The Future of LAPACK and ScaLAPACK netlib/lapack-dev

The Future of LAPACK and ScaLAPACK

www.netlib.org/lapack-dev

Jim Demmel

UC Berkeley

23 Feb 2007

Outline

• Motivation for new Sca/LAPACK

• Challenges (or research opportunities…)

• Goals of new Sca/LAPACK

• Highlights of progress– With some excursions …

Motivation• LAPACK and ScaLAPACK are widely used

– Adopted by Cray, Fujitsu, HP, IBM, IMSL, MathWorks, NAG, NEC, SGI, …

– >68M web hits @ Netlib (incl. CLAPACK, LAPACK95)• 35K hits/day

• Many ways to improve them, based on– Own algorithmic research– Enthusiastic participation of research community– User/vendor survey– Opportunities and demands of new architectures,

programming languages• New releases planned (NSF support)

Participants• UC Berkeley:

– Jim Demmel, Ming Gu, W. Kahan, Beresford Parlett, Xiaoye Li, Osni Marques, Christof Voemel, David Bindel, Yozo Hida, Jason Riedy, undergrads…

• U Tennessee, Knoxville– Jack Dongarra, Julien Langou, Julie Langou, Piotr Luszczek,

Stan Tomov, Alfredo Buttari, Jakub Kurzak• Other Academic Institutions

– UT Austin, UC Davis, Florida IT, U Kansas, U Maryland, North Carolina SU, San Jose SU, UC Santa Barbara

– TU Berlin, U Electrocomm. (Japan), FU Hagen, U Carlos III Madrid, U Manchester, U Umeå, U Wuppertal, U Zagreb

• Research Institutions– CERFACS, LBL

• Industrial Partners – Cray, HP, Intel, Interactive Supercomputing, MathWorks, NAG, SGI

Challenges: Increasing Parallelism In the Top500:

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64k-128k32k-64k16k-32k8k-16k4k-8k2049-40961025-2048513-1024257-512129-25665-12833-6417-329-165-83-421

In your Laptop (Intel just announced an 80-core, 1 Teraflop chip)

Challenges

• Increasing parallelism• Exponentially growing gaps between

– Floating point time << 1/Memory BW << Memory Latency• Improving 59%/year vs 23%/year vs 5.5%/year

– Floating point time << 1/Network BW << Network Latency• Improving 59%/year vs 26%/year vs 15%/year

• Heterogeneity (performance and semantics)• Asynchrony• Unreliability

What do users want?

• High performance, ease of use, …• Survey results at www.netlib.org/lapack-dev

– Small but interesting sample– What matrix sizes do you care about?

• 1000s: 34%• 10,000s: 26%• 100,000s or 1Ms: 26%

– How many processors, on distributed memory?• >10: 34%, >100: 31%, >1000: 19%

– Do you use more than double precision? • Sometimes or frequently: 16%

– Would Automatic Memory Allocation help?• Very useful: 72%, Not useful: 14%

Goals of next Sca/LAPACK1. Better algorithms

– Faster, more accurate

2. Expand contents – More functions, more parallel implementations

3. Automate performance tuning

4. Improve ease of use

5. Better software engineering

6. Increased community involvement

Goal 2 – Expanded Content

• Make content of ScaLAPACK mirror LAPACK as much as possible (possible class projects)

Missing Routines in Sca/LAPACKLAPACK ScaLAPACK

Linear Equations

LU

LU + iterative refine

Cholesky

LDLT

xGESV

xGESVX

xPOSV

xSYSV

PxGESV

missing

PxPOSV

missing

Least Squares (LS)

QR

QR+pivot

SVD/QR

SVD/D&C

SVD/MRRR

QR + iterative refine.

xGELS

xGELSY

xGELSS

xGELSD

missing

missing

PxGELS

missing

missing

missing (intent?)

missing

missing

Generalized LS LS + equality constr.

Generalized LM

Above + Iterative ref.

xGGLSE

xGGGLM

missing

missing

missing

missing

More missing routinesLAPACK ScaLAPACK

Symmetric EVD QR / Bisection+Invit

D&C

MRRR

xSYEV / X

xSYEVD

xSYEVR

PxSYEV / X

PxSYEVD

missing

Nonsymmetric EVD Schur form

Vectors too

xGEES / X

xGEEV /X

missing (driver)

missing

SVD QR

D&C

MRRR

Jacobi

xGESVD

xGESDD

missing

missing

PxGESVD

missing (intent?)

missing

missing

Generalized Symmetric EVD

QR / Bisection+Invit

D&C

MRRR

xSYGV / X

xSYGVD

missing

PxSYGV / X

missing (intent?)

missing

Generalized Nonsymmetric EVD

Schur form

Vectors too

xGGES / X

xGGEV / X

missing

missing

Generalized SVD Kogbetliantz

MRRR

xGGSVD

missing

missing (intent)

missing

Goal 1: Better Algorithms

• Faster– But provide “usual” accuracy, stability– … Or accurate for an important subclass

• More accurate– But provide “usual” speed– … Or at any cost

Goal 1a – Faster Algorithms (Highlights)

• MRRR algorithm for symmetric eigenproblem / SVD: – Parlett / Dhillon / Voemel / Marques / Willems

• Up to 10x faster HQR:– Byers / Mathias / Braman

• Faster Hessenberg, tridiagonal, bidiagonal reductions: – van de Geijn/Quintana, Howell / Fulton, Bischof / Lang

• Extensions to QZ: – Kågström / Kressner

• Recursive blocked layouts for packed formats: – Gustavson / Kågström / Elmroth / Jonsson/


• MRRR algorithm for symmetric eigenproblem / SVD: – Parlett / Dhillon / Voemel / Marques / Willems– Faster and more accurate than previous algorithms– SIAM SIAG/LA Prize in 2006– New sequential, first parallel versions out in 2006

Flop Counts of Eigensolvers(2.2 GHz Opteron + ACML)

Flop Count Ratios of Eigensolvers(2.2 GHz Opteron + ACML)

Run Time Ratios of Eigensolvers(2.2 GHz Opteron + ACML)

MFlop Rates of Eigensolvers(2.2 GHz Opteron + ACML)

Exploiting GPUs• Numerous emerging co-processors

– Cell, SSE, Grape, GPU, “physics coprocessor,” …

• When can we exploit them?– LIttle help if memory is bottleneck– Various attempts to use GPUs for dense linear algebra

• Bisection on GPUs for symmetric tridiagonal eigenproblem– Evaluate Count(x) = #(evals < x) for many x– Very little memory traffic– Speedups up to 100x (Volkov)

• 43 Gflops on ATI Radeon X1900 vs running on 2.8 GHz Pentium 4• Overall eigenvalue solver 6.8x faster

Parallel Runtimes of Eigensolvers(2.4 GHz Xeon Cluster + Ethernet)

Goal 1b – More Accurate Algorithms• Iterative refinement for Ax=b, least squares

– “Promise” the right answer for O(n2) additional cost

• Jacobi-based SVD– Faster than QR, can be arbitrarily more accurate

• Arbitrary precision versions of everything– Using your favorite multiple precision package


– “Promise” the right answer for O(n2) additional cost– Iterative refinement with extra-precise residuals

• Newton’s method applied to solving f(x) = A*x-b = 0

– Extra-precise BLAS needed (LAWN#165)

More Accurate: Solve Ax=bConventional Gaussian Elimination

With extra preciseiterative refinement

n1/2

Iterative Refinement: for speed

• What if double precision much slower than single?– Cell processor in Playstation 3

• 256 GFlops single, 25 GFlops double

– Pentium SSE2: single twice as fast as double

• Given Ax=b in double precision– Factor in single, do refinement in double

– If (A) < 1/single, runs at speed of single

• 1.9x speedup on Intel-based laptop• Applies to many algorithms, if difference large

Goal 2 – Expanded Content• Make content of ScaLAPACK mirror LAPACK as much as

possible• New functions (highlights)

– Updating / downdating of factorizations: • Stewart, Langou

– More generalized SVDs: • Bai , Wang

– More generalized Sylvester/Lyapunov eqns:• Kågström, Jonsson, Granat

– Structured eigenproblems • O(n2) version of roots(p)

– Gu, Chandrasekaran, Bindel et al• Selected matrix polynomials:

– Mehrmann

New algorithm for roots(p)

• To find roots of polynomial p– Roots(p) does eig(C(p))– Costs O(n3), stable, reliable

• O(n2) Alternatives– Newton, Jenkins-Traub, Laguerre, …– Stable? Reliable?

• New: Exploit “semiseparable” structure of C(p)– Low rank of any submatrix of upper triangle of C(p) preserved

under QR iteration– Complexity drops from O(n3) to O(n2), stable in practice

• Related work: Gemignani, Bini, Pan, et al• Ming Gu, Shiv Chandrasekaran, Jiang Zhu, Jianlin Xia, David Bindel,

David Garmire, Jim Demmel

-p1 -p2 … -pd

1 0 … 0 0 1 … 0 … … … … 0 … 1 0

C(p)=

Goal 3 – Automate Performance Tuning

• Widely used in performance tuning of Kernels– ATLAS (PhiPAC) – BLAS - www.netlib.org/atlas– FFTW – Fast Fourier Transform – www.fftw.org– Spiral – signal processing - www.spiral.net– OSKI – Sparse BLAS – bebop.cs.berkeley.edu/oski

Optimizing blocksizes for mat-mul

Finding a Needle in a Haystack – So Automate

Goal 3 – Automate Performance Tuning

• Widely used in performance tuning of Kernels• 1300 calls to ILAENV() to get block sizes, etc.

– Never been systematically tuned

• Extend automatic tuning techniques of ATLAS, etc. to these other parameters– Automation important as architectures evolve

• Convert ScaLAPACK data layouts on the fly– Important for ease-of-use too

ScaLAPACK Data Layouts

1D Block

1D Block Cyclic

1D Cyclic

2D Block Cyclic

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2x30

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problem size

grid shape

Execution time of PDGESV for various grid shape

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Times obtained on:

60 processors, Dual AMD Opteron 1.4GHz Cluster w/Myrinet Interconnect

2GB Memory

Speedups for using 2D processor grid range from 2x to 8xCost of redistributing from 1D to best 2D layout 1% - 10%

Conclusions

• Lots to do in Dense Linear Algebra– New numerical algorithms– Continuing architectural challenges

• Parallelism, performance tuning

– Ease of use, software engineering

• Grant support, but success depends on contributions from community

• www.netlib.org/lapack-dev• www.cs.berkeley.edu/~demmel

Extra Slides

Fast Matrix Multiplication (1) (Cohn, Kleinberg, Szegedy, Umans)

• Can think of fast convolution of polynomials p, q as– Map p (q) into group algebra i pi zi C[G] of cyclic group G = { zi } – Multiply elements of C[G] (use divide&conquer = FFT)– Extract coefficients

• For matrix multiply, need non-abelian group satisfying triple product property– There are subsets X, Y, Z of G where xyz = 1 with

x X, y Y, z Z x = y = z = 1 – Map matrix A into group algebra via xy Axy x-1y,

B into y’z By’z y’-1z.– Since x-1y y’-1z = x-1z iff y = y’ we get y Axy Byz = (AB)xz

• Search for fast algorithms reduced to search for groups with certain properties– Fastest algorithm so far is O(n2.38), same as Coppersmith/Winograd

Fast Matrix Multiplication (2)(Cohn, Kleinberg, Szegedy, Umans)

1. Embed A, B in group algebra (exact)

2. Perform FFT (roundoff)

3. Reorganize results into new matrices (exact)

4. Multiple new matrices recursively (roundoff)

5. Reorganize results into new matrices (exact)

6. Perform IFFT (roundoff)

7. Extract C = AB from group algebra (exact)

Fast Matrix Multiplication (3)(Demmel, Dumitriu, Holtz, Kleinberg)

• Thm 1: Any algorithm of this class for C = AB is “numerically stable”

– || Ccomp - C || <= c • nd • • || A || • || B || + O(– c and d are “modest” constants– Like Strassen

• Let be the exponent of matrix multiplication, i.e. no algorithm is faster than O(n).

• Thm 2: For all >0 there exists an algorithm with complexity O(n+) that is numerically stable in the sense of Thm 1.

Commodity Processor TrendsAnnual increase

Typical valuein 2006

Predicted valuein 2010


Single-chipfloating-point performance

59% 4 GFLOP/s 32 GFLOP/s 3300 GFLOP/s

Memory bus bandwidth

23% 1 GWord/s= 0.25 word/flop

3.5 GWord/s= 0.11 word/flop

27 GWord/s= 0.008 word/flop

Memory latency (5.5%) 70 ns= 280 FP ops= 70 loads

50 ns= 1600 FP ops= 170 loads

28 ns= 94,000 FP ops= 780 loads

Source: Getting Up to Speed: The Future of Supercomputing, National Research Council, 222 pages, 2004, National Academies Press, Washington DC, ISBN 0-309-09502-6.

Will our algorithms run at a high fraction of peak?

Challenges

• For all large scale computing, not just linear algebra!

• Example … your laptop• Exponentially growing gaps between

– Floating point time << 1/Memory BW << Memory Latency

– Floating point time << 1/Network BW << Network Latency

Parallel Processor TrendsAnnual increase


Predicted valuein 2010


# Processors 20 % 4,000 12,000 3300 GFLOP/s

NetworkBandwidth

26% 65 MWord/s= 0.03 word/flop

260 MWord/s= 0.008 word/flop

27 GWord/s= 0.008 word/flop

Networklatency

(15%) 5 s= 20K FP ops

2 s= 64K FP ops

28 ns= 94,000 FP ops= 780 loads

Source: Getting Up to Speed: The Future of Supercomputing, National Research Council, 222 pages, 2004, National Academies Press, Washington DC, ISBN 0-309-09502-6.

Will our algorithms scale up to more processors?


• MRRR algorithm for symmetric eigenproblem / SVD: – Parlett / Dhillon / Voemel / Marques / Willems– Faster and more accurate than previous algorithms– New sequential, first parallel versions out in 2006

Timing of Eigensolvers(1.2 GHz Athlon, only matrices where time > .1 sec)

Timing of Eigensolvers(only matrices where time > .1 sec)

Accuracy Results (old vs new MRRR)maxi ||Tqi – i qi || / ( n ) || QQT – I || / (n )



• Up to 10x faster HQR: – Byers / Mathias / Braman


• Faster Hessenberg, tridiagonal, bidiagonal reductions:

– van de Geijn/Quintana, Howell / Fulton, Bischof / Lang

– Full nonsymmetric eigenproblem: n=1500: 3.43x faster

• HQR: 5x faster, Reduction: 14% faster

– Bidiagonal Reduction (LAWN#174): n=2000: 1.32x faster

– Sequential versions out in 2006






• Extensions to QZ: – Kågström / Kressner– LAPACK Working Note (LAWN) #173– On 26 real test matrices, speedups up to 11.9x, 4.4x average

Goal 4: Improved Ease of Use

• Which do you prefer?

CALL PDGESV( N ,NRHS, A, IA, JA, DESCA, IPIV, B, IB, JB, DESCB, INFO)

A \ B

CALL PDGESVX( FACT, TRANS, N ,NRHS, A, IA, JA, DESCA, AF, IAF, JAF, DESCAF, IPIV, EQUED, R, C, B, IB, JB, DESCB, X, IX, JX, DESCX, RCOND, FERR, BERR, WORK, LWORK, IWORK, LIWORK, INFO)

Goal 4: Improved Ease of Use

• Easy interfaces vs access to details– Some users want access to all details, because

• Peak performance matters• Control over memory allocation

– Other users want “simpler” interface• Automatic allocation of workspace• No universal agreement across systems on “easiest interface”• Leave decision to higher level packages

• Keep expert driver / simple driver / computational routines• Add wrappers for other languages

– Fortran95, Java, Matlab, Python, even C– Automatic allocation of workspace

• Add wrappers to convert to “best” parallel layout

Goal 5: Better SW Engineering:What could go into Sca/LAPACK?For all linear algebra problems

For all matrix structures

For all data types

For all programming interfaces

Produce best algorithm(s) w.r.t. performance and accuracy (including condition estimates, etc)

For all architectures and networks

Need to prioritize, automate!

Goal 5: Better SW Engineering• How to map multiple SW layers to emerging HW layers?• How much better are asynchronous algorithms?• Are emerging PGAS languages better?• Statistical modeling to limit performance tuning costs, improve use

of shared clusters

• Only some things understood well enough for automation now– Telescoping languages, Bernoulli, Rose, FLAME, …

• Research Plan: explore above design space• Development Plan to deliver code (some aspects)

– Maintain core in F95 subset– Friendly wrappers for other programming environments– Use variety of source control, maintenance, development tools

Goal 6: Involve the Community

• To help identify priorities– More interesting tasks than we are funded to do– See www.netlib.org/lapack-dev for list

• To help identify promising algorithms– What have we missed?

• To help do the work– Bug reports, provide fixes– Again, more tasks than we are funded to do– Already happening: thank you!

Accuracy of Eigensolvers

maxi ||Tqi – i qi || / ( n ) || QQT – I || / (n )

Goal 2 – Expanded Content

• Make content of ScaLAPACK mirror LAPACK as much as possible

• New functions (highlights)– Updating / downdating of factorizations:

• Stewart, Langou– More generalized SVDs:

• Bai , Wang

New GSVD Algorithm

Bai et al, UC Davis PSVD, CSD on the way

Given m x n A and p x n B, factor A = U ∑a X and B = V ∑b X

Motivation• LAPACK and ScaLAPACK are widely used

– Adopted by Cray, Fujitsu, HP, IBM, IMSL, MathWorks, NAG, NEC, SGI, …

– >63M web hits @ Netlib (incl. CLAPACK, LAPACK95)• 35K hits/day

Impact (with NERSC, LBNL)

Cosmic Microwave Background Analysis,

BOOMERanG collaboration, MADCAP code (Apr. 27, 2000).

ScaLAPACK

Challenges


• Example …

Challenges


• Example … your laptop

CPU Trends

2004 2005 2006 2007 2008 2009 2010

Cores Per Processor ChipHardware Threads Per Chip

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300

Year

• Relative processing power will continue to double every 18 months

• 256 logical processors per chip in late 2010

Challenges


• Example … your laptop• Exponentially growing gaps between

– Floating point time << 1/Memory BW << Memory Latency• Improving 59%/year vs 23%/year vs 5.5%/year

Accuracy of Eigensolvers: Old vs New MRRR

maxi ||Tqi – i qi || / ( n ) || QQT – I || / (n )


• MRRR algorithm for symmetric eigenproblem / SVD: – Parlett / Dhillon / Voemel / Marques / Willems– Faster and more accurate than previous algorithms– New sequential, first parallel versions out in 2006– Numerical evidence shows DC faster if it “deflates” often,

which is hard to predict in advance. So having both algorithms is important.

– SVD still an open problem



• Up to 10x faster HQR: – Byers / Mathias / Braman– SIAM SIAG/LA Prize in 2003– Sequential version out in 2006– More on performance later

ARCH: Intel Pentium 4 ( 3.4 GHz )F77 : GNU Fortran (GCC) 3.4.4BLAS: libgoto_prescott32p-r1.00.so (one thread)

Dense nonsymmetric eigenvalue problem

No vectors All vectors

Source: Julien Langou








– Bidiagonal Reduction (LAWN#174): n=2000: 1.32x faster

– Sequential versions out in 2006






• Extensions to QZ: – Kågström / Kressner– LAPACK Working Note (LAWN) #173– On 26 real test matrices, speedups up to 11.9x, 4.4x average

Comparison of ScaLAPACK QR and new parallel multishift QZ

Execution times in secs for 4096 x 4096 random problems Ax = sx and Ax = sBx, using processor grids including 1-16 processors.

Note: work(QZ) > 2 * work(QR) butTime(// QZ) << Time (//QR)!! Times include cost for computing eigenvalues and transformation matrices.

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1 x 1 2 x 1 2 x 2 4 x 2 4 x 4

// QR

// QZ

Adlerborn-Kågström-Kressner, SIAM PP’2006







• Recursive blocked layouts for packed formats:– Gustavson / Kågström / Elmroth / Jonsson/– SIAM Review Article 2004

Recursive Layouts and Algorithms

Still merges multiple elimination steps into a few BLAS 3 operations

Best speedups for packed storage of symmetric matrices


– “Promise” the right answer for O(n2) additional cost– Iterative refinement with extra-precise residuals– Extra-precise BLAS needed (LAWN#165)– “Guarantees” based on condition number estimates

• Condition estimate < 1/(n1/2 reliable answer and tiny error bounds

• No bad bounds in 6.2M tests• Can condition estimators lie?

Can condition estimators lie?

• Yes, but rarely, unless they cost as much as matrix multiply = cost of LU factorization– Demmel/Diament/Malajovich (FCM2001)

• But what if matrix multiply costs O(n2)?– More later


– “Promise” the right answer for O(n2) additional cost– Iterative refinement with extra-precise residuals– Extra-precise BLAS needed (LAWN#165)– “Guarantees” based on condition number estimates– Get tiny componentwise bounds too

• Each xi accurate

• Slightly different condition number– Extends to Least Squares– Release in 2006


– Promise the right answer for O(n2) additional cost

• Jacobi-based SVD– Faster than QR, can be arbitrarily more accurate– LAWNS # 169, 170– Can be arbitrarily more accurate on tiny singular values– Yet faster than QR iteration!


– Promise the right answer for O(n2) additional cost

• Jacobi-based SVD– Faster than QR, can be arbitrarily more accurate

• Arbitrary precision versions of everything– Using your favorite multiple precision package– Quad, Quad-double, ARPREC, MPFR, …– Using Fortran 95 modules

Goal 2 – Expanded Content• Make content of ScaLAPACK mirror LAPACK as much as

possible• New functions (highlights)

– Updating / downdating of factorizations: • Stewart, Langou

– More generalized SVDs: • Bai , Wang

– More generalized Sylvester/Lyapunov eqns:• Kågström, Jonsson, Granat

– Structured eigenproblems • O(n2) version of roots(p)

– Gu, Chandrasekaran, Bindel et al• Selected matrix polynomials:

– Mehrmann• How should we prioritize missing functions?

The Difficulty of Tuning SpMV:Sparse Matrix Vector Multiply

// y <-- y + A*x

for all A(i,j):

y(i) += A(i,j) * x(j)

The Difficulty of Tuning SpMV

// y <-- y + A*x

for all A(i,j):

y(i) += A(i,j) * x(j)

// Compressed sparse row (CSR)

for each row i:

t = 0

for k=row[i] to row[i+1]-1:

t += A[k] * x[J[k]]

y[i] = t

• Exploit 8x8 dense blocks

Speedups on Itanium 2: The Need for Search

Reference Mflop/s (7.6%)

Mflop/s (31.1%)

Speedups on Itanium 2: The Need for Search

Reference

Best: 4x2

Mflop/s (7.6%)

Mflop/s (31.1%)

SpMV Performance—raefsky3

More Surprises tuning SpMV

• More complex example

• Example: 3x3 blocking– Logical grid of 3x3 cells

Extra Work Can Improve Efficiency

• More complex example

• Example: 3x3 blocking– Logical grid of 3x3

cells– Pad with zeros– “Fill ratio” = 1.5

• On Pentium III: 1.5x speedup! (2/3 time)

Tuning for Workloads

• BiCG - equal mix of A*x and AT*y– 3x1: Ax, ATy = 1053, 343 Mflop/s – 3x3: Ax, ATy = 806, 826 Mflop/s

• Higher-level operation - (Ax, ATy) kernel– 3x1: 757 Mflop/s– 3x3: 1400 Mflop/s

Optimizations Available in OSKI:Optimized Sparse Kernel Interface

• Optimizations for SpMV– Register blocking (RB): up to 4x over CSR– Variable block splitting: 2.1x over CSR, 1.8x over RB– Diagonals: 2x over CSR– Reordering to create dense structure + splitting: 2x over CSR– Symmetry: 2.8x over CSR, 2.6x over RB– Cache blocking: 3x over CSR– Multiple vectors (SpMM): 7x over CSR– And combinations…

• Sparse triangular solve– Hybrid sparse/dense data structure: 1.8x over CSR

• Higher-level kernels– AAT*x, ATA*x: 4x over CSR, 1.8x over RB– A*x: 2x over CSR, 1.5x over RB

• Available stand alone or integrated into PETSc

The Future of LAPACK and ScaLAPACK netlib/lapack-dev

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