2/19/97 Horst D. Simon cs.berkeley/cs267

H. Simon - CS267 - L8 04/20/23 1

CS 267 Applications of Parallel Processors

Lecture 9: Computational Electromagnetics -Large Dense Linear Systems

2/19/97

Horst D. Simon

http://www.cs.berkeley.edu/cs267

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Outline - Lecture 9

- Computational Electromagnetics

- Sources of large dense linear systems

- Review of solution of linear systems with

Gaussian elimination

- BLAS and memory hierarchy for linear algebra kernels

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Outline - Lecture 10

- Layout of matrices on distributed memory machines

- Distributed Gaussian elimination

- Speeding up with advanced algorithms

- LINPACK and LAPACK

- LINPACK benchmark

- Tflops result

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- Designing portable libraries for parallel machines

- BLACS

- ScaLAPACK for dense linear systems

- other linear algebra algorithms in ScaLAPACK

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Computational Electromagnetics

- 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), and finite differences (time domain)

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Computational Electromagnetics

image: NW Univ. Comp. Electromagnetics Laboratory http://nueml.ece.nwu.edu/

- 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

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Computational Electromagnetics (MOM)

After discretization the integral equation has the

form Z J = V

where

Z is the impedance matrix, J is the unknown vector of amplitudes, and V is the excitation vector.

Z is given as a four dimensional integral.

(see Cwik, Patterson, and Scott, Electromagnetic Scattering on the Intel Touchstone Delta, IEEE Supercomputing ‘92, pp 538 - 542)

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The main steps in the solution process are

A) computing the matrix elements

B) factoring the dense matrix

C) solving for one or more excitations

D) computing the fields scattered from the object

Computational Electromagnetics (MOM)

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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

For most scientific applications the biggest gain in performance can be obtained by focusing on one tasks.

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Results for Parallel Implementation on Delta

Task Time (hours) Performance (Gflop/s)

Fill 9.20 ~ 1.0

Factor 8.25 10.35

Solve 2.17 -

Field Calc. 0.12 3.0

The problem solved was for a matrix of size 48,672. (The world record in 1991.)

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Current Records for Solving Dense Systems

Year System Size Machine

1950's O(100) 1991 55,296 CM-2 1992 75,264 Intel 1993 75,264 Intel 1994 76,800 CM-5 1995 128,600 Paragon XP1996 215,000 ASCI Red

source: Alan Edelman http://www-math.mit.edu/~edelman/records.html

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Sources for large dense linear systems

- not many outside CEM

- even within CEM community alternatives such FD-TD are heavily debated

In many instances choices for algorithms or methods in existing scientific codes or applications are not the resultof careful planning and design. At best they are reflecting the start-of-the-art at the time, at worst they are purelycoincidental.

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Review of Gaussian Elimination

see Demmel http://HTTP.CS.Berkeley.EDU/~demmel/cs267/lecture12/lecture12.html

Gaussian elimination to solve Ax=b - start with a dense matrix - add multiples of each row to subsequent rows in order to create zeros below the diagonal- ending up with an upper triangular matrix U. Solve a linear system with U by substitution, startingwith the last variable.

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... 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 ... 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)

Review of Gaussian Elimination (cont.)

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... 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 ... 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)


= m

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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)

avoid computation of known matrix entry

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It will be convenient to store the multipliers m in the implicitly created zeros below the diagonal, so we can use them later to transform the right hand side b:

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)

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Now we use Matlab (data parallel) notation to express

the algorithm even more compactly:

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)

The inner loop consists of one vector operation, and one matrix-vector operation.Note that the loop looks elegant, but no longer intuitive.

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Lemma. (LU Factorization). If the above algorithm terminates (i.e. it did not try to divide by zero) then A = L*U.

Now we can state our complete algorithm for solving A*x=b: 1) Factorize A = L*U. 2) Solve L*y = b for y by forward substitution. 3) Solve U*x = y for x by backward substitution.

Then x is the solution we seek because A*x = L*(U*x) = L*y = b.

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Here are some obvious problems with this algorithm, which we need to address:

- If A(i,i) is zero, the algorithm cannot proceed. If A(i,i) is tiny, we will also have numerical problems. - The majority of the work is done by a rank-one update, which does not exploit a memory hierarchy as well as an operation like matrix-matrix multiplication


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Pivoting for Small A(i,i)

Why pivoting is needed?

A= [ 0 1 ] [ 1 0 ]

Even if A(i,i) is tiny, but not zero difficulties can arise(see example in Jim Demmel’s lecture notes).

This problem is resolved by partial pivoting.

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Partial Pivoting

Reordering the rows of A so that A(i,i) is large at each step of the algorithm.At step i of the algorithm, row i is swappedwith row k>i if |A(k,i)| is the largest entry among |A(i:n,i)|.

for i = 1 to n-1 find and record k where |A(k,i)| = max_{i<=j<=n} |A(j,i)| if |A(k,i)|=0, exit with a warning that A is singular, or nearly so if i != k, swap rows i and k of A 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)

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Partial Pivoting (cont.)

- for 2-by-2 example, we get a very accurate answer- several choices as to when to swap rows i and k- could use indirect addressing and not swap them at all, but this would be slow- keep permutation, then solving A*x=b only requires the additional step of permuting b

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Fast linear algebra kernels: BLAS

- Simple linear algebra kernels such as matrix-matrixmultiply (exercise) can be performed fast on memoryhierarchies.- More complicated algorithms can be built from somevery basic building blocks and kernels.- The interfaces of these kernels have been standardized as the Basic Linear Algebra Subroutinesor BLAS. - Early agreement on standard interface (around 1980) led to portable libraries for vector and shared memory parallel machines.- BLAS are classified into three categories, level 1,2,3

see Demmel http://HTTP.CS.Berkeley.EDU/~demmel/cs267/lecture02.html

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Level 1 BLAS

Operate mostly on vectors (1D arrays), or pairs of vectors; perform O(n) operations; return either a vector or a scalar. Examples saxpy y(i) = a * x(i) + y(i), for i=1 to n. Saxpy is an acronym for the operation. S stands for single precision, daxpy is for double precision, caxpy for complex, and zaxpy for double complex, sscal y = a * x, srot replaces vectors x and y by c*x+s*y and -s*x+c*y, where c and s are typically a cosine and sine. sdot computes s = sum_{i=1}^n x(i)*y(i)

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Level 2 BLAS

operate mostly on a matrix (2D array) and a vector; return a matrix or a vector; O(n^2) operations.Examples.

sgemv Matrix-vector multiplication computes y = y + A*x where A is m-by-n, x is n-by-1 and y is m-by-1.

sger rank-one update computes A = A + y*x', where A is m-by-n, y is m-by-1, x is n-by-1, x' is the transpose of x. This is a short way of saying A(i,j) = A(i,j) + y(i)*x(j) for all i,j. strsv triangular solve solves y=T*x for x, where T is a triangular matrix.

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Level 3 BLAS

operate on pairs or triples of matrices, returning a matrix;complexity is O(n**3).

Examples sgemm Matrix-matrix multiplication computes C = C + A*B, where C is m-by-n, A is m-by-k, and B is k-by-n

sgtrsm multiple triangular solve solves Y = T*X for X, where T is a triangular matrix, and X is a rectangular matrix.

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Performance of BLAS

Level 2

Level 3

Level 1

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Performance of BLAS (cont.)

- BLAS are specially optimized by the vendor (IBM)to take advantage of all features of the RS 6000/590.- Potentially a big speed advantage if an algorithm can be expressed in terms of the BLAS3 instead ofBLAS2 or BLAS1. - The top speed of the BLAS3, about 250 Mflops, is veryclose to the peak machine speed of 266 Mflops.- We will reorganize algorithms, like Gaussian elimination, so that they use BLAS3 rather than BLAS1 or BLAS2.

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Explanation of Performance of BLAS

m = number of memory references to slow memory (read + write) f = number of floating point operations q = f/m = average number of flops per slow memory reference

m justification for m f q

saxpy 3*n read x(i), y(i) ; write y(i) 2*n 2/3 sgemv n^2+O(n) read each A(i,j) once 2*n^2 2sgemm 4*n^2 read A(i,j),B(i,j),C(i,j) 2*n^3 n/2 write C(i,j) once

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CS 267 Applications of Parallel Processors

Lecture 10: Large Dense Linear Systems -Distributed Implementations

2/21/97

Horst D. Simon

http://www.cs.berkeley.edu/cs267

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Review - Lecture 9

- computational electromagnetics and linear systems

- rewritten Gaussian elimination as vector and matrix-vector operation (level 2 BLAS)

- discussed the efficiency of level 3 BLAS in terms of reducing number of memory accesses

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- Layout of matrices on distributed memory machines

- Distributed Gaussian elimination

- Speeding up with advanced algorithms

- LINPACK and LAPACK

- LINPACK benchmark

- Tflops result

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Review of Gaussian Elimination

Now we use Matlab (data parallel) notation to express

the algorithm even more compactly:

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)

The inner loop consists of one vector operation, and one matrix-vector operation.Note that the loop looks elegant, but no longer intuitive.

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Partial Pivoting

Reordering the rows of A so that A(i,i) is large at each step of the algorithm.At step i of the algorithm, row i is swappedwith row k>i if |A(k,i)| is the largest entry among |A(i:n,i)|.

for i = 1 to n-1 find and record k where |A(k,i)| = max_{i<=j<=n} |A(j,i)| if |A(k,i)|=0, exit with a warning that A is singular, or nearly so if i != k, swap rows i and k of A 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)

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How to Use Level 3 BLAS ?

The current algorithm only uses level 1 and level 2 BLAS.

Want to use level 3 BLAS because of higher performance.

The standard technique is called blocking or delayed updating.

We want to save up a sequence of level 2 operations and do them all at once.

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How to Use Level 3 BLAS in LU Decomposition

- process the matrix in blocks of b columns at a time

- b is called the block size.

- do a complete LU decomposition just of the b columns in the current block, essentially using the above BLAS2code.

- then update the remainder of the matrix doing b rank-one updates all at once, which turns out to be a single matrix-matrix multiplication of size b

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Block GE with Level 3 BLAS

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Block GE with Level 3 BLAS

Gaussian elimination with Partial Pivoting, BLAS3 implementation

... process matrix b columns at a time for ib = 1 to n-1 step b ... point to end of block of b columns end = min(ib+b-1,n)

... LU factorize A(ib:n,ib:end) with BLAS2 for i = ib to end find and record k where |A(k,i)| = max_{i<=j<=n} |A(j,i)| if |A(k,i)|=0, exit with a warning that A is singular, or nearly so if i != k, swap rows i and k of A A(i+1:n, i) = A(i+1:n, i) / A(i,i) ... only update columns i+1 to end A(i+1:n, i+1:end) = A(i+1:n, i+1:end) - A(i+1:n, i)*A(i, i+1:end) endfor

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Block GE with Level 3 BLAS (cont.)

... Let LL be the b-by-b lower triangular ... matrix whose subdiagonal entries are ... stored in A(ib:end,ib:end), and with ... 1s on the diagonal. Do delayed update ... of A(ib:end, end+1:n) by solving ... n-end triangular systems ... (A(ib:end, end+1:n) is pink below) A(ib:end, end+1:n) = LL \ A(ib:end, end+1:n)

... do delayed update of rest of matrix ... using matrix-matrix multiplication ... (A(end+1:n, end+1:n) is green below) ... (A(end+1:n, ib:end) is blue below) 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)

endfor

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Block GE with Level 3 BLAS (cont.)

- LU factorization of A(ib:n,ib:end) uses the same algorithm as before (level 2 BLAS) - Solving a system of n-end equations with triangular coefficient matrix LL is a single call to a BLAS3 subroutine(strsm) designed for that purpose. - No work or data motion is required to refer to LL; done with a pointer. - When n>>b, almost all the work is done in the final line,which multiplies an (n-end)-by-b matrix times a b-by-(n-end) matrix in a single BLAS3 call (to sgemm).

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How to select b?

b will be chosen in a machine dependent way to maximize performance. A good value of bwill have the following properties:

- b is small enough so that the b columns currently being LU-factorized fit in the fast memory (cache, say) of the machine.

- b is large enough to make matrix-matrix multiplication fast.

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LINPACK - LAPACK -ScaLAPACK

LINPACK - linear systems, least squares problems level 1 BLAS - late 70s

LAPACK - redesigned LINPACK to include eigenvalue software, level 3 BLAS for parallel and shared memory parallel machines - late 80s

ScaLAPACK - scaleable LAPACK based on BLACS for communication, distributed memory machine - mid 90s

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Efficiency on Cray C90

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Comparison of Different Machines

Machine #Procs Clock Peak Block Speed Mflops Size b (MHz)---------------------------------------------------------------------Convex C4640 1 135 810 64Convex C4640 4 135 3240 64Cray C90 1 240 952 128Cray C90 16 240 15238 128DEC Alpha 3000-500X 1 200 200 32IBM RS 6000/590 1 66 264 64SGI Power Challenge 1 75 300 64

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Efficiency of LAPACK LU, for n=1000

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LU factorization is almost as efficient as matrix-matrixmultiply for most machines, except on C90 (16 processors). (why?)

LAPACK - LU is almost as good as best vendor effort.Trade-off between performance and portability.

Vendors place a premium on LU performance - why?

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LINPACK Benchmark

- named after the LINPACK package - originally consisted of timings for 100-by-100 matrices; no vendor optimization(code changes) permitted- interesting historical record, with literally everymachine for the last 2 decades listed in decreasing order of speed, from the largest supercomputers to a hand-held calculator. - as machines grew faster 1000-by-1000 matrices wereintroduced (all code changes allowed). - a third benchmark was added for large parallel machines, which measured their speed on the largest linear system that would fit in memory, as well as the size of the system required to get half the Mflop rate ofthe largest matrix.

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Computer Num_Procs Rmax(GFlops) Nmax(order) N1/2(order) Rpeak(GFlops)--------------------------------------------- --------- ------------ ------------ ------------ -------------Intel ASCI Option Red (200 MHz Pentium Pro) 7264 1068. 215000 53400 1453CP-PACS* (150 MHz PA-RISC based CPU) 2048 368.2 103680 30720 614Intel Paragon XP/S MP (50 MHz OS=SUNMOS) 6768 281.1 128600 25700 338Intel Paragon XP/S MP (50 MHz OS=SUNMOS) 6144 256.2 122500 24300 307Numerical Wind Tunnel* (9.5 ns) 167 229.7 66132 18018 281Intel Paragon XP/S MP (50 MHz OS=SUNMOS) 5376 223.6 114500 22900 269HITACHI SR2201/1024(150MHz) 1024 220.4 138240 34560 307Fujitsu VPP500/153(10nsec) 153 200.6 62730 17000 245 Numerical Wind Tunnel* (9.5 ns) 140 195.0 60480 15730 236Intel Paragon XP/S MP (50 MHz OS=SUNMOS) 4608 191.5 106000 21000 230Numerical Wind Tunnel* (9.5 ns) 128 179.2 56832 14800 216

LINPACK Benchmark

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Data Layouts for Distributed Memory Machines

The two main issues in choosing a data layout for Gaussian elimination are

1) load balance, or splitting the work reasonably evenly among the processors 2)ability to use the BLAS3 during computations on a single processor, to account for the memory hierarchy on each processor.

Several layouts will be discussed here. All these are partof HPF. Solving linear systems served as a prototype forthese designs.

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Gaussian Elimination using BLAS 3

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Column Blocked

column i is stored on processor floor(i/c) where c=ceiling(n/p) is the maximum number of columns stored per processor.

does not permit good load balancing.

after c columns have been computedprocessor 0 is idle

row blocked has similar problemn=16 and p=4.

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Column Cyclic

each processor owns approximately 1/p-th of the square southeast corner of the matrix

good load balance

single columns are stored rather than blocks means we cannot usethe BLAS3 to update

transpose of this layout, the Row Cyclic Layout, has a similarproblem.

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Column Block Cyclic

choose a block size b, divide the columns into groups of size b, distribute these groups cyclically

for b >1, slightly worse balance than the Column Cyclic Layout;can use the BLAS2 and BLAS3

b < c, better load balancethan the Columns Blocked Layout, but can only call the BLAS on smaller subproblems, take less advantage of the local memory hierarchy

disadvantage that the factorization of A(ib:n,ib:end) will take place on perhaps juston one processor; possible serial bottleneck.

n=16, p=4 and b=2b not necessarilyBLAS3 block size

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Row and Column Block Cyclic

processors and matrix blocksare distributed in a 2d array

pcol-fold parallelismin 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 andcolumns

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Skewered Block

each row and each column is shared among all p processors

so p-fold parallelism is available for any row operation or any column operation

in contrast, the 2D block cyclic layoutcan have at most sqrt(p)-fold parallelism in all the rows and all the columns not useful for Gaussian elimination, but in a variety of other matrixoperations

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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

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Distributed GE with a 2D Block Cyclic Layout

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ScaLAPACK LU Performance Results

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Teraflop/s Performance Result

“Sorry for the delay in responding. The system had about 7000 200Mhz Pentium Pro Processors. It solved a 64bit real matrix of size 216000. It did not use Strassen. The algorithm was basically the same that Robert van de Geijn used on the Delta years ago. It does a 2D block cyclic map of the matrix and requires a power of 2 number of nodes in the vertical direction. The basicblock size was 64x64. A custom dual processor matrix multiply was written for the DGEMM call. It took a little less than 2 hours to run.”

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2/19/97 Horst D. Simon cs.berkeley/cs267