Procedures

this subroutine checks the gradients of m nonlinear functions in n variables, evaluated at a point x, for consistency with the functions themselves.

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 (ax - b) in the least squares sense, subject to the restriction that the euclidean norm of dx be at most delta.

given an n-vector x, this function calculates the euclidean norm of x.

this subroutine computes a forward-difference approximation to the n by n jacobian matrix associated with a specified problem of n functions in n variables. if the jacobian has a banded form, then function evaluations are saved by only approximating the nonzero terms.

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 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 purpose of hybrd1 is to find a zero of a system of n nonlinear functions in n variables by a modification of the powell hybrid method. this is done by using the more general nonlinear equation solver hybrd. the user must provide a subroutine which calculates the functions. the jacobian is then calculated by a forward-difference approximation.

the purpose of hybrj 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 and the jacobian.

the purpose of hybrj1 is to find a zero of a system of n nonlinear functions in n variables by a modification of the powell hybrid method. this is done by using the more general nonlinear equation solver hybrj. the user must provide a subroutine which calculates the functions and the jacobian.

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 purpose of lmder1 is to minimize the sum of the squares of m nonlinear functions in n variables by a modification of the levenberg-marquardt algorithm. this is done by using the more general least-squares solver lmder. the user must provide a subroutine which calculates the functions and the jacobian.

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 purpose of lmdif1 is to minimize the sum of the squares of m nonlinear functions in n variables by a modification of the levenberg-marquardt algorithm. this is done by using the more general least-squares solver lmdif. the user must provide a subroutine which calculates the functions. the jacobian is then calculated by a forward-difference approximation.

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

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 purpose of lmstr1 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. this is done by using the more general least-squares solver lmstr. the user must provide a subroutine which calculates the functions and the rows of the jacobian.

this subroutine proceeds from the computed qr factorization of an m by n matrix a to accumulate the m by m orthogonal matrix q from its factored form.

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 ap = qr. the householder transformation for column k, k = 1,2,...,min(m,n), is of the form

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

given an m by n matrix a, this subroutine computes aq where q is the product of 2(n - 1) transformations

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

given an n by n upper triangular matrix r, this subroutine computes the qr decomposition of the matrix formed when a row is added to r. if the row is specified by the vector w, then rwupdt determines an orthogonal matrix q such that when the n+1 by n matrix composed of r augmented by w is premultiplied by (q transpose), the resulting matrix is upper trapezoidal. the matrix (q transpose) is the product of n transformations