__minpack_attr__
void __minpack_func__(lmstr)(__minpack_decl_fcnderstr_mn__ const int *m, const int *n, real *x,
real *fvec, real *fjac, const int *ldfjac, const real *ftol,
const real *xtol, const real *gtol, const int *maxfev, real *
diag, const int *mode, const real *factor, const int *nprint, int *
info, int *nfev, int *njev, int *ipvt, real *qtf,
real *wa1, real *wa2, real *wa3, real *wa4)
{
/* Table of constant values */
const int c__1 = 1;
const int c_true = TRUE_;
/* Initialized data */
#define p1 .1
#define p5 .5
#define p25 .25
#define p75 .75
#define p0001 1e-4
/* System generated locals */
int fjac_dim1, fjac_offset, i__1, i__2;
real d__1, d__2, d__3;
/* Local variables */
int i__, j, l;
real par, sum;
int sing;
int iter;
real temp, temp1, temp2;
int iflag;
real delta;
real ratio;
real fnorm, gnorm, pnorm, xnorm = 0, fnorm1, actred, dirder,
epsmch, prered;
/* ********** */
/* subroutine lmstr */
/* the purpose of lmstr is to minimize the sum of the squares of */
/* m nonlinear functions in n variables by a modification of */
/* the levenberg-marquardt algorithm which uses minimal storage. */
/* the user must provide a subroutine which calculates the */
/* functions and the rows of the jacobian. */
/* the subroutine statement is */
/* subroutine lmstr(fcn,m,n,x,fvec,fjac,ldfjac,ftol,xtol,gtol, */
/* maxfev,diag,mode,factor,nprint,info,nfev, */
/* njev,ipvt,qtf,wa1,wa2,wa3,wa4) */
/* where */
/* fcn is the name of the user-supplied subroutine which */
/* calculates the functions and the rows of the jacobian. */
/* fcn must be declared in an external statement in the */
/* user calling program, and should be written as follows. */
/* subroutine fcn(m,n,x,fvec,fjrow,iflag) */
/* integer m,n,iflag */
/* double precision x(n),fvec(m),fjrow(n) */
/* ---------- */
/* if iflag = 1 calculate the functions at x and */
/* return this vector in fvec. */
/* if iflag = i calculate the (i-1)-st row of the */
/* jacobian at x and return this vector in fjrow. */
/* ---------- */
/* return */
/* end */
/* the value of iflag should not be changed by fcn unless */
/* the user wants to terminate execution of lmstr. */
/* in this case set iflag to a negative integer. */
/* m is a positive integer input variable set to the number */
/* of functions. */
/* n is a positive integer input variable set to the number */
/* of variables. n must not exceed m. */
/* x is an array of length n. on input x must contain */
/* an initial estimate of the solution vector. on output x */
/* contains the final estimate of the solution vector. */
/* fvec is an output array of length m which contains */
/* the functions evaluated at the output x. */
/* fjac is an output n by n array. the upper triangle of fjac */
/* contains an upper triangular matrix r such that */
/* t t t */
/* p *(jac *jac)*p = r *r, */
/* where p is a permutation matrix and jac is the final */
/* calculated jacobian. column j of p is column ipvt(j) */
/* (see below) of the identity matrix. the lower triangular */
/* part of fjac contains information generated during */
/* the computation of r. */
/* ldfjac is a positive integer input variable not less than n */
/* which specifies the leading dimension of the array fjac. */
/* ftol is a nonnegative input variable. termination */
/* occurs when both the actual and predicted relative */
/* reductions in the sum of squares are at most ftol. */
/* therefore, ftol measures the relative error desired */
/* in the sum of squares. */
/* xtol is a nonnegative input variable. termination */
/* occurs when the relative error between two consecutive */
/* iterates is at most xtol. therefore, xtol measures the */
/* relative error desired in the approximate solution. */
/* gtol is a nonnegative input variable. termination */
/* occurs when the cosine of the angle between fvec and */
/* any column of the jacobian is at most gtol in absolute */
/* value. therefore, gtol measures the orthogonality */
/* desired between the function vector and the columns */
/* of the jacobian. */
/* maxfev is a positive integer input variable. termination */
/* occurs when the number of calls to fcn with iflag = 1 */
/* has reached maxfev. */
/* diag is an array of length n. if mode = 1 (see */
/* below), diag is internally set. if mode = 2, diag */
/* must contain positive entries that serve as */
/* multiplicative scale factors for the variables. */
/* mode is an integer input variable. if mode = 1, the */
/* variables will be scaled internally. if mode = 2, */
/* the scaling is specified by the input diag. other */
/* values of mode are equivalent to mode = 1. */
/* factor is a positive input variable used in determining the */
/* initial step bound. this bound is set to the product of */
/* factor and the euclidean norm of diag*x if nonzero, or else */
/* to factor itself. in most cases factor should lie in the */
/* interval (.1,100.). 100. is a generally recommended value. */
/* nprint is an integer input variable that enables controlled */
/* printing of iterates if it is positive. in this case, */
/* fcn is called with iflag = 0 at the beginning of the first */
/* iteration and every nprint iterations thereafter and */
/* immediately prior to return, with x and fvec available */
/* for printing. if nprint is not positive, no special calls */
/* of fcn with iflag = 0 are made. */
/* info is an integer output variable. if the user has */
/* terminated execution, info is set to the (negative) */
/* value of iflag. see description of fcn. otherwise, */
/* info is set as follows. */
/* info = 0 improper input parameters. */
/* info = 1 both actual and predicted relative reductions */
/* in the sum of squares are at most ftol. */
/* info = 2 relative error between two consecutive iterates */
/* is at most xtol. */
/* info = 3 conditions for info = 1 and info = 2 both hold. */
/* info = 4 the cosine of the angle between fvec and any */
/* column of the jacobian is at most gtol in */
/* absolute value. */
/* info = 5 number of calls to fcn with iflag = 1 has */
/* reached maxfev. */
/* info = 6 ftol is too small. no further reduction in */
/* the sum of squares is possible. */
/* info = 7 xtol is too small. no further improvement in */
/* the approximate solution x is possible. */
/* info = 8 gtol is too small. fvec is orthogonal to the */
/* columns of the jacobian to machine precision. */
/* nfev is an integer output variable set to the number of */
/* calls to fcn with iflag = 1. */
/* njev is an integer output variable set to the number of */
/* calls to fcn with iflag = 2. */
/* ipvt is an integer output array of length n. ipvt */
/* defines a permutation matrix p such that jac*p = q*r, */
/* where jac is the final calculated jacobian, q is */
/* orthogonal (not stored), and r is upper triangular. */
/* column j of p is column ipvt(j) of the identity matrix. */
/* qtf is an output array of length n which contains */
/* the first n elements of the vector (q transpose)*fvec. */
/* wa1, wa2, and wa3 are work arrays of length n. */
/* wa4 is a work array of length m. */
/* subprograms called */
/* user-supplied ...... fcn */
/* minpack-supplied ... dpmpar,enorm,lmpar,qrfac,rwupdt */
/* fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, dudley v. goetschel, kenneth e. hillstrom, */
/* jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa4;
--fvec;
--wa3;
--wa2;
--wa1;
--qtf;
--ipvt;
--diag;
--x;
fjac_dim1 = *ldfjac;
fjac_offset = 1 + fjac_dim1 * 1;
fjac -= fjac_offset;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = __minpack_func__(dpmpar)(&c__1);
*info = 0;
iflag = 0;
*nfev = 0;
*njev = 0;
/* check the input parameters for errors. */
if (*n <= 0 || *m < *n || *ldfjac < *n || *ftol < 0. || *xtol < 0. ||
*gtol < 0. || *maxfev <= 0 || *factor <= 0.) {
goto L340;
}
if (*mode != 2) {
goto L20;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (diag[j] <= 0.) {
goto L340;
}
/* L10: */
}
L20:
/* evaluate the function at the starting point */
/* and calculate its norm. */
iflag = 1;
fcnderstr_mn(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
*nfev = 1;
if (iflag < 0) {
goto L340;
}
fnorm = __minpack_func__(enorm)(m, &fvec[1]);
/* initialize levenberg-marquardt parameter and iteration counter. */
par = 0.;
iter = 1;
/* beginning of the outer loop. */
L30:
/* if requested, call fcn to enable printing of iterates. */
if (*nprint <= 0) {
goto L40;
}
iflag = 0;
if ((iter - 1) % *nprint == 0) {
fcnderstr_mn(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
}
if (iflag < 0) {
goto L340;
}
L40:
/* compute the qr factorization of the jacobian matrix */
/* calculated one row at a time, while simultaneously */
/* forming (q transpose)*fvec and storing the first */
/* n components in qtf. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
qtf[j] = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
fjac[i__ + j * fjac_dim1] = 0.;
/* L50: */
}
/* L60: */
}
iflag = 2;
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
fcnderstr_mn(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
if (iflag < 0) {
goto L340;
}
temp = fvec[i__];
__minpack_func__(rwupdt)(n, &fjac[fjac_offset], ldfjac, &wa3[1], &qtf[1], &temp, &wa1[
1], &wa2[1]);
++iflag;
/* L70: */
}
++(*njev);
/* if the jacobian is rank deficient, call qrfac to */
/* reorder its columns and update the components of qtf. */
sing = FALSE_;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (fjac[j + j * fjac_dim1] == 0.) {
sing = TRUE_;
}
ipvt[j] = j;
wa2[j] = __minpack_func__(enorm)(&j, &fjac[j * fjac_dim1 + 1]);
/* L80: */
}
if (! sing) {
goto L130;
}
__minpack_func__(qrfac)(n, n, &fjac[fjac_offset], ldfjac, &c_true, &ipvt[1], n, &wa1[1], &
wa2[1], &wa3[1]);
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (fjac[j + j * fjac_dim1] == 0.) {
goto L110;
}
sum = 0.;
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
sum += fjac[i__ + j * fjac_dim1] * qtf[i__];
/* L90: */
}
temp = -sum / fjac[j + j * fjac_dim1];
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
qtf[i__] += fjac[i__ + j * fjac_dim1] * temp;
/* L100: */
}
L110:
fjac[j + j * fjac_dim1] = wa1[j];
/* L120: */
}
L130:
/* on the first iteration and if mode is 1, scale according */
/* to the norms of the columns of the initial jacobian. */
if (iter != 1) {
goto L170;
}
if (*mode == 2) {
goto L150;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
diag[j] = wa2[j];
if (wa2[j] == 0.) {
diag[j] = 1.;
}
/* L140: */
}
L150:
/* on the first iteration, calculate the norm of the scaled x */
/* and initialize the step bound delta. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa3[j] = diag[j] * x[j];
/* L160: */
}
xnorm = __minpack_func__(enorm)(n, &wa3[1]);
delta = *factor * xnorm;
if (delta == 0.) {
delta = *factor;
}
L170:
/* compute the norm of the scaled gradient. */
gnorm = 0.;
if (fnorm == 0.) {
goto L210;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = ipvt[j];
if (wa2[l] == 0.) {
goto L190;
}
sum = 0.;
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
sum += fjac[i__ + j * fjac_dim1] * (qtf[i__] / fnorm);
/* L180: */
}
/* Computing MAX */
d__2 = gnorm, d__3 = (d__1 = sum / wa2[l], abs(d__1));
gnorm = max(d__2,d__3);
L190:
/* L200: */
;
}
L210:
/* test for convergence of the gradient norm. */
if (gnorm <= *gtol) {
*info = 4;
}
if (*info != 0) {
goto L340;
}
/* rescale if necessary. */
if (*mode == 2) {
goto L230;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
d__1 = diag[j], d__2 = wa2[j];
diag[j] = max(d__1,d__2);
/* L220: */
}
L230:
/* beginning of the inner loop. */
L240:
/* determine the levenberg-marquardt parameter. */
__minpack_func__(lmpar)(n, &fjac[fjac_offset], ldfjac, &ipvt[1], &diag[1], &qtf[1], &delta,
&par, &wa1[1], &wa2[1], &wa3[1], &wa4[1]);
/* store the direction p and x + p. calculate the norm of p. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = -wa1[j];
wa2[j] = x[j] + wa1[j];
wa3[j] = diag[j] * wa1[j];
/* L250: */
}
pnorm = __minpack_func__(enorm)(n, &wa3[1]);
/* on the first iteration, adjust the initial step bound. */
if (iter == 1) {
delta = min(delta,pnorm);
}
/* evaluate the function at x + p and calculate its norm. */
iflag = 1;
fcnderstr_mn(m, n, &wa2[1], &wa4[1], &wa3[1], &iflag);
++(*nfev);
if (iflag < 0) {
goto L340;
}
fnorm1 = __minpack_func__(enorm)(m, &wa4[1]);
/* compute the scaled actual reduction. */
actred = -1.;
if (p1 * fnorm1 < fnorm) {
/* Computing 2nd power */
d__1 = fnorm1 / fnorm;
actred = 1. - d__1 * d__1;
}
/* compute the scaled predicted reduction and */
/* the scaled directional derivative. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa3[j] = 0.;
l = ipvt[j];
temp = wa1[l];
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
wa3[i__] += fjac[i__ + j * fjac_dim1] * temp;
/* L260: */
}
/* L270: */
}
temp1 = __minpack_func__(enorm)(n, &wa3[1]) / fnorm;
temp2 = sqrt(par) * pnorm / fnorm;
/* Computing 2nd power */
d__1 = temp1;
/* Computing 2nd power */
d__2 = temp2;
prered = d__1 * d__1 + d__2 * d__2 / p5;
/* Computing 2nd power */
d__1 = temp1;
/* Computing 2nd power */
d__2 = temp2;
dirder = -(d__1 * d__1 + d__2 * d__2);
/* compute the ratio of the actual to the predicted */
/* reduction. */
ratio = 0.;
if (prered != 0.) {
ratio = actred / prered;
}
/* update the step bound. */
if (ratio > p25) {
goto L280;
}
if (actred >= 0.) {
temp = p5;
}
if (actred < 0.) {
temp = p5 * dirder / (dirder + p5 * actred);
}
if (p1 * fnorm1 >= fnorm || temp < p1) {
temp = p1;
}
/* Computing MIN */
d__1 = delta, d__2 = pnorm / p1;
delta = temp * min(d__1,d__2);
par /= temp;
goto L300;
L280:
if (par != 0. && ratio < p75) {
goto L290;
}
delta = pnorm / p5;
par = p5 * par;
L290:
L300:
/* test for successful iteration. */
if (ratio < p0001) {
goto L330;
}
/* successful iteration. update x, fvec, and their norms. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
x[j] = wa2[j];
wa2[j] = diag[j] * x[j];
/* L310: */
}
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
fvec[i__] = wa4[i__];
/* L320: */
}
xnorm = __minpack_func__(enorm)(n, &wa2[1]);
fnorm = fnorm1;
++iter;
L330:
/* tests for convergence. */
if (abs(actred) <= *ftol && prered <= *ftol && p5 * ratio <= 1.) {
*info = 1;
}
if (delta <= *xtol * xnorm) {
*info = 2;
}
if (abs(actred) <= *ftol && prered <= *ftol && p5 * ratio <= 1. && *info
== 2) {
*info = 3;
}
if (*info != 0) {
goto L340;
}
/* tests for termination and stringent tolerances. */
if (*nfev >= *maxfev) {
*info = 5;
}
if (abs(actred) <= epsmch && prered <= epsmch && p5 * ratio <= 1.) {
*info = 6;
}
if (delta <= epsmch * xnorm) {
*info = 7;
}
if (gnorm <= epsmch) {
*info = 8;
}
if (*info != 0) {
goto L340;
}
/* end of the inner loop. repeat if iteration unsuccessful. */
if (ratio < p0001) {
goto L240;
}
/* end of the outer loop. */
goto L30;
L340:
/* termination, either normal or user imposed. */
if (iflag < 0) {
*info = iflag;
}
iflag = 0;
if (*nprint > 0) {
fcnderstr_mn(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
}
return;
/* last card of subroutine lmstr. */
} /* lmstr_ */
__minpack_attr__
void __minpack_func__(hybrj1)(__minpack_decl_fcnder_nn__ const int *n, real *x, real *
fvec, real *fjac, const int *ldfjac, const real *tol, int *
info, real *wa, const int *lwa, void* user_data)
{
/* Initialized data */
const real factor = 100.;
/* System generated locals */
int fjac_dim1, fjac_offset, i__1;
/* Local variables */
int j, lr, mode, nfev, njev;
real xtol;
int maxfev, nprint;
/* ********** */
/* subroutine hybrj1 */
/* the purpose of hybrj1 is to find a zero of a system of */
/* n nonlinear functions in n variables by a modification */
/* of the powell hybrid method. this is done by using the */
/* more general nonlinear equation solver hybrj. the user */
/* must provide a subroutine which calculates the functions */
/* and the jacobian. */
/* the subroutine statement is */
/* subroutine hybrj1(fcn,n,x,fvec,fjac,ldfjac,tol,info,wa,lwa) */
/* where */
/* fcn is the name of the user-supplied subroutine which */
/* calculates the functions and the jacobian. fcn must */
/* be declared in an external statement in the user */
/* calling program, and should be written as follows. */
/* subroutine fcn(n,x,fvec,fjac,ldfjac,iflag) */
/* integer n,ldfjac,iflag */
/* double precision x(n),fvec(n),fjac(ldfjac,n) */
/* ---------- */
/* if iflag = 1 calculate the functions at x and */
/* return this vector in fvec. do not alter fjac. */
/* if iflag = 2 calculate the jacobian at x and */
/* return this matrix in fjac. do not alter fvec. */
/* --------- */
/* return */
/* end */
/* the value of iflag should not be changed by fcn unless */
/* the user wants to terminate execution of hybrj1. */
/* in this case set iflag to a negative integer. */
/* n is a positive integer input variable set to the number */
/* of functions and variables. */
/* x is an array of length n. on input x must contain */
/* an initial estimate of the solution vector. on output x */
/* contains the final estimate of the solution vector. */
/* fvec is an output array of length n which contains */
/* the functions evaluated at the output x. */
/* fjac is an output n by n array which contains the */
/* orthogonal matrix q produced by the qr factorization */
/* of the final approximate jacobian. */
/* ldfjac is a positive integer input variable not less than n */
/* which specifies the leading dimension of the array fjac. */
/* tol is a nonnegative input variable. termination occurs */
/* when the algorithm estimates that the relative error */
/* between x and the solution is at most tol. */
/* info is an integer output variable. if the user has */
/* terminated execution, info is set to the (negative) */
/* value of iflag. see description of fcn. otherwise, */
/* info is set as follows. */
/* info = 0 improper input parameters. */
/* info = 1 algorithm estimates that the relative error */
/* between x and the solution is at most tol. */
/* info = 2 number of calls to fcn with iflag = 1 has */
/* reached 100*(n+1). */
/* info = 3 tol is too small. no further improvement in */
/* the approximate solution x is possible. */
/* info = 4 iteration is not making good progress. */
/* wa is a work array of length lwa. */
/* lwa is a positive integer input variable not less than */
/* (n*(n+13))/2. */
/* subprograms called */
/* user-supplied ...... fcn */
/* minpack-supplied ... hybrj */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--fvec;
--x;
fjac_dim1 = *ldfjac;
fjac_offset = 1 + fjac_dim1 * 1;
fjac -= fjac_offset;
--wa;
/* Function Body */
*info = 0;
/* check the input parameters for errors. */
if (*n <= 0 || *ldfjac < *n || *tol < 0. || *lwa < *n * (*n + 13) / 2) {
/* goto L20; */
return;
}
/* call hybrj. */
maxfev = (*n + 1) * 100;
xtol = *tol;
mode = 2;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa[j] = 1.;
/* L10: */
}
nprint = 0;
lr = *n * (*n + 1) / 2;
__minpack_func__(hybrj)(__minpack_param_fcnder_nn__ n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &xtol, &
maxfev, &wa[1], &mode, &factor, &nprint, info, &nfev, &njev, &wa[*
n * 6 + 1], &lr, &wa[*n + 1], &wa[(*n << 1) + 1], &wa[*n * 3 + 1],
&wa[(*n << 2) + 1], &wa[*n * 5 + 1], user_data);
if (*info == 5) {
*info = 4;
}
/* L20: */
return;
/* last card of subroutine hybrj1. */
} /* hybrj1_ */
__minpack_attr__
void __minpack_func__(lmdif1)(__minpack_decl_fcn_mn__ const int *m, const int *n, real *x,
real *fvec, const real *tol, int *info, int *iwa,
real *wa, const int *lwa)
{
/* Initialized data */
const real factor = 100.;
int mp5n, mode, nfev;
real ftol, gtol, xtol;
real epsfcn;
int maxfev, nprint;
/* ********** */
/* subroutine lmdif1 */
/* the purpose of lmdif1 is to minimize the sum of the squares of */
/* m nonlinear functions in n variables by a modification of the */
/* levenberg-marquardt algorithm. this is done by using the more */
/* general least-squares solver lmdif. the user must provide a */
/* subroutine which calculates the functions. the jacobian is */
/* then calculated by a forward-difference approximation. */
/* the subroutine statement is */
/* subroutine lmdif1(fcn,m,n,x,fvec,tol,info,iwa,wa,lwa) */
/* where */
/* fcn is the name of the user-supplied subroutine which */
/* calculates the functions. fcn must be declared */
/* in an external statement in the user calling */
/* program, and should be written as follows. */
/* subroutine fcn(m,n,x,fvec,iflag) */
/* integer m,n,iflag */
/* double precision x(n),fvec(m) */
/* ---------- */
/* calculate the functions at x and */
/* return this vector in fvec. */
/* ---------- */
/* return */
/* end */
/* the value of iflag should not be changed by fcn unless */
/* the user wants to terminate execution of lmdif1. */
/* in this case set iflag to a negative integer. */
/* m is a positive integer input variable set to the number */
/* of functions. */
/* n is a positive integer input variable set to the number */
/* of variables. n must not exceed m. */
/* x is an array of length n. on input x must contain */
/* an initial estimate of the solution vector. on output x */
/* contains the final estimate of the solution vector. */
/* fvec is an output array of length m which contains */
/* the functions evaluated at the output x. */
/* tol is a nonnegative input variable. termination occurs */
/* when the algorithm estimates either that the relative */
/* error in the sum of squares is at most tol or that */
/* the relative error between x and the solution is at */
/* most tol. */
/* info is an integer output variable. if the user has */
/* terminated execution, info is set to the (negative) */
/* value of iflag. see description of fcn. otherwise, */
/* info is set as follows. */
/* info = 0 improper input parameters. */
/* info = 1 algorithm estimates that the relative error */
/* in the sum of squares is at most tol. */
/* info = 2 algorithm estimates that the relative error */
/* between x and the solution is at most tol. */
/* info = 3 conditions for info = 1 and info = 2 both hold. */
/* info = 4 fvec is orthogonal to the columns of the */
/* jacobian to machine precision. */
/* info = 5 number of calls to fcn has reached or */
/* exceeded 200*(n+1). */
/* info = 6 tol is too small. no further reduction in */
/* the sum of squares is possible. */
/* info = 7 tol is too small. no further improvement in */
/* the approximate solution x is possible. */
/* iwa is an integer work array of length n. */
/* wa is a work array of length lwa. */
/* lwa is a positive integer input variable not less than */
/* m*n+5*n+m. */
/* subprograms called */
/* user-supplied ...... fcn */
/* minpack-supplied ... lmdif */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--fvec;
--iwa;
--x;
--wa;
/* Function Body */
*info = 0;
/* check the input parameters for errors. */
if (*n <= 0 || *m < *n || *tol < 0. || *lwa < *m * *n + *n * 5 + *m) {
/* goto L10; */
return;
}
/* call lmdif. */
maxfev = (*n + 1) * 200;
ftol = *tol;
xtol = *tol;
gtol = 0.;
epsfcn = 0.;
mode = 1;
nprint = 0;
mp5n = *m + *n * 5;
__minpack_func__(lmdif)(__minpack_param_fcn_mn__ m, n, &x[1], &fvec[1], &ftol, &xtol, >ol, &maxfev, &
epsfcn, &wa[1], &mode, &factor, &nprint, info, &nfev, &wa[mp5n +
1], m, &iwa[1], &wa[*n + 1], &wa[(*n << 1) + 1], &wa[*n * 3 + 1],
&wa[(*n << 2) + 1], &wa[*n * 5 + 1]);
if (*info == 8) {
*info = 4;
}
/* L10: */
return;
/* last card of subroutine lmdif1. */
} /* lmdif1_ */
/* Main program */
int main(int argc, char **argv)
{
/* Initialized data */
const int a[14] = { 0,0,0,1,0,0,0,1,0,0,0,0,1,0 };
real cp = .123;
/* Local variables */
int i, n;
real x1[10], x2[10];
int na[14], np[14];
real err[10];
int lnp;
real fjac[10*10];
const int ldfjac = 10;
real diff[10];
real fvec1[10], fvec2[10];
int nprob;
real errmin[14], errmax[14];
const int i1 = 1, i2 = 2;
for (;;) {
scanf("%5d%5d\n", &nprob, &n);
if (nprob <= 0) {
break;
}
hybipt(n,x1,nprob,1.);
for(i=0; i<n; ++i) {
x1[i] += cp;
cp = -cp;
}
printf("\n\n\n problem%5d with dimension%5d is %c\n\n", nprob, n, a[nprob-1]?'T':'F');
__minpack_func__(chkder)(&n,&n,x1,NULL,NULL,&ldfjac,x2,NULL,&i1,NULL);
vecfcn(n,x1,fvec1,nprob);
errjac(n,x1,fjac,ldfjac,nprob);
vecfcn(n,x2,fvec2,nprob);
__minpack_func__(chkder)(&n,&n,x1,fvec1,fjac,&ldfjac,NULL,fvec2,&i2,err);
errmin[nprob-1] = err[0];
errmax[nprob-1] = err[0];
for(i=0; i<n; ++i) {
diff[i] = fvec2[i] - fvec1[i];
if (errmin[nprob-1] > err[i])
errmin[nprob-1] = err[i];
if (errmax[nprob-1] < err[i])
errmax[nprob-1] = err[i];
}
np[nprob-1] = nprob;
lnp = nprob;
na[nprob-1] = n;
printf("\n first function vector \n\n");
printvec(n, fvec1);
printf("\n\n function difference vector\n\n");
printvec(n, diff);
printf("\n\n error vector\n\n");
printvec(n, err);
}
printf("\f summary of %3d tests of chkder\n", lnp);
printf("\n nprob n status errmin errmax\n\n");
for (i = 0; i < lnp; ++i) {
printf("%4d%6d %c %15.7e%15.7e\n",
np[i], na[i], a[i]?'T':'F', (double)errmin[i], (double)errmax[i]);
}
exit(0);
}
__minpack_attr__
void __minpack_func__(hybrd)(__minpack_decl_fcn_nn__ const int *n, real *x, real *
fvec, const real *xtol, const int *maxfev, const int *ml, const int *mu,
const real *epsfcn, real *diag, const int *mode, const real *
factor, const int *nprint, int *info, int *nfev, real *
fjac, const int *ldfjac, real *r__, const int *lr, real *qtf,
real *wa1, real *wa2, real *wa3, real *wa4)
{
/* Table of constant values */
const int c__1 = 1;
const int c_false = FALSE_;
/* Initialized data */
#define p1 .1
#define p5 .5
#define p001 .001
#define p0001 1e-4
/* System generated locals */
int fjac_dim1, fjac_offset, i__1, i__2;
real d__1, d__2;
/* Local variables */
int i__, j, l, jm1, iwa[1];
real sum;
int sing;
int iter;
real temp;
int msum, iflag;
real delta;
int jeval;
int ncsuc;
real ratio;
real fnorm;
real pnorm, xnorm = 0, fnorm1;
int nslow1, nslow2;
int ncfail;
real actred, epsmch, prered;
/* ********** */
/* subroutine hybrd */
/* the purpose of hybrd is to find a zero of a system of */
/* n nonlinear functions in n variables by a modification */
/* of the powell hybrid method. the user must provide a */
/* subroutine which calculates the functions. the jacobian is */
/* then calculated by a forward-difference approximation. */
/* the subroutine statement is */
/* subroutine hybrd(fcn,n,x,fvec,xtol,maxfev,ml,mu,epsfcn, */
/* diag,mode,factor,nprint,info,nfev,fjac, */
/* ldfjac,r,lr,qtf,wa1,wa2,wa3,wa4) */
/* where */
/* fcn is the name of the user-supplied subroutine which */
/* calculates the functions. fcn must be declared */
/* in an external statement in the user calling */
/* program, and should be written as follows. */
/* subroutine fcn(n,x,fvec,iflag) */
/* integer n,iflag */
/* double precision x(n),fvec(n) */
/* ---------- */
/* calculate the functions at x and */
/* return this vector in fvec. */
/* --------- */
/* return */
/* end */
/* the value of iflag should not be changed by fcn unless */
/* the user wants to terminate execution of hybrd. */
/* in this case set iflag to a negative integer. */
/* n is a positive integer input variable set to the number */
/* of functions and variables. */
/* x is an array of length n. on input x must contain */
/* an initial estimate of the solution vector. on output x */
/* contains the final estimate of the solution vector. */
/* fvec is an output array of length n which contains */
/* the functions evaluated at the output x. */
/* xtol is a nonnegative input variable. termination */
/* occurs when the relative error between two consecutive */
/* iterates is at most xtol. */
/* maxfev is a positive integer input variable. termination */
/* occurs when the number of calls to fcn is at least maxfev */
/* by the end of an iteration. */
/* ml is a nonnegative integer input variable which specifies */
/* the number of subdiagonals within the band of the */
/* jacobian matrix. if the jacobian is not banded, set */
/* ml to at least n - 1. */
/* mu is a nonnegative integer input variable which specifies */
/* the number of superdiagonals within the band of the */
/* jacobian matrix. if the jacobian is not banded, set */
/* mu to at least n - 1. */
/* epsfcn is an input variable used in determining a suitable */
/* step length for the forward-difference approximation. this */
/* approximation assumes that the relative errors in the */
/* functions are of the order of epsfcn. if epsfcn is less */
/* than the machine precision, it is assumed that the relative */
/* errors in the functions are of the order of the machine */
/* precision. */
/* diag is an array of length n. if mode = 1 (see */
/* below), diag is internally set. if mode = 2, diag */
/* must contain positive entries that serve as */
/* multiplicative scale factors for the variables. */
/* mode is an integer input variable. if mode = 1, the */
/* variables will be scaled internally. if mode = 2, */
/* the scaling is specified by the input diag. other */
/* values of mode are equivalent to mode = 1. */
/* factor is a positive input variable used in determining the */
/* initial step bound. this bound is set to the product of */
/* factor and the euclidean norm of diag*x if nonzero, or else */
/* to factor itself. in most cases factor should lie in the */
/* interval (.1,100.). 100. is a generally recommended value. */
/* nprint is an integer input variable that enables controlled */
/* printing of iterates if it is positive. in this case, */
/* fcn is called with iflag = 0 at the beginning of the first */
/* iteration and every nprint iterations thereafter and */
/* immediately prior to return, with x and fvec available */
/* for printing. if nprint is not positive, no special calls */
/* of fcn with iflag = 0 are made. */
/* info is an integer output variable. if the user has */
/* terminated execution, info is set to the (negative) */
/* value of iflag. see description of fcn. otherwise, */
/* info is set as follows. */
/* info = 0 improper input parameters. */
/* info = 1 relative error between two consecutive iterates */
/* is at most xtol. */
/* info = 2 number of calls to fcn has reached or exceeded */
/* maxfev. */
/* info = 3 xtol is too small. no further improvement in */
/* the approximate solution x is possible. */
/* info = 4 iteration is not making good progress, as */
/* measured by the improvement from the last */
/* five jacobian evaluations. */
/* info = 5 iteration is not making good progress, as */
/* measured by the improvement from the last */
/* ten iterations. */
/* nfev is an integer output variable set to the number of */
/* calls to fcn. */
/* fjac is an output n by n array which contains the */
/* orthogonal matrix q produced by the qr factorization */
/* of the final approximate jacobian. */
/* ldfjac is a positive integer input variable not less than n */
/* which specifies the leading dimension of the array fjac. */
/* r is an output array of length lr which contains the */
/* upper triangular matrix produced by the qr factorization */
/* of the final approximate jacobian, stored rowwise. */
/* lr is a positive integer input variable not less than */
/* (n*(n+1))/2. */
/* qtf is an output array of length n which contains */
/* the vector (q transpose)*fvec. */
/* wa1, wa2, wa3, and wa4 are work arrays of length n. */
/* subprograms called */
/* user-supplied ...... fcn */
/* minpack-supplied ... dogleg,dpmpar,enorm,fdjac1, */
/* qform,qrfac,r1mpyq,r1updt */
/* fortran-supplied ... dabs,dmax1,dmin1,min0,mod */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa4;
--wa3;
--wa2;
--wa1;
--qtf;
--diag;
--fvec;
--x;
fjac_dim1 = *ldfjac;
fjac_offset = 1 + fjac_dim1 * 1;
fjac -= fjac_offset;
--r__;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = __minpack_func__(dpmpar)(&c__1);
*info = 0;
iflag = 0;
*nfev = 0;
/* check the input parameters for errors. */
if (*n <= 0 || *xtol < 0. || *maxfev <= 0 || *ml < 0 || *mu < 0 || *
factor <= 0. || *ldfjac < *n || *lr < *n * (*n + 1) / 2) {
goto L300;
}
if (*mode != 2) {
goto L20;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (diag[j] <= 0.) {
goto L300;
}
/* L10: */
}
L20:
/* evaluate the function at the starting point */
/* and calculate its norm. */
iflag = 1;
fcn_nn(n, &x[1], &fvec[1], &iflag);
*nfev = 1;
if (iflag < 0) {
goto L300;
}
fnorm = __minpack_func__(enorm)(n, &fvec[1]);
/* determine the number of calls to fcn needed to compute */
/* the jacobian matrix. */
/* Computing MIN */
i__1 = *ml + *mu + 1;
msum = min(i__1,*n);
/* initialize iteration counter and monitors. */
iter = 1;
ncsuc = 0;
ncfail = 0;
nslow1 = 0;
nslow2 = 0;
/* beginning of the outer loop. */
L30:
jeval = TRUE_;
/* calculate the jacobian matrix. */
iflag = 2;
__minpack_func__(fdjac1)(__minpack_param_fcn_nn__ n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
ml, mu, epsfcn, &wa1[1], &wa2[1]);
*nfev += msum;
if (iflag < 0) {
goto L300;
}
/* compute the qr factorization of the jacobian. */
__minpack_func__(qrfac)(n, n, &fjac[fjac_offset], ldfjac, &c_false, iwa, &c__1, &wa1[1], &
wa2[1], &wa3[1]);
/* on the first iteration and if mode is 1, scale according */
/* to the norms of the columns of the initial jacobian. */
if (iter != 1) {
goto L70;
}
if (*mode == 2) {
goto L50;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
diag[j] = wa2[j];
if (wa2[j] == 0.) {
diag[j] = 1.;
}
/* L40: */
}
L50:
/* on the first iteration, calculate the norm of the scaled x */
/* and initialize the step bound delta. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa3[j] = diag[j] * x[j];
/* L60: */
}
xnorm = __minpack_func__(enorm)(n, &wa3[1]);
delta = *factor * xnorm;
if (delta == 0.) {
delta = *factor;
}
L70:
/* form (q transpose)*fvec and store in qtf. */
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
qtf[i__] = fvec[i__];
/* L80: */
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
if (fjac[j + j * fjac_dim1] == 0.) {
goto L110;
}
sum = 0.;
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
sum += fjac[i__ + j * fjac_dim1] * qtf[i__];
/* L90: */
}
temp = -sum / fjac[j + j * fjac_dim1];
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
qtf[i__] += fjac[i__ + j * fjac_dim1] * temp;
/* L100: */
}
L110:
/* L120: */
;
}
/* copy the triangular factor of the qr factorization into r. */
sing = FALSE_;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = j;
jm1 = j - 1;
if (jm1 < 1) {
goto L140;
}
i__2 = jm1;
for (i__ = 1; i__ <= i__2; ++i__) {
r__[l] = fjac[i__ + j * fjac_dim1];
l = l + *n - i__;
/* L130: */
}
L140:
r__[l] = wa1[j];
if (wa1[j] == 0.) {
sing = TRUE_;
}
/* L150: */
}
/* accumulate the orthogonal factor in fjac. */
__minpack_func__(qform)(n, n, &fjac[fjac_offset], ldfjac, &wa1[1]);
/* rescale if necessary. */
if (*mode == 2) {
goto L170;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
d__1 = diag[j], d__2 = wa2[j];
diag[j] = max(d__1,d__2);
/* L160: */
}
L170:
/* beginning of the inner loop. */
L180:
/* if requested, call fcn to enable printing of iterates. */
if (*nprint <= 0) {
goto L190;
}
iflag = 0;
if ((iter - 1) % *nprint == 0) {
fcn_nn(n, &x[1], &fvec[1], &iflag);
}
if (iflag < 0) {
goto L300;
}
L190:
/* determine the direction p. */
__minpack_func__(dogleg)(n, &r__[1], lr, &diag[1], &qtf[1], &delta, &wa1[1], &wa2[1], &wa3[
1]);
/* store the direction p and x + p. calculate the norm of p. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = -wa1[j];
wa2[j] = x[j] + wa1[j];
wa3[j] = diag[j] * wa1[j];
/* L200: */
}
pnorm = __minpack_func__(enorm)(n, &wa3[1]);
/* on the first iteration, adjust the initial step bound. */
if (iter == 1) {
delta = min(delta,pnorm);
}
/* evaluate the function at x + p and calculate its norm. */
iflag = 1;
fcn_nn(n, &wa2[1], &wa4[1], &iflag);
++(*nfev);
if (iflag < 0) {
goto L300;
}
fnorm1 = __minpack_func__(enorm)(n, &wa4[1]);
/* compute the scaled actual reduction. */
actred = -1.;
if (fnorm1 < fnorm) {
/* Computing 2nd power */
d__1 = fnorm1 / fnorm;
actred = 1. - d__1 * d__1;
}
/* compute the scaled predicted reduction. */
l = 1;
i__1 = *n;
for (i__ = 1; i__ <= i__1; ++i__) {
sum = 0.;
i__2 = *n;
for (j = i__; j <= i__2; ++j) {
sum += r__[l] * wa1[j];
++l;
/* L210: */
}
wa3[i__] = qtf[i__] + sum;
/* L220: */
}
temp = __minpack_func__(enorm)(n, &wa3[1]);
prered = 0.;
if (temp < fnorm) {
/* Computing 2nd power */
d__1 = temp / fnorm;
prered = 1. - d__1 * d__1;
}
/* compute the ratio of the actual to the predicted */
/* reduction. */
ratio = 0.;
if (prered > 0.) {
ratio = actred / prered;
}
/* update the step bound. */
if (ratio >= p1) {
goto L230;
}
ncsuc = 0;
++ncfail;
delta = p5 * delta;
goto L240;
L230:
ncfail = 0;
++ncsuc;
if (ratio >= p5 || ncsuc > 1) {
/* Computing MAX */
d__1 = delta, d__2 = pnorm / p5;
delta = max(d__1,d__2);
}
if (fabs(ratio - 1.) <= p1) {
delta = pnorm / p5;
}
L240:
/* test for successful iteration. */
if (ratio < p0001) {
goto L260;
}
/* successful iteration. update x, fvec, and their norms. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
x[j] = wa2[j];
wa2[j] = diag[j] * x[j];
fvec[j] = wa4[j];
/* L250: */
}
xnorm = __minpack_func__(enorm)(n, &wa2[1]);
fnorm = fnorm1;
++iter;
L260:
/* determine the progress of the iteration. */
++nslow1;
if (actred >= p001) {
nslow1 = 0;
}
if (jeval) {
++nslow2;
}
if (actred >= p1) {
nslow2 = 0;
}
/* test for convergence. */
if (delta <= *xtol * xnorm || fnorm == 0.) {
*info = 1;
}
if (*info != 0) {
goto L300;
}
/* tests for termination and stringent tolerances. */
if (*nfev >= *maxfev) {
*info = 2;
}
/* Computing MAX */
d__1 = p1 * delta;
if (p1 * max(d__1,pnorm) <= epsmch * xnorm) {
*info = 3;
}
if (nslow2 == 5) {
*info = 4;
}
if (nslow1 == 10) {
*info = 5;
}
if (*info != 0) {
goto L300;
}
/* criterion for recalculating jacobian approximation */
/* by forward differences. */
if (ncfail == 2) {
goto L290;
}
/* calculate the rank one modification to the jacobian */
/* and update qtf if necessary. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
sum = 0.;
i__2 = *n;
for (i__ = 1; i__ <= i__2; ++i__) {
sum += fjac[i__ + j * fjac_dim1] * wa4[i__];
/* L270: */
}
wa2[j] = (sum - wa3[j]) / pnorm;
wa1[j] = diag[j] * (diag[j] * wa1[j] / pnorm);
if (ratio >= p0001) {
qtf[j] = sum;
}
/* L280: */
}
/* compute the qr factorization of the updated jacobian. */
__minpack_func__(r1updt)(n, n, &r__[1], lr, &wa1[1], &wa2[1], &wa3[1], &sing);
__minpack_func__(r1mpyq)(n, n, &fjac[fjac_offset], ldfjac, &wa2[1], &wa3[1]);
__minpack_func__(r1mpyq)(&c__1, n, &qtf[1], &c__1, &wa2[1], &wa3[1]);
/* end of the inner loop. */
jeval = FALSE_;
goto L180;
L290:
/* end of the outer loop. */
goto L30;
L300:
/* termination, either normal or user imposed. */
if (iflag < 0) {
*info = iflag;
}
iflag = 0;
if (*nprint > 0) {
fcn_nn(n, &x[1], &fvec[1], &iflag);
}
return;
/* last card of subroutine hybrd. */
} /* hybrd_ */
int main()
{
int i, j, m, n, maxfev, mode, nprint, info, nfev, ldfjac;
int ipvt[3];
real ftol, xtol, gtol, epsfcn, factor, fnorm;
real x[3], fvec[15], diag[3], fjac[15*3], qtf[3],
wa1[3], wa2[3], wa3[3], wa4[15];
int one=1;
real covfac;
m = 15;
n = 3;
/* the following starting values provide a rough fit. */
x[1-1] = 1.;
x[2-1] = 1.;
x[3-1] = 1.;
ldfjac = 15;
/* set ftol and xtol to the square root of the machine */
/* and gtol to zero. unless high solutions are */
/* required, these are the recommended settings. */
ftol = sqrt(__minpack_func__(dpmpar)(&one));
xtol = sqrt(__minpack_func__(dpmpar)(&one));
gtol = 0.;
maxfev = 800;
epsfcn = 0.;
mode = 1;
factor = 1.e2;
nprint = 0;
__minpack_func__(lmdif)(&fcn, &m, &n, x, fvec, &ftol, &xtol, >ol, &maxfev, &epsfcn,
diag, &mode, &factor, &nprint, &info, &nfev, fjac, &ldfjac,
ipvt, qtf, wa1, wa2, wa3, wa4);
fnorm = __minpack_func__(enorm)(&m, fvec);
printf(" final l2 norm of the residuals%15.7g\n\n", (double)fnorm);
printf(" number of function evaluations%10i\n\n", nfev);
printf(" exit parameter %10i\n\n", info);
printf(" final approximate solution\n");
for (j=1; j<=n; j++) {
printf("%s%15.7g", j%3==1?"\n ":"", (double)x[j-1]);
}
printf("\n");
ftol = __minpack_func__(dpmpar)(&one);
covfac = fnorm*fnorm/(m-n);
__minpack_func__(covar)(&n, fjac, &ldfjac, ipvt, &ftol, wa1);
printf(" covariance\n");
for (i=1; i<=n; i++) {
for (j=1; j<=n; j++) {
printf("%s%15.7g", j%3==1?"\n ":"", (double)fjac[(i-1)*ldfjac+j-1]*covfac);
}
}
printf("\n");
return 0;
}
