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/************************lmdif*************************/
/*
* Solves or minimizes the sum of squares of m nonlinear
* functions of n variables.
*
* From public domain Fortran version
* of Argonne National Laboratories MINPACK
*
* C translation by Steve Moshier
*/
#include "filter.h"
#include <float.h>
extern lmfunc fcn;
#if _MSC_VER > 1000
#pragma warning(disable: 4100) // disable unreferenced formal parameter warning
#endif
// These globals are needed by MINPACK
/* resolution of arithmetic */
double MACHEP = 1.2e-16;
/* smallest nonzero number */
double DWARF = 1.0e-38;
int fdjac2(int,int,double*,double*,double*,int,int*,double,double*);
int qrfac(int,int,double*,int,int,int*,int,double*,double*,double*);
int lmpar(int,double*,int,int*,double*,double*,double,double*,double*,double*,double*,double*);
int qrsolv(int,double*,int,int*,double*,double*,double*,double*,double*);
static double enorm(int n, double x[]);
static double dmax1(double a, double b);
static double dmin1(double a, double b);
/*********************** lmdif.c ****************************/
#define BUG 0
extern double MACHEP;
int lmdif(int m, int n, double x[], double fvec[],
double ftol, double xtol, double gtol,
int maxfev, double epsfcn, double diag[],
int mode, double factor, int nprint,
int *info, int *nfev, double fjac[],
int ldfjac, int ipvt[], double qtf[],
double wa1[], double wa2[], double wa3[], double wa4[])
{
/*
* **********
*
* subroutine lmdif
*
* the purpose of lmdif 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. the jacobian is
* then calculated by a forward-difference approximation.
*
* the subroutine statement is
*
* subroutine lmdif(fcn,m,n,x,fvec,ftol,xtol,gtol,maxfev,epsfcn,
* diag,mode,factor,nprint,info,nfev,fjac,
* ldfjac,ipvt,qtf,wa1,wa2,wa3,wa4)
*
* where
*
* fcn is the name of the user-supplied subroutine which
* calculates the functions. fcn must be declared
* in an external statement in the user calling
* program, and should be written as follows.
*
* subroutine fcn(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 lmdif.
* 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.
*
* 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 is at least
* maxfev by the end of an iteration.
*
* 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 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 has reached or
* exceeded 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.
*
* 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.
*
* 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,fdjac2,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
*
* **********
*/
int i,iflag,ij,jj,iter,j,l;
double actred,delta = 1.0e-4,dirder,fnorm,fnorm1,gnorm;
double par,pnorm,prered,ratio;
double sum,temp,temp1,temp2,temp3,xnorm = 1.0e-4;
// int fcn(); /* user supplied function */
static double one = 1.0;
static double p1 = 0.1;
static double p5 = 0.5;
static double p25 = 0.25;
static double p75 = 0.75;
static double p0001 = 1.0e-4;
static double zero = 0.0;
MACHEP = DBL_EPSILON; // machine precision, was 1.2e-16;
/* smallest nonzero number */
DWARF = DBL_MIN; // was 1.0e-38;
*info = 0;
iflag = 0;
*nfev = 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 == 2 )
{ /* scaling by diag[] */
for( j=0; j<n; j++ )
{
if( diag[j] <= 0.0 )
goto L300;
}
}
#if BUG
// printf( "lmdif\n" );
#endif
/*
* evaluate the function at the starting point
* and calculate its norm.
*/
iflag = 1;
fcn(m,n,x,fvec,&iflag);
*nfev = 1;
if(iflag < 0)
goto L300;
fnorm = enorm(m,fvec);
/*
* initialize levenberg-marquardt parameter and iteration counter.
*/
par = zero;
iter = 1;
/*
* beginning of the outer loop.
*/
L30:
/*
* calculate the jacobian matrix.
*/
iflag = 2;
fdjac2(m,n,x,fvec,fjac,ldfjac,&iflag,epsfcn,wa4);
*nfev += n;
if(iflag < 0)
goto L300;
/*
* if requested, call fcn to enable printing of iterates.
*/
if( nprint > 0 )
{
iflag = 0;
if((iter-1)%nprint == 0)
{
fcn(m,n,x,fvec,&iflag);
if(iflag < 0)
goto L300;
// printf( "fnorm %.15e\n", enorm(m,fvec) );
}
}
/*
* compute the qr factorization of the jacobian.
*/
qrfac(m,n,fjac,ldfjac,1,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] == zero )
diag[j] = one;
}
}
/*
* 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 = enorm(n,wa3);
delta = factor*xnorm;
if(delta == zero)
delta = factor;
}
/*
* form (q transpose)*fvec and store the first n components in
* qtf.
*/
for( i=0; i<m; i++ )
wa4[i] = fvec[i];
jj = 0;
for( j=0; j<n; j++ )
{
temp3 = fjac[jj];
if(temp3 != zero)
{
sum = zero;
ij = jj;
for( i=j; i<m; i++ )
{
sum += fjac[ij] * wa4[i];
ij += 1; /* fjac[i+m*j] */
}
temp = -sum / temp3;
ij = jj;
for( i=j; i<m; i++ )
{
wa4[i] += fjac[ij] * temp;
ij += 1; /* fjac[i+m*j] */
}
}
fjac[jj] = wa1[j];
jj += m+1; /* fjac[j+m*j] */
qtf[j] = wa4[j];
}
/*
* compute the norm of the scaled gradient.
*/
gnorm = zero;
if(fnorm != zero)
{
jj = 0;
for( j=0; j<n; j++ )
{
l = ipvt[j];
if(wa2[l] != zero)
{
sum = zero;
ij = jj;
for( i=0; i<=j; i++ )
{
sum += fjac[ij]*(qtf[i]/fnorm);
ij += 1; /* fjac[i+m*j] */
}
gnorm = dmax1(gnorm,fabs(sum/wa2[l]));
}
jj += m;
}
}
/*
* test for convergence of the gradient norm.
*/
if(gnorm <= gtol)
*info = 4;
if( *info != 0)
goto L300;
/*
* rescale if necessary.
*/
if(mode != 2)
{
for( j=0; j<n; j++ )
diag[j] = dmax1(diag[j],wa2[j]);
}
/*
* beginning of the inner loop.
*/
L200:
/*
* determine the levenberg-marquardt parameter.
*/
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 = enorm(n,wa3);
/*
* on the first iteration, adjust the initial step bound.
*/
if(iter == 1)
delta = dmin1(delta,pnorm);
/*
* evaluate the function at x + p and calculate its norm.
*/
iflag = 1;
fcn(m,n,wa2,wa4,&iflag);
*nfev += 1;
if(iflag < 0)
goto L300;
fnorm1 = enorm(m,wa4);
#if BUG
// printf( "pnorm %.10e fnorm1 %.10e\n", pnorm, fnorm1 );
#endif
/*
* compute the scaled actual reduction.
*/
actred = -one;
if( (p1*fnorm1) < fnorm)
{
temp = fnorm1/fnorm;
actred = one - temp * temp;
}
/*
* compute the scaled predicted reduction and
* the scaled directional derivative.
*/
jj = 0;
for( j=0; j<n; j++ )
{
wa3[j] = zero;
l = ipvt[j];
temp = wa1[l];
ij = jj;
for( i=0; i<=j; i++ )
{
wa3[i] += fjac[ij]*temp;
ij += 1; /* fjac[i+m*j] */
}
jj += m;
}
temp1 = 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 = zero;
if(prered != zero)
ratio = actred/prered;
/*
* update the step bound.
*/
if(ratio <= p25)
{
if(actred >= zero)
temp = p5;
else
temp = p5*dirder/(dirder + p5*actred);
if( ((p1*fnorm1) >= fnorm)
|| (temp < p1) )
temp = p1;
delta = temp*dmin1(delta,pnorm/p1);
par = par/temp;
}
else
{
if( (par == zero) || (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 = enorm(n,wa2);
fnorm = fnorm1;
iter += 1;
}
/*
* tests for convergence.
*/
if( (fabs(actred) <= ftol)
&& (prered <= ftol)
&& (p5*ratio <= one) )
*info = 1;
if(delta <= xtol*xnorm)
*info = 2;
if( (fabs(actred) <= ftol)
&& (prered <= ftol)
&& (p5*ratio <= one)
&& ( *info == 2) )
*info = 3;
if( *info != 0)
goto L300;
/*
* tests for termination and stringent tolerances.
*/
if( *nfev >= maxfev)
*info = 5;
if( (fabs(actred) <= MACHEP)
&& (prered <= MACHEP)
&& (p5*ratio <= one) )
*info = 6;
if(delta <= MACHEP*xnorm)
*info = 7;
if(gnorm <= MACHEP)
*info = 8;
if( *info != 0)
goto L300;
/*
* end of the inner loop. repeat if iteration unsuccessful.
*/
if(ratio < p0001)
goto L200;
/*
* 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(m,n,x,fvec,&iflag);
/*
last card of subroutine lmdif.
*/
return 0;
}
/************************lmpar.c*************************/
#define BUG 0
int lmpar(int n, double r[], int ldr, int ipvt[],
double diag[], double qtb[], double delta,
double *par, double x[], double sdiag[],
double wa1[], double wa2[])
{
/* **********
*
* 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
*
* **********
*/
int i,iter,ij,jj,j,jm1,jp1,k,l,nsing;
double dxnorm,fp,gnorm,parc,parl,paru;
double sum,temp;
static double zero = 0.0;
// static double one = 1.0;
static double p1 = 0.1;
static double p001 = 0.001;
// extern double MACHEP;
extern double DWARF;
#if BUG
// printf( "lmpar\n" );
#endif
/*
* compute and store in x the gauss-newton direction. if the
* jacobian is rank-deficient, obtain a least squares solution.
*/
nsing = n;
jj = 0;
for( j=0; j<n; j++ )
{
wa1[j] = qtb[j];
if( (r[jj] == zero) && (nsing == n) )
nsing = j;
if(nsing < n)
wa1[j] = zero;
jj += ldr+1; /* [j+ldr*j] */
}
#if BUG
// printf( "nsing %d ", nsing );
#endif
if(nsing >= 1)
{
for( k=0; k<nsing; k++ )
{
j = nsing - k - 1;
wa1[j] = wa1[j]/r[j+ldr*j];
temp = wa1[j];
jm1 = j - 1;
if(jm1 >= 0)
{
ij = ldr * j;
for( i=0; i<=jm1; i++ )
{
wa1[i] -= r[ij]*temp;
ij += 1;
}
}
}
}
for( j=0; j<n; j++ )
{
l = ipvt[j];
x[l] = wa1[j];
}
/*
* initialize the iteration counter.
* evaluate the function at the origin, and test
* for acceptance of the gauss-newton direction.
*/
iter = 0;
for( j=0; j<n; j++ )
wa2[j] = diag[j]*x[j];
dxnorm = enorm(n,wa2);
fp = dxnorm - delta;
if(fp <= p1*delta)
{
#if BUG
// printf( "going to L220\n" );
#endif
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)
{
for( j=0; j<n; j++ )
{
l = ipvt[j];
wa1[j] = diag[l]*(wa2[l]/dxnorm);
}
jj = 0;
for( j=0; j<n; j++ )
{
sum = zero;
jm1 = j - 1;
if(jm1 >= 0)
{
ij = jj;
for( i=0; i<=jm1; i++ )
{
sum += r[ij]*wa1[i];
ij += 1;
}
}
wa1[j] = (wa1[j] - sum)/r[j+ldr*j];
jj += ldr; /* [i+ldr*j] */
}
temp = enorm(n,wa1);
parl = ((fp/delta)/temp)/temp;
}
/*
* calculate an upper bound, paru, for the zero of the function.
*/
jj = 0;
for( j=0; j<n; j++ )
{
sum = zero;
ij = jj;
for( i=0; i<=j; i++ )
{
sum += r[ij]*qtb[i];
ij += 1;
}
l = ipvt[j];
wa1[j] = sum/diag[l];
jj += ldr; /* [i+ldr*j] */
}
gnorm = enorm(n,wa1);
paru = gnorm/delta;
if(paru == zero)
paru = DWARF/dmin1(delta,p1);
/*
* if the input par lies outside of the interval (parl,paru),
* set par to the closer endpoint.
*/
*par = dmax1( *par,parl);
*par = dmin1( *par,paru);
if( *par == zero)
*par = gnorm/dxnorm;
#if BUG
// printf( "parl %.4e par %.4e paru %.4e\n", parl, *par, paru );
#endif
/*
* beginning of an iteration.
*/
L150:
iter += 1;
/*
* evaluate the function at the current value of par.
*/
if( *par == zero)
*par = dmax1(DWARF,p001*paru);
temp = sqrt( *par );
for( j=0; j<n; j++ )
wa1[j] = temp*diag[j];
qrsolv(n,r,ldr,ipvt,wa1,qtb,x,sdiag,wa2);
for( j=0; j<n; j++ )
wa2[j] = diag[j]*x[j];
dxnorm = enorm(n,wa2);
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( (fabs(fp) <= p1*delta)
|| ((parl == zero) && (fp <= temp) && (temp < zero))
|| (iter == 10) )
goto L220;
/*
* compute the newton correction.
*/
for( j=0; j<n; j++ )
{
l = ipvt[j];
wa1[j] = diag[l]*(wa2[l]/dxnorm);
}
jj = 0;
for( j=0; j<n; j++ )
{
wa1[j] = wa1[j]/sdiag[j];
temp = wa1[j];
jp1 = j + 1;
if(jp1 < n)
{
ij = jp1 + jj;
for( i=jp1; i<n; i++ )
{
wa1[i] -= r[ij]*temp;
ij += 1; /* [i+ldr*j] */
}
}
jj += ldr; /* ldr*j */
}
temp = enorm(n,wa1);
parc = ((fp/delta)/temp)/temp;
/*
* depending on the sign of the function, update parl or paru.
*/
if(fp > zero)
parl = dmax1(parl, *par);
if(fp < zero)
paru = dmin1(paru, *par);
/*
* compute an improved estimate for par.
*/
*par = dmax1(parl, *par + parc);
/*
* end of an iteration.
*/
goto L150;
L220:
/*
* termination.
*/
if(iter == 0)
*par = zero;
/*
* last card of subroutine lmpar.
*/
return 0;
}
/************************qrfac.c*************************/
#define BUG 0
int qrfac(int m, int n, double a[], int lda PT_UNUSED, int pivot,
int ipvt[], int lipvt PT_UNUSED, double rdiag[],
double acnorm[], double wa[])
{
/*
* **********
*
* 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
*
* argonne national laboratory. minpack project. march 1980.
* burton s. garbow, kenneth e. hillstrom, jorge j. more
*
* **********
*/
int i,ij,jj,j,jp1,k,kmax,minmn;
double ajnorm,sum,temp;
static double zero = 0.0;
static double one = 1.0;
static double p05 = 0.05;
extern double MACHEP;
/*
* compute the initial column norms and initialize several arrays.
*/
ij = 0;
for( j=0; j<n; j++ )
{
acnorm[j] = enorm(m,&a[ij]);
rdiag[j] = acnorm[j];
wa[j] = rdiag[j];
if(pivot != 0)
ipvt[j] = j;
ij += m; /* m*j */
}
#if BUG
// printf( "qrfac\n" );
#endif
/*
* reduce a to r with householder transformations.
*/
minmn = m<=n?m:n;
for( j=0; j<minmn; j++ )
{
if(pivot == 0)
goto L40;
/*
* bring the column of largest norm into the pivot position.
*/
kmax = j;
for( k=j; k<n; k++ )
{
if(rdiag[k] > rdiag[kmax])
kmax = k;
}
if(kmax == j)
goto L40;
ij = m * j;
jj = m * kmax;
for( i=0; i<m; i++ )
{
temp = a[ij]; /* [i+m*j] */
a[ij] = a[jj]; /* [i+m*kmax] */
a[jj] = temp;
ij += 1;
jj += 1;
}
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.
*/
jj = j + m*j;
ajnorm = enorm(m-j,&a[jj]);
if(ajnorm == zero)
goto L100;
if(a[jj] < zero)
ajnorm = -ajnorm;
ij = jj;
for( i=j; i<m; i++ )
{
a[ij] /= ajnorm;
ij += 1; /* [i+m*j] */
}
a[jj] += one;
/*
* apply the transformation to the remaining columns
* and update the norms.
*/
jp1 = j + 1;
if(jp1 < n )
{
for( k=jp1; k<n; k++ )
{
sum = zero;
ij = j + m*k;
jj = j + m*j;
for( i=j; i<m; i++ )
{
sum += a[jj]*a[ij];
ij += 1; /* [i+m*k] */
jj += 1; /* [i+m*j] */
}
temp = sum/a[j+m*j];
ij = j + m*k;
jj = j + m*j;
for( i=j; i<m; i++ )
{
a[ij] -= temp*a[jj];
ij += 1; /* [i+m*k] */
jj += 1; /* [i+m*j] */
}
if( (pivot != 0) && (rdiag[k] != zero) )
{
temp = a[j+m*k]/rdiag[k];
temp = dmax1( zero, one-temp*temp );
rdiag[k] *= sqrt(temp);
temp = rdiag[k]/wa[k];
if( (p05*temp*temp) <= MACHEP)
{
rdiag[k] = enorm(m-j-1,&a[jp1+m*k]);
wa[k] = rdiag[k];
}
}
}
}
L100:
rdiag[j] = -ajnorm;
}
/*
* last card of subroutine qrfac.
*/
return 0;
}
/************************qrsolv.c*************************/
#define BUG 0
int qrsolv(int n, double r[], int ldr, int ipvt[], double diag[],
double qtb[], double x[], double sdiag[], double wa[])
{
/*
* **********
*
* subroutine qrsolv
*
* given an m by n matrix a, an n by n diagonal matrix d,
* and an m-vector b, the problem is to determine an x which
* solves the system
*
* a*x = b , d*x = 0 ,
*
* in the least squares sense.
*
* 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 qrsolv expects
* the full upper triangle of r, the permutation matrix p,
* and the first n components of (q transpose)*b. the system
* a*x = b, d*x = 0, is then equivalent to
*
* t t
* r*z = q *b , p *d*p*z = 0 ,
*
* where x = p*z. if this system does not have full rank,
* then a least squares solution is obtained. on output qrsolv
* also provides an upper triangular matrix s such that
*
* t t t
* p *(a *a + d*d)*p = s *s .
*
* s is computed within qrsolv and may be of separate interest.
*
* the subroutine statement is
*
* subroutine qrsolv(n,r,ldr,ipvt,diag,qtb,x,sdiag,wa)
*
* 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.
*
* x is an output array of length n which contains the least
* squares solution of the system a*x = b, d*x = 0.
*
* sdiag is an output array of length n which contains the
* diagonal elements of the upper triangular matrix s.
*
* wa is a work array of length n.
*
* subprograms called
*
* fortran-supplied ... dabs,dsqrt
*
* argonne national laboratory. minpack project. march 1980.
* burton s. garbow, kenneth e. hillstrom, jorge j. more
*
* **********
*/
int i,ij,ik,kk,j,jp1,k,kp1,l,nsing;
double cos,cotan,qtbpj,sin,sum,tan,temp;
static double zero = 0.0;
static double p25 = 0.25;
static double p5 = 0.5;
double fabs(), sqrt();
/*
* copy r and (q transpose)*b to preserve input and initialize s.
* in particular, save the diagonal elements of r in x.
*/
kk = 0;
for( j=0; j<n; j++ )
{
ij = kk;
ik = kk;
for( i=j; i<n; i++ )
{
r[ij] = r[ik];
ij += 1; /* [i+ldr*j] */
ik += ldr; /* [j+ldr*i] */
}
x[j] = r[kk];
wa[j] = qtb[j];
kk += ldr+1; /* j+ldr*j */
}
#if BUG
// printf( "qrsolv\n" );
#endif
/*
* eliminate the diagonal matrix d using a givens rotation.
*/
for( j=0; j<n; j++ )
{
/*
* prepare the row of d to be eliminated, locating the
* diagonal element using p from the qr factorization.
*/
l = ipvt[j];
if(diag[l] == zero)
goto L90;
for( k=j; k<n; k++ )
sdiag[k] = zero;
sdiag[j] = diag[l];
/*
* the transformations to eliminate the row of d
* modify only a single element of (q transpose)*b
* beyond the first n, which is initially zero.
*/
qtbpj = zero;
for( k=j; k<n; k++ )
{
/*
* determine a givens rotation which eliminates the
* appropriate element in the current row of d.
*/
if(sdiag[k] == zero)
continue;
kk = k + ldr * k;
if(fabs(r[kk]) < fabs(sdiag[k]))
{
cotan = r[kk]/sdiag[k];
sin = p5/sqrt(p25+p25*cotan*cotan);
cos = sin*cotan;
}
else
{
tan = sdiag[k]/r[kk];
cos = p5/sqrt(p25+p25*tan*tan);
sin = cos*tan;
}
/*
* compute the modified diagonal element of r and
* the modified element of ((q transpose)*b,0).
*/
r[kk] = cos*r[kk] + sin*sdiag[k];
temp = cos*wa[k] + sin*qtbpj;
qtbpj = -sin*wa[k] + cos*qtbpj;
wa[k] = temp;
/*
* accumulate the tranformation in the row of s.
*/
kp1 = k + 1;
if( n > kp1 )
{
ik = kk + 1;
for( i=kp1; i<n; i++ )
{
temp = cos*r[ik] + sin*sdiag[i];
sdiag[i] = -sin*r[ik] + cos*sdiag[i];
r[ik] = temp;
ik += 1; /* [i+ldr*k] */
}
}
}
L90:
/*
* store the diagonal element of s and restore
* the corresponding diagonal element of r.
*/
kk = j + ldr*j;
sdiag[j] = r[kk];
r[kk] = x[j];
}
/*
* solve the triangular system for z. if the system is
* singular, then obtain a least squares solution.
*/
nsing = n;
for( j=0; j<n; j++ )
{
if( (sdiag[j] == zero) && (nsing == n) )
nsing = j;
if(nsing < n)
wa[j] = zero;
}
if(nsing < 1)
goto L150;
for( k=0; k<nsing; k++ )
{
j = nsing - k - 1;
sum = zero;
jp1 = j + 1;
if(nsing > jp1)
{
ij = jp1 + ldr * j;
for( i=jp1; i<nsing; i++ )
{
sum += r[ij]*wa[i];
ij += 1; /* [i+ldr*j] */
}
}
wa[j] = (wa[j] - sum)/sdiag[j];
}
L150:
/*
* permute the components of z back to components of x.
*/
for( j=0; j<n; j++ )
{
l = ipvt[j];
x[l] = wa[j];
}
/*
* last card of subroutine qrsolv.
*/
return 0;
}
/************************enorm.c*************************/
static double enorm(int n, double x[])
{
/*
* **********
*
* function enorm
*
* given an n-vector x, this function calculates the
* euclidean norm of x.
*
* the euclidean norm is computed by accumulating the sum of
* squares in three different sums. the sums of squares for the
* small and large components are scaled so that no overflows
* occur. non-destructive underflows are permitted. underflows
* and overflows do not occur in the computation of the unscaled
* sum of squares for the intermediate components.
* the definitions of small, intermediate and large components
* depend on two constants, rdwarf and rgiant. the main
* restrictions on these constants are that rdwarf**2 not
* underflow and rgiant**2 not overflow. the constants
* given here are suitable for every known computer.
*
* the function statement is
*
* double precision function enorm(n,x)
*
* where
*
* n is a positive integer input variable.
*
* x is an input array of length n.
*
* subprograms called
*
* fortran-supplied ... dabs,dsqrt
*
* argonne national laboratory. minpack project. march 1980.
* burton s. garbow, kenneth e. hillstrom, jorge j. more
*
* **********
*/
int i;
double agiant,floatn,s1,s2,s3,xabs,x1max,x3max;
double ans, temp;
static double rdwarf = 3.834e-20;
static double rgiant = 1.304e19;
static double zero = 0.0;
static double one = 1.0;
double fabs(), sqrt();
s1 = zero;
s2 = zero;
s3 = zero;
x1max = zero;
x3max = zero;
floatn = n;
agiant = rgiant/floatn;
for( i=0; i<n; i++ )
{
xabs = fabs(x[i]);
if( (xabs > rdwarf) && (xabs < agiant) )
{
/*
* sum for intermediate components.
*/
s2 += xabs*xabs;
continue;
}
if(xabs > rdwarf)
{
/*
* sum for large components.
*/
if(xabs > x1max)
{
temp = x1max/xabs;
s1 = one + s1*temp*temp;
x1max = xabs;
}
else
{
temp = xabs/x1max;
s1 += temp*temp;
}
continue;
}
/*
* sum for small components.
*/
if(xabs > x3max)
{
temp = x3max/xabs;
s3 = one + s3*temp*temp;
x3max = xabs;
}
else
{
if(xabs != zero)
{
temp = xabs/x3max;
s3 += temp*temp;
}
}
}
/*
* calculation of norm.
*/
if(s1 != zero)
{
temp = s1 + (s2/x1max)/x1max;
ans = x1max*sqrt(temp);
return(ans);
}
if(s2 != zero)
{
if(s2 >= x3max)
temp = s2*(one+(x3max/s2)*(x3max*s3));
else
temp = x3max*((s2/x3max)+(x3max*s3));
ans = sqrt(temp);
}
else
{
ans = x3max*sqrt(s3);
}
return(ans);
/*
* last card of function enorm.
*/
}
/************************fdjac2.c*************************/
#define BUG 0
int fdjac2(int m, int n, double x[], double fvec[], double fjac[],
int ldfjac PT_UNUSED, int *iflag, double epsfcn, double wa[])
{
/*
* **********
*
* 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
*
**********
*/
int i,j,ij;
double eps,h,temp;
static double zero = 0.0;
extern double MACHEP;
temp = dmax1(epsfcn,MACHEP);
eps = sqrt(temp);
#if BUG
// printf( "fdjac2\n" );
#endif
ij = 0;
for( j=0; j<n; j++ )
{
temp = x[j];
h = eps * fabs(temp);
if(h == zero)
h = eps;
x[j] = temp + h;
fcn(m,n,x,wa,iflag);
if( *iflag < 0)
return 0;
x[j] = temp;
for( i=0; i<m; i++ )
{
fjac[ij] = (wa[i] - fvec[i])/h;
ij += 1; /* fjac[i+m*j] */
}
}
#if BUG
pmat( m, n, fjac );
#endif
/*
* last card of subroutine fdjac2.
*/
return 0;
}
/************************lmmisc.c*************************/
static double dmax1(double a, double b)
{
if( a >= b )
return(a);
else
return(b);
}
static double dmin1(double a, double b)
{
if( a <= b )
return(a);
else
return(b);
}
static int PT_UNUSED pmat( int m, int n, double y[] PT_UNUSED)
{
int i, j, k;
k = 0;
for( i=0; i<m; i++ )
{
for( j=0; j<n; j++ )
{
// printf( "%.5e ", y[k] );
k += 1;
}
// printf( "\n" );
}
return 0;
}