## MB04OW

### Rank-one update of a Cholesky factorization (variant)

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

Purpose

      To perform the QR factorization

         ( U  ) = Q*( R ),  where  U = ( U1  U2 ),  R = ( R1  R2 ),
         ( x' )     ( 0 )              ( 0   T  )       ( 0   R3 )

      where U and R are (m+n)-by-(m+n) upper triangular matrices, x is
      an m+n element vector, U1 is m-by-m, T is n-by-n, stored
      separately, and Q is an (m+n+1)-by-(m+n+1) orthogonal matrix.

      The matrix ( U1 U2 ) must be supplied in the m-by-(m+n) upper
      trapezoidal part of the array A and this is overwritten by the
      corresponding part ( R1 R2 ) of R. The remaining upper triangular
      part of R, R3, is overwritten on the array T.

      The transformations performed are also applied to the (m+n+1)-by-p
      matrix ( B' C' d )' (' denotes transposition), where B, C, and d'
      are m-by-p, n-by-p, and 1-by-p matrices, respectively.

Specification

          SUBROUTINE MB04OW( M, N, P, A, LDA, T, LDT, X, INCX, B, LDB,
         $                   C, LDC, D, INCD )
    C     .. Scalar Arguments ..
          INTEGER            INCD, INCX, LDA, LDB, LDC, LDT, M, N, P
    C     .. Array Arguments ..
          DOUBLE PRECISION   A(LDA,*), B(LDB,*), C(LDC,*), D(*), T(LDT,*),
         $                   X(*)

Arguments

Input/Output Parameters

      M      (input) INTEGER
             The number of rows of the matrix ( U1  U2 ).  M >= 0.

      N      (input) INTEGER
             The order of the matrix T.  N >= 0.

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

      A      (input/output) DOUBLE PRECISION array, dimension (LDA,N)
             On entry, the leading M-by-(M+N) upper trapezoidal part of
             this array must contain the upper trapezoidal matrix
             ( U1 U2 ).
             On exit, the leading M-by-(M+N) upper trapezoidal part of
             this array contains the upper trapezoidal matrix ( R1 R2 ).
             The strict lower triangle of A is not referenced.

      LDA    INTEGER
             The leading dimension of the array A.  LDA >= max(1,M).

      T      (input/output) DOUBLE PRECISION array, dimension (LDT,N)
             On entry, the leading N-by-N upper triangular part of this
             array must contain the upper triangular matrix T.
             On exit, the leading N-by-N upper triangular part of this
             array contains the upper triangular matrix R3.
             The strict lower triangle of T is not referenced.

      LDT    INTEGER
             The leading dimension of the array T.  LDT >= max(1,N).

      X      (input/output) DOUBLE PRECISION array, dimension
             (1+(M+N-1)*INCX), if M+N > 0, or dimension (0), if M+N = 0.
             On entry, the incremented array X must contain the
             vector x. On exit, the content of X is changed.

      INCX   (input) INTEGER
             Specifies the increment for the elements of X.  INCX > 0.

      B      (input/output) DOUBLE PRECISION array, dimension (LDB,P)
             On entry, the leading M-by-P part of this array must
             contain the matrix B.
             On exit, the leading M-by-P part of this array contains
             the transformed matrix B.
             If M = 0 or P = 0, this array is not referenced.

      LDB    INTEGER
             The leading dimension of the array B.
             LDB >= max(1,M), if P > 0;
             LDB >= 1,        if P = 0.

      C      (input/output) DOUBLE PRECISION array, dimension (LDC,P)
             On entry, the leading N-by-P part of this array must
             contain the matrix C.
             On exit, the leading N-by-P part of this array contains
             the transformed matrix C.
             If N = 0 or P = 0, this array is not referenced.

      LDC    INTEGER
             The leading dimension of the array C.
             LDC >= max(1,N), if P > 0;
             LDC >= 1,        if P = 0.

      D      (input/output) DOUBLE PRECISION array, dimension
             (1+(P-1)*INCD), if P > 0, or dimension (0), if P = 0.
             On entry, the incremented array D must contain the
             vector d.
             On exit, this incremented array contains the transformed
             vector d.
             If P = 0, this array is not referenced.

      INCD   (input) INTEGER
             Specifies the increment for the elements of D.  INCD > 0.

Method

      Let q = m+n. The matrix Q is formed as a sequence of plane
      rotations in planes (1, q+1), (2, q+1), ..., (q, q+1), the
      rotation in the (j, q+1)th plane, Q(j), being chosen to
      annihilate the jth element of x.

Numerical Aspects

      The algorithm requires 0((M+N)*(M+N+P)) operations and is backward
      stable.

Further Comments

      For P = 0, this routine produces the same result as SLICOT Library
      routine MB04OX, but matrix T may not be stored in the array A.

Example

Program Text

      None

Program Data

      None

Program Results

      None

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