SLATEC Routines --- DMPAR ---

```*DECK DMPAR
SUBROUTINE DMPAR (N, R, LDR, IPVT, DIAG, QTB, DELTA, PAR, X,
+   SIGMA, WA1, WA2)
C***BEGIN PROLOGUE  DMPAR
C***SUBSIDIARY
C***PURPOSE  Subsidiary to DNLS1 and DNLS1E
C***LIBRARY   SLATEC
C***TYPE      DOUBLE PRECISION (LMPAR-S, DMPAR-D)
C***AUTHOR  (UNKNOWN)
C***DESCRIPTION
C
C   **** Double Precision version of LMPAR ****
C
C     Given an M by N matrix A, an N by N nonsingular DIAGONAL
C     matrix D, an M-vector B, and a positive number DELTA,
C     the problem is to determine a value for the parameter
C     PAR such that if X solves the system
C
C           A*X = B ,     SQRT(PAR)*D*X = 0 ,
C
C     in the least squares sense, and DXNORM is the Euclidean
C     norm of D*X, then either PAR is zero and
C
C           (DXNORM-DELTA) .LE. 0.1*DELTA ,
C
C     or PAR is positive and
C
C           ABS(DXNORM-DELTA) .LE. 0.1*DELTA .
C
C     This subroutine completes the solution of the problem
C     if it is provided with the necessary information from the
C     QR factorization, with column pivoting, of A. That is, if
C     A*P = Q*R, where P is a permutation matrix, Q has orthogonal
C     columns, and R is an upper triangular matrix with diagonal
C     elements of nonincreasing magnitude, then DMPAR expects
C     the full upper triangle of R, the permutation matrix P,
C     and the first N components of (Q TRANSPOSE)*B. On output
C     DMPAR also provides an upper triangular matrix S such that
C
C            T   T                   T
C           P *(A *A + PAR*D*D)*P = S *S .
C
C     S is employed within DMPAR and may be of separate interest.
C
C     Only a few iterations are generally needed for convergence
C     of the algorithm. If, however, the limit of 10 iterations
C     is reached, then the output PAR will contain the best
C     value obtained so far.
C
C     The subroutine statement is
C
C       SUBROUTINE DMPAR(N,R,LDR,IPVT,DIAG,QTB,DELTA,PAR,X,SIGMA,
C                        WA1,WA2)
C
C     where
C
C       N is a positive integer input variable set to the order of R.
C
C       R is an N by N array. On input the full upper triangle
C         must contain the full upper triangle of the matrix R.
C         On output the full upper triangle is unaltered, and the
C         strict lower triangle contains the strict upper triangle
C         (transposed) of the upper triangular matrix S.
C
C       LDR is a positive integer input variable not less than N
C         which specifies the leading dimension of the array R.
C
C       IPVT is an integer input array of length N which defines the
C         permutation matrix P such that A*P = Q*R. Column J of P
C         is column IPVT(J) of the identity matrix.
C
C       DIAG is an input array of length N which must contain the
C         diagonal elements of the matrix D.
C
C       QTB is an input array of length N which must contain the first
C         N elements of the vector (Q TRANSPOSE)*B.
C
C       DELTA is a positive input variable which specifies an upper
C         bound on the Euclidean norm of D*X.
C
C       PAR is a nonnegative variable. On input PAR contains an
C         initial estimate of the Levenberg-Marquardt parameter.
C         On output PAR contains the final estimate.
C
C       X is an output array of length N which contains the least
C         squares solution of the system A*X = B, SQRT(PAR)*D*X = 0,
C         for the output PAR.
C
C       SIGMA is an output array of length N which contains the
C         diagonal elements of the upper triangular matrix S.
C
C       WA1 and WA2 are work arrays of length N.
C