SLATEC Routines --- QRSOLV ---

*DECK QRSOLV
      SUBROUTINE QRSOLV (N, R, LDR, IPVT, DIAG, QTB, X, SIGMA, WA)
C***BEGIN PROLOGUE  QRSOLV
C***SUBSIDIARY
C***PURPOSE  Subsidiary to SNLS1 and SNLS1E
C***LIBRARY   SLATEC
C***TYPE      SINGLE PRECISION (QRSOLV-S, DQRSLV-D)
C***AUTHOR  (UNKNOWN)
C***DESCRIPTION
C
C     Given an M by N matrix A, an N by N diagonal matrix D,
C     and an M-vector B, the problem is to determine an X which
C     solves the system
C
C           A*X = B ,     D*X = 0 ,
C
C     in the least squares sense.
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 QRSOLV expects
C     the full upper triangle of R, the permutation matrix P,
C     and the first N components of (Q TRANSPOSE)*B. The system
C     A*X = B, D*X = 0, is then equivalent to
C
C                  T       T
C           R*Z = Q *B ,  P *D*P*Z = 0 ,
C
C     where X = P*Z. If this system does not have full rank,
C     then a least squares solution is obtained. On output QRSOLV
C     also provides an upper triangular matrix S such that
C
C            T   T               T
C           P *(A *A + D*D)*P = S *S .
C
C     S is computed within QRSOLV and may be of separate interest.
C
C     The subroutine statement is
C
C       SUBROUTINE QRSOLV(N,R,LDR,IPVT,DIAG,QTB,X,SIGMA,WA)
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       X is an output array of length N which contains the least
C         squares solution of the system A*X = B, D*X = 0.
C
C       SIGMA is an output array of length N which contains the
C         diagonal elements of the upper triangular matrix S.
C
C       WA is a work array of length N.
C
C***SEE ALSO  SNLS1, SNLS1E
C***ROUTINES CALLED  (NONE)
C***REVISION HISTORY  (YYMMDD)
C   800301  DATE WRITTEN
C   890831  Modified array declarations.  (WRB)
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   900326  Removed duplicate information from DESCRIPTION section.
C           (WRB)
C   900328  Added TYPE section.  (WRB)
C***END PROLOGUE  QRSOLV