MB04ID

QR factorization of a matrix with a lower left-hand side zero triangle

Purpose

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

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

  and optionally apply the transformations to an n-by-l matrix B
  (from the left). 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 information filter).

      SUBROUTINE MB04ID( 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 columns 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 P-by-MIN(P,M) lower trapezoidal/triangular part
          (if P > 0) are not referenced.
          On exit, the elements on and above the diagonal of this
          array contain the MIN(N,M)-by-M upper trapezoidal matrix
          R (R is upper triangular, if N >= M) of the QR
          factorization, and the relevant elements below 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,L)
          On entry, the leading N-by-L part of this array must
          contain the matrix B.
          On exit, the leading N-by-L 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,N) if L > 0;
          LDB >= 1        if L = 0.

  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,M-1,M-P,L).
          For optimum performance LDWORK should 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 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 (N-P+I-2)-vector.  The components of v  are stored
         i                                             i
  in the i-th column of A, beginning from the location i+1, and
  tau  is stored in TAU(i).
     i

  The algorithm is backward stable.

  None

Program Text

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