Comparisono fN umerical - Wolfram...cos (x)) x log (sinh x)) erf ( sin (x)) ⅇ x 2 1 10 100 1000 Maple time Mathematica time Map ~ map Sqrt Sin Cos Tan ArcSin ArcCos ArcTan Sec Csc

Comparison of Numerical Performance of Mathematica 11.3 and

Maple 2018Summary��

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Category Tests Median Mathematica speed

Real data operations 20 9 times faster

Real data operations (manual type override) 20 3 times faster

Complex number data operations 16 5 times faster

Complex number data operations (manual type override) 16 5 times faster

Sparse real data operations 7 80,000 times faster

Integer data operations 14 43 times faster

Integer data operations (manual type override) 13 37 times faster

Extended precision data operations (50 digits) 16 8 times faster

Extended precision data operations (1,000 digits) 16 4 times faster

Random number generation 9 7 times faster

Elementary & special functions 90 1,926 times faster

Elementary & special functions (manual type override) 66 17 times faster

Complex elementary & special functions 90 38 times faster

Complex elementary & special functions (manual type override) 90 35 times faster

Elementary & special functions (50 digits) 90 83 times faster

High-precision function evaluation 9 30 times faster

Exact functions 4 8 times faster

GPU operations 1 3 times faster

Total 587 50 times faster

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Machine-Precision Real Linear Algebra��

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Convert matrix to vector

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Covariance

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timeSort by absolute value

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Transpose

Mathematica 11.3

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Machine-Precision Complex Linear Algebra��

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timeDet

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timeConvert matrix to vector

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timeFourier

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timeLinearSolve

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timeMatrixExp

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timeTranspose

Mathematica 11.3

Maple 2018

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Sparse Data��

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timeTranspose

Mathematica 11.3

Maple 2018

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Integer Linear Algebra��

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timeMatrixExp

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timeTranspose

Mathematica 11.3

Maple 2018

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Extended-Precision Data��

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Cholesky

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timeMatrixExp

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timeTranspose

Mathematica 11.3

Maple 2018

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Sort by absolute value

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timeTranspose

Mathematica 11.3

Maple 2018

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Scalability of Data Operations��

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500000 1000000

100

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500

600× faster

Sort

500000 1000000

5×106

10×106

15×106

Fit

float

float[8]

0 100 200 300 400 500

500

1000

1500

× fasterPower

100000 200000 300000

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Mean

complex

complex[8]

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× fasterInverse

0 100 200 300 400 500 600

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Det

integer

integer[8]

100 200 300 400

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MatrixRank

500 1000 1500 2000

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Dot

integer

integer[8]

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Random Numbers��

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11×

17×

57×

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Mathematica time

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Function Evaluation��

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Cos

Tan

ArcTan

Sec

Csc Cot

Exp

Sinh

Cosh

Tanh

Erf

Gamma

ArcSinh

sin(cos(x))

erf(sin(x))

ⅇx2

500

1000

5000

104

Maple time

Mathematica time

Map

~

map

19×

21×

BesselJ

BesselK

BesselY

Zeta

0

5

10

15

Maple time

Mathematica time

451×

506×

941×

491×

510×

Sqrt

ArcSin

ArcCos

Log

Log10

ArcCosh

ArcTanh

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263×

295×

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ArcCos

ArcTan

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Log10

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Gamma

0

5

10

15

20

Maple time

Mathematica time

Map

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map

10

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Sin

Cos

Tan

ArcSin

ArcCos

ArcTan

Sec

Csc Cot

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Map

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map

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Sqrt

Sin

Cos

Tan

ArcSin

ArcCos

ArcTan

Sec

Csc Cot

Exp

Sinh

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map

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11

Sqrt

Sin

Cos

Tan

ArcSin

ArcCos

ArcTan

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Mathematica time

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High-Precision Evaluation��

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76×

69×

202×

π 2 sin(1) +1

1+sin1ⅇ log(5) tan(1+ ⅈ) K2(3) Γ

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3 erf(10)

0

10

20

30

40

Maple time

Mathematica time

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Exact Numeric Functions�� π� ��

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Function Largest test value Mathematica speed

BernoulliB 6,000 2 times faster

Fibonacci 500,000 3 times faster

HarmonicNumber 10,000 Maple cannot compute for n > 500

Zeta 100,000 13 times faster

Binomial 1014 280 times faster

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GPU Performance��

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General Testing Methodology��

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Appendix: Test Source Code

Mathematica code

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$HistoryLength = 0;steps = 10;repeats = 5;maxVector = 10^6;maxMatrix = 2000;maxSparseMatrix = 20000;

14

makeData[type_, i_] := Switchtype,

"RealVector", RandomReal[{-10, 10}, i],"RealMatrix", RandomReal[1, {i, i}],"ExtendedMatrix", RandomReal[1, {i, i}, WorkingPrecision → 50],"ExtendedMatrix1000", RandomReal[1, {i, i}, WorkingPrecision → 1000],"ExtendedVector", RandomReal[1, {i}, WorkingPrecision → 50],"ExtendedVector1000", RandomReal[1, {i}, WorkingPrecision → 1000],"IntegerMatrix", RandomInteger[100, {i, i}],"IntegerVector", RandomInteger[100, i],

"SparseMatrix", SparseArrayTable

{RandomInteger[{1, i}], RandomInteger[{1, i}]} → Random[], i^2 10000, {i, i},

"SparseVector", SparseArrayTableRandomInteger[{1, i}] → Random[],

Floori 10000 , {i},

"MediumSparseMatrix", SparseArrayTable

{RandomInteger[{1, i}], RandomInteger[{1, i}]} → Random[], i^2 200 , {i, i},

"ComplexMatrix", RandomComplex[{0, 1 + I}, {i, i}],"ComplexVector", RandomComplex[{0, 1 + I}, {i}],_, Print[type]

;

timedReport[path_, fn_, hi_, type_] := Block{data},

Export"Mathematica" <> ToString[$VersionNumber] <> path <> ".txt", Table{Floor[t],

Mean[Table[data = makeData[type, Floor[t]];AbsoluteTiming[fn[data]][[1]], {repeats}]]}

, t,hi

steps, hi,

hi

steps, "Table";

highPrecisionReport[path_, fns_, n_] :=Export["Mathematica" <> ToString[$VersionNumber] <> path <> ".txt",Map[AbsoluteTiming[N[#, n]][[1]] &, fns]]

NumericTest[path_, fns_List, n_, type_] :=Block[{data}, Export["Mathematica" <> ToString[$VersionNumber] <> path <> ".txt",

Table[data = makeData[type, n];AbsoluteTiming[i[data]][[1]], {i, fns}]]];

evaluateTest[path_, expr_Hold] :=Export["Mathematica" <> ToString[$VersionNumber] <> path <> ".txt",Apply[List, First /@ Map[AbsoluteTiming, expr]]]

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fnList = {Sqrt, Sin, Cos, Tan, ArcSin, ArcCos, ArcTan, Sec, Csc, Cot, Exp, Sinh, Cosh,Tanh, Log, Log10, Erf, Gamma, BesselJ[0, #] &, BesselK[1, #] &, BesselY[3, #] &,ArcSinh, ArcCosh, ArcTanh, Zeta, # Sin[Log[#] + 2 * Sqrt[#] + 3] + Sqrt[#] + Log[#] &,Sin[Cos[#]] &, # Log[Sinh[#]] &, Erf[Sin[#]] &, Exp[#^2] &};

15

NumericTest["ElementaryFunctions", fnList, 10 maxVector , "RealVector"];NumericTest["ElementaryFunctionsSM", fnList, maxVector , "RealVector"];NumericTest["ElementaryFunctionsComplex", fnList, maxVector , "ComplexVector"];

NumericTest"ElementaryFunctionsExtended", fnList, maxVector 10, "ExtendedVector";

NumericTest"ElementaryFunctionsExtended1000",

fnList, maxVector 100, "ExtendedVector1000";

NumericTest["ElementaryFunctionsSparse", fnList, 100000 maxVector , "SparseVector"];

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timedReport["FourierReal", Fourier, maxVector, "RealVector"];timedReport["SortReal", Sort, maxVector, "RealVector"];timedReport["SortCustomReal", SortBy[#, Abs] &, maxVector, "RealVector"];timedReport["MeanReal", Mean, 10 maxVector, "RealVector"];timedReport["DotReal", #.# &, maxMatrix , "RealMatrix"];timedReport["InverseReal", Inverse, maxMatrix, "RealMatrix"];Quiet@

timedReport["LinearSolveReal", LinearSolve[#, #[[1]]] &, maxMatrix, "RealMatrix"];timedReport["CholeskyReal", CholeskyDecomposition[Transpose[#].#] &,

maxMatrix, "RealMatrix"];timedReport["MatrixPowerReal", MatrixPower[#, 5] &, maxMatrix, "RealMatrix"];timedReport["DetReal", Det, maxMatrix, "RealMatrix"];timedReport["TransposeReal", Transpose, maxMatrix, "RealMatrix"];timedReport["FlattenReal", Flatten, maxMatrix, "RealMatrix"];

timedReport"EigenvaluesReal", Eigenvalues, maxMatrix 2, "RealMatrix";

timedReport"EigenvectorsReal", Eigenvectors, maxMatrix 2, "RealMatrix";

timedReport["LinearProgrammingReal",LinearProgramming[Abs[#[[1]]], Abs[#], Abs[#[[2]]], Method → "CLP"] &,maxMatrix, "RealMatrix"];

timedReport["ElementPowerReal", #^5 &, maxMatrix, "RealMatrix"];timedReport["MovingAverageReal", MovingAverage[#, 10] &, maxVector, "RealVector"];

timedReport"FitReal", Function[{data}, Fit[data, x, x]][Transpose[{#, #}]] &;,

4 * maxVector, "RealVector";

timedReport"MatrixExpReal", MatrixExp, maxMatrix 5, "RealMatrix";

timedReport["CovarianceReal", Covariance, maxMatrix, "RealMatrix"];timedReport["MatrixRank", MatrixRank, 3 maxMatrix, "RealMatrix"];

16

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timedReport["FourierComplex", Fourier, maxVector, "ComplexVector"];timedReport["SortCustomComplex", SortBy[#, Abs] &, maxVector, "ComplexVector"];timedReport["MeanComplex", Mean, 10 maxVector, "ComplexVector"];timedReport["DotComplex", #.# &, maxMatrix , "ComplexMatrix"];timedReport["InverseComplex", Inverse, maxMatrix, "ComplexMatrix"];timedReport["LinearSolveComplex",

LinearSolve[#, #[[1]]] &, maxMatrix, "ComplexMatrix"];timedReport["CholeskyComplex", CholeskyDecomposition[ConjugateTranspose[#].#] &,

maxMatrix, "ComplexMatrix"];

timedReport"MatrixPowerComplex", MatrixPower[#, 5] &, maxMatrix 2, "ComplexMatrix";

timedReport["DetComplex", Det, maxMatrix, "ComplexMatrix"];

timedReport"EigenvaluesComplex", Eigenvalues, maxMatrix 2, "ComplexMatrix";

timedReport"EigenvectorsComplex", Eigenvectors, maxMatrix 2, "ComplexMatrix";

timedReport["TransposeComplex", Transpose, maxMatrix, "ComplexMatrix"];timedReport["FlattenComplex", Flatten, maxMatrix, "ComplexMatrix"];timedReport["ElementPowerComplex", #^5 &, maxMatrix, "ComplexMatrix"];

timedReport"MatrixExpComplex", MatrixExp, maxMatrix 4, "ComplexMatrix";

timedReport["MatrixRankComplex", MatrixRank, 3 maxMatrix, "ComplexMatrix"]

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timedReport["DotSparse", #.# &, maxSparseMatrix, "SparseMatrix"];

Quiet@timedReport"LinearSolveSparse",

LinearSolve[#, #[[1]]] &, maxSparseMatrix 3, "MediumSparseMatrix";

timedReport["TransposeSparse", Transpose, 2 * maxSparseMatrix, "SparseMatrix"];timedReport["FlattenSparse", Flatten, maxSparseMatrix, "SparseMatrix"];timedReport["ElementPowerSparse", #^5 &, maxSparseMatrix, "SparseMatrix"];timedReport["MeanSparse", Mean, maxVector * 100, "SparseVector"];

timedReport"MatrixPowerSparse",

MatrixPower[#, 5] &, 2 * maxSparseMatrix 5, "SparseMatrix";

timedReport["MovingAverageSparse", MovingAverage[#, 10] &,10 maxVector, "SparseVector"];

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timedReport["DotInteger", #.# &, maxMatrix , "IntegerMatrix"];

timedReport"InverseInteger", Inverse, maxMatrix 25, "IntegerMatrix";

timedReport"LinearSolveInteger",

LinearSolve[#, #[[1]]] &, maxMatrix 4, "IntegerMatrix";

timedReport"MatrixPowerInteger", MatrixPower[#, 5] &, maxMatrix 4, "IntegerMatrix";

timedReport["DetInteger", Det, 600 , "IntegerMatrix"];timedReport["TransposeInteger", Transpose, maxMatrix, "IntegerMatrix"];timedReport["FlattenInteger", Flatten, maxMatrix, "IntegerMatrix"];timedReport["MeanInteger", Mean, maxVector, "IntegerVector"];timedReport["SortInteger", Sort, maxVector, "IntegerVector"];timedReport["SortCustomInteger", SortBy[#, Abs] &, maxVector, "IntegerVector"];

timedReport"EigenvaluesInteger", Eigenvalues, maxMatrix 30, "IntegerMatrix";

timedReport["ElementPowerInteger", #^5 &, maxMatrix, "IntegerMatrix"];timedReport["MatrixExpInteger", MatrixExp, 40, "IntegerMatrix"];timedReport["MatrixRankInteger", MatrixRank, maxMatrix, "IntegerMatrix"];

17

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timedReport"DotExtended", #.# &, maxMatrix 25 , "ExtendedMatrix";

timedReport"InverseExtended", Inverse, maxMatrix 25, "ExtendedMatrix";

timedReport"CholeskyExtended",

CholeskyDecomposition[Transpose[#].#] &, maxMatrix 20, "ExtendedMatrix";

timedReport["EigenvaluesExtended", Eigenvalues, 60, "ExtendedMatrix"];timedReport["EigenvectorsExtended", Eigenvectors, 80, "ExtendedMatrix"];

timedReport"ElementPowerExtended", #^5 &, maxMatrix 2, "ExtendedMatrix";

timedReport"LinearSolveExtended",

LinearSolve[#, #[[1]]] &, maxMatrix 20, "ExtendedMatrix";

timedReport"DetExtended", Det, maxMatrix 10, "ExtendedMatrix";

timedReport["TransposeExtended", Transpose, maxMatrix, "ExtendedMatrix"];timedReport["FlattenExtended", Flatten, maxMatrix, "ExtendedMatrix"];

(*timedReport"FourierExtended",Fourier,maxVector100,"ExtendedVector";

Maple converts to floats*)

timedReport"MeanExtended", Mean, maxVector 10, "ExtendedVector";

timedReport["SortExtended", Sort, 2 * maxVector, "ExtendedVector"];timedReport["SortCustomExtended", SortBy[#, Abs] &, 2 * maxVector, "ExtendedVector"];

timedReport"EigenvectorsExtended", Eigenvectors, maxMatrix 25, "ExtendedMatrix";

timedReport"MatrixPowerExtended",

MatrixPower[#, 5] &, maxMatrix 25, "ExtendedMatrix";

timedReport["MatrixExpExtended", MatrixExp, 100, "ExtendedMatrix"];

timedReport"MatrixRankExtended", MatrixRank, maxMatrix 10, "ExtendedMatrix";

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timedReport"DotExtended1000", #.# &, maxMatrix 20 , "ExtendedMatrix1000";

timedReport"InverseExtended1000", Inverse, maxMatrix 25, "ExtendedMatrix1000";

timedReport["CholeskyExtended1000",CholeskyDecomposition[Transpose[#].#] &, 100, "ExtendedMatrix1000"];

timedReport["EigenvaluesExtended1000", Eigenvalues, 80, "ExtendedMatrix1000"];timedReport["EigenvectorsExtended1000", Eigenvectors, 80, "ExtendedMatrix1000"];

timedReport"ElementPowerExtended1000", #^5 &, maxMatrix 2, "ExtendedMatrix1000";

timedReport"LinearSolveExtended1000",

LinearSolve[#, #[[1]]] &, maxMatrix 25, "ExtendedMatrix1000";

timedReport"DetExtended1000", Det, maxMatrix 10, "ExtendedMatrix1000";

timedReport["TransposeExtended1000", Transpose, maxMatrix, "ExtendedMatrix1000"];timedReport["FlattenExtended1000", Flatten, maxMatrix, "ExtendedMatrix1000"];

timedReport"MeanExtended1000", Mean, maxVector 10, "ExtendedVector1000";

timedReport["SortExtended1000", Sort, maxVector, "ExtendedVector1000"];timedReport["SortCustomExtended1000",

SortBy[#, Abs] &, maxVector, "ExtendedVector1000"];

timedReport"MatrixPowerExtended1000", MatrixPower[#, 5] &,

maxMatrix 25, "ExtendedMatrix1000";

timedReport["MatrixExpExtended1000", MatrixExp, 60, "ExtendedMatrix1000"];

timedReport"MatrixRankExtended1000", MatrixRank, maxMatrix 20, "ExtendedMatrix1000"

18

��

evaluateTest["ExactFunctions", Hold[Table[BernoulliB[i], {i, 6000}],Table[Fibonacci[i], {i, 0, 500000, 1000}],(*Table[HarmonicNumber[i],{i,10000}], Maple fails*)Table[Zeta[i], {i, 0, 100000, 10000}],Table[Binomial[i^2, i], {i, 1, 10000}]

]];

��

evaluateTest["RandomNumbers", Hold[makeData["RealMatrix", 2000],makeData["RealMatrix", 2000],makeData["IntegerMatrix", 2000],makeData["IntegerMatrix", 2000],makeData["ExtendedMatrix", 2000],makeData["ExtendedMatrix1000", 2000],RandomVariate[NormalDistribution[0, 1], 10^7],RandomVariate[PoissonDistribution[4], 10^7],RandomVariate[BinomialDistribution[10, 0.2], 10^7]

]];

��

highPrecisionReport"ManyDigits",

Pi, Sqrt[2], Sin[1] +1

1 + Sin[1], Exp[1], Log[5], Tan[1 + I], 1000000;

highPrecisionReport"FewerDigits", BesselK[2, 3], Gamma11 3, Erf[10], 5000;

Maple

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23

Documents

Comparisono fN umerical - Wolfram...cos (x)) x log (sinh x)) erf ( sin (x)) ⅇ x 2 1 10 100 1000 Maple time Mathematica time Map ~ map Sqrt Sin Cos Tan ArcSin ArcCos ArcTan Sec Csc