## MB04LD

### LQ factorization of a special structured block matrix

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

Purpose

      To calculate an LQ factorization of the first block row and apply
      the orthogonal transformations (from the right) also to the second
      block row of a structured matrix, as follows
                         _
         [ L   A ]     [ L   0 ]
         [       ]*Q = [       ]
         [ 0   B ]     [ C   D ]
                  _
      where L and L are lower triangular. The matrix A can be full or
      lower trapezoidal/triangular. The problem structure is exploited.
      This computation is useful, for instance, in combined measurement
      and time update of one iteration of the Kalman filter (square
      root covariance filter).

Specification

          SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC,
         $                   TAU, DWORK )
    C     .. Scalar Arguments ..
          CHARACTER         UPLO
          INTEGER           LDA, LDB, LDC, LDL, M, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
         $                  L(LDL,*), TAU(*)

Arguments

Mode Parameters

      UPLO    CHARACTER*1
              Indicates if the matrix A is or not triangular as follows:
              = 'L':  Matrix A is lower trapezoidal/triangular;
              = 'F':  Matrix A is full.

Input/Output Parameters

      N       (input) INTEGER                 _
              The order of the matrices L and L.  N >= 0.

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

      P       (input) INTEGER
              The number of rows of the matrices B, C and D.  P >= 0.

      L       (input/output) DOUBLE PRECISION array, dimension (LDL,N)
              On entry, the leading N-by-N lower triangular part of this
              array must contain the lower triangular matrix L.
              On exit, the leading N-by-N lower triangular part of this
                                                         _
              array contains the lower triangular matrix L.
              The strict upper triangular part of this array is not
              referenced.

      LDL     INTEGER
              The leading dimension of array L.  LDL >= MAX(1,N).

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
              On entry, if UPLO = 'F', the leading N-by-M part of this
              array must contain the matrix A. If UPLO = 'L', the
              leading N-by-MIN(N,M) part of this array must contain the
              lower trapezoidal (lower triangular if N <= M) matrix A,
              and the elements above the diagonal are not referenced.
              On exit, the leading N-by-M part (lower trapezoidal or
              triangular, if UPLO = 'L') of this array contains 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 P-by-M part of this array must
              contain the matrix B.
              On exit, the leading P-by-M part of this array contains
              the computed matrix D.

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

      C       (output) DOUBLE PRECISION array, dimension (LDC,N)
              The leading P-by-N part of this array contains the
              computed matrix C.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= MAX(1,P).

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

Workspace

      DWORK   DOUBLE PRECISION array, dimension (N)

Method

      The routine uses N Householder transformations exploiting the zero
      pattern of the block 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-vector, if UPLO = 'F', or an min(i,M)-vector, if
             i
      UPLO = 'L'.  The components of v  are stored in the i-th row of A,
                                      i
      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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