Name

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  QRSOLV

Author

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  Craig B. Markwardt, NASA/GSFC Code 662, Greenbelt, MD 20770
  craigm@lheamail.gsfc.nasa.gov

Purpose

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  Solve a linear equation after performing QR factorization

Major Topics

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  Linear Systems

Calling Sequence

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  X = QRSOLV(A, R, B, PIVOTS=IPVT)

Description

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  The procedure QRSOLV completes the solution of a linear equation,
        A ## x = B
  after the MxN matrix has been factorized by QR decomposition.
  After being factorized once using QRFAC, the matrices can be used
  for multiple righthand sides (i.e., different B's).
  The solution technique is to first compute the factorization using
  QRFAC, which yields the orthogonal matrix Q and the upper
  triangular matrix R. [ Actually, Q is represented by its
  Householder reflectors. ] Then the solution vector, X, is computed
  using QRSOLV.
  If pivoting was performed in the factorization, the permutation
  vector IPVT returned by QRFAC must also be passed to QRSOLV.
 

Parameters

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  A - upon input, the factorized matrix A, returned by QRFAC.
  R - upon input, the upper diagonal matrix R, returned by QRFAC.
  B - upon input, the righthand vector B, which fits into the
      equation, A ## x = B
  X - upon ouptut, the solution vector X, to the above linear
      equation. For an overdetermined system, X is the least
      squares solution which minimizes TOTAL( (A ## X - B)^2 ).

Keyword Parameters

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  PIVOTS - upon input, the permutation matrix IPVT returned by
            QRFAC, if pivoting is to be performed.

Example

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  Solve the equation A ## X = B, in the least squares sense, where:
    A = [[1.0,1.0,1.0,1.0,1.0,1.0],$
        [0.6,0.8,0.5,0.8,0.7,0.9],$
        [0.2,0.3,0.1,0.4,0.3,0.4]]
  and B = [0.57E,0.69,0.5,0.7,0.6,0.8]
  qrfac, a, r, ipvt, /PIVOT
  x = qrsolv(a, r, b, PIVOTS=ipvt)
  print, x
      0.0834092 0.852273 -0.179545

References

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  More', Jorge J., "The Levenberg-Marquardt Algorithm:
    Implementation and Theory," in *Numerical Analysis*, ed. Watson,
    G. A., Lecture Notes in Mathematics 630, Springer-Verlag, 1977.

Modification History

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  Written (taken from MPFIT), CM, Feb 2002
  Usage message, error checking, CM, 15 Mar 2002
  Error checking is fixed, CM, 10 May 2002
  Found error in return of permuted results, CM, 21 May 2004