/* Subroutine */ void lmstr_(void (*fcn)(const int *m, const int *n, const double *x, double *fvec,
double *fjrow, int *iflag ), const int *m, const int *n, double *x,
double *fvec, double *fjac, const int *ldfjac, const double *ftol,
const double *xtol, const double *gtol, const int *maxfev, double *
diag, const int *mode, const double *factor, const int *nprint, int *
info, int *nfev, int *njev, int *ipvt, double *qtf,
double *wa1, double *wa2, double *wa3, double *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;
double d__1, d__2, d__3;
/* Local variables */
int i__, j, l;
double par, sum;
int sing;
int iter;
double temp, temp1, temp2;
int iflag;
double delta;
double ratio;
double fnorm, gnorm, pnorm, xnorm, 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 = 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;
(*fcn)(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
*nfev = 1;
if (iflag < 0) {
goto L340;
}
fnorm = 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) {
(*fcn)(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__) {
(*fcn)(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
if (iflag < 0) {
goto L340;
}
temp = fvec[i__];
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] = enorm_(&j, &fjac[j * fjac_dim1 + 1]);
/* L80: */
}
if (! sing) {
goto L130;
}
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 = 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. */
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 = 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)(m, n, &wa2[1], &wa4[1], &wa3[1], &iflag);
++(*nfev);
if (iflag < 0) {
goto L340;
}
fnorm1 = 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 = 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 = 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) {
(*fcn)(m, n, &x[1], &fvec[1], &wa3[1], &iflag);
}
return;
/* last card of subroutine lmstr. */
} /* lmstr_ */
/*< >*/
/* Subroutine */ int lmder_(
void (*fcn)(v3p_netlib_integer*,
v3p_netlib_integer*,
v3p_netlib_doublereal*,
v3p_netlib_doublereal*,
v3p_netlib_doublereal*,
v3p_netlib_integer*,
v3p_netlib_integer*,
void*),
integer *m, integer *n, doublereal *x,
doublereal *fvec, doublereal *fjac, integer *ldfjac, doublereal *ftol,
doublereal *xtol, doublereal *gtol, integer *maxfev, doublereal *
diag, integer *mode, doublereal *factor, integer *nprint, integer *
info, integer *nfev, integer *njev, integer *ipvt, doublereal *qtf,
doublereal *wa1, doublereal *wa2, doublereal *wa3, doublereal *wa4,
void* userdata)
{
/* Initialized data */
static doublereal one = 1.; /* constant */
static doublereal p1 = .1; /* constant */
static doublereal p5 = .5; /* constant */
static doublereal p25 = .25; /* constant */
static doublereal p75 = .75; /* constant */
static doublereal p0001 = 1e-4; /* constant */
static doublereal zero = 0.; /* constant */
/* System generated locals */
integer fjac_dim1, fjac_offset, i__1, i__2;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, l;
doublereal par, sum;
integer iter;
doublereal temp=0, temp1, temp2;
integer iflag;
doublereal delta;
extern /* Subroutine */ int qrfac_(integer *, integer *, doublereal *,
integer *, logical *, integer *, integer *, doublereal *,
doublereal *, doublereal *), lmpar_(integer *, doublereal *,
integer *, integer *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
doublereal ratio;
extern doublereal enorm_(integer *, doublereal *);
doublereal fnorm, gnorm, pnorm, xnorm=0, fnorm1, actred, dirder, epsmch,
prered;
extern doublereal dpmpar_(integer *);
/*< integer m,n,ldfjac,maxfev,mode,nprint,info,nfev,njev >*/
/*< integer ipvt(n) >*/
/*< double precision ftol,xtol,gtol,factor >*/
/*< >*/
/* ********** */
/* subroutine lmder */
/* the purpose of lmder is to minimize the sum of the squares of */
/* m nonlinear functions in n variables by a modification of */
/* the levenberg-marquardt algorithm. the user must provide a */
/* subroutine which calculates the functions and the jacobian. */
/* the subroutine statement is */
/* subroutine lmder(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 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,fjac,ldfjac,iflag) */
/* integer m,n,ldfjac,iflag */
/* double precision x(n),fvec(m),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 lmder. */
/* 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 m by n array. the upper n by n submatrix */
/* of fjac contains an upper triangular matrix r with */
/* diagonal elements of nonincreasing magnitude 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 trapezoidal */
/* part of fjac contains information generated during */
/* the computation of r. */
/* ldfjac is a positive integer input variable not less than m */
/* 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, fvec, and fjac */
/* available for printing. fvec and fjac should not be */
/* altered. 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 */
/* with diagonal elements of nonincreasing magnitude. */
/* 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 */
/* fortran-supplied ... dabs,dmax1,dmin1,dsqrt,mod */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/*< integer i,iflag,iter,j,l >*/
/*< >*/
/*< double precision dpmpar,enorm >*/
/*< >*/
/* Parameter adjustments */
--wa4;
--fvec;
--wa3;
--wa2;
--wa1;
--qtf;
--ipvt;
--diag;
--x;
fjac_dim1 = *ldfjac;
fjac_offset = 1 + fjac_dim1;
fjac -= fjac_offset;
/* Function Body */
/* epsmch is the machine precision. */
/*< epsmch = dpmpar(1) >*/
epsmch = dpmpar_(&c__1);
/*< info = 0 >*/
*info = 0;
/*< iflag = 0 >*/
iflag = 0;
/*< nfev = 0 >*/
*nfev = 0;
/*< njev = 0 >*/
*njev = 0;
/* check the input parameters for errors. */
/*< >*/
if (*n <= 0 || *m < *n || *ldfjac < *m || *ftol < zero || *xtol < zero ||
*gtol < zero || *maxfev <= 0 || *factor <= zero) {
goto L300;
}
/*< if (mode .ne. 2) go to 20 >*/
if (*mode != 2) {
goto L20;
}
/*< do 10 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< if (diag(j) .le. zero) go to 300 >*/
if (diag[j] <= zero) {
goto L300;
}
/*< 10 continue >*/
/* L10: */
}
/*< 20 continue >*/
L20:
/* evaluate the function at the starting point */
/* and calculate its norm. */
/*< iflag = 1 >*/
iflag = 1;
/*< call fcn(m,n,x,fvec,fjac,ldfjac,iflag) >*/
(*fcn)(m, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
userdata);
/*< nfev = 1 >*/
*nfev = 1;
/*< if (iflag .lt. 0) go to 300 >*/
if (iflag < 0) {
goto L300;
}
/*< fnorm = enorm(m,fvec) >*/
fnorm = enorm_(m, &fvec[1]);
/* initialize levenberg-marquardt parameter and iteration counter. */
/*< par = zero >*/
par = zero;
/*< iter = 1 >*/
iter = 1;
/* beginning of the outer loop. */
/*< 30 continue >*/
L30:
/* calculate the jacobian matrix. */
/*< iflag = 2 >*/
iflag = 2;
/*< call fcn(m,n,x,fvec,fjac,ldfjac,iflag) >*/
(*fcn)(m, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
userdata);
/*< njev = njev + 1 >*/
++(*njev);
/*< if (iflag .lt. 0) go to 300 >*/
if (iflag < 0) {
goto L300;
}
/* if requested, call fcn to enable printing of iterates. */
/*< if (nprint .le. 0) go to 40 >*/
if (*nprint <= 0) {
goto L40;
}
/*< iflag = 0 >*/
iflag = 0;
/*< >*/
if ((iter - 1) % *nprint == 0) {
(*fcn)(m, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
userdata);
}
/*< if (iflag .lt. 0) go to 300 >*/
if (iflag < 0) {
goto L300;
}
/*< 40 continue >*/
L40:
/* compute the qr factorization of the jacobian. */
/*< call qrfac(m,n,fjac,ldfjac,.true.,ipvt,n,wa1,wa2,wa3) >*/
qrfac_(m, n, &fjac[fjac_offset], ldfjac, &c_true, &ipvt[1], n, &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 .ne. 1) go to 80 >*/
if (iter != 1) {
goto L80;
}
/*< if (mode .eq. 2) go to 60 >*/
if (*mode == 2) {
goto L60;
}
/*< do 50 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< diag(j) = wa2(j) >*/
diag[j] = wa2[j];
/*< if (wa2(j) .eq. zero) diag(j) = one >*/
if (wa2[j] == zero) {
diag[j] = one;
}
/*< 50 continue >*/
/* L50: */
}
/*< 60 continue >*/
L60:
/* on the first iteration, calculate the norm of the scaled x */
/* and initialize the step bound delta. */
/*< do 70 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< wa3(j) = diag(j)*x(j) >*/
wa3[j] = diag[j] * x[j];
/*< 70 continue >*/
/* L70: */
}
/*< xnorm = enorm(n,wa3) >*/
xnorm = enorm_(n, &wa3[1]);
/*< delta = factor*xnorm >*/
delta = *factor * xnorm;
/*< if (delta .eq. zero) delta = factor >*/
if (delta == zero) {
delta = *factor;
}
/*< 80 continue >*/
L80:
/* form (q transpose)*fvec and store the first n components in */
/* qtf. */
/*< do 90 i = 1, m >*/
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
/*< wa4(i) = fvec(i) >*/
wa4[i__] = fvec[i__];
/*< 90 continue >*/
/* L90: */
}
/*< do 130 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< if (fjac(j,j) .eq. zero) go to 120 >*/
if (fjac[j + j * fjac_dim1] == zero) {
goto L120;
}
/*< sum = zero >*/
sum = zero;
/*< do 100 i = j, m >*/
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
/*< sum = sum + fjac(i,j)*wa4(i) >*/
sum += fjac[i__ + j * fjac_dim1] * wa4[i__];
/*< 100 continue >*/
/* L100: */
}
/*< temp = -sum/fjac(j,j) >*/
temp = -sum / fjac[j + j * fjac_dim1];
/*< do 110 i = j, m >*/
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
/*< wa4(i) = wa4(i) + fjac(i,j)*temp >*/
wa4[i__] += fjac[i__ + j * fjac_dim1] * temp;
/*< 110 continue >*/
/* L110: */
}
/*< 120 continue >*/
L120:
/*< fjac(j,j) = wa1(j) >*/
fjac[j + j * fjac_dim1] = wa1[j];
/*< qtf(j) = wa4(j) >*/
qtf[j] = wa4[j];
/*< 130 continue >*/
/* L130: */
}
/* compute the norm of the scaled gradient. */
/*< gnorm = zero >*/
gnorm = zero;
/*< if (fnorm .eq. zero) go to 170 >*/
if (fnorm == zero) {
goto L170;
}
/*< do 160 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< l = ipvt(j) >*/
l = ipvt[j];
/*< if (wa2(l) .eq. zero) go to 150 >*/
if (wa2[l] == zero) {
goto L150;
}
/*< sum = zero >*/
sum = zero;
/*< do 140 i = 1, j >*/
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
/*< sum = sum + fjac(i,j)*(qtf(i)/fnorm) >*/
sum += fjac[i__ + j * fjac_dim1] * (qtf[i__] / fnorm);
/*< 140 continue >*/
/* L140: */
}
/*< gnorm = dmax1(gnorm,dabs(sum/wa2(l))) >*/
/* Computing MAX */
d__2 = gnorm, d__3 = (d__1 = sum / wa2[l], abs(d__1));
gnorm = max(d__2,d__3);
/*< 150 continue >*/
L150:
/*< 160 continue >*/
/* L160: */
;
}
/*< 170 continue >*/
L170:
/* test for convergence of the gradient norm. */
/*< if (gnorm .le. gtol) info = 4 >*/
if (gnorm <= *gtol) {
*info = 4;
}
/*< if (info .ne. 0) go to 300 >*/
if (*info != 0) {
goto L300;
}
/* rescale if necessary. */
/*< if (mode .eq. 2) go to 190 >*/
if (*mode == 2) {
goto L190;
}
/*< do 180 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< diag(j) = dmax1(diag(j),wa2(j)) >*/
/* Computing MAX */
d__1 = diag[j], d__2 = wa2[j];
diag[j] = max(d__1,d__2);
/*< 180 continue >*/
/* L180: */
}
/*< 190 continue >*/
L190:
/* beginning of the inner loop. */
/*< 200 continue >*/
L200:
/* determine the levenberg-marquardt parameter. */
/*< >*/
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. */
/*< do 210 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< wa1(j) = -wa1(j) >*/
wa1[j] = -wa1[j];
/*< wa2(j) = x(j) + wa1(j) >*/
wa2[j] = x[j] + wa1[j];
/*< wa3(j) = diag(j)*wa1(j) >*/
wa3[j] = diag[j] * wa1[j];
/*< 210 continue >*/
/* L210: */
}
/*< pnorm = enorm(n,wa3) >*/
pnorm = enorm_(n, &wa3[1]);
/* on the first iteration, adjust the initial step bound. */
/*< if (iter .eq. 1) delta = dmin1(delta,pnorm) >*/
if (iter == 1) {
delta = min(delta,pnorm);
}
/* evaluate the function at x + p and calculate its norm. */
/*< iflag = 1 >*/
iflag = 1;
/*< call fcn(m,n,wa2,wa4,fjac,ldfjac,iflag) >*/
(*fcn)(m, n, &wa2[1], &wa4[1], &fjac[fjac_offset], ldfjac, &iflag,
userdata);
/*< nfev = nfev + 1 >*/
++(*nfev);
/*< if (iflag .lt. 0) go to 300 >*/
if (iflag < 0) {
goto L300;
}
/*< fnorm1 = enorm(m,wa4) >*/
fnorm1 = enorm_(m, &wa4[1]);
/* compute the scaled actual reduction. */
/*< actred = -one >*/
actred = -one;
/*< if (p1*fnorm1 .lt. fnorm) actred = one - (fnorm1/fnorm)**2 >*/
if (p1 * fnorm1 < fnorm) {
/* Computing 2nd power */
d__1 = fnorm1 / fnorm;
actred = one - d__1 * d__1;
}
/* compute the scaled predicted reduction and */
/* the scaled directional derivative. */
/*< do 230 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< wa3(j) = zero >*/
wa3[j] = zero;
/*< l = ipvt(j) >*/
l = ipvt[j];
/*< temp = wa1(l) >*/
temp = wa1[l];
/*< do 220 i = 1, j >*/
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
/*< wa3(i) = wa3(i) + fjac(i,j)*temp >*/
wa3[i__] += fjac[i__ + j * fjac_dim1] * temp;
/*< 220 continue >*/
/* L220: */
}
/*< 230 continue >*/
/* L230: */
}
/*< temp1 = enorm(n,wa3)/fnorm >*/
temp1 = enorm_(n, &wa3[1]) / fnorm;
/*< temp2 = (dsqrt(par)*pnorm)/fnorm >*/
temp2 = sqrt(par) * pnorm / fnorm;
/*< prered = temp1**2 + temp2**2/p5 >*/
/* Computing 2nd power */
d__1 = temp1;
/* Computing 2nd power */
d__2 = temp2;
prered = d__1 * d__1 + d__2 * d__2 / p5;
/*< dirder = -(temp1**2 + temp2**2) >*/
/* 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 = zero >*/
ratio = zero;
/*< if (prered .ne. zero) ratio = actred/prered >*/
if (prered != zero) {
ratio = actred / prered;
}
/* update the step bound. */
/*< if (ratio .gt. p25) go to 240 >*/
if (ratio > p25) {
goto L240;
}
/*< if (actred .ge. zero) temp = p5 >*/
if (actred >= zero) {
temp = p5;
}
/*< >*/
if (actred < zero) {
temp = p5 * dirder / (dirder + p5 * actred);
}
/*< if (p1*fnorm1 .ge. fnorm .or. temp .lt. p1) temp = p1 >*/
if (p1 * fnorm1 >= fnorm || temp < p1) {
temp = p1;
}
/*< delta = temp*dmin1(delta,pnorm/p1) >*/
/* Computing MIN */
d__1 = delta, d__2 = pnorm / p1;
delta = temp * min(d__1,d__2);
/*< par = par/temp >*/
par /= temp;
/*< go to 260 >*/
goto L260;
/*< 240 continue >*/
L240:
/*< if (par .ne. zero .and. ratio .lt. p75) go to 250 >*/
if (par != zero && ratio < p75) {
goto L250;
}
/*< delta = pnorm/p5 >*/
delta = pnorm / p5;
/*< par = p5*par >*/
par = p5 * par;
/*< 250 continue >*/
L250:
/*< 260 continue >*/
L260:
/* test for successful iteration. */
/*< if (ratio .lt. p0001) go to 290 >*/
if (ratio < p0001) {
goto L290;
}
/* successful iteration. update x, fvec, and their norms. */
/*< do 270 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< x(j) = wa2(j) >*/
x[j] = wa2[j];
/*< wa2(j) = diag(j)*x(j) >*/
wa2[j] = diag[j] * x[j];
/*< 270 continue >*/
/* L270: */
}
/*< do 280 i = 1, m >*/
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
/*< fvec(i) = wa4(i) >*/
fvec[i__] = wa4[i__];
/*< 280 continue >*/
/* L280: */
}
/*< xnorm = enorm(n,wa2) >*/
xnorm = enorm_(n, &wa2[1]);
/*< fnorm = fnorm1 >*/
fnorm = fnorm1;
/*< iter = iter + 1 >*/
++iter;
/*< 290 continue >*/
L290:
/* tests for convergence. */
/*< >*/
if (abs(actred) <= *ftol && prered <= *ftol && p5 * ratio <= one) {
*info = 1;
}
/*< if (delta .le. xtol*xnorm) info = 2 >*/
if (delta <= *xtol * xnorm) {
*info = 2;
}
/*< >*/
if (abs(actred) <= *ftol && prered <= *ftol && p5 * ratio <= one && *info
== 2) {
*info = 3;
}
/*< if (info .ne. 0) go to 300 >*/
if (*info != 0) {
goto L300;
}
/* tests for termination and stringent tolerances. */
/*< if (nfev .ge. maxfev) info = 5 >*/
if (*nfev >= *maxfev) {
*info = 5;
}
/*< >*/
if (abs(actred) <= epsmch && prered <= epsmch && p5 * ratio <= one) {
*info = 6;
}
/*< if (delta .le. epsmch*xnorm) info = 7 >*/
if (delta <= epsmch * xnorm) {
*info = 7;
}
/*< if (gnorm .le. epsmch) info = 8 >*/
if (gnorm <= epsmch) {
*info = 8;
}
/*< if (info .ne. 0) go to 300 >*/
if (*info != 0) {
goto L300;
}
/* end of the inner loop. repeat if iteration unsuccessful. */
/*< if (ratio .lt. p0001) go to 200 >*/
if (ratio < p0001) {
goto L200;
}
/* end of the outer loop. */
/*< go to 30 >*/
goto L30;
/*< 300 continue >*/
L300:
/* termination, either normal or user imposed. */
/*< if (iflag .lt. 0) info = iflag >*/
if (iflag < 0) {
*info = iflag;
}
/*< iflag = 0 >*/
iflag = 0;
/*< if (nprint .gt. 0) call fcn(m,n,x,fvec,fjac,ldfjac,iflag) >*/
if (*nprint > 0) {
(*fcn)(m, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
userdata);
}
/*< return >*/
return 0;
/* last card of subroutine lmder. */
/*< end >*/
} /* lmder_ */
