CS 267 Applications of Parallel Computers
Dense Linear Algebra
James Demmel
http://www.cs.berkeley.edu/~demmel/cs267_221001.ppt
CS267 Dense Linear Algebra I.1
Demmel Fa 2001
Outline
° Motivation for Dense Linear Algebra (as opposed to sparse)
° Benchmarks
° Review Gaussian Elimination (GE) for solving Ax=b
° Optimizing GE for caches on sequential machines
• using matrix-matrix multiplication (BLAS)
° LAPACK library overview and performance
° Data layouts on parallel machines
° Parallel matrix-matrix multiplication review
° Parallel Gaussian Elimination
° ScaLAPACK library overview
° Eigenvalue problems
° Open Problems
CS267 Dense Linear Algebra I.2
Demmel Fa 2001
Motivation
° 3 Basic Linear Algebra Problems
• Linear Equations: Solve Ax=b for x
• Least Squares: Find x that minimizes S ri2 where r=Ax-b
• Eigenvalues: Find l and x where Ax = l x
• Lots of variations depending on structure of A (eg symmetry)
° Why dense A, as opposed to sparse A?
• Aren’t “most” large matrices sparse?
• Dense algorithms easier to understand
• Some applications yields large dense matrices
- Ax=b: Computational Electromagnetics
- Ax = lx: Quantum Chemistry
• Benchmarking
- “How fast is your computer?” =
“How fast can you solve dense Ax=b?”
• Large sparse matrix algorithms often yield smaller (but still large)
dense problems
CS267 Dense Linear Algebra I.3
Demmel Fa 2001
Computational Electromagnetics – Solve Ax=b
•Developed during 1980s, driven by defense applications
•Determine the RCS (radar cross section) of airplane
•Reduce signature of plane (stealth technology)
•Other applications are antenna design, medical equipment
•Two fundamental numerical approaches:
•MOM methods of moments ( frequency domain)
•Large dense matrices
•Finite differences (time domain)
•Even larger sparse matrices
CS267 Dense Linear Algebra I.4
Demmel Fa 2001
Computational Electromagnetics
- Discretize surface into triangular facets using
standard modeling tools
- Amplitude of currents on surface are
unknowns
- Integral equation is discretized into a set of linear
equations
image: NW Univ. Comp. Electromagnetics Laboratory http://nueml.ece.nwu.edu/
CS267 Dense Linear Algebra I.5
Demmel Fa 2001
Computational Electromagnetics (MOM)
After discretization the integral equation has the
form
A x = b
where
A is the (dense) impedance matrix,
x is the unknown vector of amplitudes, and
b is the excitation vector.
(see Cwik, Patterson, and Scott, Electromagnetic Scattering on the Intel Touchstone Delta,
IEEE Supercomputing ‘92, pp 538 - 542)
CS267 Dense Linear Algebra I.6
Demmel Fa 2001
Results for Parallel Implementation on Intel Delta
Task
Time (hours)
Fill (compute n2 matrix entries)
9.20
(embarrassingly parallel but slow)
Factor (Gaussian Elimination, O(n3) )
8.25
(good parallelism with right algorithm)
Solve (O(n2))
2 .17
(reasonable parallelism with right algorithm)
Field Calc. (O(n))
0.12
(embarrassingly parallel and fast)
The problem solved was for a matrix of size 48,672.
2.6 Gflops for Factor - The world record in 1991.
CS267 Dense Linear Algebra I.7
Demmel Fa 2001
Current Records for Solving Dense Systems
www.netlib.org, click on Performance DataBase Server
Year
1950's
1995
1996
1998
1998
System Size
O(100)
128,600
215,000
148,000
235,000
Machine
Intel Paragon
Intel ASCI Red
Cray T3E
Intel ASCI Red
# Procs Gflops (Peak)
6768
7264
1488
9152
281
1068
1127
1338
( 338)
(1453)
(1786)
(1830)
5040
9632
1608
2380
(2520)
(3207)
IBM ASCI White 8000
7226
(12000)
(200 MHz Ppro)
1999
2000
374,000
362,000
SGI ASCI Blue
Intel ASCI Red
(333 MHz Xeon)
2001
518,000
(375 MHz Power 3)
source: Alan Edelman http://www-math.mit.edu/~edelman/records.html
LINPACK Benchmark: http://www.netlib.org/performance/html/PDSreports.html
CS267 Dense Linear Algebra I.8
Demmel Fa 2001
Current Records for Solving Small Dense Systems
www.netlib.org, click on Performance DataBase Server
Machine
Fujitsu VPP 5000
(1 proc 300 MHz)
Cray T90
(32 proc, 450 MHz)
(4 proc, 450 MHz)
IBM Power 4
(1 proc, 1300 MHz)
…
Dell Itanium
(4 proc, 800 MHz)
(2 proc, 800 MHz)
(1 proc, 800 MHz)
Megaflops
n=100 n=1000 Peak
1156
8784
9600
1129
29360
5735
57600
7200
1074
2394
5200
600
7358
4504
2382
12800
6400
3200
source: LINPACK Benchmark: http://www.netlib.org/performance/html/PDSreports.html
CS267 Dense Linear Algebra I.9
Demmel Fa 2001
Computational Chemistry – Ax = l x
° Seek energy levels of a molecule, crystal, etc.
• Solve Schroedinger’s Equation for energy levels = eigenvalues
• Discretize to get Ax = lBx, solve for eigenvalues l and eigenvectors x
• A and B large, symmetric or Hermitian matrices (B positive definite)
• May want some or all eigenvalues and/or eigenvectors
° MP-Quest (Sandia NL)
•
•
•
•
•
Si and sapphire crystals of up to 3072 atoms
Local Density Approximation to Schroedinger Equation
A and B up to n=40000, complex Hermitian
Need all eigenvalues and eigenvectors
Need to iterate up to 20 times (for self-consistency)
° Implemented on Intel ASCI Red
•
•
•
•
•
9200 Pentium Pro 200 processors (4600 Duals, a CLUMP)
Overall application ran at 605 Gflops (out of 1800 Gflops peak),
Eigensolver ran at 684 Gflops
www.cs.berkeley.edu/~stanley/gbell/index.html
Runner-up for Gordon Bell Prize at Supercomputing 98
° Same problem arises in astrophysics...
CS267 Dense Linear Algebra I.10
Demmel Fa 2001
Review of Gaussian Elimination (GE) for solving Ax=b
° Add multiples of each row to later rows to make A upper
triangular
° Solve resulting triangular system Ux = c by substitution
… for each column i
… zero it out below the diagonal by adding multiples of row i to later rows
for i = 1 to n-1
… for each row j below row i
for j = i+1 to n
… add a multiple of row i to row j
for k = i to n
A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k)
CS267 Dense Linear Algebra I.11
Demmel Fa 2001
Refine GE Algorithm (1)
° Initial Version
… for each column i
… zero it out below the diagonal by adding multiples of row i to later rows
for i = 1 to n-1
… for each row j below row i
for j = i+1 to n
… add a multiple of row i to row j
for k = i to n
A(j,k) = A(j,k) - (A(j,i)/A(i,i)) * A(i,k)
° Remove computation of constant A(j,i)/A(i,i) from
inner loop
for i = 1 to n-1
for j = i+1 to n
m = A(j,i)/A(i,i)
for k = i to n
A(j,k) = A(j,k) - m * A(i,k)
CS267 Dense Linear Algebra I.12
Demmel Fa 2001
Refine GE Algorithm (2)
° Last version
for i = 1 to n-1
for j = i+1 to n
m = A(j,i)/A(i,i)
for k = i to n
A(j,k) = A(j,k) - m * A(i,k)
° Don’t compute what we already know:
zeros below diagonal in column i
for i = 1 to n-1
for j = i+1 to n
m = A(j,i)/A(i,i)
for k = i+1 to n
A(j,k) = A(j,k) - m * A(i,k)
CS267 Dense Linear Algebra I.13
Demmel Fa 2001
Refine GE Algorithm (3)
° Last version
for i = 1 to n-1
for j = i+1 to n
m = A(j,i)/A(i,i)
for k = i+1 to n
A(j,k) = A(j,k) - m * A(i,k)
° Store multipliers m below diagonal in zeroed entries
for later use
for i = 1 to n-1
for j = i+1 to n
A(j,i) = A(j,i)/A(i,i)
for k = i+1 to n
A(j,k) = A(j,k) - A(j,i) * A(i,k)
CS267 Dense Linear Algebra I.14
Demmel Fa 2001
Refine GE Algorithm (4)
° Last version
o Split Loop
CS267 Dense Linear Algebra I.15
for i = 1 to n-1
for j = i+1 to n
A(j,i) = A(j,i)/A(i,i)
for k = i+1 to n
A(j,k) = A(j,k) - A(j,i) * A(i,k)
for i = 1 to n-1
for j = i+1 to n
A(j,i) = A(j,i)/A(i,i)
for j = i+1 to n
for k = i+1 to n
A(j,k) = A(j,k) - A(j,i) * A(i,k)
Demmel Fa 2001
Refine GE Algorithm (5)
° Last version
for i = 1 to n-1
for j = i+1 to n
A(j,i) = A(j,i)/A(i,i)
for j = i+1 to n
for k = i+1 to n
A(j,k) = A(j,k) - A(j,i) * A(i,k)
° Express using matrix operations (BLAS)
for i = 1 to n-1
A(i+1:n,i) = A(i+1:n,i) * ( 1 / A(i,i) )
A(i+1:n,i+1:n) = A(i+1:n , i+1:n )
- A(i+1:n , i) * A(i , i+1:n)
CS267 Dense Linear Algebra I.16
Demmel Fa 2001
What GE really computes
for i = 1 to n-1
A(i+1:n,i) = A(i+1:n,i) / A(i,i)
A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) - A(i+1:n , i) * A(i , i+1:n)
° Call the strictly lower triangular matrix of multipliers
M, and let L = I+M
° Call the upper triangle of the final matrix U
° Lemma (LU Factorization): If the above algorithm
terminates (does not divide by zero) then A = L*U
° Solving A*x=b using GE
• Factorize A = L*U using GE
(cost = 2/3 n3 flops)
• Solve L*y = b for y, using substitution (cost = n2 flops)
• Solve U*x = y for x, using substitution (cost = n2 flops)
° Thus A*x = (L*U)*x = L*(U*x) = L*y = b as desired
CS267 Dense Linear Algebra I.17
Demmel Fa 2001
Problems with basic GE algorithm
° What if some A(i,i) is zero? Or very small?
• Result may not exist, or be “unstable”, so need to pivot
° Current computation all BLAS 1 or BLAS 2, but we know that
BLAS 3 (matrix multiply) is fastest (earlier lectures…)
for i = 1 to n-1
A(i+1:n,i) = A(i+1:n,i) / A(i,i)
… BLAS 1 (scale a vector)
A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) … BLAS 2 (rank-1 update)
- A(i+1:n , i) * A(i , i+1:n)
Peak
BLAS 3
BLAS 2
BLAS 1
CS267 Dense Linear Algebra I.18
Demmel Fa 2001
Pivoting in Gaussian Elimination
°
A= [0 1]
[1 0]
fails completely, even though A is “easy”
° Illustrate problems in 3-decimal digit arithmetic:
A = [ 1e-4 1 ] and
[ 1 1 ]
b = [ 1 ], correct answer to 3 places is x = [ 1 ]
[2]
[1]
° Result of LU decomposition is
L=[ 1
[ fl(1/1e-4)
U = [ 1e-4
[0
0] = [ 1
1]
[ 1e4
1
] = [ 1e-4
fl(1-1e4*1) ]
[ 0
Check if A = L*U = [ 1e-4
[ 1
1]
0]
… No roundoff error yet
0 ]
1 ]
1 ]
-1e4 ]
… Error in 4th decimal place
… (2,2) entry entirely wrong
° Algorithm “forgets” (2,2) entry, gets same L and U for all |A(2,2)|<5
° Numerical instability
° Computed solution x totally inaccurate
° Cure: Pivot (swap rows of A) so entries of L and U bounded
CS267 Dense Linear Algebra I.19
Demmel Fa 2001
Gaussian Elimination with Partial Pivoting (GEPP)
° Partial Pivoting: swap rows so that each multiplier
|L(i,j)| = |A(j,i)/A(i,i)| <= 1
for i = 1 to n-1
find and record k where |A(k,i)| = max{i <= j <= n} |A(j,i)|
… i.e. largest entry in rest of column i
if |A(k,i)| = 0
exit with a warning that A is singular, or nearly so
elseif k != i
swap rows i and k of A
end if
A(i+1:n,i) = A(i+1:n,i) / A(i,i)
… each quotient lies in [-1,1]
A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) - A(i+1:n , i) * A(i , i+1:n)
° Lemma: This algorithm computes A = P*L*U, where P is a
permutation matrix
° Since each entry of |L(i,j)| <= 1, this algorithm is considered
numerically stable
° For details see LAPACK code at www.netlib.org/lapack/single/sgetf2.f
CS267 Dense Linear Algebra I.20
Demmel Fa 2001
Converting BLAS2 to BLAS3 in GEPP
° Blocking
• Used to optimize matrix-multiplication
• Harder here because of data dependencies in GEPP
° Delayed Updates
• Save updates to “trailing matrix” from several consecutive BLAS2
updates
• Apply many saved updates simultaneously in one BLAS3
operation
° Same idea works for much of dense linear algebra
• Open questions remain
° Need to choose a block size b
• Algorithm will save and apply b updates
• b must be small enough so that active submatrix consisting of b
columns of A fits in cache
• b must be large enough to make BLAS3 fast
CS267 Dense Linear Algebra I.21
Demmel Fa 2001
Blocked GEPP (www.netlib.org/lapack/single/sgetrf.f)
for ib = 1 to n-1 step b … Process matrix b columns at a time
end = ib + b-1
… Point to end of block of b columns
apply BLAS2 version of GEPP to get A(ib:n , ib:end) = P’ * L’ * U’
… let LL denote the strict lower triangular part of A(ib:end , ib:end) + I
A(ib:end , end+1:n) = LL-1 * A(ib:end , end+1:n)
… update next b rows of U
A(end+1:n , end+1:n ) = A(end+1:n , end+1:n )
- A(end+1:n , ib:end) * A(ib:end , end+1:n)
… apply delayed updates with single matrix-multiply
… with inner dimension b
(For a correctness proof,
see on-lines notes.)
CS267 Dense Linear Algebra I.22
Demmel Fa 2001
Efficiency of Blocked GEPP
CS267 Dense Linear Algebra I.23
Demmel Fa 2001
Overview of LAPACK
° Standard library for dense/banded linear algebra
• Linear systems: A*x=b
• Least squares problems: minx || A*x-b ||2
• Eigenvalue problems: Ax = lx, Ax = lBx
• Singular value decomposition (SVD): A = USVT
° Algorithms reorganized to use BLAS3 as much as
possible
° Basis of math libraries on many computers, Matlab 6
° Many algorithmic innovations remain
•
•
•
•
Projects available
Automatic optimization
Quadtree matrix data structures for locality
New eigenvalue algorithms
CS267 Dense Linear Algebra I.24
Demmel Fa 2001
Performance of LAPACK (n=1000)
CS267 Dense Linear Algebra I.25
Demmel Fa 2001
Performance of LAPACK (n=100)
CS267 Dense Linear Algebra I.26
Demmel Fa 2001
Recursive Algorithms
° Still uses delayed updates, but organized differently
• (formulas on board)
° Can exploit recursive data layouts
• 3x speedups on least squares for tall, thin matrices
° Theoretically optimal memory hierarchy performance
° See references at
• http://lawra.uni-c.dk/lawra/index.html
• http://www.cs.berkeley.edu/~yelick/cs267f01/lectures/Lect14.html
• http://www.cs.umu.se/research/parallel/recursion/recursive-qr/
CS267 Dense Linear Algebra I.27
Demmel Fa 2001
Recursive Algorithms – Limits
° Two kinds of dense matrix compositions
° One Sided
• Sequence of simple operations applied on left of matrix
• Gaussian Elimination: A = L*U or A = P*L*U
- Symmetric Gaussian Elimination: A = L*D*LT
- Cholesky: A = L*LT
• QR Decomposition for Least Squares: A = Q*R
• Can be nearly 100% BLAS 3
• Susceptible to recursive algorithms
° Two Sided
•
•
•
•
•
Sequence of simple operations applied on both sides, alternating
Eigenvalue algorithms, SVD
At least ~25% BLAS 2
Seem impervious to recursive approach
Some recent progress on SVD (25% vs 50% BLAS2)
CS267 Dense Linear Algebra I.28
Demmel Fa 2001
Parallelizing Gaussian Elimination
° Recall parallelization steps from earlier lecture
• Decomposition: identify enough parallel work, but not too much
• Assignment: load balance work among threads
• Orchestrate: communication and synchronization
• Mapping: which processors execute which threads
° Decomposition
• In BLAS 2 algorithm nearly each flop in inner loop can be done in
parallel, so with n2 processors, need 3n parallel steps
for i = 1 to n-1
A(i+1:n,i) = A(i+1:n,i) / A(i,i)
… BLAS 1 (scale a vector)
A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) … BLAS 2 (rank-1 update)
- A(i+1:n , i) * A(i , i+1:n)
• This is too fine-grained, prefer calls to local matmuls instead
• Need to discuss parallel matrix multiplication
° Assignment
• Which processors are responsible for which submatrices?
CS267 Dense Linear Algebra I.29
Demmel Fa 2001
Different Data Layouts for Parallel GE (on 4 procs)
Bad load balance:
P0 idle after first
n/4 steps
Can trade load balance
and BLAS2/3
performance by
choosing b, but
factorization of block
column is a bottleneck
Load balanced, but can’t easily
use BLAS2 or BLAS3
The winner!
Complicated addressing
CS267 Dense Linear Algebra I.30
Demmel Fa 2001
PDGEMM = PBLAS routine
for matrix multiply
Observations:
For fixed N, as P increases
Mflops increases, but
less than 100% efficiency
For fixed P, as N increases,
Mflops (efficiency) rises
DGEMM = BLAS routine
for matrix multiply
Maximum speed for PDGEMM
= # Procs * speed of DGEMM
Observations (same as above):
Efficiency always at least 48%
For fixed N, as P increases,
efficiency drops
For fixed P, as N increases,
efficiency increases
CS267 Dense Linear Algebra I.31
Demmel Fa 2001
Review: BLAS 3 (Blocked) GEPP
for ib = 1 to n-1 step b … Process matrix b columns at a time
end = ib + b-1
… Point to end of block of b columns
apply BLAS2 version of GEPP to get A(ib:n , ib:end) = P’ * L’ * U’
… let LL denote the strict lower triangular part of A(ib:end , ib:end) + I
A(ib:end , end+1:n) = LL-1 * A(ib:end , end+1:n)
… update next b rows of U
A(end+1:n , end+1:n ) = A(end+1:n , end+1:n )
BLAS 3
- A(end+1:n , ib:end) * A(ib:end , end+1:n)
… apply delayed updates with single matrix-multiply
… with inner dimension b
CS267 Dense Linear Algebra I.32
Demmel Fa 2001
Review: Row and Column Block Cyclic Layout
processors and matrix blocks
are distributed in a 2d array
pcol-fold parallelism
in any column, and calls to the
BLAS2 and BLAS3 on matrices of
size brow-by-bcol
serial bottleneck is eased
need not be symmetric in rows and
columns
CS267 Dense Linear Algebra I.33
Demmel Fa 2001
Distributed GE with a 2D Block Cyclic Layout
block size b in the algorithm and the block sizes brow
and bcol in the layout satisfy b=brow=bcol.
shaded regions indicate busy processors or
communication performed.
unnecessary to have a barrier between each
step of the algorithm, e.g.. step 9, 10, and 11 can be
pipelined
CS267 Dense Linear Algebra I.34
Demmel Fa 2001
Distributed GE with a 2D Block Cyclic Layout
CS267 Dense Linear Algebra I.35
Demmel Fa 2001
green = green - blue * pink
Matrix multiply of
CS267 Dense Linear Algebra I.36
Demmel Fa 2001
PDGESV = ScaLAPACK
parallel LU routine
Since it can run no faster than its
inner loop (PDGEMM), we measure:
Efficiency =
Speed(PDGESV)/Speed(PDGEMM)
Observations:
Efficiency well above 50% for large
enough problems
For fixed N, as P increases,
efficiency decreases
(just as for PDGEMM)
For fixed P, as N increases
efficiency increases
(just as for PDGEMM)
From bottom table, cost of solving
Ax=b about half of matrix multiply
for large enough matrices.
From the flop counts we would
expect it to be (2*n3)/(2/3*n3) = 3
times faster, but communication
makes it a little slower.
CS267 Dense Linear Algebra I.37
Demmel Fa 2001
CS267 Dense Linear Algebra I.38
Demmel Fa 2001
CS267 Dense Linear Algebra I.39
Demmel Fa 2001
Scales well,
nearly full machine speed
CS267 Dense Linear Algebra I.40
Demmel Fa 2001
Old version,
pre 1998 Gordon Bell Prize
Still have ideas to accelerate
Project Available!
Old Algorithm,
plan to abandon
CS267 Dense Linear Algebra I.41
Demmel Fa 2001
The “Holy Grail” of Eigensolvers for Symmetric matrices
To be propagated throughout LAPACK and ScaLAPACK
CS267 Dense Linear Algebra I.42
Demmel Fa 2001
Have good ideas to speedup
Project available!
Hardest of all to parallelize
CS267 Dense Linear Algebra I.43
Demmel Fa 2001
Out-of-core means
matrix lives on disk;
too big for main mem
Much harder to hide
latency of disk
QR much easier than LU
because no pivoting
needed for QR
CS267 Dense Linear Algebra I.44
Demmel Fa 2001
A small software project ...
CS267 Dense Linear Algebra I.45
Demmel Fa 2001
Extra Slides
CS267 Dense Linear Algebra I.46
Demmel Fa 2001
Parallelizing Gaussian Elimination
° Recall parallelization steps from Lecture 3
• Decomposition: identify enough parallel work, but not too much
• Assignment: load balance work among threads
• Orchestrate: communication and synchronization
• Mapping: which processors execute which threads
° Decomposition
• In BLAS 2 algorithm nearly each flop in inner loop can be done in
parallel, so with n2 processors, need 3n parallel steps
for i = 1 to n-1
A(i+1:n,i) = A(i+1:n,i) / A(i,i)
… BLAS 1 (scale a vector)
A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) … BLAS 2 (rank-1 update)
- A(i+1:n , i) * A(i , i+1:n)
• This is too fine-grained, prefer calls to local matmuls instead
CS267 Dense Linear Algebra I.47
Demmel Fa 2001
Assignment of parallel work in GE
° Think of assigning submatrices to threads, where
each thread responsible for updating submatrix it
owns
• “owner computes” rule natural because of locality
° What should submatrices look like to achieve load
balance?
CS267 Dense Linear Algebra I.48
Demmel Fa 2001
Different Data Layouts for Parallel GE (on 4 procs)
Bad load balance:
P0 idle after first
n/4 steps
Can trade load balance
and BLAS2/3
performance by
choosing b, but
factorization of block
column is a bottleneck
Load balanced, but can’t easily
use BLAS2 or BLAS3
The winner!
Complicated addressing
CS267 Dense Linear Algebra I.49
Demmel Fa 2001
Computational Electromagnetics (MOM)
The main steps in the solution process are
Fill:
computing the matrix elements of A
Factor:
factoring the dense matrix A
Solve:
solving for one or more excitations b
Field Calc: computing the fields scattered from the objec
CS267 Dense Linear Algebra I.50
Demmel Fa 2001
Analysis of MOM for Parallel Implementation
Task
Work
Parallelism
Parallel Speed
Fill
O(n**2)
embarrassing
Factor
O(n**3)
moderately diff.
very high
Solve
O(n**2)
moderately diff.
high
O(n)
embarrassing
high
Field Calc.
CS267 Dense Linear Algebra I.51
low
Demmel Fa 2001
BLAS 3 (Blocked) GEPP, using Delayed Updates
for ib = 1 to n-1 step b … Process matrix b columns at a time
end = ib + b-1
… Point to end of block of b columns
apply BLAS2 version of GEPP to get A(ib:n , ib:end) = P’ * L’ * U’
… let LL denote the strict lower triangular part of A(ib:end , ib:end) + I
A(ib:end , end+1:n) = LL-1 * A(ib:end , end+1:n)
… update next b rows of U
A(end+1:n , end+1:n ) = A(end+1:n , end+1:n )
BLAS 3
- A(end+1:n , ib:end) * A(ib:end , end+1:n)
… apply delayed updates with single matrix-multiply
… with inner dimension b
CS267 Dense Linear Algebra I.52
Demmel Fa 2001
BLAS2 version of Gaussian Elimination with Partial Pivoting (GEPP)
for i = 1 to n-1
find and record k where |A(k,i)| = max{i <= j <= n} |A(j,i)|
… i.e. largest entry in rest of column i
if |A(k,i)| = 0
exit with a warning that A is singular, or nearly so
elseif k != i
swap rows i and k of A
end if
A(i+1:n,i) = A(i+1:n,i) / A(i,i)
… each quotient lies in [-1,1]
… BLAS 1
A(i+1:n,i+1:n) = A(i+1:n , i+1:n ) - A(i+1:n , i) * A(i , i+1:n)
… BLAS 2, most work in this line
CS267 Dense Linear Algebra I.53
Demmel Fa 2001
How to proceed:
° Consider basic parallel matrix multiplication
algorithms on simple layouts
• Performance modeling to choose best one
- Time (message) = latency + #words * time-per-word
= a + n*b
° Briefly discuss block-cyclic layout
° PBLAS = Parallel BLAS
CS267 Dense Linear Algebra I.54
Demmel Fa 2001
Parallel Matrix Multiply
° Computing C=C+A*B
° Using basic algorithm: 2*n3 Flops
° Variables are:
• Data layout
• Topology of machine
• Scheduling communication
° Use of performance models for algorithm design
CS267 Dense Linear Algebra I.55
Demmel Fa 2001
1D Layout
° Assume matrices are n x n and n is divisible by p
p0 p1 p2 p3 p4 p5 p6 p7
° A(i) refers to the n by n/p block column that
processor i owns (similiarly for B(i) and C(i))
° B(i,j) is the n/p by n/p sublock of B(i)
• in rows j*n/p through (j+1)*n/p
° Algorithm uses the formula
C(i) = C(i) + A*B(i) = C(i) + S A(j)*B(j,i)
j
CS267 Dense Linear Algebra I.56
Demmel Fa 2001
Matrix Multiply: 1D Layout on Bus or Ring
° Algorithm uses the formula
C(i) = C(i) + A*B(i) = C(i) + S A(j)*B(j,i)
j
° First consider a bus-connected machine without
broadcast: only one pair of processors can
communicate at a time (ethernet)
° Second consider a machine with processors on a
ring: all processors may communicate with nearest
neighbors simultaneously
CS267 Dense Linear Algebra I.57
Demmel Fa 2001
Naïve MatMul for 1D layout on Bus without Broadcast
Naïve algorithm:
C(myproc) = C(myproc) + A(myproc)*B(myproc,myproc)
for i = 0 to p-1
for j = 0 to p-1 except i
if (myproc == i) send A(i) to processor j
if (myproc == j)
receive A(i) from processor i
C(myproc) = C(myproc) + A(i)*B(i,myproc)
barrier
Cost of inner loop:
computation: 2*n*(n/p)2 = 2*n3/p2
communication: a + b*n2 /p
CS267 Dense Linear Algebra I.58
Demmel Fa 2001
Naïve MatMul (continued)
Cost of inner loop:
computation: 2*n*(n/p)2 = 2*n3/p2
communication: a + b*n2 /p
… approximately
Only 1 pair of processors (i and j) are active on any iteration,
an of those, only i is doing computation
=> the algorithm is almost entirely serial
Running time: (p*(p-1) + 1)*computation + p*(p-1)*communication
~= 2*n3 + p2*a + p*n2*b
this is worse than the serial time and grows with p
CS267 Dense Linear Algebra I.59
Demmel Fa 2001
Better Matmul for 1D layout on a Processor Ring
° Proc
i can communicate with Proc(i-1) and Proc(i+1) simultaneously for all i
Copy A(myproc) into Tmp
C(myproc) = C(myproc) + T*B(myproc , myproc)
for j = 1 to p-1
Send Tmp to processor myproc+1 mod p
Receive Tmp from processor myproc-1 mod p
C(myproc) = C(myproc) + Tmp*B( myproc-j mod p , myproc)
° Same idea as for gravity in simple sharks and fish algorithm
° Time of inner loop = 2*(a + b*n2/p) + 2*n*(n/p)2
° Total Time = 2*n* (n/p)2 + (p-1) * Time of inner loop
~ 2*n3/p + 2*p*a + 2*b*n2
° Optimal for 1D layout on Ring or Bus, even with with Broadcast:
Perfect speedup for arithmetic
A(myproc) must move to each other processor, costs at least
(p-1)*cost of sending n*(n/p) words
° Parallel Efficiency = 2*n3 / (p * Total Time) = 1/(1 + a * p2/(2*n3) + b * p/(2*n) )
= 1/ (1 + O(p/n))
Grows to 1 as n/p increases (or a and b shrink)
CS267 Dense Linear Algebra I.60
Demmel Fa 2001
MatMul with 2D Layout
° Consider processors in 2D grid (physical or logical)
° Processors can communicate with 4 nearest
neighbors
• Broadcast along rows and columns
p(0,0)
p(0,1)
p(0,2)
p(1,0)
p(1,1)
p(1,2)
p(2,0)
p(2,1)
p(2,2)
CS267 Dense Linear Algebra I.61
Demmel Fa 2001
Cannon’s Algorithm
… C(i,j) = C(i,j) + S A(i,k)*B(k,j)
k
… assume s = sqrt(p) is an integer
forall i=0 to s-1
… “skew” A
left-circular-shift row i of A by i
… so that A(i,j) overwritten by A(i,(j+i)mod s)
forall i=0 to s-1
… “skew” B
up-circular-shift B column i of B by i
… so that B(i,j) overwritten by B((i+j)mod s), j)
for k=0 to s-1
… sequential
forall i=0 to s-1 and j=0 to s-1
… all processors in parallel
C(i,j) = C(i,j) + A(i,j)*B(i,j)
left-circular-shift each row of A by 1
up-circular-shift each row of B by 1
CS267 Dense Linear Algebra I.62
Demmel Fa 2001
Communication in Cannon
C(1,2) = A(1,0) * B(0,2) + A(1,1) * B(1,2) + A(1,2) * B(2,2)
CS267 Dense Linear Algebra I.63
Demmel Fa 2001
Cost of Cannon’s Algorithm
forall i=0 to s-1
… recall s = sqrt(p)
left-circular-shift row i of A by i
… cost = s*(a + b*n2/p)
forall i=0 to s-1
up-circular-shift B column i of B by i … cost = s*(a + b*n2/p)
for k=0 to s-1
forall i=0 to s-1 and j=0 to s-1
C(i,j) = C(i,j) + A(i,j)*B(i,j) … cost = 2*(n/s)3 = 2*n3/p3/2
left-circular-shift each row of A by 1 … cost = a + b*n2/p
up-circular-shift each row of B by 1
… cost = a + b*n2/p
° Total Time = 2*n3/p + 4* s*a + 4*b*n2/s
° Parallel Efficiency = 2*n3 / (p * Total Time)
= 1/( 1 + a * 2*(s/n)3 + b * 2*(s/n) )
= 1/(1 + O(sqrt(p)/n))
° Grows to 1 as n/s = n/sqrt(p) = sqrt(data per processor) grows
° Better than 1D layout, which had Efficiency = 1/(1 + O(p/n))
CS267 Dense Linear Algebra I.64
Demmel Fa 2001
Drawbacks to Cannon
° Hard to generalize for
• p not a perfect square
•
•
•
•
A and B not square
Dimensions of A, B not perfectly divisible by s=sqrt(p)
A and B not “aligned” in the way they are stored on processors
block-cyclic layouts
° Memory hog (extra copies of local matrices)
CS267 Dense Linear Algebra I.65
Demmel Fa 2001
SUMMA = Scalable Universal Matrix Multiply Algorithm
° Slightly less efficient, but simpler and easier to
generalize
° Presentation from van de Geijn and Watts
• www.netlib.org/lapack/lawns/lawn96.ps
• Similar ideas appeared many times
° Used in practice in PBLAS = Parallel BLAS
• www.netlib.org/lapack/lawns/lawn100.ps
CS267 Dense Linear Algebra I.66
Demmel Fa 2001
SUMMA
k
J
B(k,J)
k
*
I
=
C(I,J)
A(I,k)
I, J represent all rows, columns owned by a processor
° k is a single row or column (or a block of b rows or columns)
° C(I,J) = C(I,J) + Sk A(I,k)*B(k,J)
° Assume a pr by pc processor grid (pr = pc = 4 above)
For k=0 to n-1 … or n/b-1 where b is the block size
… = # cols in A(I,k) and # rows in B(k,J)
for all I = 1 to pr … in parallel
owner of A(I,k) broadcasts it to whole processor row
for all J = 1 to pc … in parallel
owner of B(k,J) broadcasts it to whole processor column
Receive A(I,k) into Acol
Receive B(k,J) into Brow
C( myproc , myproc ) = C( myproc , myproc) + Acol * Brow
°
CS267 Dense Linear Algebra I.67
Demmel Fa 2001
SUMMA performance
For k=0 to n/b-1
for all I = 1 to s … s = sqrt(p)
owner of A(I,k) broadcasts it to whole processor row
… time = log s *( a + b * b*n/s), using a tree
for all J = 1 to s
owner of B(k,J) broadcasts it to whole processor column
… time = log s *( a + b * b*n/s), using a tree
Receive A(I,k) into Acol
Receive B(k,J) into Brow
C( myproc , myproc ) = C( myproc , myproc) + Acol * Brow
… time = 2*(n/s)2*b
° Total time = 2*n3/p + a * log p * n/b + b * log p * n2 /s
° Parallel Efficiency = 1/(1 + a * log p * p / (2*b*n2) + b * log p * s/(2*n) )
° ~Same b term as Cannon, except for log p factor
log p grows slowly so this is ok
° Latency (a) term can be larger, depending on b
When b=1, get a * log p * n
As b grows to n/s, term shrinks to a * log p * s (log p times Cannon)
° Temporary storage grows like 2*b*n/s
° Can change b to tradeoff latency cost with memory
CS267 Dense Linear Algebra I.68
Demmel Fa 2001
Summary of Parallel Matrix Multiply Algorithms
° 1D Layout
• Bus without broadcast - slower than serial
• Nearest neighbor communication on a ring (or bus with
broadcast): Efficiency = 1/(1 + O(p/n))
° 2D Layout
• Cannon
- Efficiency = 1/(1+O(p1/2 /n))
- Hard to generalize for general p, n, block cyclic, alignment
• SUMMA
-
Efficiency = 1/(1 + O(log p * p / (b*n2) + log p * p1/2 /n))
- Very General
- b small => less memory, lower efficiency
- b large => more memory, high efficiency
• Gustavson et al
- Efficiency = 1/(1 + O(p1/3 /n) ) ??
CS267 Dense Linear Algebra I.69
Demmel Fa 2001