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java.lang.Object optimization.Minpack_f77
public class Minpack_f77
This class contains Java translations of the MINPACK nonlinear least squares routines. As of November 2000, it does not yet contain the MINPACK solvers of systems of nonlinear equations. They should be added in the Spring of 2001.
The original FORTRAN MINPACK package was produced by Burton S. Garbow, Kenneth E. Hillstrom, and Jorge J. More as part of the Argonne National Laboratory MINPACK project, March 1980.
IMPORTANT: The "_f77" suffixes indicate that these routines use FORTRAN style indexing. For example, you will see
for (i = 1; i <= n; i++)rather than
for (i = 0; i < n; i++)To use the "_f77" routines you will have to declare your vectors and matrices to be one element larger (e.g., v[101] rather than v[100], and a[101][101] rather than a[100][100]), and you will have to fill elements 1 through n rather than elements 0 through n  1. Versions of these programs that use C/Java style indexing might eventually be available. They would end with the suffix "_j".
This class was translated by a statistician from FORTRAN versions of lmder and lmdif. It is NOT an official translation. It wastes memory by failing to use the first elements of vectors. When public domain Java optimization routines become available from the people who produced MINPACK, then THE CODE PRODUCED BY THE NUMERICAL ANALYSTS SHOULD BE USED.
Meanwhile, if you have suggestions for improving this code, please contact Steve Verrill at steve@ws13.fpl.fs.fed.us.
Constructor Summary  

Minpack_f77()

Method Summary  

static double 
enorm_f77(int n,
double[] x)
The enorm_f77 method calculates the Euclidean norm of a vector. 
static void 
fdjac2_f77(optimization.Lmdif_fcn nlls,
int m,
int n,
double[] x,
double[] fvec,
double[][] fjac,
int[] iflag,
double epsfcn,
double[] wa)
The fdjac2 method computes a forwarddifference approximation to the m by n Jacobian matrix associated with a specified problem of m functions in n variables. 
static void 
lmder_f77(optimization.Lmder_fcn nlls,
int m,
int n,
double[] x,
double[] fvec,
double[][] fjac,
double ftol,
double xtol,
double gtol,
int maxfev,
double[] diag,
int mode,
double factor,
int nprint,
int[] info,
int[] nfev,
int[] njev,
int[] ipvt,
double[] qtf)
The lmder_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. 
static void 
lmder1_f77(optimization.Lmder_fcn nlls,
int m,
int n,
double[] x,
double[] fvec,
double[][] fjac,
double tol,
int[] info,
int[] ipvt)
The lmder1_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. 
static void 
lmdif_f77(optimization.Lmdif_fcn nlls,
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[] ipvt,
double[] qtf)
The lmdif_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. 
static void 
lmdif1_f77(optimization.Lmdif_fcn nlls,
int m,
int n,
double[] x,
double[] fvec,
double tol,
int[] info)
The lmdif1_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. 
static void 
lmpar_f77(int n,
double[][] r,
int[] ipvt,
double[] diag,
double[] qtb,
double delta,
double[] par,
double[] x,
double[] sdiag,
double[] wa1,
double[] wa2)
Given an m by n matrix A, an n by n nonsingular diagonal matrix D, an mvector b, and a positive number delta, the problem is to determine a value for the parameter par such that if x solves the system 
static void 
qrfac_f77(int m,
int n,
double[][] a,
boolean pivot,
int[] ipvt,
double[] rdiag,
double[] acnorm,
double[] wa)
The qrfac_f77 method uses Householder transformations with column pivoting (optional) to compute a QR factorization of the m by n matrix A. 
static void 
qrsolv_f77(int n,
double[][] r,
int[] ipvt,
double[] diag,
double[] qtb,
double[] x,
double[] sdiag,
double[] wa)
Given an m by n matrix A, an n by n diagonal matrix D, and an mvector b, the problem is to determine an x which solves the system 
Methods inherited from class java.lang.Object 

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait 
Constructor Detail 

public Minpack_f77()
Method Detail 

public static void lmder1_f77(optimization.Lmder_fcn nlls, int m, int n, double[] x, double[] fvec, double[][] fjac, double tol, int[] info, int[] ipvt)
The lmder1_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. This is done by using the more general leastsquares solver lmder_f77. The user must provide a method which calculates the functions and the Jacobian.
Translated by Steve Verrill on November 17, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
nlls
 A class that implements the Lmder_fcn interface (see the
definition in Lmder_fcn.java). See LmderTest_f77.java for an
example of such a class. The class must define a method, fcn,
that must have the form
public static void fcn(int m, int n, double x[], double
fvec[], double fjac[][], int iflag[])
If iflag[1] equals 1, fcn calculates the values of the m
functions [residuals] at x and returns this vector in fvec. If
iflag[1] equals 2, fcn calculates the Jacobian at x and
returns this matrix in fjac (and does not alter fvec).
The value of iflag[1] should not be changed by fcn unless the
user wants to terminate execution of lmder_f77. In this case
set iflag[1] to a negative integer.m
 A positive integer set to the number of functions [number of
observations]n
 A positive integer set to the number of variables [number of
parameters]. n must not exceed m.x
 On input, it contains the initial estimate of the solution
vector [the least squares parameters]. On output it contains
the final estimate of the solution vector.fvec
 An output vector that contains the m functions [residuals]
evaluated at x.fjac
 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] of the identity matrix. The lower trapezoidal part of fjac contains information generated during the computation of R.
tol
 tol is a nonnegative input variable. Termination occurs when
the algorithm estimates either that the relative error in the
sum of squares is at most tol or that the relative error
between x and the solution is at most tol.info
 An integer output variable. If the user has terminated
execution, info is set to the (negative) value of iflag[1].
See description of fcn. Otherwise, info is set as follows.
info = 0 improper input parameters.
info = 1 algorithm estimates that the relative error in the
sum of squares is at most tol.
info = 2 algorithm estimates that the relative error between x
and the solution is at most tol.
info = 3 conditions for info = 1 and info = 2 both hold.
info = 4 fvec is orthogonal to the columns of the Jacobian to
machine precision.
info = 5 number of calls to fcn with iflag[1] = 1 has reached
100*(n+1).
info = 6 tol is too small. No further reduction in the sum of
squares is possible.
info = 7 tol is too small. No further improvement in the
approximate solution x is possible.ipvt
 An integer output array of length n. ipvt defines a
permutation matrix P such that jac*P = QR, 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.public static void lmder_f77(optimization.Lmder_fcn nlls, int m, int n, double[] x, double[] fvec, double[][] fjac, double ftol, double xtol, double gtol, int maxfev, double[] diag, int mode, double factor, int nprint, int[] info, int[] nfev, int[] njev, int[] ipvt, double[] qtf)
The lmder_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. The user must provide a method which calculates the functions and the Jacobian.
Translated by Steve Verrill on November 3, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
nlls
 A class that implements the Lmder_fcn interface (see the
definition in Lmder_fcn.java). See LmderTest_f77.java for an
example of such a class. The class must define a method, fcn,
that must have the form
public static void fcn(int m, int n, double x[], double
fvec[], double fjac[][], int iflag[])
If iflag[1] equals 1, fcn calculates the values of the m
functions [residuals] at x and returns this vector in fvec. If
iflag[1] equals 2, fcn calculates the Jacobian at x and
returns this matrix in fjac (and does not alter fvec).
The value of iflag[1] should not be changed by fcn unless the
user wants to terminate execution of lmder_f77. In this case
set iflag[1] to a negative integer.m
 A positive integer set to the number of functions [number of
observations]n
 A positive integer set to the number of variables [number of
parameters]. n must not exceed m.x
 On input, it contains the initial estimate of the solution
vector [the least squares parameters]. On output it contains
the final estimate of the solution vector.fvec
 An output vector that contains the m functions [residuals]
evaluated at x.fjac
 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] of the identity matrix. The lower trapezoidal part of fjac contains information generated during the computation of R.
ftol
 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
 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
 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
 A positive integer input variable. Termination occurs when the
number of calls to fcn with iflag[1] = 1 has reached maxfev.diag
 An vector 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
 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
 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
 An integer input variable that enables controlled printing of
iterates if it is positive. In this case, fcn is called with
iflag[1] = 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[1] = 0 are made.info
 An integer output variable. If the user has terminated
execution, info is set to the (negative) value of iflag[1].
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] = 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
 An integer output variable set to the number of calls to fcn
with iflag[1] = 1.njev
 An integer output variable set to the number of calls to fcn
with iflag[1] = 2.ipvt
 An integer output array of length n. ipvt defines a
permutation matrix P such that jac*P = QR, 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
 An output array of length n which contains the first n
elements of the vector (Q transpose)fvec.public static double enorm_f77(int n, double[] x)
The enorm_f77 method calculates the Euclidean norm of a vector.
Translated by Steve Verrill on November 14, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
n
 The length of the vector, x.x
 The vector whose Euclidean norm is to be calculated.public static void qrfac_f77(int m, int n, double[][] a, boolean pivot, int[] ipvt, double[] rdiag, double[] acnorm, double[] wa)
The qrfac_f77 method uses Householder transformations with column pivoting (optional) to compute a QR factorization of the m by n matrix A. That is, qrfac_f77 determines an orthogonal matrix Q, a permutation matrix P, and an upper trapezoidal matrix R with diagonal elements of nonincreasing magnitude, such that AP = QR.
Translated by Steve Verrill on November 17, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
m
 The number of rows of A.n
 The number of columns of A.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.pivot
 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
 ipvt is an integer output array. 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.rdiag
 rdiag is an output array of length n which contains the
diagonal elements of R.acnorm
 acnorm is an output array of length n which contains the norms
of the corresponding columns of the input matrix A.wa
 wa is a work array of length n.public static void qrsolv_f77(int n, double[][] r, int[] ipvt, double[] diag, double[] qtb, double[] x, double[] sdiag, double[] wa)
Given an m by n matrix A, an n by n diagonal matrix D, and an mvector b, the problem is to determine an x which solves the system
Ax = b , Dx = 0 ,in the least squares sense.
This method 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 AP = QR, 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_f77 expects the full upper triangle of R, the permutation matrix P, and the first n components of (Q transpose)b. The system
Ax = b, Dx = 0, is then equivalent to t t Rz = Q b, P DPz = 0 ,where x = Pz. If this system does not have full rank, then a least squares solution is obtained. On output qrsolv_f77 also provides an upper triangular matrix S such that
t t t P (A A + DD)P = S S .S is computed within qrsolv_f77 and may be of separate interest.
Translated by Steve Verrill on November 17, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
n
 The order of r.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.ipvt
 ipvt is an integer input array of length n which defines the
permutation matrix P such that AP = QR. Column j of P is
column ipvt[j] of the identity matrix.diag
 diag is an input array of length n which must contain the
diagonal elements of the matrix D.qtb
 qtb is an input array of length n which must contain the first
n elements of the vector (Q transpose)b.x
 x is an output array of length n which contains the least
squares solution of the system Ax = b, Dx = 0.sdiag
 sdiag is an output array of length n which contains the
diagonal elements of the upper triangular matrix S.wa
 wa is a work array of length n.public static void lmpar_f77(int n, double[][] r, int[] ipvt, double[] diag, double[] qtb, double delta, double[] par, double[] x, double[] sdiag, double[] wa1, double[] wa2)
Given an m by n matrix A, an n by n nonsingular diagonal matrix D, an mvector 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 = 0in the least squares sense, and dxnorm is the Euclidean norm of D*x, then either par is zero and
(dxnormdelta) <= 0.1*delta ,or par is positive and
abs(dxnormdelta) <= 0.1*delta .This method (lmpar_f77) 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 AP = QR, 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_f77 expects the full upper triangle of R, the permutation matrix P, and the first n components of (Q transpose)b. On output lmpar_f77 also provides an upper triangular matrix S such that
t t t P (A A + par*DD)P = S S .S is employed within lmpar_f77 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.
Translated by Steve Verrill on November 17, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
n
 The order of r.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.ipvt
 ipvt is an integer input array of length n which defines the
permutation matrix P such that AP = QR. Column j of P is
column ipvt[j] of the identity matrix.diag
 diag is an input array of length n which must contain the
diagonal elements of the matrix D.qtb
 qtb is an input array of length n which must contain the first
n elements of the vector (Q transpose)b.delta
 delta is a positive input variable which specifies an upper
bound on the Euclidean norm of Dx.par
 par is a nonnegative variable. On input par contains an
initial estimate of the LevenbergMarquardt parameter. On
output par contains the final estimate.x
 x is an output array of length n which contains the least
squares solution of the system Ax = b, sqrt(par)*Dx = 0, for
the output par.sdiag
 sdiag is an output array of length n which contains the
diagonal elements of the upper triangular matrix S.wa1
 wa1 is a work array of length n.wa2
 wa2 is a work array of length n.public static void lmdif1_f77(optimization.Lmdif_fcn nlls, int m, int n, double[] x, double[] fvec, double tol, int[] info)
The lmdif1_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. This is done by using the more general leastsquares solver lmdif. The user must provide a method that calculates the functions. The Jacobian is then calculated by a forwarddifference approximation.
Translated by Steve Verrill on November 24, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
nlls
 A class that implements the Lmdif_fcn interface (see the
definition in Lmdif_fcn.java). See LmdifTest_f77.java for an
example of such a class. The class must define a method, fcn,
that must have the form
public static void fcn(int m, int n, double x[], double
fvec[], int iflag[])
The value of iflag[1] should not be changed by fcn unless the
user wants to terminate execution of lmdif_f77. In this case
set iflag[1] to a negative integer.m
 A positive integer set to the number of functions [number of
observations]n
 A positive integer set to the number of variables [number of
parameters]. n must not exceed m.x
 On input, it contains the initial estimate of the solution
vector [the least squares parameters]. On output it contains
the final estimate of the solution vector.fvec
 An output vector that contains the m functions [residuals]
evaluated at x.tol
 tol is a nonnegative input variable. Termination occurs when
the algorithm estimates either that the relative error in the
sum of squares is at most tol or that the relative error
between x and the solution is at most tol.info
 An integer output variable. If the user has terminated
execution, info is set to the (negative) value of iflag[1].
See description of fcn. Otherwise, info is set as follows.
info = 0 improper input parameters.
info = 1 algorithm estimates that the relative error in the
sum of squares is at most tol.
info = 2 algorithm estimates that the relative error between x
and the solution is at most tol.
info = 3 conditions for info = 1 and info = 2 both hold.
info = 4 fvec is orthogonal to the columns of the Jacobian to
machine precision.
info = 5 number of calls to fcn has reached or exceeded
200*(n+1).
info = 6 tol is too small. No further reduction in the sum of
squares is possible.
info = 7 tol is too small. No further improvement in the
approximate solution x is possible.public static void lmdif_f77(optimization.Lmdif_fcn nlls, 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[] ipvt, double[] qtf)
The lmdif_f77 method minimizes the sum of the squares of m nonlinear functions in n variables by a modification of the LevenbergMarquardt algorithm. The user must provide a method that calculates the functions. The Jacobian is then calculated by a forwarddifference approximation.
Translated by Steve Verrill on November 20, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
nlls
 A class that implements the Lmdif_fcn interface (see the
definition in Lmdif_fcn.java). See LmdifTest_f77.java for an
example of such a class. The class must define a method, fcn,
that must have the form
public static void fcn(int m, int n, double x[], double
fvec[], int iflag[])
The value of iflag[1] should not be changed by fcn unless the
user wants to terminate execution of lmdif_f77. In this case
set iflag[1] to a negative integer.m
 A positive integer set to the number of functions [number of
observations]n
 A positive integer set to the number of variables [number of
parameters]. n must not exceed m.x
 On input, it contains the initial estimate of the solution
vector [the least squares parameters]. On output it contains
the final estimate of the solution vector.fvec
 An output vector that contains the m functions [residuals]
evaluated at x.ftol
 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
 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
 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
 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
 An input variable used in determining a suitable step length
for the forwarddifference 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
 An vector 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
 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
 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
 An integer input variable that enables controlled printing of
iterates if it is positive. In this case, fcn is called with
iflag[1] = 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[1] = 0 are made.info
 An integer output variable. If the user has terminated
execution, info is set to the (negative) value of iflag[1].
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] = 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
 An integer output variable set to the number of calls to fcn.fjac
 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.ipvt
 An integer output array of length n. ipvt defines a
permutation matrix P such that jac*P = QR, 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
 An output array of length n which contains the first n
elements of the vector (Q transpose)fvec.public static void fdjac2_f77(optimization.Lmdif_fcn nlls, int m, int n, double[] x, double[] fvec, double[][] fjac, int[] iflag, double epsfcn, double[] wa)
The fdjac2 method computes a forwarddifference approximation to the m by n Jacobian matrix associated with a specified problem of m functions in n variables.
Translated by Steve Verrill on November 24, 2000 from the FORTRAN MINPACK source produced by Garbow, Hillstrom, and More.
nlls
 A class that implements the Lmdif_fcn interface (see the
definition in Lmdif_fcn.java). See LmdifTest_f77.java for an
example of such a class. The class must define a method, fcn,
that must have the form
public static void fcn(int m, int n, double x[], double
fvec[], int iflag[])
The value of iflag[1] should not be changed by fcn unless the
user wants to terminate execution of fdjac2_f77. In this case
iflag[1] should be set to a negative integer.m
 A positive integer set to the number of functions [number of
observations]n
 A positive integer set to the number of variables [number of
parameters]. n must not exceed m.x
 An input array.fvec
 An input array that contains the functions evaluated at x.fjac
 An output m by n array that contains the approximation to the
Jacobian matrix evaluated at x.iflag
 An integer variable that can be used to terminate the
execution of fdjac2. See the description of nlls.epsfcn
 An input variable used in determining a suitable step length
for the forwarddifference 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
 A work array.

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