dogleg Subroutine

private subroutine dogleg(n, r, lr, diag, qtb, delta, x)

Finds the minimizing combination of Gauss-Newton and gradient steps.

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.

This function completes the solution of the problem if it is provided with the necessary information from the QR factorization of A. That is, if A = QR, 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'B.

Arguments

Type	Intent	Optional	Attributes		Name
integer				::	n	In - order of the matrix R
real(kind=dp)				::	r(lr)	In - upper triangular matrix R stored by rows.
integer				::	lr	In - size of the R array, which must be no less than (N*(N+1))/2.
real(kind=dp)				::	diag(n)	In - diagonal elements of the matrix D.
real(kind=dp)				::	qtb(n)	In - first N elements of the vector Q'* B.
real(kind=dp)				::	delta	In - positive upper bound on the euclidean norm of D*X(1:N).
real(kind=dp)				::	x(n)	Out - desired convex combination of the Gauss-Newton direction and the scaled gradient direction.