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CS 267 Applications of Parallel Computers Dense Linear Algebra

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### CS 267 Applications of Parallel ComputersDense Linear Algebra

James Demmel

http://www.cs.berkeley.edu/~demmel/cs267_221001.ppt

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

Motivation

- 3 Basic Linear Algebra Problems
- Linear Equations: Solve Ax=b for x
- Least Squares: Find x that minimizes S ri2where r=Ax-b
- Eigenvalues: Findland 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

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

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/

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)

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.

Current Records for Solving Dense Systems

www.netlib.org, click on Performance DataBase Server

Year System Size Machine # Procs Gflops (Peak)

1950\'s O(100)

1995 128,600 Intel Paragon 6768 281 ( 338)

1996 215,000 Intel ASCI Red 7264 1068 (1453)

1998 148,000 Cray T3E 1488 1127 (1786)

1998 235,000 Intel ASCI Red 9152 1338 (1830)

(200 MHz Ppro)

1999 374,000 SGI ASCI Blue 5040 1608 (2520)

2000 362,000 Intel ASCI Red 9632 2380 (3207)

(333 MHz Xeon)

2001518,000IBM ASCI White80007226(12000)

(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

Current Records for Solving Small Dense Systems

www.netlib.org, click on Performance DataBase Server

Megaflops

Machine n=100 n=1000 Peak

Fujitsu VPP 5000 1156 8784 9600

(1 proc 300 MHz)

Cray T90

(32 proc, 450 MHz) 29360 57600

(4 proc, 450 MHz) 1129 5735 7200

IBM Power 4

(1 proc, 1300 MHz) 1074 2394 5200

…

Dell Itanium

(4 proc, 800 MHz) 7358 12800

(2 proc, 800 MHz) 4504 6400

(1 proc, 800 MHz) 600 2382 3200

source: LINPACK Benchmark: http://www.netlib.org/performance/html/PDSreports.html

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...

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)

Refine GE Algorithm (1)

- Initial Version
- Remove computation of constant A(j,i)/A(i,i) from inner loop

… 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)

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)

Refine GE Algorithm (2)

- Last version
- 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 to n

A(j,k) = A(j,k) - m * A(i,k)

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)

Refine GE Algorithm (3)

- Last version
- Store multipliers m below diagonal in zeroed entries for later use

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)

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)

Refine GE Algorithm (4)

- Last version

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)

- Split Loop

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)

Refine GE Algorithm (5)

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)

- Last version
- 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)

What GE really computes

- 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

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)

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

Pivoting in Gaussian Elimination

- A = [ 0 1 ] fails completely, even though A is “easy”
- [ 1 0 ]
- Illustrate problems in 3-decimal digit arithmetic:
- A = [ 1e-4 1 ] and b = [ 1 ], correct answer to 3 places is x = [ 1 ]
- [ 1 1 ] [ 2 ] [ 1 ]
- Result of LU decomposition is
- L = [ 1 0 ] = [ 1 0 ] … No roundoff error yet
- [ fl(1/1e-4) 1 ] [ 1e4 1 ]
- U = [ 1e-4 1 ] = [ 1e-4 1 ] … Error in 4th decimal place
- [ 0 fl(1-1e4*1) ] [ 0 -1e4 ]
- Check if A = L*U = [ 1e-4 1 ] … (2,2) entry entirely wrong
- [ 1 0 ]
- 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

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

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

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.)

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

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/

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)

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
- 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?

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)

Different Data Layouts for Parallel GE (on 4 procs)

Load balanced, but can’t easily

use BLAS2 or BLAS3

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

The winner!

Complicated addressing

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

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 )

- A(end+1:n , ib:end) * A(ib:end , end+1:n)

… apply delayed updates with single matrix-multiply

… with inner dimension b

BLAS 3

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

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

green = green - blue * pink

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.

nearly full machine speed

pre 1998 Gordon Bell Prize

Still have ideas to accelerate

Project Available!

Old Algorithm,

plan to abandon

The “Holy Grail” of Eigensolvers for Symmetric matrices

To be propagated throughout LAPACK and ScaLAPACK

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

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
- This is too fine-grained, prefer calls to local matmuls instead

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)

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?

Different Data Layouts for Parallel GE (on 4 procs)

Load balanced, but can’t easily

use BLAS2 or BLAS3

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

The winner!

Complicated addressing

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 object

Analysis of MOM for Parallel Implementation

Task Work Parallelism Parallel Speed

Fill O(n**2) embarrassing low

Factor O(n**3) moderately diff. very high

Solve O(n**2) moderately diff. high

Field Calc. O(n) embarrassing high

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 )

- A(end+1:n , ib:end) * A(ib:end , end+1:n)

… apply delayed updates with single matrix-multiply

… with inner dimension b

BLAS 3

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

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

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

p1

p2

p3

p4

p5

p6

p7

1D Layout- Assume matrices are n x n and n is divisible by p
- 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

Matrix Multiply: 1D Layout on Bus or Ring

- Algorithm uses the formula

C(i) = C(i) + A*B(i) = C(i) + SA(j)*B(j,i)

- 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

j

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

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

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)

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)

Cannon’s Algorithm

… C(i,j) = C(i,j) + S A(i,k)*B(k,j)

… 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

k

Communication in Cannon

C(1,2) = A(1,0) * B(0,2) + A(1,1) * B(1,2) + A(1,2) * B(2,2)

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))

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)

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

SUMMA

B(k,J)

k

J

k

*

=

C(I,J)

I

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

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

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) ) ??

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