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dqrank

  1. C DQRANK    SOURCE    PV090527  23/02/07    11:58:36     11592          
  2.       subroutine dqrank ( a, lda, m, n, tol, kr, jpvt, qraux, work )
  3.  
  4. c*****************************************************************************80
  5. c
  6. c! DQRANK computes the QR factorization of a rectangular matrix.
  7. c
  8. c  Discussion:
  9. c
  10. c    This routine is used in conjunction with sqrlss to solve
  11. c    overdetermined, underdetermined and singular linear systems
  12. c    in a least squares sense.
  13. c
  14. c    DQRANK uses the LINPACK subroutine DQRDC to compute the QR
  15. c    factorization, with column pivoting, of an M by N matrix A.
  16. c    The numerical rank is determined using the tolerance TOL.
  17. c
  18. c    Note that on output, ABS ( A(1,1) ) / ABS ( A(KR,KR) ) is an estimate
  19. c    of the condition number of the matrix of independent columns,
  20. c    and of R.  This estimate will be .le. 1/TOL.
  21. c
  22. c  Reference:
  23. c
  24. c    Jack Dongarra, Cleve Moler, Jim Bunch, Pete Stewart,
  25. c    LINPACK User's Guide,
  26. c    SIAM, 1979,
  27. c    ISBN13: 978-0-898711-72-1,
  28. c    LC: QA214.L56.
  29. c
  30. c  Parameters:
  31. c
  32. c    Input/output, real*8 A(LDA,N).  On input, the matrix whose
  33. c    decomposition is to be computed.  On output, the information from DQRDC.
  34. c    The triangular matrix R of the QR factorization is contained in the
  35. c    upper triangle and information needed to recover the orthogonal
  36. c    matrix Q is stored below the diagonal in A and in the vector QRAUX.
  37. c
  38. c    Input, integer LDA, the leading dimension of A, which must
  39. c    be at least M.
  40. c
  41. c    Input, integer M, the number of rows of A.
  42. c
  43. c    Input, integer N, the number of columns of A.
  44. c
  45. c    Input, real*8 TOL, a relative tolerance used to determine the
  46. c    numerical rank.  The problem should be scaled so that all the elements
  47. c    of A have roughly the same absolute accuracy, EPS.  Then a reasonable
  48. c    value for TOL is roughly EPS divided by the magnitude of the largest
  49. c    element.
  50. c
  51. c    Output, integer KR, the numerical rank.
  52. c
  53. c    Output, integer JPVT(N), the pivot information from DQRDC.
  54. c    Columns JPVT(1), ..., JPVT(KR) of the original matrix are linearly
  55. c    independent to within the tolerance TOL and the remaining columns
  56. c    are linearly dependent.
  57. c
  58. c    Output, real*8 QRAUX(N), will contain extra information defining
  59. c    the QR factorization.
  60. c
  61. c    Workspace, real*8 WORK(N).
  62. c
  63.         implicit real*8 (a-h,o-z)
  64.         implicit integer (i-n)
  65.  
  66.         integer lda
  67.         integer n
  68.  
  69.         real*8 a(lda,n)
  70.         integer j
  71.         integer job
  72.         integer jpvt(n)
  73.         integer k
  74.         integer kr
  75.         integer m
  76.         real*8 qraux(n)
  77.         real*8 tol
  78.         real*8 work(n)
  79.  
  80.         jpvt(1:n) = 0
  81.         job = 1
  82.  
  83.         call dqrdc ( a, lda, m, n, qraux, jpvt, work, job )
  84.  
  85.         kr = 0
  86.         k = min ( m, n )
  87.  
  88.         do j = 1, k
  89.           if ( abs ( a(j,j) ) .le. tol * abs ( a(1,1) ) ) then
  90.             return
  91.           end if
  92.           kr = j
  93.         end do
  94.  
  95.         return
  96.       end
  97.  
  98.  

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