update ast 8.6.2

1 files changed, 526 insertions, 0 deletions

diff --git a/ast/cminpack/lmder.c b/ast/cminpack/lmder.c
new file mode 100644
index 0000000..7f57428
--- /dev/null
+++ b/ast/cminpack/lmder.c
@@ -0,0 +1,526 @@
+#include "cminpack.h"
+#include <math.h>
+#include "cminpackP.h"
+
+__cminpack_attr__
+int __cminpack_func__(lmder)(__cminpack_decl_fcnder_mn__ void *p, int m, int n, real *x, 
+	real *fvec, real *fjac, int ldfjac, real ftol,
+	real xtol, real gtol, int maxfev, real *
+	diag, int mode, real factor, int nprint,
+	int *nfev, int *njev, int *ipvt, real *qtf, 
+	real *wa1, real *wa2, real *wa3, real *wa4)
+{
+    /* Initialized data */
+
+#define p1 .1
+#define p5 .5
+#define p25 .25
+#define p75 .75
+#define p0001 1e-4
+
+    /* System generated locals */
+    real d1, d2;
+
+    /* Local variables */
+    int i, j, l;
+    real par, sum;
+    int iter;
+    real temp, temp1, temp2;
+    int iflag;
+    real delta = 0.;
+    real ratio;
+    real fnorm, gnorm, pnorm, xnorm = 0., fnorm1, actred, dirder, 
+	    epsmch, prered;
+    int info;
+
+/*     ********** */
+
+/*     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 */
+
+/*     ********** */
+
+/*     epsmch is the machine precision. */
+
+    epsmch = __cminpack_func__(dpmpar)(1);
+
+    info = 0;
+    iflag = 0;
+    *nfev = 0;
+    *njev = 0;
+
+/*     check the input parameters for errors. */
+
+    if (n <= 0 || m < n || ldfjac < m || ftol < 0. || xtol < 0. || 
+	    gtol < 0. || maxfev <= 0 || factor <= 0.) {
+	goto TERMINATE;
+    }
+    if (mode == 2) {
+        for (j = 0; j < n; ++j) {
+            if (diag[j] <= 0.) {
+                goto TERMINATE;
+            }
+        }
+    }
+
+/*     evaluate the function at the starting point */
+/*     and calculate its norm. */
+
+    iflag = fcnder_mn(p, m, n, x, fvec, fjac, ldfjac, 1);
+    *nfev = 1;
+    if (iflag < 0) {
+	goto TERMINATE;
+    }
+    fnorm = __cminpack_enorm__(m, fvec);
+
+/*     initialize levenberg-marquardt parameter and iteration counter. */
+
+    par = 0.;
+    iter = 1;
+
+/*     beginning of the outer loop. */
+
+    for (;;) {
+
+/*        calculate the jacobian matrix. */
+
+        iflag = fcnder_mn(p, m, n, x, fvec, fjac, ldfjac, 2);
+        ++(*njev);
+        if (iflag < 0) {
+            goto TERMINATE;
+        }
+
+/*        if requested, call fcn to enable printing of iterates. */
+
+        if (nprint > 0) {
+            iflag = 0;
+            if ((iter - 1) % nprint == 0) {
+                iflag = fcnder_mn(p, m, n, x, fvec, fjac, ldfjac, 0);
+            }
+            if (iflag < 0) {
+                goto TERMINATE;
+            }
+        }
+
+/*        compute the qr factorization of the jacobian. */
+
+        __cminpack_func__(qrfac)(m, n, fjac, ldfjac, TRUE_, ipvt, n,
+              wa1, wa2, wa3);
+
+/*        on the first iteration and if mode is 1, scale according */
+/*        to the norms of the columns of the initial jacobian. */
+
+        if (iter == 1) {
+            if (mode != 2) {
+                for (j = 0; j < n; ++j) {
+                    diag[j] = wa2[j];
+                    if (wa2[j] == 0.) {
+                        diag[j] = 1.;
+                    }
+                }
+            }
+
+/*        on the first iteration, calculate the norm of the scaled x */
+/*        and initialize the step bound delta. */
+
+            for (j = 0; j < n; ++j) {
+                wa3[j] = diag[j] * x[j];
+            }
+            xnorm = __cminpack_enorm__(n, wa3);
+            delta = factor * xnorm;
+            if (delta == 0.) {
+                delta = factor;
+            }
+        }
+
+/*        form (q transpose)*fvec and store the first n components in */
+/*        qtf. */
+
+        for (i = 0; i < m; ++i) {
+            wa4[i] = fvec[i];
+        }
+        for (j = 0; j < n; ++j) {
+            if (fjac[j + j * ldfjac] != 0.) {
+                sum = 0.;
+                for (i = j; i < m; ++i) {
+                    sum += fjac[i + j * ldfjac] * wa4[i];
+                }
+                temp = -sum / fjac[j + j * ldfjac];
+                for (i = j; i < m; ++i) {
+                    wa4[i] += fjac[i + j * ldfjac] * temp;
+                }
+            }
+            fjac[j + j * ldfjac] = wa1[j];
+            qtf[j] = wa4[j];
+        }
+
+/*        compute the norm of the scaled gradient. */
+
+        gnorm = 0.;
+        if (fnorm != 0.) {
+            for (j = 0; j < n; ++j) {
+                l = ipvt[j]-1;
+                if (wa2[l] != 0.) {
+                    sum = 0.;
+                    for (i = 0; i <= j; ++i) {
+                        sum += fjac[i + j * ldfjac] * (qtf[i] / fnorm);
+                    }
+                    /* Computing MAX */
+                    d1 = fabs(sum / wa2[l]);
+                    gnorm = max(gnorm,d1);
+                }
+            }
+        }
+
+/*        test for convergence of the gradient norm. */
+
+        if (gnorm <= gtol) {
+            info = 4;
+        }
+        if (info != 0) {
+            goto TERMINATE;
+        }
+
+/*        rescale if necessary. */
+
+        if (mode != 2) {
+            for (j = 0; j < n; ++j) {
+                /* Computing MAX */
+                d1 = diag[j], d2 = wa2[j];
+                diag[j] = max(d1,d2);
+            }
+        }
+
+/*        beginning of the inner loop. */
+
+        do {
+
+/*           determine the levenberg-marquardt parameter. */
+
+            __cminpack_func__(lmpar)(n, fjac, ldfjac, ipvt, diag, qtf, delta,
+                  &par, wa1, wa2, wa3, wa4);
+
+/*           store the direction p and x + p. calculate the norm of p. */
+
+            for (j = 0; j < n; ++j) {
+                wa1[j] = -wa1[j];
+                wa2[j] = x[j] + wa1[j];
+                wa3[j] = diag[j] * wa1[j];
+            }
+            pnorm = __cminpack_enorm__(n, wa3);
+
+/*           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 = fcnder_mn(p, m, n, wa2, wa4, fjac, ldfjac, 1);
+            ++(*nfev);
+            if (iflag < 0) {
+                goto TERMINATE;
+            }
+            fnorm1 = __cminpack_enorm__(m, wa4);
+
+/*           compute the scaled actual reduction. */
+
+            actred = -1.;
+            if (p1 * fnorm1 < fnorm) {
+                /* Computing 2nd power */
+                d1 = fnorm1 / fnorm;
+                actred = 1. - d1 * d1;
+            }
+
+/*           compute the scaled predicted reduction and */
+/*           the scaled directional derivative. */
+
+            for (j = 0; j < n; ++j) {
+                wa3[j] = 0.;
+                l = ipvt[j]-1;
+                temp = wa1[l];
+                for (i = 0; i <= j; ++i) {
+                    wa3[i] += fjac[i + j * ldfjac] * temp;
+                }
+            }
+            temp1 = __cminpack_enorm__(n, wa3) / fnorm;
+            temp2 = (sqrt(par) * pnorm) / fnorm;
+            prered = temp1 * temp1 + temp2 * temp2 / p5;
+            dirder = -(temp1 * temp1 + temp2 * temp2);
+
+/*           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) {
+                if (actred >= 0.) {
+                    temp = p5;
+                } else {
+                    temp = p5 * dirder / (dirder + p5 * actred);
+                }
+                if (p1 * fnorm1 >= fnorm || temp < p1) {
+                    temp = p1;
+                }
+                /* Computing MIN */
+                d1 = pnorm / p1;
+                delta = temp * min(delta,d1);
+                par /= temp;
+            } else {
+                if (par == 0. || ratio >= p75) {
+                    delta = pnorm / p5;
+                    par = p5 * par;
+                }
+            }
+
+/*           test for successful iteration. */
+
+            if (ratio >= p0001) {
+
+/*           successful iteration. update x, fvec, and their norms. */
+
+                for (j = 0; j < n; ++j) {
+                    x[j] = wa2[j];
+                    wa2[j] = diag[j] * x[j];
+                }
+                for (i = 0; i < m; ++i) {
+                    fvec[i] = wa4[i];
+                }
+                xnorm = __cminpack_enorm__(n, wa2);
+                fnorm = fnorm1;
+                ++iter;
+            }
+
+/*           tests for convergence. */
+
+            if (fabs(actred) <= ftol && prered <= ftol && p5 * ratio <= 1.) {
+                info = 1;
+            }
+            if (delta <= xtol * xnorm) {
+                info = 2;
+            }
+            if (fabs(actred) <= ftol && prered <= ftol && p5 * ratio <= 1. && info == 2) {
+                info = 3;
+            }
+            if (info != 0) {
+                goto TERMINATE;
+            }
+
+/*           tests for termination and stringent tolerances. */
+
+            if (*nfev >= maxfev) {
+                info = 5;
+            }
+            if (fabs(actred) <= epsmch && prered <= epsmch && p5 * ratio <= 1.) {
+                info = 6;
+            }
+            if (delta <= epsmch * xnorm) {
+                info = 7;
+            }
+            if (gnorm <= epsmch) {
+                info = 8;
+            }
+            if (info != 0) {
+                goto TERMINATE;
+            }
+
+/*           end of the inner loop. repeat if iteration unsuccessful. */
+
+        } while (ratio < p0001);
+
+/*        end of the outer loop. */
+
+    }
+TERMINATE:
+
+/*     termination, either normal or user imposed. */
+
+    if (iflag < 0) {
+	info = iflag;
+    }
+    if (nprint > 0) {
+	fcnder_mn(p, m, n, x, fvec, fjac, ldfjac, 0);
+    }
+    return info;
+
+/*     last card of subroutine lmder. */
+
+} /* lmder_ */
+