void main ()
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void chldecbnd(real_t [], int, int, real_t []);
void chlsolbnd(real_t [], int, int, real_t []);
real_t chldetermbnd(real_t [], int, int);
int i;
real_t *symband,*right,*aux;
symband=allocate_real_vector(1,9);
right=allocate_real_vector(1,5);
aux=allocate_real_vector(2,3);
for (i=1; i<=9; i++)
symband[i] = ((i/2)*2 < i) ? 2.0 : -1.0;
right[1]=right[5]=1.0;
right[2]=right[3]=right[4]=0.0;
aux[2]=1.0e-12;
chldecbnd(symband,5,1,aux);
if (aux[3] == 5) {
chlsolbnd(symband,5,1,right);
printf("Delivers: %8.4f %8.4f %8.4f %8.4f %8.4f\n"
"Determinant is %e\n",right[1],right[2],right[3],
right[4],right[5],chldetermbnd(symband,5,1));
}
free_real_vector(symband,1);
free_real_vector(right,1);
free_real_vector(aux,2);
}
void main ()
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void chldecsol1(real_t [], int, real_t [], real_t []);
real_t chldeterm1(real_t [], int);
void chldecinv1(real_t [], int, real_t []);
int i,j,jj;
real_t determinant,*pascal1,*b,*aux;
pascal1=allocate_real_vector(1,((4+1)*4)/2);
b=allocate_real_vector(1,4);
aux=allocate_real_vector(2,3);
jj=1;
for (j=1; j<=4; j++) {
pascal1[jj]=1.0;
for (i=2; i<=j; i++)
pascal1[jj+i-1] = (i == j) ?
pascal1[jj+i-2]*2.0 : pascal1[jj+i-2]+pascal1[jj+i-j];
b[j]=pow(2.0,j);
jj += j;
}
aux[2]=1.0e-11;
chldecsol1(pascal1,4,aux,b);
if (aux[3] == 4)
determinant=chldeterm1(pascal1,4);
else
printf("Matrix not positive definite");
printf("Solution with CHLDECSOL1:\n %e %e %e %e\n",
b[1],b[2],b[3],b[4]);
printf("\nDeterminant with CHLDETERM1: %e\n",determinant);
jj=1;
for (j=1; j<=4; j++) {
pascal1[jj]=1.0;
for (i=2; i<=j; i++)
pascal1[jj+i-1] = (i == j) ?
pascal1[jj+i-2]*2.0 : pascal1[jj+i-2]+pascal1[jj+i-j];
jj += j;
}
chldecinv1(pascal1,4,aux);
printf("\nInverse matrix with CHLDECINV1:\n");
for (i=1; i<=4; i++) {
for (j=1; j<=4; j++)
if (j < i)
printf(" ");
else
printf("%11.5f",pascal1[((j-1)*j)/2+i]);
printf("\n");
}
free_real_vector(pascal1,1);
free_real_vector(b,1);
free_real_vector(aux,2);
}
void main ()
{
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 chldecsol2(real_t **, int, real_t [], real_t []);
real_t chldeterm2(real_t **, int);
void chldecinv2(real_t **, int, real_t []);
int i,j;
real_t determinant,**pascal2,*b,*aux;
pascal2=allocate_real_matrix(1,4,1,4);
b=allocate_real_vector(1,4);
aux=allocate_real_vector(2,3);
for (j=1; j<=4; j++) {
pascal2[1][j]=1.0;
for (i=2; i<=j; i++)
pascal2[i][j] = (i == j) ?
pascal2[i-1][j]*2.0 : pascal2[i][j-1]+pascal2[i-1][j];
b[j]=pow(2.0,j);
}
aux[2]=1.0e-11;
chldecsol2(pascal2,4,aux,b);
if (aux[3] == 4)
determinant=chldeterm2(pascal2,4);
else
printf("Matrix not positive definite");
printf("Solution with CHLDECSOL2:\n %e %e %e %e\n",
b[1],b[2],b[3],b[4]);
printf("\nDeterminant with CHLDETERM2: %e\n",determinant);
for (j=1; j<=4; j++) {
pascal2[1][j]=1.0;
for (i=2; i<=j; i++)
pascal2[i][j] = (i == j) ?
pascal2[i-1][j]*2.0 : pascal2[i][j-1]+pascal2[i-1][j];
}
chldecinv2(pascal2,4,aux);
printf("\nInverse matrix with CHLDECINV2:\n");
for (i=1; i<=4; i++) {
for (j=1; j<=4; j++)
if (j < i)
printf(" ");
else
printf("%11.5f",pascal2[i][j]);
printf("\n");
}
free_real_matrix(pascal2,1,4,1);
free_real_vector(b,1);
free_real_vector(aux,2);
}
void main ()
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void ark(real_t *, real_t *, int *, int *, real_t [],
void (*)(int *, int *, real_t *, real_t[]), real_t [],
void (*)(int *, int *, real_t *, real_t *,
real_t [], real_t []));
int m0,m,i;
static real_t dat1[13]={3.0, 3.0, 1.0, 1.0, 1.0e-3, 1.0e-6,
1.0e-6, 0.0, 0.0, 0.0, 1.0, 0.5, 1.0/6.0};
static real_t dat2[14]={4.0, 3.0, 0.0, 500.0/3.0, 0.0,
-1.0, -1.0, 0.0, 0.0, 0.0, 1.0, 0.5, 1.0/6.0, 1.0/24.0};
real_t t,te,y[2],*u,data[15];
u=allocate_real_vector(-150,150);
for (i=1; i<=13; i++) data[i]=dat1[i-1];
t=0.0;
y[1]=1.0;
te=1.0;
m0=m=1;
ark(&t,&te,&m0,&m,y,der1,data,out1);
for (i=1; i<=14; i++) data[i]=dat2[i-1];
data[3]=sqrt(8.0);
data[5]=data[3]/data[4];
m0 = -150;
m=150;
t=0.0;
u[0]=1.0;
for (i=1; i<=m; i++) u[i]=u[-i]=exp(-(0.003*i)*(0.003*i));
te=0.6;
ark(&t,&te,&m0,&m,u,der2,data,out2);
free_real_vector(u,-150);
}
void chlinv1(real_t a[], int n)
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
real_t seqvec(int, int, int, int, real_t [], real_t []);
real_t symmatvec(int, int, int, real_t [], real_t []);
int i,ii,i1,j,ij;
real_t r,*u;
u=allocate_real_vector(1,n);
ii=((n+1)*n)/2;
for (i=n; i>=1; i--) {
r=1.0/a[ii];
i1=i+1;
ij=ii+i;
for (j=i1; j<=n; j++) {
u[j]=a[ij];
ij += j;
}
for (j=n; j>=i1; j--) {
ij -= j;
a[ij] = -symmatvec(i1,n,j,a,u)*r;
}
a[ii]=(r-seqvec(i1,n,ii+i,0,a,u))*r;
ii -= i;
}
free_real_vector(u,1);
}
void eigvalhrm(real_t **a, int n, int numval, real_t val[], real_t em[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void hshhrmtrival(real_t **, int, real_t [], real_t [], real_t []);
void valsymtri(real_t [], real_t [], int, int, int,
real_t [], real_t []);
real_t *d,*bb;
d=allocate_real_vector(1,n);
bb=allocate_real_vector(1,n-1);
hshhrmtrival(a,n,d,bb,em);
valsymtri(d,bb,n,1,numval,val,em);
free_real_vector(d,1);
free_real_vector(bb,1);
}
void hshdecmul(int n, real_t **a, real_t **b, real_t dwarf)
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
real_t tammat(int, int, int, int, real_t **, real_t **);
void hshvecmat(int, int, int, int, real_t, real_t [], real_t **);
int j,k,k1,n1;
real_t r,t,c,*v;
v=allocate_real_vector(1,n);
k=1;
n1=n+1;
for (k1=2; k1<=n1; k1++) {
r=tammat(k1,n,k,k,b,b);
if (r > dwarf) {
r = (b[k][k] < 0.0) ? -sqrt(r+b[k][k]*b[k][k]) :
sqrt(r+b[k][k]*b[k][k]);
t=b[k][k]+r;
c = -t/r;
b[k][k] = -r;
v[k]=1.0;
for (j=k1; j<=n; j++) v[j]=b[j][k]/t;
hshvecmat(k,n,k1,n,c,v,b);
hshvecmat(k,n,1,n,c,v,a);
}
k=k1;
}
free_real_vector(v,1);
}
int LLL(long n, double **b)
{
/* Lattice reduction algorithm. */
double *B = allocate_real_vector(n);
double **bs = allocate_real_matrix(n, n);
double **mu = allocate_real_matrix(n, n);
double C, t, temp, x, y;
long i, j, k, l;
for (i = 0; i < n; i++) bs[0][i] = b[0][i];
B[0] = Scalar(n, bs[0], bs[0]);
for (i = 1; i < n; i++) {
for (j = 0; j < n; j++) bs[i][j] = b[i][j];
for (j = 0; j < i; j++) {
mu[i][j] = Scalar(n, b[i], bs[j]) / B[j];
for (k = 0; k < n; k++)
bs[i][k] -= mu[i][j] * bs[j][k];
}
B[i] = Scalar(n, bs[i], bs[i]);
}
L3:
k = 1;
L4:
l = k - 1;
Reduce(k, l, n, b, mu);
x = mu[k][l];
y = 0.75 - x * x;
if (B[k] < y * B[l]) {
C = B[k] + x * x * B[l];
mu[k][l] = x * B[l] / C;
B[k] *= B[l] / C;
B[l] = C;
for (i = 0; i < n; i++) {
temp = b[k][i];
b[k][i] = b[l][i];
b[l][i] = temp;
}
if (k > 1) {
for (j = 0; j < k - 1; j++) {
temp = mu[k][j];
mu[k][j] = mu[l][j];
mu[l][j] = temp;
}
}
for (i = k + 1; i < n; i++) {
t = mu[i][k];
mu[i][k] = mu[i][l] - x * t;
mu[i][l] = t + mu[k][l] * mu[i][k];
}
k = max(1, k - 1);
goto L4;
}
for (l = k - 2; l >= 0; l--) Reduce(k, l, n, b, mu);
k++;
if (k < n) goto L4;
free_real_matrix(bs, n);
free_real_matrix(mu, n);
free_real_vector(B);
return 1;
}
void control(real_t *tp, real_t t, real_t h, real_t hnew, real_t **y,
real_t err[], int n, real_t tend)
{
int i;
real_t c[6],*x,s,s2,s3,s4;
x=allocate_real_vector(1,n);
while (1) {
s=(t-(*tp))/h;
s2=s*s;
s3=s2*s;
s4=s3*s;
c[3]=(s2-s)/2.0;
c[4] = -s3/6.0+s2/2.0-s/3.0;
c[5]=s4/24.0-s3/4.0+11.0*s2/24.0-s/4.0;
for (i=1; i<=n; i++)
x[i]=y[1][i]-s*y[2][i]+c[3]*y[3][i]+
c[4]*y[4][i]+c[5]*y[5][i];
printf(" %6.2f %7.2e %e %e %4d %3d\n",
*tp,err[3],x[1],x[2],nfe,nje);
if (*tp >= tend) break;
point++;
*tp = print[point];
if (*tp > t) break;
}
free_real_vector(x,1);
}
void hsh2row3(int l, int ua, int ub, int ux, int j, real_t a1, real_t a2,
real_t **a, real_t **b, real_t **x)
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void hshvectam(int, int, int, int, real_t, real_t [], real_t **);
real_t *v,d1,d2,s1,s2,r,d,c;
if (a2 != 0.0) {
v=allocate_real_vector(j,j+1);
d1=fabs(a1);
d2=fabs(a2);
s1 = (a1 >= 0.0) ? 1.0 : -1.0;
s2 = (a2 >= 0.0) ? 1.0 : -1.0;
if (d2 <= d1) {
r=d2/d1;
d=sqrt(1.0+r*r);
c = -1.0-1.0/d;
v[j]=s1*s2*r/(1.0+d);
} else {
r=d1/d2;
d=sqrt(1.0+r*r);
c = -1.0-r/d;
v[j]=s1*s2/(r+d);
}
v[j+1]=1.0;
hshvectam(l,ua,j,j+1,c,v,a);
hshvectam(l,ub,j,j+1,c,v,b);
hshvectam(1,ux,j,j+1,c,v,x);
free_real_vector(v,j);
}
}
void eigvalsym1(real_t a[], int n, int numval, real_t val[], real_t em[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void tfmsymtri1(real_t [], int, real_t [], real_t [], real_t [],
real_t []);
void valsymtri(real_t [], real_t [], int, int, int,
real_t [], real_t []);
real_t *b,*bb,*d;
b=allocate_real_vector(1,n);
bb=allocate_real_vector(1,n);
d=allocate_real_vector(1,n);
tfmsymtri1(a,n,d,b,bb,em);
valsymtri(d,bb,n,1,numval,val,em);
free_real_vector(b,1);
free_real_vector(bb,1);
free_real_vector(d,1);
}
int main(void)
{
long i, n = 8;
double *a = allocate_real_vector(n);
double *x = allocate_real_vector(n);
double s;
srand(time(NULL));
printf("\n");
for (i = 0; i < n; i++) a[i] = pow(2, i);
s = a[rand() % n] + a[rand() % n];
if (SubsetSum(n, s, a, x)) {
printf("sum: %f\n\n", s);
printf("x[i]\t\ta[i]\n\n");
for (i = 0; i < n; i++)
printf("%f\t%f\n", x[i], a[i]);
}
else printf("subset sum has no solution\n");
free_real_vector(a);
free_real_vector(x);
return 0;
}
void alllagzer(int n, real_t alfa, real_t zer[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void allzerortpol(int, real_t [], real_t [], real_t [], real_t []);
int i;
real_t *a,*b,em[6];
a=allocate_real_vector(0,n);
b=allocate_real_vector(0,n);
b[0]=0.0;
a[n-1]=n+n+alfa-1.0;
for (i=1; i<=n-1; i++) {
a[i-1]=i+i+alfa-1.0;
b[i]=i*(i+alfa);
}
em[0]=FLT_MIN;
em[2]=FLT_EPSILON;
em[4]=6*n;
allzerortpol(n,a,b,zer,em);
free_real_vector(a,0);
free_real_vector(b,0);
}
void main ()
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void quanewbnd1(int, int, int, real_t [], real_t [],
int (*)(int, int, int, real_t[], real_t[]),
real_t [], real_t []);
int i;
real_t *x,*f,in[6],out[6];
x=allocate_real_vector(1,600);
f=allocate_real_vector(1,600);
for (i=1; i<=600; i++) x[i] = -1.0;
in[0]=1.0e-6; in[1]=in[2]=in[3]=1.0e-5; in[4]=20000.0;
in[5]=0.001;
quanewbnd1(600,1,1,x,f,fun,in,out);
printf("Norm Residual vector: %e\n"
"Length of last step: %e\n"
"Number of function component evaluations: %6.0f\n"
"Number of iterations: %3.0f\nReport: %3.0f\n",
out[2],out[1],out[3],out[4],out[5]);
free_real_vector(x,1);
free_real_vector(f,1);
}
int homsol(real_t **a, int m, int n, real_t **v, real_t em[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
int qrisngvaldec(real_t **, int, int, real_t [], real_t **, real_t []);
void homsolsvd(real_t **, real_t [], real_t **, int, int);
int i;
real_t *val;
val=allocate_real_vector(1,n);
i=qrisngvaldec(a,m,n,val,v,em);
if (i == 0) homsolsvd(a,val,v,m,n);
free_real_vector(val,1);
return i;
}
int qrivalhrm(real_t **a, int n, real_t val[], real_t em[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void hshhrmtrival(real_t **, int, real_t [], real_t [], real_t []);
int qrivalsymtri(real_t [], real_t [], int, real_t []);
int i;
real_t *bb;
bb=allocate_real_vector(1,n);
hshhrmtrival(a,n,val,bb,em);
bb[n]=0.0;
i=qrivalsymtri(val,bb,n,em);
free_real_vector(bb,1);
return i;
}
void hsh3row2(int l, int u, int j, real_t a1, real_t a2,
real_t a3, real_t **a, real_t **b)
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void hshvectam(int, int, int, int, real_t, real_t [], real_t **);
real_t *v,c,d1,d2,d3,s1,s2,s3,r1,r2,r3,d;
if (a2 != 0.0 || a3 != 0.0) {
v=allocate_real_vector(j,j+2);
d1=fabs(a1);
d2=fabs(a2);
d3=fabs(a3);
s1 = (a1 >= 0.0) ? 1.0 : -1.0;
s2 = (a2 >= 0.0) ? 1.0 : -1.0;
s3 = (a3 >= 0.0) ? 1.0 : -1.0;
if (d1 >= d2 && d1 >= d3) {
r2=d2/d1;
r3=d3/d1;
d=sqrt(1.0+r2*r2+r3*r3);
c = -1.0-(1.0/d);
d=1.0/(1.0+d);
v[j+1]=s1*s2*r2*d;
v[j]=s1*s3*r3*d;
} else if (d2 >= d1 && d2 >= d3) {
r1=d1/d2;
r3=d3/d2;
d=sqrt(1.0+r1*r1+r3*r3);
c = -1.0-(s1*r1/d);
d=1.0/(r1+d);
v[j+1]=s1*s2*d;
v[j]=s1*s3*r3*d;
} else {
r1=d1/d3;
r2=d2/d3;
d=sqrt(1.0+r1*r1+r2*r2);
c = -1.0-(s1*r1/d);
d=1.0/(r1+d);
v[j+1]=s1*s2*r2*d;
v[j]=s1*s3*d;
}
v[j+2]=1.0;
hshvectam(l,u,j,j+2,c,v,a);
hshvectam(l,u,j,j+2,c,v,b);
free_real_vector(v,j);
}
}
void ixqfix(real_t x, real_t p, real_t q, int nmax, real_t eps,
real_t i[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
real_t incbeta(real_t, real_t, real_t, real_t);
void forward(real_t, real_t, real_t, real_t, real_t, int, real_t []);
void backward(real_t, real_t, real_t, real_t, int, real_t, real_t []);
int m,mmax;
real_t s,iq0,iq1,q0,*iq;
m=floor(q);
s=q-m;
q0 = (s > 0.0) ? s : s+1.0;
mmax = (s > 0.0) ? m : m-1;
iq0=incbeta(x,p,q0,eps);
if (mmax > 0) iq1=incbeta(x,p,q0+1.0,eps);
iq=allocate_real_vector(0,mmax);
forward(x,p,q0,iq0,iq1,mmax,iq);
backward(x,p,q,iq[mmax],nmax,eps,i);
free_real_vector(iq,0);
}
int Ti_Optimization::qrisngvaldec(
double **a, // the given matrix, exit: the matrix U in the singular value decomposition UDV'
int m, // entry : the number of rows of a
int n, // entry : the number of columns of a, n should satisfy n<=m
double val[], // exit: the singular values
double **v, // exit: the transpose of matrix V in the singular value decomposition
double em[8]
// entry:
// em[0]: the machine precision
// em[2]: the relative precision in the singular values
// em[4]: the maximal number of interations to be performed
// em[6]: the minimal non-neglectable singular value;
// exit:
// em[1]: the infinity norm of the matrix
// em[3]: the maximal neglected superdiagonal element;
// em[5]: the number of iterations performed;
// em[7]: the numerical rank of the matrix; i.e. the number of singular values greater than or equal to em[6]
)
{
/*double *allocate_real_vector(int, int);
void free_real_vector(double *, int);
void hshreabid(double **, int, int, double [], double [], double []);
void psttfmmat(double **, int, double **, double []);
void pretfmmat(double **, int, int, double []);
int qrisngvaldecbid(double [], double [], int, int, double **,
double **, double []);*/
int i = 0;
double *b;
b=allocate_real_vector(1,n);
hshreabid(a,m,n,val,b,em);
psttfmmat(a,n,v,b);
pretfmmat(a,m,n,val);
i=qrisngvaldecbid(val,b,m,n,a,v,em);
free_real_vector(b,1);
return i;
}
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];
}
void rk3n(real_t *x, real_t a, real_t b, real_t y[], real_t ya[],
real_t z[], real_t za[],
real_t (*fxyj)(int, int, real_t, real_t[]),
real_t e[], real_t d[], int fi, int n)
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
int j,jj,last,first,reject,test,ta,tb;
real_t xl,h,hmin,ind,hl,absh,fhm,discry,discrz,toly,tolz,mu,
mu1,fhy,fhz,*yl,*zl,*k0,*k1,*k2,*k3,*k4,*k5,*ee;
yl=allocate_real_vector(1,n);
zl=allocate_real_vector(1,n);
k0=allocate_real_vector(1,n);
k1=allocate_real_vector(1,n);
k2=allocate_real_vector(1,n);
k3=allocate_real_vector(1,n);
k4=allocate_real_vector(1,n);
k5=allocate_real_vector(1,n);
ee=allocate_real_vector(1,4*n);
if (fi) {
d[3]=a;
for (jj=1; jj<=n; jj++) {
d[jj+3]=ya[jj];
d[n+jj+3]=za[jj];
}
}
d[1]=0.0;
xl=d[3];
for (jj=1; jj<=n; jj++) {
yl[jj]=d[jj+3];
zl[jj]=d[n+jj+3];
}
if (fi) d[2]=b-d[3];
absh=h=fabs(d[2]);
if (b-xl < 0.0) h = -h;
ind=fabs(b-xl);
hmin=ind*e[1]+e[2];
for (jj=2; jj<=2*n; jj++) {
hl=ind*e[2*jj-1]+e[2*jj];
if (hl < hmin) hmin=hl;
}
for (jj=1; jj<=4*n; jj++) ee[jj]=e[jj]/ind;
first=reject=1;
test=1;
if (fi) {
last=1;
test=0;
}
while (1) {
if (test) {
absh=fabs(h);
if (absh < hmin) {
h = (h > 0.0) ? hmin : -hmin;
absh=hmin;
}
ta=(h >= b-xl);
tb=(h >= 0.0);
if ((ta && tb) || (!(ta || tb))) {
d[2]=h;
last=1;
h=b-xl;
absh=fabs(h);
} else
last=0;
}
test=1;
if (reject) {
*x=xl;
for (jj=1; jj<=n; jj++) y[jj]=yl[jj];
for (j=1; j<=n; j++) k0[j]=(*fxyj)(n,j,*x,y)*h;
} else {
fhy=h/hl;
for (jj=1; jj<=n; jj++) k0[jj]=k5[jj]*fhy;
}
*x=xl+0.276393202250021*h;
for (jj=1; jj<=n; jj++)
y[jj]=yl[jj]+(zl[jj]*0.276393202250021+
k0[jj]*0.038196601125011)*h;
for (j=1; j<=n; j++) k1[j]=(*fxyj)(n,j,*x,y)*h;
*x=xl+0.723606797749979*h;
for (jj=1; jj<=n; jj++)
y[jj]=yl[jj]+(zl[jj]*0.723606797749979+
k1[jj]*0.261803398874989)*h;
for (j=1; j<=n; j++) k2[j]=(*fxyj)(n,j,*x,y)*h;
*x=xl+h*0.5;
for (jj=1; jj<=n; jj++)
y[jj]=yl[jj]+(zl[jj]*0.5+k0[jj]*0.046875+k1[jj]*
0.079824155839840-k2[jj]*0.001699155839840)*h;
for (j=1; j<=n; j++) k4[j]=(*fxyj)(n,j,*x,y)*h;
*x = (last ? b : xl+h);
for (jj=1; jj<=n; jj++)
y[jj]=yl[jj]+(zl[jj]+k0[jj]*0.309016994374947+
k2[jj]*0.190983005625053)*h;
for (j=1; j<=n; j++) k3[j]=(*fxyj)(n,j,*x,y)*h;
for (jj=1; jj<=n; jj++)
y[jj]=yl[jj]+(zl[jj]+k0[jj]*0.083333333333333+k1[jj]*
0.301502832395825+k2[jj]*0.115163834270842)*h;
for (j=1; j<=n; j++) k5[j]=(*fxyj)(n,j,*x,y)*h;
reject=0;
fhm=0.0;
for (jj=1; jj<=n; jj++) {
discry=fabs((-k0[jj]*0.5+k1[jj]*1.809016994374947+
k2[jj]*0.690983005625053-k4[jj]*2.0)*h);
discrz=fabs((k0[jj]-k3[jj])*2.0-(k1[jj]+k2[jj])*10.0+
k4[jj]*16.0+k5[jj]*4.0);
toly=absh*(fabs(zl[jj])*ee[2*jj-1]+ee[2*jj]);
tolz=fabs(k0[jj])*ee[2*(jj+n)-1]+absh*ee[2*(jj+n)];
reject=((discry > toly) || (discrz > tolz) || reject);
fhy=discry/toly;
fhz=discrz/tolz;
if (fhz > fhy) fhy=fhz;
if (fhy > fhm) fhm=fhy;
}
mu=1.0/(1.0+fhm)+0.45;
if (reject) {
if (absh <= hmin) {
d[1] += 1.0;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj];
z[jj]=zl[jj];
}
first=1;
if (b == *x) break;
xl = *x;
for (jj=1; jj<=n; jj++) {
yl[jj]=y[jj];
zl[jj]=z[jj];
}
} else
h *= mu;
} else {
if (first) {
first=0;
hl=h;
h *= mu;
} else {
fhy=mu*h/hl+mu-mu1;
hl=h;
h *= fhy;
}
mu1=mu;
for (jj=1; jj<=n; jj++)
z[jj]=zl[jj]+(k0[jj]+k3[jj])*0.083333333333333+
(k1[jj]+k2[jj])*0.416666666666667;
if (b == *x) break;
xl = *x;
for (jj=1; jj<=n; jj++) {
yl[jj]=y[jj];
zl[jj]=z[jj];
}
}
}
if (!last) d[2]=h;
d[3] = *x;
for (jj=1; jj<=n; jj++) {
d[jj+3]=y[jj];
d[n+jj+3]=z[jj];
}
free_real_vector(yl,1);
free_real_vector(zl,1);
free_real_vector(k0,1);
free_real_vector(k1,1);
free_real_vector(k2,1);
free_real_vector(k3,1);
free_real_vector(k4,1);
free_real_vector(k5,1);
free_real_vector(ee,1);
}
void lsqdecomp(real_t **a, int n, int m, int n1, real_t aux[],
real_t aid[], int ci[])
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
real_t matmat(int, int, int, int, real_t **, real_t **);
real_t tammat(int, int, int, int, real_t **, real_t **);
void elmcol(int, int, int, int, real_t **, real_t **, real_t);
void ichcol(int, int, int, int, real_t **);
int j,k,kpiv,nr,s,fsum;
real_t beta,sigma,norm,aidk,akk,w,eps,temp,*sum;
sum=allocate_real_vector(1,m);
norm=0.0;
aux[3]=m;
nr=n1;
fsum=1;
for (k=1; k<=m; k++) {
if (k == n1+1) {
fsum=1;
nr=n;
}
if (fsum)
for (j=k; j<=m; j++) {
w=sum[j]=tammat(k,nr,j,j,a,a);
if (w > norm) norm=w;
}
fsum=0;
eps=aux[2]*sqrt(norm);
sigma=sum[k];
kpiv=k;
for (j=k+1; j<=m; j++)
if (sum[j] > sigma) {
sigma=sum[j];
kpiv=j;
}
if (kpiv != k) {
sum[kpiv]=sum[k];
ichcol(1,n,k,kpiv,a);
}
ci[k]=kpiv;
akk=a[k][k];
sigma=tammat(k,nr,k,k,a,a);
w=sqrt(sigma);
aidk=aid[k]=((akk < 0.0) ? w : -w);
if (w < eps) {
aux[3]=k-1;
break;
}
beta=1.0/(sigma-akk*aidk);
a[k][k]=akk-aidk;
for (j=k+1; j<=m; j++) {
elmcol(k,nr,j,k,a,a,-beta*tammat(k,nr,k,j,a,a));
temp=a[k][j];
sum[j] -= temp*temp;
}
if (k == n1)
for (j=n1+1; j<=n; j++)
for (s=1; s<=m; s++) {
nr = (s > n1) ? n1 : s-1;
w=a[j][s]-matmat(1,nr,j,s,a,a);
a[j][s] = (s > n1) ? w : w/aid[s];
}
}
free_real_vector(sum,1);
}
void rk2n(real_t *x, real_t a, real_t b, real_t y[], real_t ya[],
real_t z[], real_t za[],
real_t (*fxyzj)(int, int, real_t, real_t [], real_t []),
real_t e[], real_t d[], int fi, int n)
{
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
int j,jj,last,first,reject,test,ta,tb;
real_t xl,h,ind,hmin,hl,absh,fhm,discry,discrz,toly,tolz,
mu,mu1,fhy,fhz,*yl,*zl,*k0,*k1,*k2,*k3,*k4,*k5,*ee;
yl=allocate_real_vector(1,n);
zl=allocate_real_vector(1,n);
k0=allocate_real_vector(1,n);
k1=allocate_real_vector(1,n);
k2=allocate_real_vector(1,n);
k3=allocate_real_vector(1,n);
k4=allocate_real_vector(1,n);
k5=allocate_real_vector(1,n);
ee=allocate_real_vector(1,4*n);
if (fi) {
d[3]=a;
for (jj=1; jj<=n; jj++) {
d[jj+3]=ya[jj];
d[n+jj+3]=za[jj];
}
}
d[1]=0.0;
xl=d[3];
for (jj=1; jj<=n; jj++) {
yl[jj]=d[jj+3];
zl[jj]=d[n+jj+3];
}
if (fi) d[2]=b-d[3];
absh=h=fabs(d[2]);
if (b-xl < 0.0) h = -h;
ind=fabs(b-xl);
hmin=ind*e[1]+e[2];
for (jj=2; jj<=2*n; jj++) {
hl=ind*e[2*jj-1]+e[2*jj];
if (hl < hmin) hmin=hl;
}
for (jj=1; jj<=4*n; jj++) ee[jj]=e[jj]/ind;
first=1;
test=1;
if (fi) {
last=1;
test=0;
}
while (1) {
if (test) {
absh=fabs(h);
if (absh < hmin) {
h = (h > 0.0) ? hmin : -hmin;
absh=fabs(h);
}
ta=(h >= b-xl);
tb=(h >= 0.0);
if ((ta && tb) || (!(ta || tb))) {
d[2]=h;
last=1;
h=b-xl;
absh=fabs(h);
} else
last=0;
}
test=1;
*x=xl;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj];
z[jj]=zl[jj];
}
for (j=1; j<=n; j++) k0[j]=(*fxyzj)(n,j,*x,y,z)*h;
*x=xl+h/4.5;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*18.0+k0[jj]*2.0)/81.0*h;
z[jj]=zl[jj]+k0[jj]/4.5;
}
for (j=1; j<=n; j++) k1[j]=(*fxyzj)(n,j,*x,y,z)*h;
*x=xl+h/3.0;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*6.0+k0[jj])/18.0*h;
z[jj]=zl[jj]+(k0[jj]+k1[jj]*3.0)/12.0;
}
for (j=1; j<=n; j++) k2[j]=(*fxyzj)(n,j,*x,y,z)*h;
*x=xl+h*0.5;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*8.0+k0[jj]+k2[jj])/16.0*h;
z[jj]=zl[jj]+(k0[jj]+k2[jj]*3.0)/8.0;
}
for (j=1; j<=n; j++) k3[j]=(*fxyzj)(n,j,*x,y,z)*h;
*x=xl+h*0.8;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*100.0+k0[jj]*12.0+
k3[jj]*28.0)/125.0*h;
z[jj]=zl[jj]+(k0[jj]*53.0-k1[jj]*135.0+k2[jj]*126.0+
k3[jj]*56.0)/125.0;
}
for (j=1; j<=n; j++) k4[j]=(*fxyzj)(n,j,*x,y,z)*h;
*x = (last ? b : xl+h);
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*336.0+k0[jj]*21.0+k2[jj]*92.0+
k4[jj]*55.0)/336.0*h;
z[jj]=zl[jj]+(k0[jj]*133.0-k1[jj]*378.0+k2[jj]*276.0+
k3[jj]*112.0+k4[jj]*25.0)/168.0;
}
for (j=1; j<=n; j++) k5[j]=(*fxyzj)(n,j,*x,y,z)*h;
reject=0;
fhm=0.0;
for (jj=1; jj<=n; jj++) {
discry=fabs((-k0[jj]*21.0+k2[jj]*108.0-k3[jj]*112.0+
k4[jj]*25.0)/56.0*h);
discrz=fabs(k0[jj]*21.0-k2[jj]*162.0+k3[jj]*224.0-
k4[jj]*125.0+k5[jj]*42.0)/14.0;
toly=absh*(fabs(zl[jj])*ee[2*jj-1]+ee[2*jj]);
tolz=fabs(k0[jj])*ee[2*(jj+n)-1]+absh*ee[2*(jj+n)];
reject=((discry > toly) || (discrz > tolz) || reject);
fhy=discry/toly;
fhz=discrz/tolz;
if (fhz > fhy) fhy=fhz;
if (fhy > fhm) fhm=fhy;
}
mu=1.0/(1.0+fhm)+0.45;
if (reject) {
if (absh <= hmin) {
d[1] += 1.0;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj];
z[jj]=zl[jj];
}
first=1;
if (b == *x) break;
xl = *x;
for (jj=1; jj<=n; jj++) {
yl[jj] = y[jj];
zl[jj] = z[jj];
}
} else
h *= mu;
} else {
if (first) {
first=0;
hl=h;
h *= mu;
} else {
fhm=mu*h/hl+mu-mu1;
hl=h;
h *= fhm;
}
mu1=mu;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*56.0+k0[jj]*7.0+k2[jj]*36.0-
k4[jj]*15.0)/56.0*hl;
z[jj]=zl[jj]+(-k0[jj]*63.0+k1[jj]*189.0-k2[jj]*36.0-
k3[jj]*112.0+k4[jj]*50.0)/28.0;
}
for (j=1; j<=n; j++) k5[j]=(*fxyzj)(n,j,*x,y,z)*hl;
for (jj=1; jj<=n; jj++) {
y[jj]=yl[jj]+(zl[jj]*336.0+k0[jj]*35.0+k2[jj]*108.0+
k4[jj]*25.0)/336.0*hl;
z[jj]=zl[jj]+(k0[jj]*35.0+k2[jj]*162.0+k4[jj]*125.0+
k5[jj]*14.0)/336.0;
}
if (b == *x) break;
xl = *x;
for (jj=1; jj<=n; jj++) {
yl[jj] = y[jj];
zl[jj] = z[jj];
}
}
}
if (!last) d[2]=h;
d[3] = *x;
for (jj=1; jj<=n; jj++) {
d[jj+3]=y[jj];
d[n+jj+3]=z[jj];
}
free_real_vector(yl,1);
free_real_vector(zl,1);
free_real_vector(k0,1);
free_real_vector(k1,1);
free_real_vector(k2,1);
free_real_vector(k3,1);
free_real_vector(k4,1);
free_real_vector(k5,1);
free_real_vector(ee,1);
}
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 enx(real_t x, int n1, int n2, real_t a[])
{
if (x <= 1.5) {
real_t ei(real_t);
int i;
real_t w,e;
w = -ei(-x);
if (n1 == 1) a[1]=w;
if (n2 > 1) e=exp(-x);
for (i=2; i<=n2; i++) {
w=(e-x*w)/(i-1);
if (i >= n1) a[i]=w;
}
} else {
int i,n;
real_t w,e,an;
n=ceil(x);
if (n <= 10) {
real_t f,w1,t,h,p[20];
p[2] =0.37534261820491e-1; p[11]=0.135335283236613;
p[3] =0.89306465560228e-2; p[12]=0.497870683678639e-1;
p[4] =0.24233983686581e-2; p[13]=0.183156388887342e-1;
p[5] =0.70576069342458e-3; p[14]=0.673794699908547e-2;
p[6] =0.21480277819013e-3; p[15]=0.247875217666636e-2;
p[7] =0.67375807781018e-4; p[16]=0.911881965554516e-3;
p[8] =0.21600730159975e-4; p[17]=0.335462627902512e-3;
p[9] =0.70411579854292e-5; p[18]=0.123409804086680e-3;
p[10]=0.23253026570282e-5; p[19]=0.453999297624848e-4;
f=w=p[n];
e=p[n+9];
w1=t=1.0;
h=x-n;
i=n-1;
do {
f=(e-i*f)/n;
t = -h*t/(n-i);
w1=t*f;
w += w1;
i--;
} while (fabs(w1) > 1.0e-15*w);
} else {
real_t *allocate_real_vector(int, int);
void free_real_vector(real_t *, int);
void nonexpenx(real_t, int, int, real_t []);
real_t *b;
b=allocate_real_vector(n,n);
nonexpenx(x,n,n,b);
w=b[n]*exp(-x);
free_real_vector(b,n);
}
if (n1 == n2 && n1 == n)
a[n]=w;
else {
e=exp(-x);
an=w;
if (n <= n2 && n >= n1) a[n]=w;
for (i=n-1; i>=n1; i--) {
w=(e-i*w)/x;
if (i <= n2) a[i]=w;
}
w=an;
for (i=n+1; i<=n2; i++) {
w=(e-x*w)/(i-1);
if (i >= n1) a[i]=w;
}
}
}
}
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 efsirk(real_t *x, real_t xe, int m, real_t y[],
real_t *delta, void (*derivative)(int, real_t[], real_t *),
void (*jacobian)(int, real_t **, real_t [], real_t *),
real_t **j, int *n, real_t aeta, real_t reta, real_t hmin,
real_t hmax, int linear,
void (*output)(real_t, real_t, int, real_t [],
real_t, real_t **, int))
{
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 []);
real_t matmat(int, int, int, int, real_t **, real_t **);
real_t matvec(int, int, int, real_t **, real_t []);
void gsselm(real_t **, int, real_t [], int [], int []);
void solelm(real_t **, int, int [], int [], real_t []);
int k,l,lin,*ri,*ci;
real_t step,h,mu0,mu1,mu2,theta0,theta1,nu1,nu2,nu3,yk,fk,c1,c2,
d,*f,*k0,*labda,**j1,aux[8],discr,eta,s,z1,z2,e,alpha1,a,b;
ri=allocate_integer_vector(1,m);
ci=allocate_integer_vector(1,m);
f=allocate_real_vector(1,m);
k0=allocate_real_vector(1,m);
labda=allocate_real_vector(1,m);
j1=allocate_real_matrix(1,m,1,m);
aux[2]=FLT_EPSILON;
aux[4]=8.0;
for (k=1; k<=m; k++) f[k]=y[k];
*n = 0;
(*output)(*x,xe,m,y,*delta,j,*n);
step=0.0;
do {
(*n)++;
/* difference scheme */
(*derivative)(m,f,delta);
/* step size */
if (linear)
s=h=hmax;
else
if (*n == 1 || hmin == hmax)
s=h=hmin;
else {
eta=aeta+reta*sqrt(vecvec(1,m,0,y,y));
c1=nu3*step;
for (k=1; k<=m; k++) labda[k] += c1*f[k]-y[k];
discr=sqrt(vecvec(1,m,0,labda,labda));
s=h=(eta/(0.75*(eta+discr))+0.33)*h;
if (h < hmin)
s=h=hmin;
else
if (h > hmax) s=h=hmax;
}
if ((*x)+s > xe) s=xe-(*x);
lin=((step == s) && linear);
step=s;
if (!linear || *n == 1) (*jacobian)(m,j,y,delta);
if (!lin) {
/* coefficient */
z1=step*(*delta);
if (*n == 1) z2=z1+z1;
if (fabs(z2-z1) > 1.0e-6*fabs(z1) || z2 > -1.0) {
a=z1*z1+12.0;
b=6.0*z1;
if (fabs(z1) < 0.1)
alpha1=(z1*z1/140.0-1.0)*z1/30.0;
else if (z1 < 1.0e-14)
alpha1=1.0/3.0;
else if (z1 < -33.0)
alpha1=(a+b)/(3.0*z1*(2.0+z1));
else {
e=((z1 < 230.0) ? exp(z1) : FLT_MAX);
alpha1=((a-b)*e-a-b)/(((2.0-z1)*e-2.0-z1)*3.0*z1);
}
mu2=(1.0/3.0+alpha1)*0.25;
mu1 = -(1.0+alpha1)*0.5;
mu0=(6.0*mu1+2.0)/9.0;
theta0=0.25;
theta1=0.75;
a=3.0*alpha1;
nu3=(1.0+a)/(5.0-a)*0.5;
a=nu3+nu3;
nu1=0.5-a;
nu2=(1.0+a)*0.75;
z2=z1;
}
c1=step*mu1;
d=step*step*mu2;
for (k=1; k<=m; k++) {
for (l=1; l<=m; l++)
j1[k][l]=d*matmat(1,m,k,l,j,j)+c1*j[k][l];
j1[k][k] += 1.0;
}
gsselm(j1,m,aux,ri,ci);
}
c1=step*step*mu0;
d=step*2.0/3.0;
for (k=1; k<=m; k++) {
k0[k]=fk=f[k];
labda[k]=d*fk+c1*matvec(1,m,k,j,f);
}
solelm(j1,m,ri,ci,labda);
for (k=1; k<=m; k++) f[k]=y[k]+labda[k];
(*derivative)(m,f,delta);
c1=theta0*step;
c2=theta1*step;
d=nu1*step;
for (k=1; k<=m; k++) {
yk=y[k];
fk=f[k];
labda[k]=yk+d*fk+nu2*labda[k];
y[k]=f[k]=yk+c1*k0[k]+c2*fk;
}
(*x) += step;
(*output)(*x,xe,m,y,*delta,j,*n);
} while (*x < xe);
free_integer_vector(ri,1);
free_integer_vector(ci,1);
free_real_vector(f,1);
free_real_vector(k0,1);
free_real_vector(labda,1);
free_real_matrix(j1,1,m,1);
}