MB02QY

Minimum-norm solution to a linear least squares problem, given a rank-revealing QR factorization

Purpose

  To determine the minimum-norm solution to a real linear least
  squares problem:

      minimize || A * X - B ||,

  using the rank-revealing QR factorization of a real general
  M-by-N matrix  A,  computed by SLICOT Library routine  MB03OD.

      SUBROUTINE MB02QY( M, N, NRHS, RANK, A, LDA, JPVT, B, LDB, TAU,
     $                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDB, LDWORK, M, N, NRHS, RANK
C     .. Array Arguments ..
      INTEGER            JPVT( * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DWORK( * ), TAU( * )

Input/Output Parameters

  M       (input) INTEGER
          The number of rows of the matrices A and B.  M >= 0.

  N       (input) INTEGER
          The number of columns of the matrix A.  N >= 0.

  NRHS    (input) INTEGER
          The number of columns of the matrix B.  NRHS >= 0.

  RANK    (input) INTEGER
          The effective rank of  A,  as returned by SLICOT Library
          routine  MB03OD.  min(M,N) >= RANK >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension
          ( LDA, N )
          On entry, the leading min(M,N)-by-N upper trapezoidal
          part of this array contains the triangular factor  R,  as
          returned by SLICOT Library routine  MB03OD.  The strict
          lower trapezoidal part of  A  is not referenced.
          On exit, if  RANK < N,  the leading  RANK-by-RANK  upper
          triangular part of this array contains the upper
          triangular matrix  R  of the complete orthogonal
          factorization of  A,  and the submatrix  (1:RANK,RANK+1:N)
          of this array, with the array  TAU,  represent the
          orthogonal matrix  Z  (of the complete orthogonal
          factorization of  A),  as a product of  RANK  elementary
          reflectors.
          On exit, if  RANK = N,  this array is unchanged.

  LDA     INTEGER
          The leading dimension of the array A.  LDA >= max(1,M).

  JPVT    (input) INTEGER array, dimension ( N )
          The recorded permutations performed by SLICOT Library
          routine  MB03OD;  if  JPVT(i) = k,  then the i-th column
          of  A*P  was the k-th column of the original matrix  A.

  B       (input/output) DOUBLE PRECISION array, dimension
          ( LDB, NRHS )
          On entry, if  NRHS > 0,  the leading M-by-NRHS part of
          this array must contain the matrix  B  (corresponding to
          the transformed matrix  A,  returned by SLICOT Library
          routine  MB03OD).
          On exit, if  NRHS > 0,  the leading N-by-NRHS part of this
          array contains the solution matrix X.
          If  M >= N  and  RANK = N,  the residual sum-of-squares
          for the solution in the i-th column is given by the sum
          of squares of elements  N+1:M  in that column.
          If  NRHS = 0,  the array  B  is not referenced.

  LDB     INTEGER
          The leading dimension of the array B.
          LDB >= max(1,M,N),  if  NRHS > 0.
          LDB >= 1,           if  NRHS = 0.

  TAU     (output) DOUBLE PRECISION array, dimension ( min(M,N) )
          The scalar factors of the elementary reflectors.
          If  RANK = N,  the array  TAU  is not referenced.

  DWORK   DOUBLE PRECISION array, dimension ( LDWORK )
          On exit, if  INFO = 0,  DWORK(1) returns the optimal value
          of LDWORK.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= max( 1, N, NRHS ).
          For good performance,  LDWORK  should sometimes be larger.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

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

  The routine uses a QR factorization with column pivoting:

     A * P = Q * R = Q * [ R11 R12 ],
                         [  0  R22 ]

  where  R11  is an upper triangular submatrix of estimated rank
  RANK,  the effective rank of  A.  The submatrix  R22  can be
  considered as negligible.

  If  RANK < N,  then  R12  is annihilated by orthogonal
  transformations from the right, arriving at the complete
  orthogonal factorization:

     A * P = Q * [ T11 0 ] * Z.
                 [  0  0 ]

  The minimum-norm solution is then

     X = P * Z' [ inv(T11)*Q1'*B ],
                [        0       ]

  where Q1 consists of the first  RANK  columns of Q.

  The input data for  MB02QY  are the transformed matrices  Q' * A
  (returned by SLICOT Library routine  MB03OD)  and  Q' * B.
  Matrix  Q  is not needed.

  The implemented method is numerically stable.

  None

Program Text

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