Comparison of Numerical Performance of Mathematica 11.2 and Maple

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1 Comparison of Numerical Performance of Mathematica. and Maple 07. Summary Category Tests Median Mathematica speed Real data operations 0 s faster Real data operations (manual type override) 0 s faster Complex number data operations 6 s faster Complex number data operations (manual type override) 6 s faster Sparse real data operations 7 7,000 s faster Integer data operations 0 s faster Integer data operations (manual type override) 0 s faster Extended precision data operations (0 digits) 6 s faster Extended precision data operations (,000 digits) 6 s faster Random number generation 9 7 s faster Elementary & special functions 90,69 s faster Elementary & special functions (manual type override) 66 6 s faster Complex elementary & special functions 90 6 s faster Complex elementary & special functions (manual type override) 90 s faster Elementary & special functions (0 digits) 90 9 s faster High-precision function evaluation 9 s faster Exact functions 9 s faster GPU operations s faster Total 87 s faster

2 Machine-Precision Real Linear Algebra.0. Cholesky size Convert matrix to vector size.0. Covariance size Det size Dot size Eigenvalues size Eigenvectors Fit. Fourier size size size Inverse LinearProgramming 0.8 LinearSolve size size size MatrixExp MatrixPower MatrixRank size size size

3 Mean Power Sort size size size. Sort by absolute value 0.0 Transpose Mathematica. Maple size size Machine-Precision Complex Linear Algebra.0. Det..0. Dot 8 6 Eigenvalues size size size Eigenvectors Power. Convert matrix to vector size size size Fourier Inverse LinearSolve size size size

4 .0 MatrixExp. MatrixPower 00 MatrixRank size size size.0. Mean..0. Sort by absolute value Transpose size size size Mathematica. Maple 07. Sparse Data Convert matrix to vector Dot LinearSolve size size size MatrixPower Mean Power size size size Transpose Mathematica. Maple size

5 Integer Linear Algebra Det Dot Eigenvalues size size size.0. Power 0.0 Convert matrix to vector Inverse size size size LinearSolve MatrixExp 0 0 MatrixPower size size size MatrixRank Mean Sort size size size Sort by absolute value Transpose Mathematica. Maple size size

6 6 Extended-Precision Data Cholesky 8 Det Dot size size size 8 Eigenvalues 0 Eigenvectors 0 Power size size size 8 6 Convert matrix to vector Inverse LinearSolve size size size MatrixExp MatrixPower MatrixRank size size size Mean Sort Sort by absolute value size size size Transpose 8 6 Mathematica. Maple size

7 7 Cholesky Det Dot size size size 0 Eigenvalues 00 Eigenvectors 0 Power size size size Convert matrix to vector 8 6 Inverse LinearSolve size size size MatrixExp 0 MatrixPower 0 MatrixRank size size size Mean 0 Sort Sort by absolute value size size size Transpose Mathematica. Maple size

8 8 Scalability of Data Operations faster 600 Sort 00 Power real Fit Sort by absolute value real faster Inverse Det integer integer[8] MatrixRank Dot integer integer[8]

9 9 Random Numbers Maple Mathematica Function Evaluation

10 Maple Mathematica 000 Map ~ map Sin Cos Tan ArcTan Sec Csc Cot Exp 9 BesselJ BesselK BesselY Zeta Sinh Cosh Tanh Erf Gamma ArcSinh sin(cos(x)) erf(sin(x)) e x Maple Mathematica Sqrt ArcSin ArcCos Log Log ArcCosh ArcTanh + ( ) ( ) x log(sinh(x)) Map ~ map 0 Maple Mathematica Map ~ map 0 Sqrt Sin Cos Tan ArcSin ArcCos ArcTan Sec Csc Cot Exp Sinh Cosh Tanh Log Log Erf Gamma

11 Maple Mathematica Sqrt Sin Cos Tan ArcSin ArcCos ArcTan Sec Csc Cot Exp Sinh Cosh Tanh Log Log Erf Gamma BesselJ BesselK BesselY ArcSinh ArcCosh ArcTanh Zeta + ( ) ( ) sin(cos(x)) x log(sinh(x)) erf(sin(x)) e x Map ~ map Maple Mathematica Sqrt Sin Cos Tan ArcSin ArcCos ArcTan Sec Csc Cot Exp Sinh Cosh Tanh Log Log Erf Gamma BesselJ BesselK BesselY ArcSinh ArcCosh ArcTanh Zeta + ( ) ( ) sin(cos(x)) x log(sinh(x)) erf(sin(x)) e x Map ~ map

12 Maple Mathematica 0 0 Map ~ map Sqrt Sin Cos Tan ArcSin ArcCos ArcTan Sec Csc Cot Exp Sinh Cosh Tanh Log Log Erf Gamma BesselJ BesselK BesselY ArcSinh ArcCosh ArcTanh Zeta + ( ) ( ) sin(cos(x)) x log(sinh(x)) erf(sin(x)) e x High-Precision Evaluation Maple Mathematica π sin() + +sin e log() tan( + i) K () Γ erf() Exact Numeric Functions π

13 Function Largest test value Mathematica speed BernoulliB 6,000 s faster Fibonacci 00,000 s faster HarmonicNumber,000 Maple cannot compute for n > 00 Zeta 0,000 s faster Binomial 7 s faster GPU Performance 8 6 Dot size General Testing Methodology +

14 Revision Notes Appendix: Test Source Code Mathematica code $HistoryLength = 0; steps = ; repeats = ; maxvector = ^6; maxmatrix = 000; maxsparsematrix = 0 000; makedata[type_, i_] := Switch type, "RealVector", RandomReal[{-, }, i], "RealMatrix", RandomReal[, {i, i}], "ExtendedMatrix", RandomReal[, {i, i}, WorkingPrecision 0], "ExtendedMatrix00", RandomReal[, {i, i}, WorkingPrecision 00], "ExtendedVector", RandomReal[, {i}, WorkingPrecision 0], "ExtendedVector00", RandomReal[, {i}, WorkingPrecision 00], "IntegerMatrix", RandomInteger[0, {i, i}], "IntegerVector", RandomInteger[0, i], "SparseMatrix", SparseArray Table {RandomInteger[{, i}], RandomInteger[{, i}]} Random[], i^ 000, {i, i}, "SparseVector", SparseArray Table RandomInteger[{, i}] Random[], Floor i 000, {i}, "MediumSparseMatrix", SparseArray Table {RandomInteger[{, i}], RandomInteger[{, i}]} Random[], i^ 00, {i, i}, "ComplexMatrix", RandomComplex[{0, + I}, {i, i}], "ComplexVector", RandomComplex[{0, + I}, {i}], _, Print[type] ;

15 dreport[path_, fn_, hi_, type_] := Block {data}, Export "Mathematica" <> ToString[$VersionNumber] <> path <> ".txt", Table {Floor[t], Mean[Table[ data = makedata[type, Floor[t]]; AbsoluteTiming[fn[data]][[]], {repeats}]]} hi, t, steps, hi, hi, "Table" ; steps highprecisionreport[path_, fns_, n_] := Export["Mathematica" <> ToString[$VersionNumber] <> path <> ".txt", Map[AbsoluteTiming[N[#, n]][[]] &, fns]] NumericTest[path_, fns_list, n_, type_] := Block[{data}, Export["Mathematica" <> ToString[$VersionNumber] <> path <> ".txt", Table[data = makedata[type, n]; AbsoluteTiming[i[data]][[]], {i, fns}]]]; evaluatetest[path_, expr_hold] := Export["Mathematica" <> ToString[$VersionNumber] <> path <> ".txt", Apply[List, First /@ Map[AbsoluteTiming, expr]]] fnlist = {Sqrt, Sin, Cos, Tan, ArcSin, ArcCos, ArcTan, Sec, Csc, Cot, Exp, Sinh, Cosh, Tanh, Log, Log, Erf, Gamma, BesselJ[0, #] &, BesselK[, #] &, BesselY[, #] &, ArcSinh, ArcCosh, ArcTanh, Zeta, # Sin[Log[#] + * Sqrt[#] + ] + Sqrt[#] + Log[#] &, Sin[Cos[#]] &, # Log[Sinh[#]] &, Erf[Sin[#]] &, Exp[#^] &}; NumericTest["ElementaryFunctions", fnlist, maxvector, "RealVector"]; NumericTest["ElementaryFunctionsComplex", fnlist, maxvector, "ComplexVector"]; NumericTest "ElementaryFunctionsExtended", fnlist, maxvector, "ExtendedVector" ; NumericTest "ElementaryFunctionsExtended00", fnlist, maxvector 0, "ExtendedVector00" ; NumericTest["ElementaryFunctionsSparse", fnlist, maxvector, "SparseVector"];

16 6 dreport["fourierreal", Fourier, maxvector, "RealVector"]; dreport["sortreal", Sort, maxvector, "RealVector"]; dreport["sortcustomreal", SortBy[#, Abs] &, maxvector, "RealVector"]; dreport["meanreal", Mean, maxvector, "RealVector"]; dreport["dotreal", #.# &, maxmatrix, "RealMatrix"]; dreport["inversereal", Inverse, maxmatrix, "RealMatrix"]; dreport["linearsolvereal", LinearSolve[#, #[[]]] &, maxmatrix, "RealMatrix"]; dreport["choleskyreal", CholeskyDecomposition[Transpose[#].#] &, maxmatrix, "RealMatrix"]; dreport["matrixpowerreal", MatrixPower[#, ] &, maxmatrix, "RealMatrix"]; dreport["detreal", Det, maxmatrix, "RealMatrix"]; dreport["transposereal", Transpose, maxmatrix, "RealMatrix"]; dreport["flattenreal", Flatten, maxmatrix, "RealMatrix"]; dreport "EigenvaluesReal", Eigenvalues, maxmatrix, "RealMatrix" ; dreport "EigenvectorsReal", Eigenvectors, maxmatrix, "RealMatrix" ; dreport["linearprogrammingreal", LinearProgramming[Abs[#[[]]], Abs[#], Abs[#[[]]], Method "CLP"] &, maxmatrix, "RealMatrix"]; dreport["elementpowerreal", # ^ &, maxmatrix, "RealMatrix"]; dreport["movingaveragereal", MovingAverage[#, ] &, maxvector, "RealVector"]; dreport "FitReal", Function[{data}, Fit[data, x, x]][transpose[{#, #}]] &;, * maxvector, "RealVector" ; dreport "MatrixExpReal", MatrixExp, maxmatrix, "RealMatrix" ; dreport["covariancereal", Covariance, maxmatrix, "RealMatrix"]; dreport["matrixrank", MatrixRank, maxmatrix, "RealMatrix"]; dreport["fouriercomplex", Fourier, maxvector, "ComplexVector"]; dreport["sortcustomcomplex", SortBy[#, Abs] &, maxvector, "ComplexVector"]; dreport["meancomplex", Mean, maxvector, "ComplexVector"]; dreport["dotcomplex", #.# &, maxmatrix, "ComplexMatrix"]; dreport["inversecomplex", Inverse, maxmatrix, "ComplexMatrix"]; dreport["linearsolvecomplex", LinearSolve[#, #[[]]] &, maxmatrix, "ComplexMatrix"]; dreport["choleskycomplex", CholeskyDecomposition[ConjugateTranspose[#].#] &, maxmatrix, "ComplexMatrix"]; dreport "MatrixPowerComplex", MatrixPower[#, ] &, maxmatrix, "ComplexMatrix" ; dreport["detcomplex", Det, maxmatrix, "ComplexMatrix"]; dreport "EigenvaluesComplex", Eigenvalues, maxmatrix, "ComplexMatrix" ; dreport "EigenvectorsComplex", Eigenvectors, maxmatrix, "ComplexMatrix" ; dreport["transposecomplex", Transpose, maxmatrix, "ComplexMatrix"]; dreport["flattencomplex", Flatten, maxmatrix, "ComplexMatrix"]; dreport["elementpowercomplex", # ^ &, maxmatrix, "ComplexMatrix"]; dreport "MatrixExpComplex", MatrixExp, maxmatrix, "ComplexMatrix" ; dreport["matrixrankcomplex", MatrixRank, maxmatrix, "ComplexMatrix"]

17 7 dreport["dotsparse", #.# &, maxsparsematrix, "SparseMatrix"]; "LinearSolveSparse", LinearSolve[#, #[[]]] &, maxsparsematrix, "MediumSparseMatrix" ; dreport["transposesparse", Transpose, * maxsparsematrix, "SparseMatrix"]; dreport["flattensparse", Flatten, maxsparsematrix, "SparseMatrix"]; dreport["elementpowersparse", # ^ &, maxsparsematrix, "SparseMatrix"]; dreport["meansparse", Mean, maxvector * 0, "SparseVector"]; dreport "MatrixPowerSparse", MatrixPower[#, ] &, * maxsparsematrix, "SparseMatrix" ; dreport["movingaveragesparse", MovingAverage[#, ] &, maxvector, "SparseVector"]; dreport["dotinteger", #.# &, maxmatrix, "IntegerMatrix"]; dreport "InverseInteger", Inverse, maxmatrix, "IntegerMatrix" ; dreport "LinearSolveInteger", LinearSolve[#, #[[]]] &, maxmatrix, "IntegerMatrix" ; dreport "MatrixPowerInteger", MatrixPower[#, ] &, maxmatrix, "IntegerMatrix" ; dreport["detinteger", Det, 600, "IntegerMatrix"]; dreport["transposeinteger", Transpose, maxmatrix, "IntegerMatrix"]; dreport["flatteninteger", Flatten, maxmatrix, "IntegerMatrix"]; dreport["meaninteger", Mean, maxvector, "IntegerVector"]; dreport["sortinteger", Sort, maxvector, "IntegerVector"]; dreport["sortcustominteger", SortBy[#, Abs] &, maxvector, "IntegerVector"]; dreport "EigenvaluesInteger", Eigenvalues, maxmatrix 0, "IntegerMatrix" ; dreport["elementpowerinteger", # ^ &, maxmatrix, "IntegerMatrix"]; dreport["matrixexpinteger", MatrixExp, 0, "IntegerMatrix"]; dreport["matrixrankinteger", MatrixRank, maxmatrix, "IntegerMatrix"]; dreport "DotExtended", #.# &, maxmatrix, "ExtendedMatrix" ; dreport "InverseExtended", Inverse, maxmatrix, "ExtendedMatrix" ; dreport "CholeskyExtended", CholeskyDecomposition[Transpose[#].#] &, maxmatrix 0, "ExtendedMatrix" ; dreport["eigenvaluesextended", Eigenvalues, 60, "ExtendedMatrix"]; dreport["eigenvectorsextended", Eigenvectors, 80, "ExtendedMatrix"]; dreport "ElementPowerExtended", #^ &, maxmatrix, "ExtendedMatrix" ; dreport "LinearSolveExtended", LinearSolve[#, #[[]]] &, maxmatrix 0, "ExtendedMatrix" ; dreport "DetExtended", Det, maxmatrix, "ExtendedMatrix" ; dreport["transposeextended", Transpose, maxmatrix, "ExtendedMatrix"]; dreport["flattenextended", Flatten, maxmatrix, "ExtendedMatrix"]; (*dreport "FourierExtended",Fourier,maxVector 0,"ExtendedVector" ; Maple converts to floats*) dreport "MeanExtended", Mean, maxvector, "ExtendedVector" ; dreport["sortextended", Sort, * maxvector, "ExtendedVector"]; dreport["sortcustomextended", SortBy[#, Abs] &, * maxvector, "ExtendedVector"]; dreport "EigenvectorsExtended", Eigenvectors, maxmatrix, "ExtendedMatrix" ; dreport "MatrixPowerExtended", MatrixPower[#, ] &, maxmatrix, "ExtendedMatrix" ; dreport["matrixexpextended", MatrixExp, 0, "ExtendedMatrix"]; dreport "MatrixRankExtended", MatrixRank, maxmatrix, "ExtendedMatrix" ;

18 8 dreport "DotExtended00", #.# &, maxmatrix 0, "ExtendedMatrix00" ; dreport "InverseExtended00", Inverse, maxmatrix, "ExtendedMatrix00" ; dreport "CholeskyExtended00", CholeskyDecomposition[Transpose[#].#] &, maxmatrix 0, "ExtendedMatrix00" ; dreport["eigenvaluesextended00", Eigenvalues, 80, "ExtendedMatrix00"]; dreport["eigenvectorsextended00", Eigenvectors, 80, "ExtendedMatrix00"]; dreport "ElementPowerExtended00", #^ &, maxmatrix, "ExtendedMatrix00" ; dreport "LinearSolveExtended00", LinearSolve[#, #[[]]] &, maxmatrix, "ExtendedMatrix00" ; dreport "DetExtended00", Det, maxmatrix, "ExtendedMatrix00" ; dreport["transposeextended00", Transpose, maxmatrix, "ExtendedMatrix00"]; dreport["flattenextended00", Flatten, maxmatrix, "ExtendedMatrix00"]; dreport "MeanExtended00", Mean, maxvector, "ExtendedVector00" ; dreport["sortextended00", Sort, maxvector, "ExtendedVector00"]; dreport["sortcustomextended00", SortBy[#, Abs] &, maxvector, "ExtendedVector00"]; dreport "MatrixPowerExtended00", MatrixPower[#, ] &, maxmatrix, "ExtendedMatrix00" ; dreport["matrixexpextended00", MatrixExp, 60, "ExtendedMatrix00"]; dreport "MatrixRankExtended00", MatrixRank, maxmatrix 0, "ExtendedMatrix00" evaluatetest["exactfunctions", Hold[ Table[BernoulliB[i], {i, 6000}], Table[Fibonacci[i], {i, 0, , 00}], (*Table[HarmonicNumber[i],{i,000}], Maple fails*) Table[Zeta[i], {i, 0, 0 000, 000}], Table[Binomial[i^, i], {i,, 000}] ]]; evaluatetest["randomnumbers", Hold[ makedata["realmatrix", 000], makedata["realmatrix", 000], makedata["integermatrix", 000], makedata["integermatrix", 000], makedata["extendedmatrix", 000], makedata["extendedmatrix00", 000], RandomVariate[NormalDistribution[0, ], ^ 7], RandomVariate[PoissonDistribution[], ^ 7], RandomVariate[BinomialDistribution[, 0.], ^ 7] ]]; highprecisionreport "ManyDigits", Pi, Sqrt[], Sin[] +, Exp[], Log[], Tan[ + I], ; + Sin[] highprecisionreport "FewerDigits", BesselK[, ], Gamma, Erf[], 000 ;

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