MB04OW

Rank-one update of a Cholesky factorization (variant)

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.

      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(*)

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.

  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.

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

  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.

Program Text

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