## MB04KD

### QR factorization of a special structured block matrix

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

Purpose

      To calculate a QR factorization of the first block column and
      apply the orthogonal transformations (from the left) also to the
      second block column of a structured matrix, as follows
                           _
             [ R   0 ]   [ R   C ]
        Q' * [       ] = [       ]
             [ A   B ]   [ 0   D ]
                  _
      where R and R are upper triangular. The matrix A can be full or
      upper 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 information filter).

Specification

          SUBROUTINE MB04KD( UPLO, N, M, P, R, LDR, A, LDA, B, LDB, C, LDC,
         $                   TAU, DWORK )
    C     .. Scalar Arguments ..
          CHARACTER         UPLO
          INTEGER           LDA, LDB, LDC, LDR, M, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
         $                  R(LDR,*), TAU(*)

Arguments

Mode Parameters

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

Input/Output Parameters

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

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

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

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

      LDR     INTEGER
              The leading dimension of array R.  LDR >= MAX(1,N).

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, if UPLO = 'F', the leading P-by-N part of this
              array must contain the matrix A. If UPLO = 'U', the
              leading MIN(P,N)-by-N part of this array must contain the
              upper trapezoidal (upper triangular if P >= N) matrix A,
              and the elements below the diagonal are not referenced.
              On exit, the leading P-by-N part (upper trapezoidal or
              triangular, if UPLO = 'U') 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,P).

      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,M)
              The leading N-by-M part of this array contains the
              computed matrix C.

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

      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 a P-vector, if UPLO = 'F', or an min(i,P)-vector, if
             i
      UPLO = 'U'.  The components of v  are stored in the i-th column
                                      i
      of A, 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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