**Purpose**

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, where A is an m-by-n matrix, D is an n-by-n nonsingular diagonal matrix, and b is an m-vector, and if DELTA is a positive number, 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 . It is assumed that a QR factorization, with column pivoting, of A is available, that is, 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. The routine needs the full upper triangle of R, the permutation matrix P, and the first n components of Q'*b (' denotes the transpose). On output, MD03BY also provides an upper triangular matrix S such that P'*(A'*A + PAR*D*D)*P = S'*S . Matrix S is used in the solution process.

SUBROUTINE MD03BY( COND, N, R, LDR, IPVT, DIAG, QTB, DELTA, PAR, $ RANK, X, RX, TOL, DWORK, LDWORK, INFO ) C .. Scalar Arguments .. CHARACTER COND INTEGER INFO, LDR, LDWORK, N, RANK DOUBLE PRECISION DELTA, PAR, TOL C .. Array Arguments .. INTEGER IPVT(*) DOUBLE PRECISION DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)

**Mode Parameters**

COND CHARACTER*1 Specifies whether the condition of the matrices R and S should be estimated, as follows: = 'E' : use incremental condition estimation for R and S; = 'N' : do not use condition estimation, but check the diagonal entries of R and S for zero values; = 'U' : use the rank already stored in RANK (for R).

N (input) INTEGER The order of the matrix R. N >= 0. R (input/output) DOUBLE PRECISION array, dimension (LDR, N) On entry, the leading N-by-N upper triangular part of this array must contain the upper triangular matrix R. On exit, the full upper triangle is unaltered, and the strict lower triangle contains the strict upper triangle (transposed) of the upper triangular matrix S. LDR INTEGER The leading dimension of array R. LDR >= MAX(1,N). IPVT (input) INTEGER array, dimension (N) This array must define the permutation matrix P such that A*P = Q*R. Column j of P is column IPVT(j) of the identity matrix. DIAG (input) DOUBLE PRECISION array, dimension (N) This array must contain the diagonal elements of the matrix D. DIAG(I) <> 0, I = 1,...,N. QTB (input) DOUBLE PRECISION array, dimension (N) This array must contain the first n elements of the vector Q'*b. DELTA (input) DOUBLE PRECISION An upper bound on the Euclidean norm of D*x. DELTA > 0. PAR (input/output) DOUBLE PRECISION On entry, PAR must contain an initial estimate of the Levenberg-Marquardt parameter. PAR >= 0. On exit, it contains the final estimate of this parameter. RANK (input or output) INTEGER On entry, if COND = 'U', this parameter must contain the (numerical) rank of the matrix R. On exit, this parameter contains the numerical rank of the matrix S. X (output) DOUBLE PRECISION array, dimension (N) This array contains the least squares solution of the system A*x = b, sqrt(PAR)*D*x = 0. RX (output) DOUBLE PRECISION array, dimension (N) This array contains the matrix-vector product -R*P'*x.

TOL DOUBLE PRECISION If COND = 'E', the tolerance to be used for finding the rank of the matrices R and S. If the user sets TOL > 0, then the given value of TOL is used as a lower bound for the reciprocal condition number; a (sub)matrix whose estimated condition number is less than 1/TOL is considered to be of full rank. If the user sets TOL <= 0, then an implicitly computed, default tolerance, defined by TOLDEF = N*EPS, is used instead, where EPS is the machine precision (see LAPACK Library routine DLAMCH). This parameter is not relevant if COND = 'U' or 'N'.

DWORK DOUBLE PRECISION array, dimension (LDWORK) On exit, the first N elements of this array contain the diagonal elements of the upper triangular matrix S. LDWORK INTEGER The length of the array DWORK. LDWORK >= 4*N, if COND = 'E'; LDWORK >= 2*N, if COND <> 'E'.

INFO INTEGER = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

The algorithm computes the Gauss-Newton direction. A least squares solution is found if the Jacobian is rank deficient. If the Gauss- Newton direction is not acceptable, then an iterative algorithm obtains improved lower and upper bounds for the parameter PAR. Only a few iterations are generally needed for convergence of the algorithm. If, however, the limit of ITMAX = 10 iterations is reached, then the output PAR will contain the best value obtained so far. If the Gauss-Newton step is acceptable, it is stored in x, and PAR is set to zero, hence S = R.

[1] More, J.J., Garbow, B.S, and Hillstrom, K.E. User's Guide for MINPACK-1. Applied Math. Division, Argonne National Laboratory, Argonne, Illinois, Report ANL-80-74, 1980.

2 The algorithm requires 0(N ) operations and is backward stable.

This routine is a LAPACK-based modification of LMPAR from the MINPACK package [1], and with optional condition estimation. The option COND = 'U' is useful when dealing with several right-hand side vectors, but RANK should be reset. If COND = 'E', but the matrix S is guaranteed to be nonsingular and well conditioned relative to TOL, i.e., rank(R) = N, and min(DIAG) > 0, then its condition is not estimated.

**Program Text**

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