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<H1><A NAME="SECTION03330000000000000000"></A><A NAME="secblockalg"></A><A NAME="7631"></A>
<BR>
Block Algorithms and their
Derivation
</H1>
<P>
It is comparatively straightforward to recode many of the algorithms in
LINPACK and EISPACK so that they call Level 2 BLAS<A NAME="7632"></A>.
Indeed, in the simplest
cases the same floating-point operations are performed, possibly even in
the same order: it is just a matter of reorganizing the software. To
illustrate
this point we derive the Cholesky factorization algorithm that is used in
the
LINPACK<A NAME="7633"></A> routine SPOFA<A NAME="7634"></A>, which
factorizes a symmetric positive definite matrix
as <B><I>A</I> = <I>U</I><SUP><I>T</I></SUP> <I>U</I></B>. Writing these equations as:
<P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\left( \begin{array}{ccc}
A_{11} & a_j & A_{13} \\
. & a_{jj} & \alpha_{j}^T \\
. & . & A_{33} \\
\end{array} \right) =
\left( \begin{array}{ccc}
U_{11}^T & 0 & 0 \\
u_{j}^T & u_{jj} & 0 \\
U_{13}^T & \mu_j & U_{33}^T \\
\end{array} \right)
\left( \begin{array}{ccc}
U_{11} & u_j & U_{13} \\
0 & u_{jj} & \mu_{j}^T \\
0 & 0 & U_{33} \\
\end{array} \right)
\end{displaymath}
-->
<IMG
WIDTH="473" HEIGHT="73" BORDER="0"
SRC="img214.png"
ALT="\begin{displaymath}
\left( \begin{array}{ccc}
A_{11} & a_j & A_{13} \\
. & a_{j...
... u_{jj} & \mu_{j}^T \\
0 & 0 & U_{33} \\
\end{array} \right)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
and equating coefficients of the <B><I>j</I><SUP><I>th</I></SUP></B> column, we obtain:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="154" HEIGHT="60" BORDER="0"
SRC="img215.png"
ALT="\begin{eqnarray*}
a_j & = & U_{11}^T u_j \\
a_{jj} & = & u_{j}^T u_j + u_{jj}^2.
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
<P>
Hence, if <B><I>U</I><SUB>11</SUB></B> has already been computed, we can compute <B><I>u</I><SUB><I>j</I></SUB></B> and
<B><I>u</I><SUB><I>jj</I></SUB></B>
from the equations:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="173" HEIGHT="60" BORDER="0"
SRC="img216.png"
ALT="\begin{eqnarray*}
U_{11}^T u_j & = & a_j \\
u_{jj}^2 & = & a_{jj} - u_{j}^T u_j.
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
<P>
Here is the body of the code of the LINPACK routine SPOFA<A NAME="7674"></A>,
which implements the above method:
<P>
<PRE>
DO 30 J = 1, N
INFO = J
S = 0.0E0
JM1 = J - 1
IF (JM1 .LT. 1) GO TO 20
DO 10 K = 1, JM1
T = A(K,J) - SDOT(K-1,A(1,K),1,A(1,J),1)
T = T/A(K,K)
A(K,J) = T
S = S + T*T
10 CONTINUE
20 CONTINUE
S = A(J,J) - S
C ......EXIT
IF (S .LE. 0.0E0) GO TO 40
A(J,J) = SQRT(S)
30 CONTINUE
</PRE>
<P>
And here is the same computation recoded in ``LAPACK-style''
to use the Level 2 BLAS<A NAME="7677"></A> routine
STRSV (which solves a triangular system of equations). The call to STRSV
has replaced the loop over K which made several calls to the
Level 1 BLAS routine SDOT. (For reasons given below, this is not the actual
code used in LAPACK -- hence the term ``LAPACK-style''.)
<P>
<PRE>
DO 10 J = 1, N
CALL STRSV( 'Upper', 'Transpose', 'Non-unit', J-1, A, LDA,
$ A(1,J), 1 )
S = A(J,J) - SDOT( J-1, A(1,J), 1, A(1,J), 1 )
IF( S.LE.ZERO ) GO TO 20
A(J,J) = SQRT( S )
10 CONTINUE
</PRE>
<P>
This change by itself is sufficient to make big gains in performance
on machines like the CRAY C-90.
<P>
But on many machines such as an IBM RISC Sys/6000-550 (using double
precision)
there is virtually no difference in performance between
the LINPACK-style and the LAPACK Level 2 BLAS style code.
Both styles run at a megaflop rate far below its peak performance for
matrix-matrix multiplication.
To exploit the faster speed of Level 3 BLAS<A NAME="7680"></A>, the
algorithms must undergo a deeper level of restructuring, and be re-cast as a
<B>block algorithm</B> -- that is, an algorithm that operates on <B>blocks</B>
or submatrices of the original matrix.
<P>
To derive a block form of Cholesky
factorization<A NAME="7683"></A>, we write the
defining equation in partitioned form thus:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\left( \begin{array}{ccc}
A_{11} & A_{12} & A_{13}\\
. & A_{22} & A_{23}\\
. & . & A_{33}\\
\end{array} \right) =
\left( \begin{array}{ccc}
U_{11}^T & 0 & 0\\
U_{12}^T & U_{22}^T & 0\\
U_{13}^T & U_{23}^T & U_{33}^T\\
\end{array} \right)
\left( \begin{array}{ccc}
U_{11} & U_{12} & U_{13}\\
0 & U_{22} & U_{23}\\
0 & 0 & U_{33}\\
\end{array} \right).
\end{displaymath}
-->
<IMG
WIDTH="495" HEIGHT="73" BORDER="0"
SRC="img217.png"
ALT="\begin{displaymath}
\left( \begin{array}{ccc}
A_{11} & A_{12} & A_{13}\\
. & A_...
...
0 & U_{22} & U_{23}\\
0 & 0 & U_{33}\\
\end{array} \right).
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
Equating submatrices in the second block of columns, we obtain:
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="203" HEIGHT="58" BORDER="0"
SRC="img218.png"
ALT="\begin{eqnarray*}
A_{12} & = & U_{11}^T U_{12} \\
A_{22} & = & U_{12}^T U_{12} + U_{22}^T U_{22}.
\end{eqnarray*}">
</DIV><P></P>
<BR CLEAR="ALL">
<P>
Hence, if <B><I>U</I><SUB>11</SUB></B> has already been computed, we can compute <B><I>U</I><SUB>12</SUB></B> as
the solution to the equation
<B>
<I>U</I><SUB>11</SUB><SUP><I>T</I></SUP> <I>U</I><SUB>12</SUB> = <I>A</I><SUB>12</SUB>
</B>
<BR CLEAR="ALL"><P></P>
by a call to the Level 3 BLAS routine STRSM; and then we can compute
<B><I>U</I><SUB>22</SUB></B>
from
<B>
<I>U</I><SUB>22</SUB><SUP><I>T</I></SUP> <I>U</I><SUB>22</SUB> = <I>A</I><SUB>22</SUB> - <I>U</I><SUB>12</SUB><SUP><I>T</I></SUP> <I>U</I><SUB>12</SUB>.
</B>
<BR CLEAR="ALL"><P></P>
This involves first updating the symmetric submatrix <B><I>A</I><SUB>22</SUB></B> by a call to
the
Level 3 BLAS routine SSYRK, and then computing its Cholesky factorization.
Since Fortran does not allow recursion, a separate routine must be called
(using Level 2 BLAS rather than Level 3), named SPOTF2 in the code below.
In this way successive blocks of columns of <B><I>U</I></B> are computed.
Here is LAPACK-style code for the block algorithm. In this code-fragment
<TT>NB</TT> denotes the width<A NAME="7738"></A> of the blocks.
<P>
<PRE>
DO 10 J = 1, N, NB
JB = MIN( NB, N-J+1 )
CALL STRSM( 'Left', 'Upper', 'Transpose', 'Non-unit', J-1, JB,
$ ONE, A, LDA, A( 1, J ), LDA )
CALL SSYRK( 'Upper', 'Transpose', JB, J-1, -ONE, A( 1, J ), LDA,
$ ONE, A( J, J ), LDA )
CALL SPOTF2( 'Upper', JB, A( J, J ), LDA, INFO )
IF( INFO.NE.0 ) GO TO 20
10 CONTINUE
</PRE>
<P>
But that is not the end of the story, and the code given above is
not the code that is actually used in the LAPACK routine
SPOTRF<A NAME="7741"></A>.
We mentioned in subsection <A HREF="node62.html#subsecvectorize">3.1.1</A> that for many
linear algebra computations there
are several vectorizable variants, often referred to as <B><I>i</I></B>-, <B><I>j</I></B>- and
<B><I>k</I></B>-variants, according to a convention introduced in [<A
HREF="node151.html#Dongarra84a">45</A>]
and used
in [<A
HREF="node151.html#GVL2">55</A>]. The same is true of the corresponding block algorithms.
<P>
It turns out that the <B><I>j</I></B>-variant
that was chosen for LINPACK, and used in the above
examples, is not the fastest on many machines, because it is based on
solving triangular
systems of equations, which can be significantly slower than matrix-matrix
multiplication.
The variant actually used in LAPACK is the <B><I>i</I></B>-variant, which does
rely on matrix-matrix multiplication.
<P>
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<ADDRESS>
<I>Susan Blackford</I>
<BR><I>1999-10-01</I>
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