control_systems_torbox 0.2.1

Control systems toolbox
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<HTML>
<HEAD><TITLE>MB04LD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>

<H2><A Name="MB04LD">MB04LD</A></H2>
<H3>
LQ factorization of a special structured block matrix
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>

<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
  To calculate an LQ 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
                     _
     [ L   A ]     [ L   0 ]
     [       ]*Q = [       ]
     [ 0   B ]     [ C   D ]
              _
  where L and L are lower triangular. The matrix A can be full or
  lower 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 covariance filter).

</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
      SUBROUTINE MB04LD( UPLO, N, M, P, L, LDL, A, LDA, B, LDB, C, LDC,
     $                   TAU, DWORK )
C     .. Scalar Arguments ..
      CHARACTER         UPLO
      INTEGER           LDA, LDB, LDC, LDL, M, N, P
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*),
     $                  L(LDL,*), TAU(*)

</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>

<B>Mode Parameters</B>
<PRE>
  UPLO    CHARACTER*1
          Indicates if the matrix A is or not triangular as follows:
          = 'L':  Matrix A is lower trapezoidal/triangular;
          = 'F':  Matrix A is full.

</PRE>
<B>Input/Output Parameters</B>
<PRE>
  N       (input) INTEGER                 _
          The order of the matrices L and L.  N &gt;= 0.

  M       (input) INTEGER
          The number of columns of the matrices A, B and D.  M &gt;= 0.

  P       (input) INTEGER
          The number of rows of the matrices B, C and D.  P &gt;= 0.

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

  LDL     INTEGER
          The leading dimension of array L.  LDL &gt;= MAX(1,N).

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,M)
          On entry, if UPLO = 'F', the leading N-by-M part of this
          array must contain the matrix A. If UPLO = 'L', the
          leading N-by-MIN(N,M) part of this array must contain the
          lower trapezoidal (lower triangular if N &lt;= M) matrix A,
          and the elements above the diagonal are not referenced.
          On exit, the leading N-by-M part (lower trapezoidal or
          triangular, if UPLO = 'L') 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 &gt;= MAX(1,N).

  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 &gt;= MAX(1,P).

  C       (output) DOUBLE PRECISION array, dimension (LDC,N)
          The leading P-by-N part of this array contains the
          computed matrix C.

  LDC     INTEGER
          The leading dimension of array C.  LDC &gt;= MAX(1,P).

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

</PRE>
<B>Workspace</B>
<PRE>
  DWORK   DOUBLE PRECISION array, dimension (N)

</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
  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 an M-vector, if UPLO = 'F', or an min(i,M)-vector, if
         i
  UPLO = 'L'.  The components of v  are stored in the i-th row of A,
                                  i
  and tau  is stored in TAU(i).
         i

</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
  The algorithm is backward stable.

</PRE>

<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
  None
</PRE>

<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
  None
</PRE>
<B>Program Data</B>
<PRE>
  None
</PRE>
<B>Program Results</B>
<PRE>
  None
</PRE>

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