Factorizations for Solving Linear Equations

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## Factorizations for Solving Linear Equations

The well-known LU and Cholesky factorizations are the simplest block algorithms to derive. No extra floating-point operations nor extra working storage are required.

Table 3.3 illustrates the speed of the LAPACK routine for LU factorization of a real matrix , SGETRF  in single precision on CRAY machines, and DGETRF  in double precision on all other machines. This corresponds to 64-bit floating-point arithmetic on all machines tested. A block size of 1 means that the unblocked algorithm is used, since it is faster than - or at least as fast as - a blocked algorithm.

```---------------------------------------------------
No. of    Block    Values of n
processors   size    100     1000
---------------------------------------------------
CONVEX C-4640          1        64     274      711
CONVEX C-4640          4        64     379     2588
CRAY C90               1       128     375      863
CRAY C90              16       128     386     7412
DEC 3000-500X Alpha    1        32      53       91
IBM POWER2 model 590   1        32     110      168
IBM RISC Sys/6000-550  1        32      33       56
SGI POWER CHALLENGE    1        64      81      201
SGI POWER CHALLENGE    4        64      79      353
---------------------------------------------------
```
Table 3.3: Speed in megaflops of SGETRF/DGETRF for square matrices of order n

Table 3.4 gives similar results for Cholesky factorization  .

```---------------------------------------------------
No. of    Block    Values of n
processors   size    100     1000
---------------------------------------------------
CONVEX C-4640          1        64     120      546
CONVEX C-4640          4        64     150     1521
CRAY C90               1       128     324      859
CRAY C90              16       128     453     9902
DEC 3000-500X Alpha    1        32      37       83
IBM POWER2 model 590   1        32     102      247
IBM RISC Sys/6000-550  1        32      40       72
SGI POWER CHALLENGE    1        64      74      199
SGI POWER CHALLENGE    4        64      69      424
---------------------------------------------------
```
Table 3.4: Speed in megaflops of SPOTRF/DPOTRF for matrices of order n with UPLO = `U'

LAPACK, like LINPACK, provides a factorization for symmetric indefinite  matrices, so that A is factorized as , where P is a permutation matrix, and D is block diagonal with blocks of order 1 or 2. A block form of this algorithm has been derived, and is implemented in the LAPACK routine SSYTRF /DSYTRF . It has to duplicate a little of the computation in order to ``look ahead'' to determine the necessary row and column interchanges, but the extra work can be more than compensated for by the greater speed of updating the matrix by blocks as is illustrated in Table 3.5 .

```-------------------
Block   Values of n
size    100    1000
-------------------
1       62      86
64       68     165
-------------------
```
Table 3.5: Speed in megaflops of DSYTRF for matrices of order n with UPLO = `U' on an IBM POWER2 model 590

LAPACK, like LINPACK, provides LU and Cholesky factorizations of band matrices. The LINPACK algorithms can easily be restructured to use Level 2 BLAS, though that has little effect on performance for matrices of very narrow bandwidth. It is also possible to use Level 3 BLAS, at the price of doing some extra work with zero elements outside the band [39]. This becomes worthwhile for matrices of large order and semi-bandwidth greater than 100 or so.

Next: Factorization Up: Examples of Block Previous: Examples of Block

Tue Nov 29 14:03:33 EST 1994