We continue to compare our toolbox with Maple and Symbolic Math Toolbox (VPA). This time we focus on main operations in dense linear algebra – solvers and factorizations.

Every case/routine is tested on a random `100 x 100`

matrix. Average timing over ten tests is used.

Matrix Type | Timing (sec) | Speed-up (times) | |||
---|---|---|---|---|---|

VPA | Maple | Advanpix | Over VPA | Over Maple | |

General Real | 0.99 | 0.58 | 0.02 | 42.22 | 24.67 |

Symmetric Positive Definite | 0.99 | 0.63 | 0.01 | 77.10 | 49.31 |

General Complex | 3.13 | 2.43 | 0.08 | 38.34 | 29.74 |

Hermitian Positive Definite | 3.72 | 2.56 | 0.05 | 70.85 | 48.82 |

In average Advanpix solver is faster by **60** times over VPA and **40** times over Maple.

Factorization | Timing (sec) | Speed-up (times) | |||
---|---|---|---|---|---|

VPA | Maple | Advanpix | Over VPA | Over Maple | |

A is a general real matrix: | |||||

[L,U] = lu(A) | 1.67 | 0.61 | 0.02 | 78.26 | 28.29 |

[Q,R] = qr(A) | 3.46 | 2.60 | 0.08 | 44.44 | 33.46 |

[U,S,V] = svd(A) | 30.85 | 20.99 | 0.30 | 101.66 | 69.17 |

S = svd(A) | 15.88 | 2.51 | 0.07 | 223.54 | 35.35 |

[V,D] = eig(A) | 57.25 | 29.54 | 0.72 | 79.20 | 40.86 |

lambda = eig(A) | 21.21 | 26.90 | 0.34 | 62.31 | 79.01 |

A is a general complex matrix: | |||||

[L,U] = lu(A) | 2.74 | 2.56 | 0.07 | 41.63 | 38.89 |

[Q,R] = qr(A) | 10.99 | 11.39 | 0.25 | 43.38 | 44.99 |

[U,S,V] = svd(A) | 65.56 | 57.53 | 0.81 | 80.96 | 71.04 |

S = svd(A) | 25.28 | 10.22 | 0.26 | 98.24 | 39.71 |

[V,D] = eig(A) | 89.30 | 129.54 | 2.85 | 31.34 | 45.46 |

lambda = eig(A) | 33.14 | 113.99 | 1.29 | 25.67 | 88.32 |

As for factorizations, Advanpix toolbox outperforms VPA by **75** times and Maple by **50** times in average. Worth to note the massive speed gain of singular value decomposition (SVD) in particular.

Our test environment and scripts:

`Core i7-3960X Extreme Edition, 3900 MHz, 64GB, Windows 7 64-bit`

.`MATLAB R2013a, Multiprecision Computing Toolbox 3.6.1`

.`Maple 17 64-bit`

.- Maple test script: maple_test.mw.
- Matlab test script for Advanpix vs. VPA: advanpix-vs-vpa.zip.

Sincere gratitude to AndrĂ© Van Moer for ongoing support, fruitful discussions and for running the tests on his powerful computer.

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