### Recursive Computational Formulae:

• Determinants: fln(det) = n*(fln-1(det)+2)-1 flops when computing nth 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.

• Cramer's Rule: fln(cr) (n+1)*(fln(det)+1)-1 flops for solving nth 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).

### Asymptotic Analysis:

• The main idea is assuming the fln(det) ~ cn*n! for some bounded factor cn with c1 := 0, using this relation in the determinant recursion relation, solving the resulting recursion for cn and finding that it is the first (n-1) terms of the exponential series of e1 := e with the remainder going quickly to zero as n ---> infinity by Taylor's theorem with remainder and an exponential generalization of Stirling's approximation given below.

• Determinant Computational Work fln(det) ~ e*n! flops as n ---> infinity when computing nth order determinants.

• Cramer's Rule Computational Work fln(cr) ~ e*(n+1)! flops as n ---> infinity when solving nth order algebraic systems of equations.

### Assumptions:

1. MegaFlop Machine at 1µsec/flop or Tn(cr) = fln(cr)*10(-6).

2. Precision with chopping to the nearest 1 digit decimal.

### Computational Estimate as Function of Problem Size n:

Problem
Order n
fln(cr)
(flops)
Tn(cr)
(time units)

11F1µs F = flops, microseconds = µs = 10(-6)s
210F10µs"
360F60µs"
4300F300µs"
52KF2ms KF = KiloFlops = 103flops, milliseconds = ms = 10(-3)s
610KF10ms"
7100KF100ms"
81MF1s MF = MegaFlops = 106flops, seconds = s
910MF10s"
10100MF2min"
111GF20min GF = GigaFlops = 109flops
1220GF5hours"
13200GF3days"
144TF6weeks TF = TeraFlops = 1012flops
1560TF2years"
161PF30decades PF = PetaFlops = 1015flops
1720PF6centuries"
18300PF10Kyears 1Kyear = 103years = 1 Millenium
197EF200Kyears EF = ExaFlops = 1018flops
20100EF4MyearsMyears = Million years

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

### Computational Estimate as Function of Machine Performance:

Order Machine
Performance
Problem
10×10
Problem
20×20

100KFLOPS20min40Myrs old PC;M = Mega := 106, K = Kilo := 103
MegaFLOPS 2min4Myrstypical workstation; M = Mega := 106
GigaFLOPS 0.1sec4Kyrs typical supercomputer; G = Giga := 109
TeraFLOPS 0.1msec4yrs ultracomputer for ca. 1995; Tera := 1012

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)*n3 FLOPS, n ---> infinity, so a 20×20 can be solved in time ~ 5msec., on a MFLOPS workstation.

100×100 case data: J. J. Dongarra, LINPACK Benchmark: Performance of Various Computers Using Standard Linear Equations Software, Supercomputing Review, vol. 5, pp. 54-63, March 1992.

Compare with Hanson-Stirling's exponential-formula: n! ~ sqrt{2*Pi*n}e-nnn as n ---> infinity.

F. B. Hanson
MCS 471 Coordinator
Web Source: http://www.math.uic.edu/~hanson/cramers.html

Email Comments or Questions to  hanson@uic.edu