int fit( double *p0, int *info){
int m, n, lwa, iwa[4], one=1;
double tol, fnorm, x[4], fvec[64], wa[512];
extern void fcn();
/*Fits one set of intensities from threshold scan. ud is 1 for lower threshold scan,
-1 for upper threshold scan. Returns intensity, centroid and width of best fit to
erf() function. x is array of DAC values for intensities y_meas.
p0[0...3] is initial guess for solution.*/
tol = sqrt(dpmpar_(&one));
m=NPOINTS;
n=4;
lwa=512;
lmdif1_(&fcn, &m, &n, p0, fvec, &tol, info, iwa, wa, &lwa);
}
int main()
{
int m, n, ldfjac, info, lwa, ipvt[3], one=1;
double tol, fnorm;
double x[3], fvec[15], fjac[9], wa[30];
m = 15;
n = 3;
/* the following starting values provide a rough fit. */
x[0] = 1.;
x[1] = 1.;
x[2] = 1.;
ldfjac = 3;
lwa = 30;
/* set tol to the square root of the machine precision.
unless high precision solutions are required,
this is the recommended setting. */
tol = sqrt(dpmpar_(&one));
lmstr1_(&fcn, &m, &n,
x, fvec, fjac, &ldfjac,
&tol, &info, ipvt, wa, &lwa);
fnorm = enorm_(&m, fvec);
printf(" FINAL L2 NORM OF THE RESIDUALS%15.7g\n\n", fnorm);
printf(" EXIT PARAMETER %10i\n\n", info);
printf(" FINAL APPROXIMATE SOLUTION\n\n%15.7g%15.7g%15.7g\n",
x[0], x[1], x[2]);
return 0;
}
/* Subroutine */ int r1updt_(integer *m, integer *n, doublereal *s, integer *
ls, doublereal *u, doublereal *v, doublereal *w, logical *sing)
{
/* Initialized data */
static doublereal one = 1.;
static doublereal p5 = .5;
static doublereal p25 = .25;
static doublereal zero = 0.;
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer i__, j, l, jj, nm1;
static doublereal tan__;
static integer nmj;
static doublereal cos__, sin__, tau, temp, giant, cotan;
extern doublereal dpmpar_(integer *);
/* ********** */
/* subroutine r1updt */
/* given an m by n lower trapezoidal matrix s, an m-vector u, */
/* and an n-vector v, the problem is to determine an */
/* orthogonal matrix q such that */
/* t */
/* (s + u*v )*q */
/* is again lower trapezoidal. */
/* this subroutine determines q as the product of 2*(n - 1) */
/* transformations */
/* gv(n-1)*...*gv(1)*gw(1)*...*gw(n-1) */
/* where gv(i), gw(i) are givens rotations in the (i,n) plane */
/* which eliminate elements in the i-th and n-th planes, */
/* respectively. q itself is not accumulated, rather the */
/* information to recover the gv, gw rotations is returned. */
/* the subroutine statement is */
/* subroutine r1updt(m,n,s,ls,u,v,w,sing) */
/* where */
/* m is a positive integer input variable set to the number */
/* of rows of s. */
/* n is a positive integer input variable set to the number */
/* of columns of s. n must not exceed m. */
/* s is an array of length ls. on input s must contain the lower */
/* trapezoidal matrix s stored by columns. on output s contains */
/* the lower trapezoidal matrix produced as described above. */
/* ls is a positive integer input variable not less than */
/* (n*(2*m-n+1))/2. */
/* u is an input array of length m which must contain the */
/* vector u. */
/* v is an array of length n. on input v must contain the vector */
/* v. on output v(i) contains the information necessary to */
/* recover the givens rotation gv(i) described above. */
/* w is an output array of length m. w(i) contains information */
/* necessary to recover the givens rotation gw(i) described */
/* above. */
/* sing is a logical output variable. sing is set true if any */
/* of the diagonal elements of the output s are zero. otherwise */
/* sing is set false. */
/* subprograms called */
/* minpack-supplied ... dpmpar */
/* fortran-supplied ... dabs,dsqrt */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more, */
/* john l. nazareth */
/* ********** */
/* Parameter adjustments */
--w;
--u;
--v;
--s;
/* Function Body */
/* giant is the largest magnitude. */
giant = dpmpar_(&c__3);
/* initialize the diagonal element pointer. */
jj = *n * ((*m << 1) - *n + 1) / 2 - (*m - *n);
/* move the nontrivial part of the last column of s into w. */
l = jj;
i__1 = *m;
for (i__ = *n; i__ <= i__1; ++i__) {
w[i__] = s[l];
++l;
/* L10: */
}
/* rotate the vector v into a multiple of the n-th unit vector */
/* in such a way that a spike is introduced into w. */
nm1 = *n - 1;
if (nm1 < 1) {
goto L70;
}
i__1 = nm1;
for (nmj = 1; nmj <= i__1; ++nmj) {
j = *n - nmj;
jj -= *m - j + 1;
w[j] = zero;
if (v[j] == zero) {
goto L50;
}
/* determine a givens rotation which eliminates the */
/* j-th element of v. */
if ((d__1 = v[*n], abs(d__1)) >= (d__2 = v[j], abs(d__2))) {
goto L20;
}
cotan = v[*n] / v[j];
/* Computing 2nd power */
d__1 = cotan;
sin__ = p5 / sqrt(p25 + p25 * (d__1 * d__1));
cos__ = sin__ * cotan;
tau = one;
if (abs(cos__) * giant > one) {
tau = one / cos__;
}
goto L30;
L20:
tan__ = v[j] / v[*n];
/* Computing 2nd power */
d__1 = tan__;
cos__ = p5 / sqrt(p25 + p25 * (d__1 * d__1));
sin__ = cos__ * tan__;
tau = sin__;
L30:
/* apply the transformation to v and store the information */
/* necessary to recover the givens rotation. */
v[*n] = sin__ * v[j] + cos__ * v[*n];
v[j] = tau;
/* apply the transformation to s and extend the spike in w. */
l = jj;
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
temp = cos__ * s[l] - sin__ * w[i__];
w[i__] = sin__ * s[l] + cos__ * w[i__];
s[l] = temp;
++l;
/* L40: */
}
L50:
/* L60: */
;
}
L70:
/* add the spike from the rank 1 update to w. */
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
w[i__] += v[*n] * u[i__];
/* L80: */
}
/* eliminate the spike. */
*sing = FALSE_;
if (nm1 < 1) {
goto L140;
}
i__1 = nm1;
for (j = 1; j <= i__1; ++j) {
if (w[j] == zero) {
goto L120;
}
/* determine a givens rotation which eliminates the */
/* j-th element of the spike. */
if ((d__1 = s[jj], abs(d__1)) >= (d__2 = w[j], abs(d__2))) {
goto L90;
}
cotan = s[jj] / w[j];
/* Computing 2nd power */
d__1 = cotan;
sin__ = p5 / sqrt(p25 + p25 * (d__1 * d__1));
cos__ = sin__ * cotan;
tau = one;
if (abs(cos__) * giant > one) {
tau = one / cos__;
}
goto L100;
L90:
tan__ = w[j] / s[jj];
/* Computing 2nd power */
d__1 = tan__;
cos__ = p5 / sqrt(p25 + p25 * (d__1 * d__1));
sin__ = cos__ * tan__;
tau = sin__;
L100:
/* apply the transformation to s and reduce the spike in w. */
l = jj;
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
temp = cos__ * s[l] + sin__ * w[i__];
w[i__] = -sin__ * s[l] + cos__ * w[i__];
s[l] = temp;
++l;
/* L110: */
}
/* store the information necessary to recover the */
/* givens rotation. */
w[j] = tau;
L120:
/* test for zero diagonal elements in the output s. */
if (s[jj] == zero) {
*sing = TRUE_;
}
jj += *m - j + 1;
/* L130: */
}
L140:
/* move w back into the last column of the output s. */
l = jj;
i__1 = *m;
for (i__ = *n; i__ <= i__1; ++i__) {
s[l] = w[i__];
++l;
/* L150: */
}
if (s[jj] == zero) {
*sing = TRUE_;
}
return 0;
/* last card of subroutine r1updt. */
} /* r1updt_ */
/* ********** */
/* Main program */ int MAIN__(void)
{
/* Initialized data */
static integer nread = 5;
static integer nwrite = 6;
static doublereal one = 1.;
static doublereal ten = 10.;
/* Format strings */
static char fmt_50[] = "(3i5)";
static char fmt_60[] = "(////5x,\002 PROBLEM\002,i5,5x,\002 DIMENSION"
"\002,i5,5x//)";
static char fmt_70[] = "(5x,\002 INITIAL L2 NORM OF THE RESIDUALS\002,d1"
"5.7//5x,\002 FINAL L2 NORM OF THE RESIDUALS \002,d15.7//5x,\002"
" NUMBER OF FUNCTION EVALUATIONS \002,i10//5x,\002 EXIT PARAMETER"
"\002,18x,i10//5x,\002 FINAL APPROXIMATE SOLUTION\002//(5x,5d15.7"
"))";
static char fmt_80[] = "(\0021SUMMARY OF \002,i3,\002 CALLS TO HYBRD1"
"\002/)";
static char fmt_90[] = "(\002 NPROB N NFEV INFO FINAL L2 NORM\002"
"/)";
static char fmt_100[] = "(i4,i6,i7,i6,1x,d15.7)";
/* System generated locals */
integer i__1, i__2;
/* Builtin functions */
double sqrt(doublereal);
integer s_rsfe(cilist *), do_fio(integer *, char *, ftnlen), e_rsfe(void),
s_wsfe(cilist *), e_wsfe(void);
/* Subroutine */ int s_stop(char *, ftnlen);
/* Local variables */
static integer i__, k, n;
static doublereal x[40];
static integer ic, na[60], nf[60];
static doublereal wa[2660];
static integer np[60], nx[60];
extern /* Subroutine */ int fcn_();
static doublereal fnm[60];
static integer lwa;
static doublereal tol, fvec[40];
static integer info;
extern doublereal enorm_(integer *, doublereal *);
extern /* Subroutine */ int hybrd1_(U_fp, integer *, doublereal *,
doublereal *, doublereal *, integer *, doublereal *, integer *);
static doublereal fnorm1, fnorm2;
extern /* Subroutine */ int vecfcn_(integer *, doublereal *, doublereal *,
integer *);
static doublereal factor;
extern doublereal dpmpar_(integer *);
static integer ntries;
extern /* Subroutine */ int initpt_(integer *, doublereal *, integer *,
doublereal *);
/* Fortran I/O blocks */
static cilist io___8 = { 0, 0, 0, fmt_50, 0 };
static cilist io___16 = { 0, 0, 0, fmt_60, 0 };
static cilist io___25 = { 0, 0, 0, fmt_70, 0 };
static cilist io___27 = { 0, 0, 0, fmt_80, 0 };
static cilist io___28 = { 0, 0, 0, fmt_90, 0 };
static cilist io___29 = { 0, 0, 0, fmt_100, 0 };
/* LOGICAL INPUT UNIT IS ASSUMED TO BE NUMBER 5. */
/* LOGICAL OUTPUT UNIT IS ASSUMED TO BE NUMBER 6. */
tol = sqrt(dpmpar_(&c__1));
lwa = 2660;
ic = 0;
L10:
io___8.ciunit = nread;
s_rsfe(&io___8);
do_fio(&c__1, (char *)&refnum_1.nprob, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&ntries, (ftnlen)sizeof(integer));
e_rsfe();
if (refnum_1.nprob <= 0) {
goto L30;
}
factor = one;
i__1 = ntries;
for (k = 1; k <= i__1; ++k) {
++ic;
initpt_(&n, x, &refnum_1.nprob, &factor);
vecfcn_(&n, x, fvec, &refnum_1.nprob);
fnorm1 = enorm_(&n, fvec);
io___16.ciunit = nwrite;
s_wsfe(&io___16);
do_fio(&c__1, (char *)&refnum_1.nprob, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&n, (ftnlen)sizeof(integer));
e_wsfe();
refnum_1.nfev = 0;
hybrd1_((U_fp)fcn_, &n, x, fvec, &tol, &info, wa, &lwa);
fnorm2 = enorm_(&n, fvec);
np[ic - 1] = refnum_1.nprob;
na[ic - 1] = n;
nf[ic - 1] = refnum_1.nfev;
nx[ic - 1] = info;
fnm[ic - 1] = fnorm2;
io___25.ciunit = nwrite;
s_wsfe(&io___25);
do_fio(&c__1, (char *)&fnorm1, (ftnlen)sizeof(doublereal));
do_fio(&c__1, (char *)&fnorm2, (ftnlen)sizeof(doublereal));
do_fio(&c__1, (char *)&refnum_1.nfev, (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&info, (ftnlen)sizeof(integer));
i__2 = n;
for (i__ = 1; i__ <= i__2; ++i__) {
do_fio(&c__1, (char *)&x[i__ - 1], (ftnlen)sizeof(doublereal));
}
e_wsfe();
factor = ten * factor;
/* L20: */
}
goto L10;
L30:
io___27.ciunit = nwrite;
s_wsfe(&io___27);
do_fio(&c__1, (char *)&ic, (ftnlen)sizeof(integer));
e_wsfe();
io___28.ciunit = nwrite;
s_wsfe(&io___28);
e_wsfe();
i__1 = ic;
for (i__ = 1; i__ <= i__1; ++i__) {
io___29.ciunit = nwrite;
s_wsfe(&io___29);
do_fio(&c__1, (char *)&np[i__ - 1], (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&na[i__ - 1], (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&nf[i__ - 1], (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&nx[i__ - 1], (ftnlen)sizeof(integer));
do_fio(&c__1, (char *)&fnm[i__ - 1], (ftnlen)sizeof(doublereal));
e_wsfe();
/* L40: */
}
s_stop("", (ftnlen)0);
/* LAST CARD OF DRIVER. */
return 0;
} /* MAIN__ */
/* 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 qrfac_(integer *m, integer *n, doublereal *a, integer *
lda, logical *pivot, integer *ipvt, integer *lipvt, doublereal *rdiag,
doublereal *acnorm, doublereal *wa)
{
/* Initialized data */
static doublereal one = 1.;
static doublereal p05 = .05;
static doublereal zero = 0.;
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer i__, j, k, jp1;
static doublereal sum;
static integer kmax;
static doublereal temp;
static integer minmn;
extern doublereal enorm_(integer *, doublereal *);
static doublereal epsmch;
extern doublereal dpmpar_(integer *);
static doublereal ajnorm;
/* ********** */
/* subroutine qrfac */
/* this subroutine uses householder transformations with column */
/* pivoting (optional) to compute a qr factorization of the */
/* m by n matrix a. that is, qrfac determines an orthogonal */
/* matrix q, a permutation matrix p, and an upper trapezoidal */
/* matrix r with diagonal elements of nonincreasing magnitude, */
/* such that a*p = q*r. the householder transformation for */
/* column k, k = 1,2,...,min(m,n), is of the form */
/* t */
/* i - (1/u(k))*u*u */
/* where u has zeros in the first k-1 positions. the form of */
/* this transformation and the method of pivoting first */
/* appeared in the corresponding linpack subroutine. */
/* the subroutine statement is */
/* subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa) */
/* where */
/* m is a positive integer input variable set to the number */
/* of rows of a. */
/* n is a positive integer input variable set to the number */
/* of columns of a. */
/* a is an m by n array. on input a contains the matrix for */
/* which the qr factorization is to be computed. on output */
/* the strict upper trapezoidal part of a contains the strict */
/* upper trapezoidal part of r, and the lower trapezoidal */
/* part of a contains a factored form of q (the non-trivial */
/* elements of the u vectors described above). */
/* lda is a positive integer input variable not less than m */
/* which specifies the leading dimension of the array a. */
/* pivot is a logical input variable. if pivot is set true, */
/* then column pivoting is enforced. if pivot is set false, */
/* then no column pivoting is done. */
/* ipvt is an integer output array of length lipvt. ipvt */
/* defines the permutation matrix p such that a*p = q*r. */
/* column j of p is column ipvt(j) of the identity matrix. */
/* if pivot is false, ipvt is not referenced. */
/* lipvt is a positive integer input variable. if pivot is false, */
/* then lipvt may be as small as 1. if pivot is true, then */
/* lipvt must be at least n. */
/* rdiag is an output array of length n which contains the */
/* diagonal elements of r. */
/* acnorm is an output array of length n which contains the */
/* norms of the corresponding columns of the input matrix a. */
/* if this information is not needed, then acnorm can coincide */
/* with rdiag. */
/* wa is a work array of length n. if pivot is false, then wa */
/* can coincide with rdiag. */
/* subprograms called */
/* minpack-supplied ... dpmpar,enorm */
/* fortran-supplied ... dmax1,dsqrt,min0 */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa;
--acnorm;
--rdiag;
a_dim1 = *lda;
a_offset = 1 + a_dim1 * 1;
a -= a_offset;
--ipvt;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = dpmpar_(&c__1);
/* compute the initial column norms and initialize several arrays. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
acnorm[j] = enorm_(m, &a[j * a_dim1 + 1]);
rdiag[j] = acnorm[j];
wa[j] = rdiag[j];
if (*pivot) {
ipvt[j] = j;
}
/* L10: */
}
/* reduce a to r with householder transformations. */
minmn = min(*m,*n);
i__1 = minmn;
for (j = 1; j <= i__1; ++j) {
if (! (*pivot)) {
goto L40;
}
/* bring the column of largest norm into the pivot position. */
kmax = j;
i__2 = *n;
for (k = j; k <= i__2; ++k) {
if (rdiag[k] > rdiag[kmax]) {
kmax = k;
}
/* L20: */
}
if (kmax == j) {
goto L40;
}
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
temp = a[i__ + j * a_dim1];
a[i__ + j * a_dim1] = a[i__ + kmax * a_dim1];
a[i__ + kmax * a_dim1] = temp;
/* L30: */
}
rdiag[kmax] = rdiag[j];
wa[kmax] = wa[j];
k = ipvt[j];
ipvt[j] = ipvt[kmax];
ipvt[kmax] = k;
L40:
/* compute the householder transformation to reduce the */
/* j-th column of a to a multiple of the j-th unit vector. */
i__2 = *m - j + 1;
ajnorm = enorm_(&i__2, &a[j + j * a_dim1]);
if (ajnorm == zero) {
goto L100;
}
if (a[j + j * a_dim1] < zero) {
ajnorm = -ajnorm;
}
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
a[i__ + j * a_dim1] /= ajnorm;
/* L50: */
}
a[j + j * a_dim1] += one;
/* apply the transformation to the remaining columns */
/* and update the norms. */
jp1 = j + 1;
if (*n < jp1) {
goto L100;
}
i__2 = *n;
for (k = jp1; k <= i__2; ++k) {
sum = zero;
i__3 = *m;
for (i__ = j; i__ <= i__3; ++i__) {
sum += a[i__ + j * a_dim1] * a[i__ + k * a_dim1];
/* L60: */
}
temp = sum / a[j + j * a_dim1];
i__3 = *m;
for (i__ = j; i__ <= i__3; ++i__) {
a[i__ + k * a_dim1] -= temp * a[i__ + j * a_dim1];
/* L70: */
}
if (! (*pivot) || rdiag[k] == zero) {
goto L80;
}
temp = a[j + k * a_dim1] / rdiag[k];
/* Computing MAX */
/* Computing 2nd power */
d__3 = temp;
d__1 = zero, d__2 = one - d__3 * d__3;
rdiag[k] *= sqrt((max(d__1,d__2)));
/* Computing 2nd power */
d__1 = rdiag[k] / wa[k];
if (p05 * (d__1 * d__1) > epsmch) {
goto L80;
}
i__3 = *m - j;
rdiag[k] = enorm_(&i__3, &a[jp1 + k * a_dim1]);
wa[k] = rdiag[k];
L80:
/* L90: */
;
}
L100:
rdiag[j] = -ajnorm;
/* L110: */
}
return 0;
/* last card of subroutine qrfac. */
} /* qrfac_ */
/* Subroutine */ int fdjac2_(S_fp fcn, integer *m, integer *n, doublereal *x,
doublereal *fvec, doublereal *fjac, integer *ldfjac, integer *iflag,
doublereal *epsfcn, doublereal *wa)
{
/* Initialized data */
static doublereal zero = 0.;
/* System generated locals */
integer fjac_dim1, fjac_offset, i__1, i__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static doublereal h__;
static integer i__, j;
static doublereal eps, temp, epsmch;
extern doublereal dpmpar_(integer *);
/* ********** */
/* subroutine fdjac2 */
/* this subroutine computes a forward-difference approximation */
/* to the m by n jacobian matrix associated with a specified */
/* problem of m functions in n variables. */
/* the subroutine statement is */
/* subroutine fdjac2(fcn,m,n,x,fvec,fjac,ldfjac,iflag,epsfcn,wa) */
/* 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 fdjac2. */
/* 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 input array of length n. */
/* fvec is an input array of length m which must contain the */
/* functions evaluated at x. */
/* fjac is an output m by n array which contains the */
/* approximation to the jacobian matrix evaluated at x. */
/* ldfjac is a positive integer input variable not less than m */
/* which specifies the leading dimension of the array fjac. */
/* iflag is an integer variable which can be used to terminate */
/* the execution of fdjac2. see description of fcn. */
/* 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. */
/* wa is a work array of length m. */
/* subprograms called */
/* user-supplied ...... fcn */
/* minpack-supplied ... dpmpar */
/* fortran-supplied ... dabs,dmax1,dsqrt */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa;
--fvec;
--x;
fjac_dim1 = *ldfjac;
fjac_offset = fjac_dim1 + 1;
fjac -= fjac_offset;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = dpmpar_(&c__1);
eps = sqrt((max(*epsfcn,epsmch)));
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
temp = x[j];
h__ = eps * abs(temp);
if (h__ == zero) {
h__ = eps;
}
x[j] = temp + h__;
(*fcn)(m, n, &x[1], &wa[1], iflag);
if (*iflag < 0) {
goto L30;
}
x[j] = temp;
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
fjac[i__ + j * fjac_dim1] = (wa[i__] - fvec[i__]) / h__;
/* L10: */
}
/* L20: */
}
L30:
return 0;
/* last card of subroutine fdjac2. */
} /* fdjac2_ */
/* Subroutine */ int lmpar_(integer *n, doublereal *r__, integer *ldr,
integer *ipvt, doublereal *diag, doublereal *qtb, doublereal *delta,
doublereal *par, doublereal *x, doublereal *sdiag, doublereal *wa1,
doublereal *wa2)
{
/* Initialized data */
static doublereal p1 = .1;
static doublereal p001 = .001;
static doublereal zero = 0.;
/* System generated locals */
integer r_dim1, r_offset, i__1, i__2;
doublereal d__1, d__2;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer i__, j, k, l;
static doublereal fp;
static integer jm1, jp1;
static doublereal sum, parc, parl;
static integer iter;
static doublereal temp, paru, dwarf;
static integer nsing;
extern doublereal enorm_(integer *, doublereal *);
static doublereal gnorm;
extern doublereal dpmpar_(integer *);
static doublereal dxnorm;
extern /* Subroutine */ int qrsolv_(integer *, doublereal *, integer *,
integer *, doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *);
/* ********** */
/* subroutine lmpar */
/* given an m by n matrix a, an n by n nonsingular diagonal */
/* matrix d, an m-vector b, and a positive number delta, */
/* the problem is to determine a value for the parameter */
/* par such that if x solves the system */
/* a*x = b , sqrt(par)*d*x = 0 , */
/* in the least squares sense, and dxnorm is the euclidean */
/* norm of d*x, then either par is zero and */
/* (dxnorm-delta) .le. 0.1*delta , */
/* or par is positive and */
/* abs(dxnorm-delta) .le. 0.1*delta . */
/* this subroutine completes the solution of the problem */
/* if it is provided with the necessary information from the */
/* qr factorization, with column pivoting, of a. that is, if */
/* a*p = q*r, where p is a permutation matrix, q has orthogonal */
/* columns, and r is an upper triangular matrix with diagonal */
/* elements of nonincreasing magnitude, then lmpar expects */
/* the full upper triangle of r, the permutation matrix p, */
/* and the first n components of (q transpose)*b. on output */
/* lmpar also provides an upper triangular matrix s such that */
/* t t t */
/* p *(a *a + par*d*d)*p = s *s . */
/* s is employed within lmpar and may be of separate interest. */
/* only a few iterations are generally needed for convergence */
/* of the algorithm. if, however, the limit of 10 iterations */
/* is reached, then the output par will contain the best */
/* value obtained so far. */
/* the subroutine statement is */
/* subroutine lmpar(n,r,ldr,ipvt,diag,qtb,delta,par,x,sdiag, */
/* wa1,wa2) */
/* where */
/* n is a positive integer input variable set to the order of r. */
/* r is an n by n array. on input the full upper triangle */
/* must contain the full upper triangle of the matrix r. */
/* on output the full upper triangle is unaltered, and the */
/* strict lower triangle contains the strict upper triangle */
/* (transposed) of the upper triangular matrix s. */
/* ldr is a positive integer input variable not less than n */
/* which specifies the leading dimension of the array r. */
/* ipvt is an integer input array of length n which defines the */
/* permutation matrix p such that a*p = q*r. column j of p */
/* is column ipvt(j) of the identity matrix. */
/* diag is an input array of length n which must contain the */
/* diagonal elements of the matrix d. */
/* qtb is an input array of length n which must contain the first */
/* n elements of the vector (q transpose)*b. */
/* delta is a positive input variable which specifies an upper */
/* bound on the euclidean norm of d*x. */
/* par is a nonnegative variable. on input par contains an */
/* initial estimate of the levenberg-marquardt parameter. */
/* on output par contains the final estimate. */
/* x is an output array of length n which contains the least */
/* squares solution of the system a*x = b, sqrt(par)*d*x = 0, */
/* for the output par. */
/* sdiag is an output array of length n which contains the */
/* diagonal elements of the upper triangular matrix s. */
/* wa1 and wa2 are work arrays of length n. */
/* subprograms called */
/* minpack-supplied ... dpmpar,enorm,qrsolv */
/* fortran-supplied ... dabs,dmax1,dmin1,dsqrt */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa2;
--wa1;
--sdiag;
--x;
--qtb;
--diag;
--ipvt;
r_dim1 = *ldr;
r_offset = r_dim1 + 1;
r__ -= r_offset;
/* Function Body */
/* dwarf is the smallest positive magnitude. */
dwarf = dpmpar_(&c__2);
/* compute and store in x the gauss-newton direction. if the */
/* jacobian is rank-deficient, obtain a least squares solution. */
nsing = *n;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = qtb[j];
if (r__[j + j * r_dim1] == zero && nsing == *n) {
nsing = j - 1;
}
if (nsing < *n) {
wa1[j] = zero;
}
/* L10: */
}
if (nsing < 1) {
goto L50;
}
i__1 = nsing;
for (k = 1; k <= i__1; ++k) {
j = nsing - k + 1;
wa1[j] /= r__[j + j * r_dim1];
temp = wa1[j];
jm1 = j - 1;
if (jm1 < 1) {
goto L30;
}
i__2 = jm1;
for (i__ = 1; i__ <= i__2; ++i__) {
wa1[i__] -= r__[i__ + j * r_dim1] * temp;
/* L20: */
}
L30:
/* L40: */
;
}
L50:
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = ipvt[j];
x[l] = wa1[j];
/* L60: */
}
/* initialize the iteration counter. */
/* evaluate the function at the origin, and test */
/* for acceptance of the gauss-newton direction. */
iter = 0;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa2[j] = diag[j] * x[j];
/* L70: */
}
dxnorm = enorm_(n, &wa2[1]);
fp = dxnorm - *delta;
if (fp <= p1 * *delta) {
goto L220;
}
/* if the jacobian is not rank deficient, the newton */
/* step provides a lower bound, parl, for the zero of */
/* the function. otherwise set this bound to zero. */
parl = zero;
if (nsing < *n) {
goto L120;
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = ipvt[j];
wa1[j] = diag[l] * (wa2[l] / dxnorm);
/* L80: */
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
sum = zero;
jm1 = j - 1;
if (jm1 < 1) {
goto L100;
}
i__2 = jm1;
for (i__ = 1; i__ <= i__2; ++i__) {
sum += r__[i__ + j * r_dim1] * wa1[i__];
/* L90: */
}
L100:
wa1[j] = (wa1[j] - sum) / r__[j + j * r_dim1];
/* L110: */
}
temp = enorm_(n, &wa1[1]);
parl = fp / *delta / temp / temp;
L120:
/* calculate an upper bound, paru, for the zero of the function. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
sum = zero;
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
sum += r__[i__ + j * r_dim1] * qtb[i__];
/* L130: */
}
l = ipvt[j];
wa1[j] = sum / diag[l];
/* L140: */
}
gnorm = enorm_(n, &wa1[1]);
paru = gnorm / *delta;
if (paru == zero) {
paru = dwarf / min(*delta,p1);
}
/* if the input par lies outside of the interval (parl,paru), */
/* set par to the closer endpoint. */
*par = max(*par,parl);
*par = min(*par,paru);
if (*par == zero) {
*par = gnorm / dxnorm;
}
/* beginning of an iteration. */
L150:
++iter;
/* evaluate the function at the current value of par. */
if (*par == zero) {
/* Computing MAX */
d__1 = dwarf, d__2 = p001 * paru;
*par = max(d__1,d__2);
}
temp = sqrt(*par);
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = temp * diag[j];
/* L160: */
}
qrsolv_(n, &r__[r_offset], ldr, &ipvt[1], &wa1[1], &qtb[1], &x[1], &sdiag[
1], &wa2[1]);
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa2[j] = diag[j] * x[j];
/* L170: */
}
dxnorm = enorm_(n, &wa2[1]);
temp = fp;
fp = dxnorm - *delta;
/* if the function is small enough, accept the current value */
/* of par. also test for the exceptional cases where parl */
/* is zero or the number of iterations has reached 10. */
if (abs(fp) <= p1 * *delta || (parl == zero && fp <= temp && temp < zero) ||
iter == 10) {
goto L220;
}
/* compute the newton correction. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
l = ipvt[j];
wa1[j] = diag[l] * (wa2[l] / dxnorm);
/* L180: */
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] /= sdiag[j];
temp = wa1[j];
jp1 = j + 1;
if (*n < jp1) {
goto L200;
}
i__2 = *n;
for (i__ = jp1; i__ <= i__2; ++i__) {
wa1[i__] -= r__[i__ + j * r_dim1] * temp;
/* L190: */
}
L200:
/* L210: */
;
}
temp = enorm_(n, &wa1[1]);
parc = fp / *delta / temp / temp;
/* depending on the sign of the function, update parl or paru. */
if (fp > zero) {
parl = max(parl,*par);
}
if (fp < zero) {
paru = min(paru,*par);
}
/* compute an improved estimate for par. */
/* Computing MAX */
d__1 = parl, d__2 = *par + parc;
*par = max(d__1,d__2);
/* end of an iteration. */
goto L150;
L220:
/* termination. */
if (iter == 0) {
*par = zero;
}
return 0;
/* last card of subroutine lmpar. */
} /* lmpar_ */
/*< subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa) >*/
/* Subroutine */ int qrfac_(integer *m, integer *n, doublereal *a, integer *
lda, logical *pivot, integer *ipvt, integer *lipvt, doublereal *rdiag,
doublereal *acnorm, doublereal *wa)
{
/* Initialized data */
static doublereal one = 1.; /* constant */
static doublereal p05 = .05; /* constant */
static doublereal zero = 0.; /* constant */
/* System generated locals */
integer a_dim1, a_offset, i__1, i__2, i__3;
doublereal d__1, d__2, d__3;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
integer i__, j, k, jp1;
doublereal sum;
integer kmax;
doublereal temp;
integer minmn;
extern doublereal enorm_(integer *, doublereal *);
doublereal epsmch;
extern doublereal dpmpar_(integer *);
doublereal ajnorm;
(void)lipvt;
/*< integer m,n,lda,lipvt >*/
/*< integer ipvt(lipvt) >*/
/*< logical pivot >*/
/*< double precision a(lda,n),rdiag(n),acnorm(n),wa(n) >*/
/* ********** */
/* subroutine qrfac */
/* this subroutine uses householder transformations with column */
/* pivoting (optional) to compute a qr factorization of the */
/* m by n matrix a. that is, qrfac determines an orthogonal */
/* matrix q, a permutation matrix p, and an upper trapezoidal */
/* matrix r with diagonal elements of nonincreasing magnitude, */
/* such that a*p = q*r. the householder transformation for */
/* column k, k = 1,2,...,min(m,n), is of the form */
/* t */
/* i - (1/u(k))*u*u */
/* where u has zeros in the first k-1 positions. the form of */
/* this transformation and the method of pivoting first */
/* appeared in the corresponding linpack subroutine. */
/* the subroutine statement is */
/* subroutine qrfac(m,n,a,lda,pivot,ipvt,lipvt,rdiag,acnorm,wa) */
/* where */
/* m is a positive integer input variable set to the number */
/* of rows of a. */
/* n is a positive integer input variable set to the number */
/* of columns of a. */
/* a is an m by n array. on input a contains the matrix for */
/* which the qr factorization is to be computed. on output */
/* the strict upper trapezoidal part of a contains the strict */
/* upper trapezoidal part of r, and the lower trapezoidal */
/* part of a contains a factored form of q (the non-trivial */
/* elements of the u vectors described above). */
/* lda is a positive integer input variable not less than m */
/* which specifies the leading dimension of the array a. */
/* pivot is a logical input variable. if pivot is set true, */
/* then column pivoting is enforced. if pivot is set false, */
/* then no column pivoting is done. */
/* ipvt is an integer output array of length lipvt. ipvt */
/* defines the permutation matrix p such that a*p = q*r. */
/* column j of p is column ipvt(j) of the identity matrix. */
/* if pivot is false, ipvt is not referenced. */
/* lipvt is a positive integer input variable. if pivot is false, */
/* then lipvt may be as small as 1. if pivot is true, then */
/* lipvt must be at least n. */
/* rdiag is an output array of length n which contains the */
/* diagonal elements of r. */
/* acnorm is an output array of length n which contains the */
/* norms of the corresponding columns of the input matrix a. */
/* if this information is not needed, then acnorm can coincide */
/* with rdiag. */
/* wa is a work array of length n. if pivot is false, then wa */
/* can coincide with rdiag. */
/* subprograms called */
/* minpack-supplied ... dpmpar,enorm */
/* fortran-supplied ... dmax1,dsqrt,min0 */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/*< integer i,j,jp1,k,kmax,minmn >*/
/*< double precision ajnorm,epsmch,one,p05,sum,temp,zero >*/
/*< double precision dpmpar,enorm >*/
/*< data one,p05,zero /1.0d0,5.0d-2,0.0d0/ >*/
/* Parameter adjustments */
--wa;
--acnorm;
--rdiag;
a_dim1 = *lda;
a_offset = 1 + a_dim1;
a -= a_offset;
--ipvt;
/* Function Body */
/* epsmch is the machine precision. */
/*< epsmch = dpmpar(1) >*/
epsmch = dpmpar_(&c__1);
/* compute the initial column norms and initialize several arrays. */
/*< do 10 j = 1, n >*/
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
/*< acnorm(j) = enorm(m,a(1,j)) >*/
acnorm[j] = enorm_(m, &a[j * a_dim1 + 1]);
/*< rdiag(j) = acnorm(j) >*/
rdiag[j] = acnorm[j];
/*< wa(j) = rdiag(j) >*/
wa[j] = rdiag[j];
/*< if (pivot) ipvt(j) = j >*/
if (*pivot) {
ipvt[j] = j;
}
/*< 10 continue >*/
/* L10: */
}
/* reduce a to r with householder transformations. */
/*< minmn = min0(m,n) >*/
minmn = min(*m,*n);
/*< do 110 j = 1, minmn >*/
i__1 = minmn;
for (j = 1; j <= i__1; ++j) {
/*< if (.not.pivot) go to 40 >*/
if (! (*pivot)) {
goto L40;
}
/* bring the column of largest norm into the pivot position. */
/*< kmax = j >*/
kmax = j;
/*< do 20 k = j, n >*/
i__2 = *n;
for (k = j; k <= i__2; ++k) {
/*< if (rdiag(k) .gt. rdiag(kmax)) kmax = k >*/
if (rdiag[k] > rdiag[kmax]) {
kmax = k;
}
/*< 20 continue >*/
/* L20: */
}
/*< if (kmax .eq. j) go to 40 >*/
if (kmax == j) {
goto L40;
}
/*< do 30 i = 1, m >*/
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
/*< temp = a(i,j) >*/
temp = a[i__ + j * a_dim1];
/*< a(i,j) = a(i,kmax) >*/
a[i__ + j * a_dim1] = a[i__ + kmax * a_dim1];
/*< a(i,kmax) = temp >*/
a[i__ + kmax * a_dim1] = temp;
/*< 30 continue >*/
/* L30: */
}
/*< rdiag(kmax) = rdiag(j) >*/
rdiag[kmax] = rdiag[j];
/*< wa(kmax) = wa(j) >*/
wa[kmax] = wa[j];
/*< k = ipvt(j) >*/
k = ipvt[j];
/*< ipvt(j) = ipvt(kmax) >*/
ipvt[j] = ipvt[kmax];
/*< ipvt(kmax) = k >*/
ipvt[kmax] = k;
/*< 40 continue >*/
L40:
/* compute the householder transformation to reduce the */
/* j-th column of a to a multiple of the j-th unit vector. */
/*< ajnorm = enorm(m-j+1,a(j,j)) >*/
i__2 = *m - j + 1;
ajnorm = enorm_(&i__2, &a[j + j * a_dim1]);
/*< if (ajnorm .eq. zero) go to 100 >*/
if (ajnorm == zero) {
goto L100;
}
/*< if (a(j,j) .lt. zero) ajnorm = -ajnorm >*/
if (a[j + j * a_dim1] < zero) {
ajnorm = -ajnorm;
}
/*< do 50 i = j, m >*/
i__2 = *m;
for (i__ = j; i__ <= i__2; ++i__) {
/*< a(i,j) = a(i,j)/ajnorm >*/
a[i__ + j * a_dim1] /= ajnorm;
/*< 50 continue >*/
/* L50: */
}
/*< a(j,j) = a(j,j) + one >*/
a[j + j * a_dim1] += one;
/* apply the transformation to the remaining columns */
/* and update the norms. */
/*< jp1 = j + 1 >*/
jp1 = j + 1;
/*< if (n .lt. jp1) go to 100 >*/
if (*n < jp1) {
goto L100;
}
/*< do 90 k = jp1, n >*/
i__2 = *n;
for (k = jp1; k <= i__2; ++k) {
/*< sum = zero >*/
sum = zero;
/*< do 60 i = j, m >*/
i__3 = *m;
for (i__ = j; i__ <= i__3; ++i__) {
/*< sum = sum + a(i,j)*a(i,k) >*/
sum += a[i__ + j * a_dim1] * a[i__ + k * a_dim1];
/*< 60 continue >*/
/* L60: */
}
/*< temp = sum/a(j,j) >*/
temp = sum / a[j + j * a_dim1];
/*< do 70 i = j, m >*/
i__3 = *m;
for (i__ = j; i__ <= i__3; ++i__) {
/*< a(i,k) = a(i,k) - temp*a(i,j) >*/
a[i__ + k * a_dim1] -= temp * a[i__ + j * a_dim1];
/*< 70 continue >*/
/* L70: */
}
/*< if (.not.pivot .or. rdiag(k) .eq. zero) go to 80 >*/
if (! (*pivot) || rdiag[k] == zero) {
goto L80;
}
/*< temp = a(j,k)/rdiag(k) >*/
temp = a[j + k * a_dim1] / rdiag[k];
/*< rdiag(k) = rdiag(k)*dsqrt(dmax1(zero,one-temp**2)) >*/
/* Computing MAX */
/* Computing 2nd power */
d__3 = temp;
d__1 = zero, d__2 = one - d__3 * d__3;
rdiag[k] *= sqrt((max(d__1,d__2)));
/*< if (p05*(rdiag(k)/wa(k))**2 .gt. epsmch) go to 80 >*/
/* Computing 2nd power */
d__1 = rdiag[k] / wa[k];
if (p05 * (d__1 * d__1) > epsmch) {
goto L80;
}
/*< rdiag(k) = enorm(m-j,a(jp1,k)) >*/
i__3 = *m - j;
rdiag[k] = enorm_(&i__3, &a[jp1 + k * a_dim1]);
/*< wa(k) = rdiag(k) >*/
wa[k] = rdiag[k];
/*< 80 continue >*/
L80:
/*< 90 continue >*/
/* L90: */
;
}
/*< 100 continue >*/
L100:
/*< rdiag(j) = -ajnorm >*/
rdiag[j] = -ajnorm;
/*< 110 continue >*/
/* L110: */
}
/*< return >*/
return 0;
/* last card of subroutine qrfac. */
/*< end >*/
} /* qrfac_ */
/*< >*/
/* 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_ */
/* Subroutine */ int _omc_hybrd_(S_fp fcn, integer *n, doublereal *x, doublereal *
fvec, doublereal *xtol, integer *maxfev, integer *ml, integer *mu,
doublereal *epsfcn, doublereal *diag, integer *mode, doublereal *
factor, integer *nprint, integer *info, integer *nfev, doublereal *
fjac, doublereal * fjacobian, integer *ldfjac, doublereal *r__, integer *lr, doublereal *qtf,
doublereal *wa1, doublereal *wa2, doublereal *wa3, doublereal *wa4, void* userdata)
{
/* Initialized data */
static doublereal one = 1.;
static doublereal p1 = .1;
static doublereal p5 = .5;
static doublereal p001 = .001;
static doublereal p0001 = 1e-4;
static doublereal zero = 0.;
/* System generated locals */
integer fjac_dim1, fjac_offset, i__1, i__2;
doublereal d__1, d__2;
/* Local variables */
integer i__, j, l, jm1, iwa[1];
doublereal sum;
logical sing;
integer iter;
doublereal temp;
integer msum, iflag;
doublereal delta;
extern /* Subroutine */ int qrfac_(integer *, integer *, doublereal *,
integer *, logical *, integer *, integer *, doublereal *,
doublereal *, doublereal *);
logical jeval;
integer ncsuc;
doublereal ratio;
extern doublereal enorm_(integer *, doublereal *);
doublereal fnorm;
extern /* Subroutine */ int qform_(integer *, integer *, doublereal *,
integer *, doublereal *), fdjac1_(S_fp, integer *, doublereal *,
doublereal *, doublereal *, integer *, integer *, integer *,
integer *, doublereal *, doublereal *, doublereal *, void *);
doublereal pnorm, xnorm, fnorm1;
extern /* Subroutine */ int r1updt_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *, doublereal *, logical *);
integer nslow1, nslow2;
extern /* Subroutine */ int r1mpyq_(integer *, integer *, doublereal *,
integer *, doublereal *, doublereal *);
integer ncfail;
extern /* Subroutine */ int dogleg_(integer *, doublereal *, integer *,
doublereal *, doublereal *, doublereal *, doublereal *,
doublereal *, doublereal *);
doublereal actred, epsmch, prered;
extern doublereal dpmpar_(integer *);
/* ********** */
/* 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,fjac, */
/* ldfjac,r,lr,qtf,wa1,wa2,wa3,wa4, userdata) */
/* 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. */
/* fjacobian is an output n by n array which contains the */
/* 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;
fjac -= fjac_offset;
--fjacobian;
--r__;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = dpmpar_(&c__1);
*info = 0;
iflag = 0;
*nfev = 0;
/* check the input parameters for errors. */
if(*n <= 0 || *xtol < zero || *maxfev <= 0 || *ml < 0 || *mu < 0 || *
factor <= zero || *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] <= zero) {
goto L300;
}
/* L10: */
}
L20:
/* evaluate the function at the starting point */
/* and calculate its norm. */
iflag = 1;
(*fcn)(n, &x[1], &fvec[1], &iflag, userdata);
*nfev = 1;
if(iflag < 0) {
goto L300;
}
fnorm = 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;
fdjac1_((S_fp)fcn, n, &x[1], &fvec[1], &fjac[fjac_offset], ldfjac, &iflag,
ml, mu, epsfcn, &wa1[1], &wa2[1], userdata);
*nfev += msum;
/* store the calculate jacobain matrix */
/* added by wbraun for scaling residuals */
{
int l = fjac_offset;
int k = 1;
for(j = 1; j <= *n; ++j){
for(i__ = 1; i__<= *n; ++i__, l++, k++){
fjacobian[k] = fjac[l];
}
}
}
if(iflag < 0) {
goto L300;
}
/* compute the qr factorization of the jacobian. */
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] == zero) {
diag[j] = one;
}
/* 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 = enorm_(n, &wa3[1]);
delta = *factor * xnorm;
if(delta == zero) {
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] == zero) {
goto L110;
}
sum = zero;
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] == zero) {
sing = TRUE_;
}
/* L150: */
}
/* accumulate the orthogonal factor in fjac. */
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)(n, &x[1], &fvec[1], &iflag, userdata);
}
if(iflag < 0) {
goto L300;
}
L190:
/* determine the direction p. */
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 = 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)(n, &wa2[1], &wa4[1], &iflag, userdata);
++(*nfev);
/* Scaling Residual vector */
/* added by wbraun */
/*
{
for(i__=1;i__<*n;i__++)
wa4[i__] = diagres[i__] * wa4[i__];
}
*/
if(iflag < 0) {
goto L300;
}
fnorm1 = enorm_(n, &wa4[1]);
/* compute the scaled actual reduction. */
actred = -one;
if(fnorm1 < fnorm) {
/* Computing 2nd power */
d__1 = fnorm1 / fnorm;
actred = one - d__1 * d__1;
}
/* compute the scaled predicted reduction. */
l = 1;
i__1 = *n;
for(i__ = 1; i__ <= i__1; ++i__) {
sum = zero;
i__2 = *n;
for(j = i__; j <= i__2; ++j) {
sum += r__[l] * wa1[j];
++l;
/* L210: */
}
wa3[i__] = qtf[i__] + sum;
/* L220: */
}
temp = enorm_(n, &wa3[1]);
prered = zero;
if(temp < fnorm) {
/* Computing 2nd power */
d__1 = temp / fnorm;
prered = one - d__1 * d__1;
}
/* compute the ratio of the actual to the predicted */
/* reduction. */
ratio = zero;
if(prered > zero) {
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((d__1 = ratio - one, abs(d__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 = 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 == zero) {
*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 = zero;
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. */
r1updt_(n, n, &r__[1], lr, &wa1[1], &wa2[1], &wa3[1], &sing);
r1mpyq_(n, n, &fjac[fjac_offset], ldfjac, &wa2[1], &wa3[1]);
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)(n, &x[1], &fvec[1], &iflag, userdata);
}
return 0;
/* last card of subroutine hybrd. */
} /* _omc_hybrd_ */
/* Subroutine */ int dogleg_(integer *n, doublereal *r__, integer *lr,
doublereal *diag, doublereal *qtb, doublereal *delta, doublereal *x,
doublereal *wa1, doublereal *wa2)
{
/* Initialized data */
static doublereal one = 1.;
static doublereal zero = 0.;
/* System generated locals */
integer i__1, i__2;
doublereal d__1, d__2, d__3, d__4;
/* Builtin functions */
double sqrt(doublereal);
/* Local variables */
static integer i__, j, k, l, jj, jp1;
static doublereal sum, temp, alpha, bnorm;
extern doublereal enorm_(integer *, doublereal *);
static doublereal gnorm, qnorm, epsmch;
extern doublereal dpmpar_(integer *);
static doublereal sgnorm;
/* ********** */
/* subroutine dogleg */
/* given an m by n matrix a, an n by n nonsingular diagonal */
/* matrix d, an m-vector b, and a positive number delta, the */
/* problem is to determine the convex combination x of the */
/* gauss-newton and scaled gradient directions that minimizes */
/* (a*x - b) in the least squares sense, subject to the */
/* restriction that the euclidean norm of d*x be at most delta. */
/* this subroutine completes the solution of the problem */
/* if it is provided with the necessary information from the */
/* qr factorization of a. that is, if a = q*r, where q has */
/* orthogonal columns and r is an upper triangular matrix, */
/* then dogleg expects the full upper triangle of r and */
/* the first n components of (q transpose)*b. */
/* the subroutine statement is */
/* subroutine dogleg(n,r,lr,diag,qtb,delta,x,wa1,wa2) */
/* where */
/* n is a positive integer input variable set to the order of r. */
/* r is an input array of length lr which must contain the upper */
/* triangular matrix r stored by rows. */
/* lr is a positive integer input variable not less than */
/* (n*(n+1))/2. */
/* diag is an input array of length n which must contain the */
/* diagonal elements of the matrix d. */
/* qtb is an input array of length n which must contain the first */
/* n elements of the vector (q transpose)*b. */
/* delta is a positive input variable which specifies an upper */
/* bound on the euclidean norm of d*x. */
/* x is an output array of length n which contains the desired */
/* convex combination of the gauss-newton direction and the */
/* scaled gradient direction. */
/* wa1 and wa2 are work arrays of length n. */
/* subprograms called */
/* minpack-supplied ... dpmpar,enorm */
/* fortran-supplied ... dabs,dmax1,dmin1,dsqrt */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--wa2;
--wa1;
--x;
--qtb;
--diag;
--r__;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = dpmpar_(&c__1);
/* first, calculate the gauss-newton direction. */
jj = *n * (*n + 1) / 2 + 1;
i__1 = *n;
for (k = 1; k <= i__1; ++k) {
j = *n - k + 1;
jp1 = j + 1;
jj -= k;
l = jj + 1;
sum = zero;
if (*n < jp1) {
goto L20;
}
i__2 = *n;
for (i__ = jp1; i__ <= i__2; ++i__) {
sum += r__[l] * x[i__];
++l;
/* L10: */
}
L20:
temp = r__[jj];
if (temp != zero) {
goto L40;
}
l = j;
i__2 = j;
for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
d__2 = temp, d__3 = (d__1 = r__[l], abs(d__1));
temp = max(d__2,d__3);
l = l + *n - i__;
/* L30: */
}
temp = epsmch * temp;
if (temp == zero) {
temp = epsmch;
}
L40:
x[j] = (qtb[j] - sum) / temp;
/* L50: */
}
/* test whether the gauss-newton direction is acceptable. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = zero;
wa2[j] = diag[j] * x[j];
/* L60: */
}
qnorm = enorm_(n, &wa2[1]);
if (qnorm <= *delta) {
goto L140;
}
/* the gauss-newton direction is not acceptable. */
/* next, calculate the scaled gradient direction. */
l = 1;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
temp = qtb[j];
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
wa1[i__] += r__[l] * temp;
++l;
/* L70: */
}
wa1[j] /= diag[j];
/* L80: */
}
/* calculate the norm of the scaled gradient and test for */
/* the special case in which the scaled gradient is zero. */
gnorm = enorm_(n, &wa1[1]);
sgnorm = zero;
alpha = *delta / qnorm;
if (gnorm == zero) {
goto L120;
}
/* calculate the point along the scaled gradient */
/* at which the quadratic is minimized. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
wa1[j] = wa1[j] / gnorm / diag[j];
/* L90: */
}
l = 1;
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
sum = zero;
i__2 = *n;
for (i__ = j; i__ <= i__2; ++i__) {
sum += r__[l] * wa1[i__];
++l;
/* L100: */
}
wa2[j] = sum;
/* L110: */
}
temp = enorm_(n, &wa2[1]);
sgnorm = gnorm / temp / temp;
/* test whether the scaled gradient direction is acceptable. */
alpha = zero;
if (sgnorm >= *delta) {
goto L120;
}
/* the scaled gradient direction is not acceptable. */
/* finally, calculate the point along the dogleg */
/* at which the quadratic is minimized. */
bnorm = enorm_(n, &qtb[1]);
temp = bnorm / gnorm * (bnorm / qnorm) * (sgnorm / *delta);
/* Computing 2nd power */
d__1 = sgnorm / *delta;
/* Computing 2nd power */
d__2 = temp - *delta / qnorm;
/* Computing 2nd power */
d__3 = *delta / qnorm;
/* Computing 2nd power */
d__4 = sgnorm / *delta;
temp = temp - *delta / qnorm * (d__1 * d__1) + sqrt(d__2 * d__2 + (one -
d__3 * d__3) * (one - d__4 * d__4));
/* Computing 2nd power */
d__1 = sgnorm / *delta;
alpha = *delta / qnorm * (one - d__1 * d__1) / temp;
L120:
/* form appropriate convex combination of the gauss-newton */
/* direction and the scaled gradient direction. */
temp = (one - alpha) * min(sgnorm,*delta);
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
x[j] = temp * wa1[j] + alpha * x[j];
/* L130: */
}
L140:
return 0;
/* last card of subroutine dogleg. */
} /* dogleg_ */
/* Subroutine */ void chkder_(const int *m, const int *n, const double *x,
double *fvec, double *fjac, const int *ldfjac, double *xp,
double *fvecp, const int *mode, double *err)
{
/* Initialized data */
const int c__1 = 1;
/* System generated locals */
int fjac_dim1, fjac_offset, i__1, i__2;
/* Local variables */
int i__, j;
double eps, epsf, temp, epsmch;
double epslog;
/* ********** */
/* subroutine chkder */
/* this subroutine checks the gradients of m nonlinear functions */
/* in n variables, evaluated at a point x, for consistency with */
/* the functions themselves. the user must call chkder twice, */
/* first with mode = 1 and then with mode = 2. */
/* mode = 1. on input, x must contain the point of evaluation. */
/* on output, xp is set to a neighboring point. */
/* mode = 2. on input, fvec must contain the functions and the */
/* rows of fjac must contain the gradients */
/* of the respective functions each evaluated */
/* at x, and fvecp must contain the functions */
/* evaluated at xp. */
/* on output, err contains measures of correctness of */
/* the respective gradients. */
/* the subroutine does not perform reliably if cancellation or */
/* rounding errors cause a severe loss of significance in the */
/* evaluation of a function. therefore, none of the components */
/* of x should be unusually small (in particular, zero) or any */
/* other value which may cause loss of significance. */
/* the subroutine statement is */
/* subroutine chkder(m,n,x,fvec,fjac,ldfjac,xp,fvecp,mode,err) */
/* where */
/* 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. */
/* x is an input array of length n. */
/* fvec is an array of length m. on input when mode = 2, */
/* fvec must contain the functions evaluated at x. */
/* fjac is an m by n array. on input when mode = 2, */
/* the rows of fjac must contain the gradients of */
/* the respective functions evaluated at x. */
/* ldfjac is a positive integer input parameter not less than m */
/* which specifies the leading dimension of the array fjac. */
/* xp is an array of length n. on output when mode = 1, */
/* xp is set to a neighboring point of x. */
/* fvecp is an array of length m. on input when mode = 2, */
/* fvecp must contain the functions evaluated at xp. */
/* mode is an integer input variable set to 1 on the first call */
/* and 2 on the second. other values of mode are equivalent */
/* to mode = 1. */
/* err is an array of length m. on output when mode = 2, */
/* err contains measures of correctness of the respective */
/* gradients. if there is no severe loss of significance, */
/* then if err(i) is 1.0 the i-th gradient is correct, */
/* while if err(i) is 0.0 the i-th gradient is incorrect. */
/* for values of err between 0.0 and 1.0, the categorization */
/* is less certain. in general, a value of err(i) greater */
/* than 0.5 indicates that the i-th gradient is probably */
/* correct, while a value of err(i) less than 0.5 indicates */
/* that the i-th gradient is probably incorrect. */
/* subprograms called */
/* minpack supplied ... dpmpar */
/* fortran supplied ... dabs,dlog10,dsqrt */
/* argonne national laboratory. minpack project. march 1980. */
/* burton s. garbow, kenneth e. hillstrom, jorge j. more */
/* ********** */
/* Parameter adjustments */
--err;
--fvecp;
--fvec;
--xp;
--x;
fjac_dim1 = *ldfjac;
fjac_offset = 1 + fjac_dim1 * 1;
fjac -= fjac_offset;
/* Function Body */
/* epsmch is the machine precision. */
epsmch = dpmpar_(&c__1);
eps = sqrt(epsmch);
if (*mode == 2) {
goto L20;
}
/* mode = 1. */
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
temp = eps * fabs(x[j]);
if (temp == 0.) {
temp = eps;
}
xp[j] = x[j] + temp;
/* L10: */
}
/* goto L70; */
return;
L20:
/* mode = 2. */
epsf = factor * epsmch;
epslog = log10e * log(eps);
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
err[i__] = 0.;
/* L30: */
}
i__1 = *n;
for (j = 1; j <= i__1; ++j) {
temp = fabs(x[j]);
if (temp == 0.) {
temp = 1.;
}
i__2 = *m;
for (i__ = 1; i__ <= i__2; ++i__) {
err[i__] += temp * fjac[i__ + j * fjac_dim1];
/* L40: */
}
/* L50: */
}
i__1 = *m;
for (i__ = 1; i__ <= i__1; ++i__) {
temp = 1.;
if (fvec[i__] != 0. && fvecp[i__] != 0. && fabs(fvecp[i__] -
fvec[i__]) >= epsf * fabs(fvec[i__]))
{
temp = eps * fabs((fvecp[i__] - fvec[i__]) / eps - err[i__])
/ (fabs(fvec[i__]) +
fabs(fvecp[i__]));
}
err[i__] = 1.;
if (temp > epsmch && temp < eps) {
err[i__] = (log10e * log(temp) - epslog) / epslog;
}
if (temp >= eps) {
err[i__] = 0.;
}
/* L60: */
}
/* L70: */
/* return 0; */
/* last card of subroutine chkder. */
} /* chkder_ */
