## MB04ID

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

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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).

Specification

          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(*)

Arguments

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.

Workspace

      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.

Error Indicator

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

Method

      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

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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