup,&toldwn,&tolconv,
&a0,&decompose);
if (!(*deriv)(n,m,par,y,x,df)) {
save[-3]=3.0;
goto Finish;
}
s=FLT_MIN;
for (i=1; i<=n; i++) {
aa=matvec(1,n,i,fy,df)/ymax[i];
s += aa*aa;
}
h=sqrt(2.0*eps/sqrt(s));
if (h > hmax)
h=hmax;
else
if (h < hmin) h=hmin;
xold=x;
hold=h;
ch=1.0;
for (i=1; i<=n; i++) {
save[i]=y[i];
save[n+i]=y[n+i]=df[i]*h;
}
fails=0;
while (x < xend) {
if (x+h <= xend)
x += h;
else {
h=xend-x;
x=xend;
ch=h/hold;
c=1.0;
for (j=n; j<=k*n; j += n) {
c *= ch;
for (i=j+1; i<=j+n; i++) y[i] *= c;
}
same = (same < 3) ? 3 : same+1;
}
/* prediction */
for (l=1; l<=n; l++) {
for (i=l; i<=(k-1)*n+l; i += n)
for (j=(k-1)*n+l; j>=i; j -= n) y[j] += y[j+n];
delta[l]=0.0;
}
evaluated=0;
/* correction and estimation local error */
for (l=1; l<=3; l++) {
if (!(*deriv)(n,m,par,y,x,df)) {
save[-3]=3;
goto Finish;
}
for (i=1; i<=n; i++) df[i]=df[i]*h-y[n+i];
if (evaluate) {
/* evaluate jacobian */
evaluate=0;
decompose=evaluated=1;
if (!(*jacdfdy)(n,m,par,y,x,fy)) {
save[-3]=4.0;
goto Finish;
}
}
if (decompose) {
/* decompose jacobian */
decompose=0;
c = -a0*h;
for (j=1; j<=n; j++) {
for (i=1; i<=n; i++) jacob[i][j]=fy[i][j]*c;
jacob[j][j] += 1.0;
}
dec(jacob,n,aux,p);
}
sol(jacob,n,p,df);
conv=1;
for (i=1; i<=n; i++) {
dfi=df[i];
y[i] += a0*dfi;
y[n+i] += dfi;
delta[i] += dfi;
conv=(conv && (fabs(dfi) < tolconv*ymax[i]));
}
if (conv) {
s=FLT_MIN;
for (i=1; i<=n; i++) {
aa=delta[i]/ymax[i];
s += aa*aa;
}
error=s;
break;
}
}
/* acceptance or rejection */
if (!conv) {
if (!evaluated)
evaluate=1;
else {
ch /= 4.0;
if (h < 4.0*hmin) {
save[-1] += 10.0;
hmin /= 10.0;
if (save[-1] > 40.0) goto Finish;
}
}
peidereset(n,k,hmin,hmax,hold,xold,y,save,&ch,&x,
&h,&decompose);
} else if (error > tol) {
fails++;
if (h > 1.1*hmin) {
if (fails > 2) {
peidereset(n,k,hmin,hmax,hold,xold,y,save,&ch,&x,
&h,&decompose);
goto Newstart;
} else {
/* calculate step and order */
peidestep(n,k,fails,tolup,toldwn,tol,error,delta,
lastdelta,y,ymax,&knew,&chnew);
if (knew != k) {
k=knew;
peideorder(n,k,eps,a,save,&tol,&tolup,
&toldwn,&tolconv,&a0,&decompose);
}
ch *= chnew;
peidereset(n,k,hmin,hmax,hold,xold,y,save,&ch,&x,
&h,&decompose);
}
} else {
if (k == 1) {
/* violate eps criterion */
save[-2] += 1.0;
same=4;
goto Errortestok;
}
k=1;
peidereset(n,k,hmin,hmax,hold,xold,y,save,&ch,&x,
&h,&decompose);
peideorder(n,k,eps,a,save,&tol,&tolup,
&toldwn,&tolconv,&a0,&decompose);
same=2;
}
} else {
Errortestok:
fails=0;
for (i=1; i<=n; i++) {
c=delta[i];
for (l=2; l<=k; l++) y[l*n+i] += a[l]*c;
if (fabs(y[i]) > ymax[i]) ymax[i]=fabs(y[i]);
}
same--;
if (same == 1)
dupvec(1,n,0,lastdelta,delta);
else if (same == 0) {
/* calculate step and order */
peidestep(n,k,fails,tolup,toldwn,tol,error,delta,
lastdelta,y,ymax,&knew,&chnew);
if (chnew > 1.1) {
if (k != knew) {
if (knew > k)
mulvec(knew*n+1,knew*n+n,-knew*n,y,delta,
a[k]/knew);
k=knew;
peideorder(n,k,eps,a,save,&tol,&tolup,
&toldwn,&tolconv,&a0,&decompose);
}
same=k+1;
if (chnew*h > hmax) chnew=hmax/h;
h *= chnew;
c=1.0;
for (j=n; j<=k*n; j += n) {
c *= chnew;
mulvec(j+1,j+n,0,y,y,c);
}
decompose=1;
} else
same=10;
}
(*nis)++;
/* start of an integration step of yp */
if (clean) {
hold=h;
xold=x;
kpold=k;
ch=1.0;
dupvec(1,k*n+n,0,save,y);
} else {
if (h != hold) {
ch=h/hold;
c=1.0;
for (j=n6+nnpar; j<=kpold*nnpar+n6; j += nnpar) {
c *= ch;
for (i=j+1; i<=j+nnpar; i++) y[i] *= c;
}
hold=h;
}
if (k > kpold)
inivec(n6+k*nnpar+1,n6+k*nnpar+nnpar,y,0.0);
xold=x;
kpold=k;
ch=1.0;
dupvec(1,k*n+n,0,save,y);
/* evaluate jacobian */
evaluate=0;
decompose=evaluated=1;
if (!(*jacdfdy)(n,m,par,y,x,fy)) {
save[-3]=4.0;
goto Finish;
}
/* decompose jacobian */
decompose=0;
c = -a0*h;
for (j=1; j<=n; j++) {
for (i=1; i<=n; i++) jacob[i][j]=fy[i][j]*c;
jacob[j][j] += 1.0;
}
dec(jacob,n,aux,p);
if (!(*jacdfdp)(n,m,par,y,x,fp)) {
save[-3]=5.0;
goto Finish;
}
if (npar > m) inimat(1,n,m+1,npar,fp,0.0);
/* prediction */
for (l=0; l<=k-1; l++)
for (j=k-1; j>=l; j--)
elmvec(j*nnpar+n6+1,j*nnpar+n6+nnpar,nnpar,
y,y,1.0);
/* correction */
for (j=1; j<=npar; j++) {
j5n=(j+5)*n;
dupvec(1,n,j5n,y0,y);
for (i=1; i<=n; i++)
df[i]=h*(fp[i][j]+matvec(1,n,i,fy,y0))-
y[nnpar+j5n+i];
sol(jacob,n,p,df);
for (l=0; l<=k; l++) {
i=l*nnpar+j5n;
elmvec(i+1,i+n,-i,y,df,a[l]);
}
}
}
while (x >= t) {
/* calculate a row of the jacobian matrix and an
element of the residual vector */
tobsdif=(tobs[ii]-x)/h;
cobsii=cobs[ii];
res[ii]=peideinterpol(cobsii,n,k,tobsdif,y)-obs[ii];
if (!clean) {
for (i=1; i<=npar; i++)
yp[ii][i]=peideinterpol(cobsii+(i+5)*n,nnpar,k,
tobsdif,y);
/* introducing break-points */
if (bp[jj] != ii) {
} else if (first && fabs(res[ii]) < eps1) {
(*nbp)--;
for (i=jj; i<=(*nbp); i++) bp[i]=bp[i+1];
bp[*nbp+1]=0;
} else {
extra++;
if (first) par[m+jj]=obs[ii];
/* introducing a jacobian row and a residual
vector element for continuity requirements */
yp[nobs+jj][m+jj] = -weight;
mulrow(1,npar,nobs+jj,ii,yp,yp,weight);
res[nobs+jj]=weight*(res[ii]+obs[ii]-par[m+jj]);
}
}
if (ii == nobs)
goto Finish;
else {
t=tobs[ii+1];
if (bp[jj] == ii && jj < *nbp) jj++;
hmax=t-tobs[ii];
hmin=hmax*in[1];
ii++;
}
}
/* break-points introduce new initial values for y & yp */
if (extra > 0) {
for (i=1; i<=n; i++) {
y[i]=peideinterpol(i,n,k,tobsdif,y);
for (j=1; j<=npar; j++)
y[i+(j+5)*n]=peideinterpol(i+(j+5)*n,nnpar,
k,tobsdif,y);
}
for (l=1; l<=extra; l++) {
cobsii=cobs[bp[npar-m+l]];
y[cobsii]=par[npar+l];
for (i=1; i<=npar+extra; i++) y[cobsii+(5+i)*n]=0.0;
inivec(1+nnpar+(l+5)*n,nnpar+(l+6)*n,y,0.0);
y[cobsii+(5+npar+l)*n]=1.0;
}
npar += extra;
extra=0;
x=tobs[ii-1];
/* evaluate jacobian */
evaluate=0;
decompose=evaluated=1;
if (!(*jacdfdy)(n,m,par,y,x,fy)) {
save[-3]=4.0;
goto Finish;
}
nnpar=n*npar;
goto Newstart;
}
}
}
Finish:
if (save[-2] > *max) *max=save[-2];
if (!first) (*monitor)(1,ncol,nrow,par,res,weight,*nis);
free_integer_vector(p,1);
free_real_vector(delta,1);
free_real_vector(lastdelta,1);
free_real_vector(df,1);
free_real_vector(y0,1);
free_real_matrix(jacob,1,n,1);
return (save[-1] <= 40.0 && save[-3] == 0.0);
}
void peide(int n, int m, int nobs, int *nbp, real_t par[],
real_t res[], int bp[], real_t **jtjinv,
real_t in[], real_t out[],
int (*deriv)(int,int,real_t [],real_t [],real_t,real_t []),
int (*jacdfdy)(int,int,real_t [],real_t [],real_t,real_t **),
int (*jacdfdp)(int,int,real_t [],real_t [],real_t,real_t **),
void (*callystart)(int,int,real_t [],real_t [],real_t[]),
void (*data)(int,real_t [],real_t [],int[]),
void (*monitor)(int,int,int,real_t [],real_t [],int,int))
{
int i,j,weight,ncol,nrow,away,max,nfe,nis,*cobs,
first,sec,clean,nbpold,maxfe,fe,it,err,emergency;
real_t eps1,res1,in3,in4,fac3,fac4,aux[4],*obs,*save,*tobs,
**yp,*ymax,*y,**fy,**fp,w,**aid,temp,
vv,ww,w2,mu,res2,fpar,fparpres,lambda,lambdamin,p,pw,
reltolres,abstolres,em[8],*val,*b,*bb,*parpres,**jaco;
static real_t save1[35]={1.0, 1.0, 9.0, 4.0, 0.0, 2.0/3.0, 1.0,
1.0/3.0, 36.0, 20.25, 1.0, 6.0/11.0, 1.0, 6.0/11.0,
1.0/11.0, 84.028, 53.778, 0.25, 0.48, 1.0, 0.7, 0.2,
0.02, 156.25, 108.51, 0.027778, 120.0/274.0, 1.0,
225.0/274.0, 85.0/274.0, 15.0/274.0, 1.0/274.0, 0.0,
187.69, 0.0047361};
nbpold=(*nbp);
cobs=allocate_integer_vector(1,nobs);
obs=allocate_real_vector(1,nobs);
save=allocate_real_vector(-38,6*n);
tobs=allocate_real_vector(0,nobs);
ymax=allocate_real_vector(1,n);
y=allocate_real_vector(1,6*n*(nbpold+m+1));
yp=allocate_real_matrix(1,nbpold+nobs,1,nbpold+m);
fy=allocate_real_matrix(1,n,1,n);
fp=allocate_real_matrix(1,n,1,m+nbpold);
aid=allocate_real_matrix(1,m+nbpold,1,m+nbpold);
for (i=0; i<=34; i++) save[-38+i]=save1[i];
(*data)(nobs,tobs,obs,cobs);
weight=1;
first=sec=0;
clean=(*nbp > 0);
aux[2]=FLT_EPSILON;
eps1=1.0e10;
out[1]=0.0;
bp[0]=max=0;
/* smooth integration without break-points */
if (!peidefunct(nobs,m,par,res,
n,m,nobs,nbp,first,&sec,&max,&nis,eps1,weight,bp,
save,ymax,y,yp,fy,fp,cobs,tobs,obs,in,aux,clean,deriv,
jacdfdy,jacdfdp,callystart,monitor)) goto Escape;
res1=sqrt(vecvec(1,nobs,0,res,res));
nfe=1;
if (in[5] == 1.0) {
out[1]=1.0;
goto Escape;
}
if (clean) {
first=1;
clean=0;
fac3=sqrt(sqrt(in[3]/res1));
fac4=sqrt(sqrt(in[4]/res1));
eps1=res1*fac4;
if (!peidefunct(nobs,m,par,res,
n,m,nobs,nbp,first,&sec,&max,&nis,eps1,weight,bp,
save,ymax,y,yp,fy,fp,cobs,tobs,obs,in,aux,clean,deriv,
jacdfdy,jacdfdp,callystart,monitor)) goto Escape;
first=0;
} else
nfe=0;
ncol=m+(*nbp);
nrow=nobs+(*nbp);
sec=1;
in3=in[3];
in4=in[4];
in[3]=res1;
weight=away=0;
out[4]=out[5]=w=0.0;
temp=sqrt(weight)+1.0;
weight=temp*temp;
while (weight != 16 && *nbp > 0) {
if (away == 0 && w != 0.0) {
/* if no break-points were omitted then one function
function evaluation is saved */
w=weight/w;
for (i=nobs+1; i<=nrow; i++) {
for (j=1; j<=ncol; j++) yp[i][j] *= w;
res[i] *= w;
}
sec=1;
nfe--;
}
in[3] *= fac3*weight;
in[4]=eps1;
(*monitor)(2,ncol,nrow,par,res,weight,nis);
/* marquardt's method */
val=allocate_real_vector(1,ncol);
b=allocate_real_vector(1,ncol);
bb=allocate_real_vector(1,ncol);
parpres=allocate_real_vector(1,ncol);
jaco=allocate_real_matrix(1,nrow,1,ncol);
vv=10.0;
w2=0.5;
mu=0.01;
ww = (in[6] < 1.0e-7) ? 1.0e-8 : 1.0e-1*in[6];
em[0]=em[2]=em[6]=in[0];
em[4]=10*ncol;
reltolres=in[3];
abstolres=in[4]*in[4];
maxfe=in[5];
err=0;
fe=it=1;
p=fpar=res2=0.0;
pw = -log(ww*in[0])/2.30;
if (!peidefunct(nrow,ncol,par,res,
n,m,nobs,nbp,first,&sec,&max,&nis,eps1,
weight,bp,save,ymax,y,yp,fy,fp,cobs,tobs,obs,
in,aux,clean,deriv,jacdfdy,jacdfdp,
callystart,monitor))
err=3;
else {
fpar=vecvec(1,nrow,0,res,res);
out[3]=sqrt(fpar);
emergency=0;
it=1;
do {
dupmat(1,nrow,1,ncol,jaco,yp);
i=qrisngvaldec(jaco,nrow,ncol,val,aid,em);
if (it == 1)
lambda=in[6]*vecvec(1,ncol,0,val,val);
else
if (p == 0.0) lambda *= w2;
for (i=1; i<=ncol; i++)
b[i]=val[i]*tamvec(1,nrow,i,jaco,res);
while (1) {
for (i=1; i<=ncol; i++)
bb[i]=b[i]/(val[i]*val[i]+lambda);
for (i=1; i<=ncol; i++)
parpres[i]=par[i]-matvec(1,ncol,i,aid,bb);
fe++;
if (fe >= maxfe)
err=1;
else
if (!peidefunct(nrow,ncol,parpres,res,
n,m,nobs,nbp,first,&sec,&max,&nis,
eps1,weight,bp,save,ymax,y,yp,fy,fp,
cobs,tobs,obs,in,aux,clean,deriv,
jacdfdy,jacdfdp,callystart,monitor))
err=2;
if (err != 0) {
emergency=1;
break;
}
fparpres=vecvec(1,nrow,0,res,res);
res2=fpar-fparpres;
if (res2 < mu*vecvec(1,ncol,0,b,bb)) {
p += 1.0;
lambda *= vv;
if (p == 1.0) {
lambdamin=ww*vecvec(1,ncol,0,val,val);
if (lambda < lambdamin) lambda=lambdamin;
}
if (p >= pw) {
err=4;
emergency=1;
break;
}
} else {
dupvec(1,ncol,0,par,parpres);
fpar=fparpres;
break;
}
}
if (emergency) break;
it++;
} while (fpar>abstolres &&
res2>reltolres*fpar+abstolres);
for (i=1; i<=ncol; i++)
mulcol(1,ncol,i,i,jaco,aid,1.0/(val[i]+in[0]));
for (i=1; i<=ncol; i++)
for (j=1; j<=i; j++)
aid[i][j]=aid[j][i]=mattam(1,ncol,i,j,jaco,jaco);
lambda=lambdamin=val[1];
for (i=2; i<=ncol; i++)
if (val[i] > lambda)
lambda=val[i];
else
if (val[i] < lambdamin) lambdamin=val[i];
temp=lambda/(lambdamin+in[0]);
out[7]=temp*temp;
out[2]=sqrt(fpar);
out[6]=sqrt(res2+fpar)-out[2];
}
out[4]=fe;
out[5]=it-1;
out[1]=err;
free_real_vector(val,1);
free_real_vector(b,1);
free_real_vector(bb,1);
free_real_vector(parpres,1);
free_real_matrix(jaco,1,nrow,1);
if (out[1] > 0.0) goto Escape;
/* the relative starting value of lambda is adjusted
to the last value of lambda used */
away=out[4]-out[5]-1.0;
in[6] *= pow(5.0,away)*pow(2.0,away-out[5]);
nfe += out[4];
w=weight;
temp=sqrt(weight)+1.0;
eps1=temp*temp*in[4]*fac4;
away=0;
/* omit useless break-points */
for (j=1; j<=(*nbp); j++)
if (fabs(obs[bp[j]]+res[bp[j]]-par[j+m]) < eps1) {
(*nbp)--;
for (i=j; i<=(*nbp); i++) bp[i]=bp[i+1];
dupvec(j+m,(*nbp)+m,1,par,par);
j--;
away++;
bp[*nbp+1]=0;
}
ncol -= away;
nrow -= away;
temp=sqrt(weight)+1.0;
weight=temp*temp;
}
in[3]=in3;
in[4]=in4;
*nbp=0;
weight=1;
(*monitor)(2,m,nobs,par,res,weight,nis);
/* marquardt's method */
val=allocate_real_vector(1,m);
b=allocate_real_vector(1,m);
bb=allocate_real_vector(1,m);
parpres=allocate_real_vector(1,m);
jaco=allocate_real_matrix(1,nobs,1,m);
vv=10.0;
w2=0.5;
mu=0.01;
ww = (in[6] < 1.0e-7) ? 1.0e-8 : 1.0e-1*in[6];
em[0]=em[2]=em[6]=in[0];
em[4]=10*m;
reltolres=in[3];
abstolres=in[4]*in[4];
maxfe=in[5];
err=0;
fe=it=1;
p=fpar=res2=0.0;
pw = -log(ww*in[0])/2.30;
if (!peidefunct(nobs,m,par,res,
n,m,nobs,nbp,first,&sec,&max,&nis,eps1,weight,bp,
save,ymax,y,yp,fy,fp,cobs,tobs,obs,in,aux,clean,
deriv,jacdfdy,jacdfdp,callystart,monitor))
err=3;
else {
fpar=vecvec(1,nobs,0,res,res);
out[3]=sqrt(fpar);
emergency=0;
it=1;
do {
dupmat(1,nobs,1,m,jaco,yp);
i=qrisngvaldec(jaco,nobs,m,val,jtjinv,em);
if (it == 1)
lambda=in[6]*vecvec(1,m,0,val,val);
else
if (p == 0.0) lambda *= w2;
for (i=1; i<=m; i++)
b[i]=val[i]*tamvec(1,nobs,i,jaco,res);
while (1) {
for (i=1; i<=m; i++)
bb[i]=b[i]/(val[i]*val[i]+lambda);
for (i=1; i<=m; i++)
parpres[i]=par[i]-matvec(1,m,i,jtjinv,bb);
fe++;
if (fe >= maxfe)
err=1;
else
if (!peidefunct(nobs,m,parpres,res,
n,m,nobs,nbp,first,&sec,&max,&nis,eps1,
weight,bp,save,ymax,y,yp,fy,fp,cobs,tobs,
obs,in,aux,clean,deriv,jacdfdy,jacdfdp,
callystart,monitor))
err=2;
if (err != 0) {
emergency=1;
break;
}
fparpres=vecvec(1,nobs,0,res,res);
res2=fpar-fparpres;
if (res2 < mu*vecvec(1,m,0,b,bb)) {
p += 1.0;
lambda *= vv;
if (p == 1.0) {
lambdamin=ww*vecvec(1,m,0,val,val);
if (lambda < lambdamin) lambda=lambdamin;
}
if (p >= pw) {
err=4;
emergency=1;
break;
}
} else {
dupvec(1,m,0,par,parpres);
fpar=fparpres;
break;
}
}
if (emergency) break;
it++;
} while (fpar>abstolres && res2>reltolres*fpar+abstolres);
for (i=1; i<=m; i++)
mulcol(1,m,i,i,jaco,jtjinv,1.0/(val[i]+in[0]));
for (i=1; i<=m; i++)
for (j=1; j<=i; j++)
jtjinv[i][j]=jtjinv[j][i]=mattam(1,m,i,j,jaco,jaco);
lambda=lambdamin=val[1];
for (i=2; i<=m; i++)
if (val[i] > lambda)
lambda=val[i];
else
if (val[i] < lambdamin) lambdamin=val[i];
temp=lambda/(lambdamin+in[0]);
out[7]=temp*temp;
out[2]=sqrt(fpar);
out[6]=sqrt(res2+fpar)-out[2];
}
out[4]=fe;
out[5]=it-1;
out[1]=err;
free_real_vector(val,1);
free_real_vector(b,1);
free_real_vector(bb,1);
free_real_vector(parpres,1);
free_real_matrix(jaco,1,nobs,1);
nfe += out[4];
Escape:
if (out[1] == 3.0)
out[1]=2.0;
else
if (out[1] == 4.0) out[1]=6.0;
if (save[-3] != 0.0) out[1]=save[-3];
out[3]=res1;
out[4]=nfe;
out[5]=max;
free_integer_vector(cobs,1);
free_real_vector(obs,1);
free_real_vector(save,-38);
free_real_vector(tobs,0);
free_real_vector(ymax,1);
free_real_vector(y,1);
free_real_matrix(yp,1,nbpold+nobs,1);
free_real_matrix(fy,1,n,1);
free_real_matrix(fp,1,n,1);
free_real_matrix(aid,1,m+nbpold,1);
}
real_t flemin(int n, real_t x[], real_t g[], real_t h[],
real_t (*funct)(int, real_t[], real_t[]),
real_t in[], real_t out[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
real_t vecvec(int, int, int, real_t [], real_t []);
void elmvec(int, int, int, real_t [], real_t [], real_t);
real_t symmatvec(int, int, int, real_t [], real_t []);
void inivec(int, int, real_t [], real_t);
void inisymd(int, int, int, real_t [], real_t);
void mulvec(int, int, int, real_t [], real_t [], real_t);
void dupvec(int, int, int, real_t [], real_t []);
void linemin(int, real_t [], real_t [], real_t, real_t *, real_t [],
real_t (*)(int, real_t[], real_t[]), real_t, real_t *,
real_t, real_t *, int *, int, real_t []);
void davupd(real_t [], int, real_t [], real_t [], real_t, real_t);
void fleupd(real_t [], int, real_t [], real_t [], real_t, real_t);
int i,it,cntl,evl,evlmax;
real_t f,f0,fmin,mu,dg,dg0,nrmdelta,alfa,reltol,abstol,eps,tolg,
aid,*v,*delta,*s;
v=allocate_real_vector(1,n);
delta=allocate_real_vector(1,n);
s=allocate_real_vector(1,n);
reltol=in[1];
abstol=in[2];
mu=in[3];
tolg=in[4];
fmin=in[5];
alfa=in[6];
evlmax=in[7];
out[4]=0.0;
it=0;
f=(*funct)(n,x,g);
evl=1;
cntl=0;
if (alfa > 0.0) {
inivec(1,(n*(n+1))/2,h,0.0);
inisymd(1,n,0,h,alfa);
}
for (i=1; i<=n; i++) delta[i] = -symmatvec(1,n,i,h,g);
dg=sqrt(vecvec(1,n,0,g,g));
nrmdelta=sqrt(vecvec(1,n,0,delta,delta));
eps=sqrt(vecvec(1,n,0,x,x))*reltol+abstol;
dg0=vecvec(1,n,0,delta,g);
it++;
while ((nrmdelta > eps || dg > tolg) && (evl < evlmax)) {
dupvec(1,n,0,s,x);
dupvec(1,n,0,v,g);
if (it >= n)
alfa=1.0;
else {
if (it != 1)
alfa /= nrmdelta;
else {
alfa=2.0*(fmin-f)/dg0;
if (alfa > 1.0) alfa=1.0;
}
}
elmvec(1,n,0,x,delta,alfa);
f0=f;
f=(*funct)(n,x,g);
evl++;
dg=vecvec(1,n,0,delta,g);
if (it == 1 || f0-f < -mu*dg0*alfa) {
/* line minimization */
i=evlmax-evl;
cntl++;
linemin(n,s,delta,nrmdelta,&alfa,g,funct,f0,&f,
dg0,&dg,&i,0,in);
evl += i;
dupvec(1,n,0,x,s);
}
if (alfa != 1.0) mulvec(1,n,0,delta,delta,alfa);
mulvec(1,n,0,v,v,-1.0);
elmvec(1,n,0,v,g,1.0);
for (i=1; i<=n; i++) s[i]=symmatvec(1,n,i,h,v);
aid=vecvec(1,n,0,v,s);
dg=(dg-dg0)*alfa;
if (dg > 0.0)
if (dg >= aid)
fleupd(h,n,delta,s,1.0/dg,(1.0+aid/dg)/dg);
else
davupd(h,n,delta,s,1.0/dg,1.0/aid);
for (i=1; i<=n; i++) delta[i] = -symmatvec(1,n,i,h,g);
alfa *= nrmdelta;
nrmdelta=sqrt(vecvec(1,n,0,delta,delta));
eps=sqrt(vecvec(1,n,0,x,x))*reltol+abstol;
dg=sqrt(vecvec(1,n,0,g,g));
dg0=vecvec(1,n,0,delta,g);
if (dg0 > 0.0) {
out[4] = -1.0;
break;
}
it++;
}
out[0]=nrmdelta;
out[1]=dg;
out[2]=evl;
out[3]=cntl;
free_real_vector(v,1);
free_real_vector(delta,1);
free_real_vector(s,1);
return f;
}
void ark(real_t *t, real_t *te, int *m0, int *m, real_t u[],
void (*derivative)(int *, int *, real_t *, real_t[]),
real_t data[],
void (*out)(int *, int *, real_t *, real_t *, real_t [],
real_t []))
{
real_t *allocate_real_vector(int, int);
real_t **allocate_real_matrix(int, int, int, int);
void free_real_vector(real_t *, int);
void free_real_matrix(real_t **, int, int, int);
void inivec(int, int, real_t [], real_t);
void mulvec(int, int, int, real_t [], real_t [], real_t);
void dupvec(int, int, int, real_t [], real_t []);
real_t vecvec(int, int, int, real_t [], real_t []);
void elmvec(int, int, int, real_t [], real_t [], real_t);
void decsol(real_t **, int, real_t [], real_t []);
real_t arkmui(int, int, int, real_t []);
real_t arklabda(int, int, int, int, real_t []);
static real_t th1[8] = {1.0, 0.5, 1.0/6.0, 1.0/3.0, 1.0/24.0,
1.0/12.0, 0.125, 0.25};
static real_t ec0,ec1,ec2,tau0,tau1,tau2,taus,t2;
int p,n,q,start,step1,last,i,j,k,l,n1,m00;
real_t thetanm1,tau,betan,qinv,eta,*mu,*lambda,*thetha,*ro,*r,
**alfa,th[9],aux[4],s,ss,theta0,tauacc,taustab,
aa,bb,cc,ec,mt,lt;
n=data[1];
m00=(*m0);
mu=allocate_real_vector(1,n);
lambda=allocate_real_vector(1,n);
thetha=allocate_real_vector(0,n);
ro=allocate_real_vector(m00,*m);
r=allocate_real_vector(m00,*m);
alfa=allocate_real_matrix(1,8,1,n+1);
p=data[2];
ec1=ec2=0.0;
betan=data[3];
thetanm1 = (p == 3) ? 0.75 : 1.0;
theta0=1.0-thetanm1;
s=1.0;
for (j=n-1; j>=1; j--) {
s = -s*theta0+data[n+10-j];
mu[j]=data[n+11-j]/s;
lambda[j]=mu[j]-theta0;
}
for (i=1; i<=8; i++)
for (j=0; j<=n; j++)
if (i == 1) alfa[i][j+1]=1.0;
else if (j == 0) alfa[i][j+1]=0.0;
else if (i == 2 || i == 4 || i == 8)
alfa[i][j+1]=pow(arkmui(j,n,p,lambda),(i+2)/3);
else if ((i == 3 || i == 6) && j > 1) {
s=0.0;
for (l=1; l<=j-1; l++)
s += arklabda(j,l,n,p,lambda)*
pow(arkmui(l,n,p,lambda),i/3);
alfa[i][j+1]=s;
}
else if (i == 5 && j > 2) {
s=0.0;
for (l=2; l<=j-1; l++) {
ss=0.0;
for (k=1; k<=l-1; k++)
ss += arklabda(l,k,n,p,lambda)*
arkmui(k,n,p,lambda);
s += arklabda(j,l,n,p,lambda)*ss;
}
alfa[i][j+1]=s;
}
else if (i == 7 && j > 1) {
s=0.0;
for (l=1; l<=j-1; l++)
s += arklabda(j,l,n,p,lambda)*arkmui(l,n,p,lambda);
alfa[i][j+1]=s*arkmui(j,n,p,lambda);
}
else alfa[i][j+1]=0.0;
n1 = ((n < 4) ? n+1 : ((n < 7) ? 4 : 8));
for (i=1; i<=8; i++) th[i]=th1[i-1];
if (p == 3 && n < 7) th[1]=th[2]=0.0;
aux[2]=FLT_EPSILON;
decsol(alfa,n1,aux,th);
inivec(0,n,thetha,0.0);
dupvec(0,n1-1,1,thetha,th);
if (!(p == 3 && n < 7)) {
thetha[0] -= theta0;
thetha[n-1] -= thetanm1;
q=p+1;
} else
q=3;
qinv=1.0/q;
start=(data[8] == 0.0);
data[10]=0.0;
last=0;
dupvec(*m0,*m,0,r,u);
(*derivative)(m0,m,t,r);
do {
/* stepsize */
eta=sqrt(vecvec(*m0,*m,0,u,u))*data[7]+data[6];
if (eta > 0.0) {
if (start) {
if (data[8] == 0) {
tauacc=data[5];
step1=1;
} else
if (step1) {
tauacc=pow(eta/ec2,qinv);
if (tauacc > 10.0*tau2)
tauacc=10.0*tau2;
else
step1=0;
} else {
bb=(ec2-ec1)/tau1;
cc = -bb*t2+ec2;
ec=bb*(*t)+cc;
tauacc = (ec < 0.0) ? tau2 : pow(eta/ec,qinv);
start=0;
}
} else {
aa=((ec0-ec1)/tau0+(ec2-ec1)/tau1)/(tau1+tau0);
bb=(ec2-ec1)/tau1-(2.0*t2-tau1)*aa;
cc = -(aa*t2+bb)*t2+ec2;
ec=(aa*(*t)+bb)*(*t)+cc;
tauacc = ((ec < 0.0) ? taus : pow(eta/ec,qinv));
if (tauacc > 2.0*taus) tauacc=2.0*taus;
if (tauacc < taus/2.0) tauacc=taus/2.0;
}
} else
tauacc=data[5];
if (tauacc < data[5]) tauacc=data[5];
taustab=betan/data[4];
if (taustab < data[5]) {
data[10]=1.0;
break;
}
tau = ((tauacc > taustab) ? taustab : tauacc);
taus=tau;
if (tau >= (*te)-(*t)) {
tau=(*te)-(*t);
last=1;
}
tau0=tau1;
tau1=tau2;
tau2=tau;
/* difference scheme */
mulvec(*m0,*m,0,ro,r,thetha[0]);
if (p == 3) elmvec(*m0,*m,0,u,r,0.25*tau);
for (i=1; i<=n-1; i++) {
mt=mu[i]*tau;
lt=lambda[i]*tau;
for (j=(*m0); j<=(*m); j++) r[j]=lt*r[j]+u[j];
s=(*t)+mt;
(*derivative)(m0,m,&s,r);
if (thetha[i] != 0.0) elmvec(*m0,*m,0,ro,r,thetha[i]);
if (i == n) {
data[9]=sqrt(vecvec(*m0,*m,0,ro,ro))*tau;
ec0=ec1;
ec1=ec2;
ec2=data[9]/pow(tau,q);
}
}
elmvec(*m0,*m,0,u,r,thetanm1*tau);
dupvec(*m0,*m,0,r,u);
s=(*t)+tau;
(*derivative)(m0,m,&s,r);
if (thetha[n] != 0.0) elmvec(*m0,*m,0,ro,r,thetha[n]);
data[9]=sqrt(vecvec(*m0,*m,0,ro,ro))*tau;
ec0=ec1;
ec1=ec2;
ec2=data[9]/pow(tau,q);
t2=(*t);
if (last) {
last=0;
(*t)=(*te);
} else
(*t) += tau;
data[8] += 1.0;
(*out)(m0,m,t,te,u,data);
} while ((*t) != (*te));
free_real_vector(mu,1);
free_real_vector(lambda,1);
free_real_vector(thetha,0);
free_real_vector(ro,m00);
free_real_vector(r,m00);
free_real_matrix(alfa,1,8,1);
}
void lupzerortpol(int n, int m, real_t b[], real_t c[], real_t zer[],
real_t em[])
{
real_t infnrmvec(int, int, int *, real_t []);
void dupvec(int, int, int, real_t [], real_t []);
int i,posdef,j,k,t,converge;
real_t nrm,dlam,eps,delta,e,ep,err,p,q,qp,r,s,tot;
nrm=fabs(b[0]);
for (i=1; i<=n-2; i++)
if (c[i]+fabs(b[i]) > nrm) nrm=c[i]+fabs(b[i]);
if (n > 1)
nrm = (nrm+1 >= c[n-1]+fabs(b[n-1])) ? nrm+1.0 :
(c[n-1]+fabs(b[n-1]));
em[1]=nrm;
for (i=n; i>=1; i--) b[i]=b[i-1];
for (i=n; i>=2; i--) c[i]=c[i-1];
posdef = (em[6] == 1.0);
dlam=em[2];
eps=em[0];
c[1]=err=q=s=0.0;
tot=b[1];
for (i=n; i>=1; i--) {
p=q;
q=sqrt(c[i]);
e=b[i]-p-q;
if (e < tot) tot=e;
}
if (posdef && (tot < 0.0))
tot=0.0;
else
for(i=1; i<=n; i++) b[i] -= tot;
t=0;
for (k=1; k<=m; k++) {
converge=0;
/* next qr transformation */
do {
t++;
tot += s;
delta=b[n]-s;
i=n;
e=fabs(eps*tot);
if (dlam < e) dlam=e;
if (delta <= dlam) {
converge=1;
break;
}
e=c[n]/delta;
qp=delta+e;
p=1.0;
for (i=n-1; i>=k; i--) {
q=b[i]-s-e;
r=q/qp;
p=p*r+1.0;
ep=e*r;
b[i+1]=qp+ep;
delta=q-ep;
if (delta <= dlam) {
converge=1;
break;
}
e=c[i]/q;
qp=delta+e;
c[i+1]=qp*ep;
}
if (converge) break;
b[k]=qp;
s=qp/p;
} while (tot+s > tot); /* end of qr transformation */
if (!converge) {
/* irregular end of iteration,
deflate minimum diagonal element */
s=0.0;
i=k;
delta=qp;
for (j=k+1; j<=n; j++)
if (b[j] < delta) {
i=j;
delta=b[j];
}
}
/* convergence */
if (i < n) c[i+1]=c[i]*e/qp;
for (j=i-1; j>=k; j--) {
b[j+1]=b[j]-s;
c[j+1]=c[j];
}
b[k]=tot;
c[k] = err += fabs(delta);
}
em[5]=t;
em[3]=infnrmvec(1,m,&t,c);
dupvec(1,m,0,zer,b);
}
void quanewbnd(int n, int lw, int rw,
real_t x[], real_t f[], real_t jac[],
int (*funct)(int, int, int, real_t[], real_t[]),
real_t in[], real_t out[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
real_t vecvec(int, int, int, real_t [], real_t []);
void elmvec(int, int, int, real_t [], real_t [], real_t);
void mulvec(int, int, int, real_t [], real_t [], real_t);
void dupvec(int, int, int, real_t [], real_t []);
void decsolbnd(real_t [], int, int, int, real_t [], real_t []);
int l,it,fcnt,fmax,err,b,i,j,k,r,m;
real_t macheps,reltol,abstol,tolres,nd,mz,res,*delta,mul,crit,
*pp,*s,aux[6],*lu;
delta=allocate_real_vector(1,n);
nd=0.0;
macheps=in[0];
reltol=in[1];
abstol=in[2];
tolres=in[3];
fmax=in[4];
mz=macheps*macheps;
it=fcnt=0;
b=lw+rw;
l=(n-1)*b+n;
b++;
res=sqrt(vecvec(1,n,0,f,f));
err=0;
while (1) {
if (err != 0 || (res < tolres &&
sqrt(nd) < sqrt(vecvec(1,n,0,x,x))*reltol+abstol)) break;
it++;
if (it != 1) {
/* update jac */
pp=allocate_real_vector(1,n);
s=allocate_real_vector(1,n);
crit=nd*mz;
for (i=1; i<=n; i++) pp[i]=delta[i]*delta[i];
r=k=1;
m=rw+1;
for (i=1; i<=n; i++) {
mul=0.0;
for (j=r; j<=m; j++) mul += pp[j];
j=r-k;
if (fabs(mul) > crit) elmvec(k,m-j,j,jac,delta,f[i]/mul);
k += b;
if (i > lw)
r++;
else
k--;
if (m < n) m++;
}
free_real_vector(pp,1);
free_real_vector(s,1);
}
/* direction */
lu=allocate_real_vector(1,l);
aux[2]=macheps;
mulvec(1,n,0,delta,f,-1.0);
dupvec(1,l,0,lu,jac);
decsolbnd(lu,n,lw,rw,aux,delta);
free_real_vector(lu,1);
if (aux[3] != n) {
err=3;
break;
} else {
elmvec(1,n,0,x,delta,1.0);
nd=vecvec(1,n,0,delta,delta);
/* evaluate */
fcnt += n;
if (!((*funct)(n,1,n,x,f))) {
err=2;
break;
}
if (fcnt > fmax) err=1;
res=sqrt(vecvec(1,n,0,f,f));
}
}
out[1]=sqrt(nd);
out[2]=res;
out[3]=fcnt;
out[4]=it;
out[5]=err;
free_real_vector(delta,1);
}
/*-------------------------------------------------------------------------------
calculate the least squares solution of an overdetermined system of nonlinear equations
with Marquardt's method
-------------------------------------------------------------------------------*/
void Ti_Optimization::MarquardtforCylinderFitting(
int m,
int n,
double**g_pnt,
double* const par,
double*& g,
double**v,
int (*funct)(int m, int n, double* const par, double* g,double**g_pnt),
void (*jacobian)(int m, int n, double* const par, double*& g, double **jac,double**g_pnt),
double in[],
double out[]
)
{
int maxfe,fe,it,i,j,err,emergency;
double vv,ww,w,mu,res,fpar,fparpres,lambda,lambdamin,p,pw,reltolres,
abstolres,em[8],*val,*b,*bb,*parpres,**jac,temp;
val = allocate_real_vector(1,n);
b = allocate_real_vector(1,n);
bb = allocate_real_vector(1,n);
parpres = allocate_real_vector(1,n);
jac = allocate_real_matrix(1,m,1,n);
assert( (val != NULL) &&
(b != NULL) &&
(bb != NULL) &&
(parpres!= NULL)&&
(jac != NULL)
);
vv = 10.0;
w = 0.5;
mu = 0.01;
ww = (in[6] < 1.0e-7) ? 1.0e-8 : 1.0e-1*in[6];
em[0] = em[2] = em[6] = in[0];
em[4] = 10*n;
reltolres =in[3];
abstolres=in[4]*in[4];
maxfe=(int)in[5];
err=0;
fe=it=1;
p=fpar=res=0.0;
pw = -log(ww*in[0])/2.30;
if (!(*funct)(m,n,par,g,g_pnt))
{
err=3;
out[4]=fe;
out[5]=it-1;
out[1]=err;
free_real_vector(val,1);
free_real_vector(b,1);
free_real_vector(bb,1);
free_real_vector(parpres,1);
free_real_matrix(jac,1,m,1);
return;
}
fpar=vecvec(1,m,0,g,g);// norm of residual vector
out[3]=sqrt(fpar);
emergency=0;
it=1;
do {
(*jacobian)(m,n,par,g,jac,g_pnt);
i = qrisngvaldec(jac,m,n,val,v,em);
if (it == 1)
lambda = in[6]*vecvec(1,n,0,val,val);
else
if (p == 0.0)
lambda *= w;
for (i=1; i<=n; i++)
b[i] = val[i]*tamvec(1,m,i,jac,g);
while (1)
{
for (i=1; i<=n; i++)
bb[i]=b[i]/(val[i]*val[i]+lambda);
for (i=1; i<=n; i++)
parpres[i]=par[i]-matvec(1,n,i,v,bb);
//normalization ,this section only used for cylinder fitting,
//when it is used in other situations,it should be removed
temp = sqrt(parpres[4]*parpres[4]+parpres[5]*parpres[5]+parpres[6]*parpres[6]);
parpres[4] /= temp;
parpres[5] /= temp;
parpres[6] /= temp;
//end normalization
fe++;
if (fe >= maxfe)
err=1;
else
if (!(*funct)(m,n,parpres,g,g_pnt))
err=2;
if (err != 0)
{
emergency = 1;
break;
}
fparpres=vecvec(1,m,0,g,g);
res=fpar-fparpres;
if (res < mu*vecvec(1,n,0,b,bb))
{
p += 1.0;
lambda *= vv;
if (p == 1.0)
{
lambdamin=ww*vecvec(1,n,0,val,val);
if (lambda < lambdamin)
lambda=lambdamin;
}
if (p >= pw)
{
err=4;
emergency=1;
break;
}
} // end if
else
{
dupvec(1,n,0,par,parpres);
fpar=fparpres;
break;
} // end else
} // end while
if (emergency)
break;
it++;
}
while (
(fpar > abstolres) &&
(res > reltolres*fpar+abstolres)
);
for (i=1; i<=n; i++)
mulcol(1,n,i,i,jac,v,1.0/(val[i]+in[0]));
for (i=1; i<=n; i++)
{
for (j=1; j<=i; j++)
v[i][j]=v[j][i]=mattam(1,n,i,j,jac,jac);
lambda=lambdamin=val[1];
}
for (i=2; i<=n; i++)
{
if (val[i] > lambda)
lambda=val[i];
else
{
if (val[i] < lambdamin)
lambdamin=val[i];
}
}
temp=lambda/(lambdamin+in[0]);
out[7]=temp*temp;
out[2]=sqrt(fpar);
out[6]=sqrt(res+fpar)-out[2];
out[4]=fe;
out[5]=it-1;
out[1]=err;
if(val != NULL)
{
free_real_vector(val,1);
val = NULL;
}
if (b != NULL)
{
free_real_vector(b,1);
b = NULL;
}
if(bb!=NULL)
{
free_real_vector(bb,1);
bb = NULL;
}
if(parpres != NULL)
{
free_real_vector(parpres,1);
parpres = NULL;
}
if (jac != NULL)
{
free_real_matrix(jac,1,m,1);
jac = NULL;
}
}
