## MB04ND

### RQ factorization of a special structured block matrix

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

Purpose

      To calculate an RQ 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
                               _
        [ A   R ]        [ 0   R ]
        [       ] * Q' = [ _   _ ]
        [ C   B ]        [ C   B ]
                  _
      where R and R are upper triangular. The matrix A can be full or
      upper trapezoidal/triangular. The problem structure is exploited.

Specification

          SUBROUTINE MB04ND( 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 rows of the matrices B and C.  M >= 0.

      P       (input) INTEGER
              The number of columns of the matrices A and C.  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,P)
              On entry, if UPLO = 'F', the leading N-by-P part of this
              array must contain the matrix A. For UPLO = 'U', if
              N <= P, the upper triangle of the subarray A(1:N,P-N+1:P)
              must contain the N-by-N upper triangular matrix A, and if
              N >= P, the elements on and above the (N-P)-th subdiagonal
              must contain the N-by-P upper trapezoidal matrix A.
              On exit, if UPLO = 'F', the leading N-by-P part of this
              array contains the trailing components (the vectors v, see
              METHOD) of the elementary reflectors used in the
              factorization. If UPLO = 'U', the upper triangle of the
              subarray A(1:N,P-N+1:P) (if N <= P), or the elements on
              and above the (N-P)-th subdiagonal (if N >= P), contain
              the trailing components (the vectors v, see METHOD) of the
              elementary reflectors used in the factorization.
              The remaining elements are not referenced.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,N)
              On entry, the leading M-by-N part of this array must
              contain the matrix B.
              On exit, the leading M-by-N part of this array contains
                                  _
              the computed matrix B.

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

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,P)
              On entry, the leading M-by-P part of this array must
              contain the matrix C.
              On exit, the leading M-by-P part of this array contains
                                  _
              the computed matrix C.

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

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

Workspace

      DWORK   DOUBLE PRECISION array, dimension (MAX(N-1,M))

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 a min(N-i+1,P)-vector,
             i
      if UPLO = 'U'.  The components of v  are stored in the i-th row
                                         i
      of A, and tau  is stored in TAU(i), i = N,N-1,...,1.
                   i
      In-line code for applying Householder transformations is used
      whenever possible (see MB04NY routine).

Numerical Aspects

      The algorithm is backward stable.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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