- Determinants:
flops when computing*fl*_{n}^{(det)}= n*(fl_{n-1}^{(det)}+2)-1th order determinants (each nth order determinant can be expanded as a sum of (n-1)st order determinants each multiplied by the elements of the row or column expanded by.*n* - Cramer's Rule:
flops for solving*fl*_{n}^{(cr)}(n+1)*(fl_{n}^{(det)}+1)-1th order algebraic systems of equations (A*x=b) (there are n components of the solution and each component by Cramer's rule is the quotient of the nth order determinant of the co-factor matrix that is found by replace the ith column of the coefficient matrix by the right hand side vector b then dividing the co-factor determinant by the determinant of A).*n*

- The main idea is assuming the
for some bounded factor*fl*_{n}^{(det)}~ c_{n}*n!with*c*_{n}, using this relation in the determinant recursion relation, solving the resulting recursion for*c*_{1}:= 0and finding that it is the first (n-1) terms of the exponential series of*c*_{n}with the remainder going quickly to zero as*e*^{1}:= eby Taylor's theorem with remainder and an exponential generalization of Stirling's approximation given below.*n ---> infinity* - Determinant Computational Work
flops as*fl*_{n}^{(det)}~ e*n!when computing*n ---> infinity*th order determinants.*n* - Cramer's Rule Computational Work
flops as*fl*_{n}^{(cr)}~ e*(n+1)!when solving*n ---> infinity*th order algebraic systems of equations.*n*

- MegaFlop Machine at
or*1µsec/flop*.*T*_{n}^{(cr)}= fl_{n}^{(cr)}*10^{(-6)} - Precision with chopping to the nearest 1 digit decimal.

Problem Order n |
fl_{n}^{(cr)}(flops) |
T_{n}^{(cr)}(time units) |
Comments |
---|---|---|---|

1 | 1F | 1µs | F = flops, microseconds = µs = 10^{(-6)}s |

2 | 10F | 10µs | " |

3 | 60F | 60µs | " |

4 | 300F | 300µs | " |

5 | 2KF | 2ms | KF = KiloFlops = 10^{3}flops, milliseconds = ms = 10^{(-3)}s |

6 | 10KF | 10ms | " |

7 | 100KF | 100ms | " |

8 | 1MF | 1s | MF = MegaFlops = 10^{6}flops, seconds = s |

9 | 10MF | 10s | " |

10 | 100MF | 2min | " |

11 | 1GF | 20min | GF = GigaFlops = 10^{9}flops |

12 | 20GF | 5hours | " |

13 | 200GF | 3days | " |

14 | 4TF | 6weeks | TF = TeraFlops = 10^{12}flops |

15 | 60TF | 2years | " |

16 | 1PF | 30decades | PF = PetaFlops = 10^{15}flops |

17 | 20PF | 6centuries | " |

18 | 300PF | 10Kyears | 1Kyear = 10^{3}years = 1 Millenium |

19 | 7EF | 200Kyears | EF = ExaFlops = 10^{18}flops |

20 | 100EF | 4Myears | Myears = Million years |

flops := Floating point operations (i.e., ** +, -, *, /**)

Order Machine Performance |
Problem 10×10 |
Problem 20×20 |
Comments |
---|---|---|---|

100KFLOPS | 20min | 40Myrs | old PC;M = Mega := 10^{6}, K = Kilo := 10^{3} |

MegaFLOPS | 2min | 4Myrs | typical workstation; M = Mega := 10^{6} |

GigaFLOPS | 0.1sec | 4Kyrs | typical supercomputer; G = Giga := 10^{9} |

TeraFLOPS | 0.1msec | 4yrs | ultracomputer for ca. 1995; Tera := 10^{12} |

Mileage will vary with computers, but do not even think about computing with
Cramer's rule for ** n>4**, say.

flops := FLoating point operations (i.e., ** +, -, *, /**)

FLOPS := FLoating point OPerations per Second (i.e., ** +, -, *, /**
per sec)

Gaussian Elimination with Back Substitution
** ~ (2/3)*n^{3}** FLOPS,

** 100×100** case data: J. J. Dongarra,

Compare with Hanson-Stirling's exponential-formula:

F. B. Hanson

MCS 471 Coordinator

**Email Comments or Questions to
hanson@uic.edu
**