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Plamen Fiziev BLTF, JINR, Dubna Parallel computations using Maple 17 Mathematical Modeling and Computational Physics Dubna, 2013

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The point of the talk is to present some of the achievements and technical problems in parallel calculations using last versions of Maple package. In particular, we intend to describe the simplicity of Parallel computations with Maple Linear Algebra Package DEtools Package MathematicalFunctions Package New methods for generalized Heun’s functions THE HEUN PROJECT

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Some of the basic Maple 17 packages for doing Math and Computational Physics: 1.Linear Algebra Package 2.DEtools Package 3.MathematicalFunctions Package ( including Heun’s functions ) 4.IntegrationTools Package 5.DifferentialGeometry Package 6.DiscreteTransforms Package 7.MultiSeries package 8.DynamicSystems Package 9.PDEtools Package 10.ScientificErrorAnalysis Package 11.Physics Package

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Supermicro 4U/tower SYS-7047A-TH 2 X CPU Intel Xeon E Processor (20M Cache, 2.60 GHz, 8.00 GT/s Intel® QPI) 16 core (32 logical) 2 X HDD SAS 300GB 15000RPM HDD SATA 2TB 7200RPM Enterprise Memory 128 GB DDR3-1600MHz ECC REG OS: Scientific Linux release 5.4 x 64 WARNING: using Scientific Linux instead of Windows seems to be critical for successful parallel computations on this and similar platforms using Maple 16/17. Software: 1.Maple 17 2.Fortran 90 3.MPI

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Lenovo W520 Memory 16 GB DDR3 SDRAM -1333MHz Two Intel(R) Core (TM), i QM 2.20 GHz 4 core (8 logical) OS: 1.Windows 7 x 64 Software: 1.Maple 17.2

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Linear Algebra is already almost fully paralellezed using the threads technology with(LinearAlgebra): Neq:=1200; A0 := RandomMatrix(Neq,Neq,datatype=float[8]): nrmA:=sqrt(add(add(A0[j,i]^2,i=1..Neq),j=1..Neq)): A:=A0/nrmA: EV:=Eigenvectors(A, output='list'): - Time=2 sec - Time=10 sec - for a Neq X Neq rendom matrix A ……………………………………………………………. Eigenvalues:

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Eigenvalues: Time versus Matrix size For such simple computations is CRITICAL the amount of RAM for one core ! The same statement is true for the other Linear Algebra operations.

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Linear systems of ODEs with rendom coefficients and rendom initial conditions: parallelSyssolver:=proc() uses Grid; local me,numNodes,nrm,Xin,r,i,rply; global syss, NK, Neq, IC; me:= MyNode(); numNodes:= NumNodes(); Xin:=seq(x[i](0)=IC[i,me+1]/nrm,i=1..Neq); r:=syssolver(me+1); if me <> 0 then Send(0, r); else rply := r; for i from 1 to (numNodes-1) do r := Receive(i); rply := rply, r; end do; return [rply]; end if: end proc: Neq := Number of equations A0 := RandomMatrix(Neq,Neq,datatype=float[8]): nrmA:=sqrt(add(add(A0[j,i]^2,i=1..Neq),j=1..Neq)) :A:=A0/nrmA: f:= (j) -> x[j](t):X:= Vector(Neq,f):df:=(j) -> diff(x[j](t),t):dX:=Vector(Neq,df): F:=A.X: dsys := seq(dX[m]=F[m],m=1..Neq): IC0:=RandomMatrix(Neq,NK,datatype=float [8]);nrm:=sqrt(add(add(IC0[i,j]^2,i=1..Neq), j=1..NK)): IC:=IC0/nrm: result:=Grid:-Launch(parallelSyssolver,imports=['syssolver','dsys','IC','NK','Neq'],numnodes=NK): syssolver:=proc(k) local ssys, nrm,Xin; global NK, Neq, IC; UseHardwareFloats:=true: nrm:=sqrt(add(IC[i,k]^2,i=1..Neq)); Xin:=seq(x[i](0)=IC[i,k],i=1..Neq); ssys:= dsolve({dsys,Xin},numeric);ssys(500): end proc:

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500 ODEs => parallel 1.7 s sequential 23.5 s => 13.6 times (An almost maximal usage of all cores is reached via the inner Grid technology) An example for a parallelized procedure - Computing a Convex Hull from Maple help. ( => acceleration about 2 times)

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Critical is the amount of RAM for one core !

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Examples of Linear ODEs with essential singular points 1. Euler’s example: Bi -section method Parallel Comp + Multi -section: M = # Cores+1 Asymptotic Series

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2. Uniformed Neznamov et. al. system of ODEs for Dirac particle in Schwarzschild metric (arXiv: ): Examples of Linear ODEs with essential singular points Unknown functions: and

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1. Regularization (no zeros in the denominator): 2. Iniformization (no roots): 3. Polar variables:

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The vector field on the torus {Φ, η} : -The relief of the vector field - Separatrix valley

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The phase portrait on the torus {Φ, η} : Node at Saddle point at => Bounded states => Unbounded states

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Discrete spectrum of the bounded states of Dirac particle in Schwarzschild metric Solving by parallel computing the solutions of the corresponding ODEs one obtains: From Then, according to Neznamov et.al. arXiv: , the spectral condition is

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Another formulation of the same problem A Schrödinger like form with quite complicated quasipotential W(ρ,ε) which yields a second order ODE with five singular points – even not a Heun equation.

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The Heun Project Particular case: the Heun equation

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The confluent Heun equation: Huge amount of applications in all scientific areas: physics, astrophysics, astronomy, chemistry, biology, economics, e.t.c Maple Maple 17.2 Conclusion: NEW CODES ARE NEEDED. The above example gives some idea what we really need.

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Thank you for your attention !

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