Gregory Wright gwright at antiope.com
Thu Aug 2 22:27:21 UTC 2018

```Hi,

Something Haskell has lacked for a long time is a good medium-duty
linear system solver based on the LU decomposition.  There are bindings
to the usual C/Fortran libraries, but not one in pure Haskell.  (An
example "LU factorization" routine that does not do partial pivoting has
been around for years, but lacking pivoting it can fail unexpectedly on
well-conditioned inputs.  Another Haskell LU decomposition using partial
pivoting is around, but it uses an inefficient representation of the
pivot matrix, so it's not suited to solving systems of more than 100 x
100, say.)

By medium duty I mean a linear system solver that can handle systems of
(1000s) x (1000s) and uses Crout's efficient in-place algorithm.  In
short, a program does everything short of exploiting SIMD vector
instructions for solving small subproblems.

implements this.  It contains an LU factorization function and an LU
system solver.  The LU factorization also returns the parity of the
pivots ( = (-1)^(number of row swaps) ) so it can be used to calculate
determinants.  I used Gustavson's recursive (imperative) version of
Crout's method.  The implementation is quite simple and deserves to be
better known by people using functional languages to do numeric work.
https://github.com/gwright83/luSolve

The performance scales as expected (as n^3, a linear system 10 times
larger in each dimension takes a 1000 times longer to solve):

Benchmark luSolve-bench: RUNNING...
benchmarking LUSolve/luFactor 100 x 100 matrix
time                 1.944 ms   (1.920 ms .. 1.980 ms)
0.996 R²   (0.994 R² .. 0.998 R²)
mean                 1.981 ms   (1.958 ms .. 2.009 ms)
std dev              85.64 μs   (70.21 μs .. 107.7 μs)
variance introduced by outliers: 30% (moderately inflated)

benchmarking LUSolve/luFactor 500 x 500 matrix
time                 204.3 ms   (198.1 ms .. 208.2 ms)
1.000 R²   (0.999 R² .. 1.000 R²)
mean                 203.3 ms   (201.2 ms .. 206.2 ms)
std dev              3.619 ms   (2.307 ms .. 6.231 ms)
variance introduced by outliers: 14% (moderately inflated)

benchmarking LUSolve/luFactor 1000 x 1000 matrix
time                 1.940 s    (1.685 s .. 2.139 s)
0.998 R²   (0.993 R² .. 1.000 R²)
mean                 1.826 s    (1.696 s .. 1.880 s)
std dev              93.63 ms   (5.802 ms .. 117.8 ms)
variance introduced by outliers: 19% (moderately inflated)

Benchmark luSolve-bench: FINISH

The puzzle is why the overall performance is so poor.  When I solve
random 1000 x 1000 systems using the linsys.c example file from the
Recursive LAPACK (ReLAPACK) library -- which implements the same
algorithm -- the average time is only 26 ms.  (I have tweaked
ReLAPACK's  dgetrf.c so that it doesn't use optimized routines for small
matrices.  As near as I can make it, the C and haskell versions should
be doing the same thing.)

The haskell version runs 75 times slower.  This is the puzzle.

My haskell version uses a mutable, matrix of unboxed doubles (from Kai
library).  Matrix reads and writes are unsafe, so there is no overhead
from bounds checking.

Let's look at the result of profiling:

Tue Jul 31 21:07 2018 Time and Allocation Profiling Report  (Final)

luSolve-hspec +RTS -N -p -RTS

total time  =     7665.31 secs   (7665309 ticks @ 1000 us, 1
processor)
total alloc = 10,343,030,811,040 bytes  (excludes profiling

COST CENTRE           MODULE
SRC                                                      %time %alloc

unsafeWrite           Data.Matrix.Dense.Generic.Mutable
src/Data/Matrix/Dense/Generic/Mutable.hs:(38,5)-(39,38)   17.7   29.4
basicUnsafeWrite      Data.Vector.Primitive.Mutable
Data/Vector/Primitive/Mutable.hs:115:3-69                 14.7   13.0
src/Data/Matrix/Dense/Generic/Mutable.hs:(34,5)-(35,38)   14.2   20.7
matrixMultiply.\.\.\  Numeric.LinearAlgebra.LUSolve
src/Numeric/LinearAlgebra/LUSolve.hs:(245,54)-(249,86)    13.4   13.5
Data/Primitive/Types.hs:184:30-132                         9.0   15.5
Data/Vector/Primitive/Mutable.hs:112:3-63                  8.8    0.1
triangularSolve.\.\.\ Numeric.LinearAlgebra.LUSolve
src/Numeric/LinearAlgebra/LUSolve.hs:(382,45)-(386,58)     5.2    4.5
matrixMultiply.\.\    Numeric.LinearAlgebra.LUSolve
src/Numeric/LinearAlgebra/LUSolve.hs:(244,54)-(249,86)     4.1    0.3
Data/Vector/Unboxed/Base.hs:278:813-868                    3.3    0.0
basicUnsafeWrite      Data.Vector.Unboxed.Base
Data/Vector/Unboxed/Base.hs:278:872-933                    1.5    0.0
triangularSolve.\.\   Numeric.LinearAlgebra.LUSolve
src/Numeric/LinearAlgebra/LUSolve.hs:(376,33)-(386,58)     1.3    0.1

<snip>

A large amount of time is spent on the invocations of unsafeRead and
unsafeWrite.  This is a bit suspicious -- it looks as if these call may
not be inlined.  In the Data.Vector.Unboxed.Mutable library, which
provides the underlying linear vector of storage locations, the
unsafeRead and unsafeWrite functions are declared INLINE.  Could this be
a failure of the 'matrices' library to mark its unsafeRead/Write
functions as INLINE or SPECIALIZABLE as well?

On the other hand, looking at the core (.dump-simpl) of the library
doesn't show any dictionary passing, and the access to matrix seem to be

If this program took three to five times longer, I would not be
concerned, but a factor of seventy five indicates that I've missed
something important.  Can anyone tell me what it is?

Best Wishes,

Greg

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