MB04KD

QR factorization of a special structured block matrix

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).

      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(*)

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.

  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.

  DWORK   DOUBLE PRECISION array, dimension (N)

  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

  The algorithm is backward stable.

  None

Program Text

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