MB04JD

LQ factorization of a matrix with an upper right-hand side zero triangle

Purpose

  To compute an LQ factorization of an n-by-m matrix A (A = L * Q),
  having a min(n,p)-by-p zero triangle in the upper right-hand side
  corner, as shown below, for n = 8, m = 7, and p = 2:

         [ x x x x x 0 0 ]
         [ x x x x x x 0 ]
         [ x x x x x x x ]
         [ x x x x x x x ]
     A = [ x x x x x x x ],
         [ x x x x x x x ]
         [ x x x x x x x ]
         [ x x x x x x x ]

  and optionally apply the transformations to an l-by-m matrix B
  (from the right). The problem structure is exploited. This
  computation is useful, for instance, in combined measurement and
  time update of one iteration of the time-invariant Kalman filter
  (square root covariance filter).

      SUBROUTINE MB04JD( N, M, P, L, A, LDA, B, LDB, TAU, DWORK, LDWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      INTEGER           INFO, L, LDA, LDB, LDWORK, M, N, P
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), DWORK(*), TAU(*)

Input/Output Parameters

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

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

  P       (input) INTEGER
          The order of the zero triagle.  P >= 0.

  L       (input) INTEGER
          The number of rows of the matrix B.  L >= 0.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
          On entry, the leading N-by-M part of this array must
          contain the matrix A. The elements corresponding to the
          zero MIN(N,P)-by-P upper trapezoidal/triangular part
          (if P > 0) are not referenced.
          On exit, the elements on and below the diagonal of this
          array contain the N-by-MIN(N,M) lower trapezoidal matrix
          L (L is lower triangular, if N <= M) of the LQ
          factorization, and the relevant elements above the
          diagonal contain the trailing components (the vectors v,
          see Method) of the elementary reflectors used in the
          factorization.

  LDA     INTEGER
          The leading dimension of array A.  LDA >= MAX(1,N).

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading L-by-M part of this array must
          contain the matrix B.
          On exit, the leading L-by-M part of this array contains
          the updated matrix B.
          If L = 0, this array is not referenced.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,L).

  TAU     (output) DOUBLE PRECISION array, dimension MIN(N,M)
          The scalar factors of the elementary reflectors used.

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

  LDWORK  The length of the array DWORK.
          LDWORK >= MAX(1,N-1,N-P,L).
          For optimum performance LDWORK should be larger.

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

  The routine uses min(N,M) Householder transformations exploiting
  the zero pattern of the matrix.  A Householder matrix has the form

                                  ( 1 ),
     H  = I - tau *u *u',    u  = ( v )
      i          i  i  i      i   (  i)

  where v  is an (M-P+I-2)-vector.  The components of v  are stored
         i                                             i
  in the i-th row of A, beginning from the location i+1, and tau
                                                                i
  is stored in TAU(i).

  The algorithm is backward stable.

  None

Program Text

  None

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