[6.63] Faster Numerical Matrix Inverses - Texas Instruments TI-89 Tip List

The plot shows the difference between the two function on the interval x = [5, 100]. The cursor readout
shows that we could use x = 60 as the transition point between the functions. The function value at this
point is about 0.989999885775, and the difference between the functions is about 6.3E-6. If we only
need 5 or 6 significant digits in this range, the approximation is good enough. If we need more
significant digits, we could increase the number of terms in the asymptotic expansion, but adding four
more terms does not decrease the difference between the functions. We would need a lot more terms
to increase the accuracy.
For more details on asymptotic expansions, you might start with section 18.6 of Advanced Engineering
Mathematics, 6th edition, by Erwin Krezszig, Wiley, 1988. The entry for asymptotic series at Eric
Weisstein's World of Mathematics site (http://mathworld.wolfram.com/AsymptoticSeries.html) also has
a good bibliography.

[6.63] Faster numerical matrix inverses

The calculator has a built-in matrix inverse operator, ^-1, but the C program inv(), by Rafael Humberto
Padilla Velazquez, is much faster for inverses of matrices with floating-point elements. The call syntax
is
inv(m)
where m is the matrix or matrix name. The matrix inverse is saved in the global variable _asminfo.
This table shows some typical execution times for random integer matrices of various sizes.
Matrix size
5 x 5
10 x 10
15 x 15
20 x 20
inv() is considerably faster than the built-in inversion, and the residuals slightly better, as well. The
residual is the element with the largest absolute value of matrix E, where
E = M $ M −1 − I
M is the original matrix and I is the identity matrix. So, the residual is not actually the error in the
inverse, but instead the 'round-trip' error in finding the inverse and evaluating the identity above.
m^-1 Maximum residual
2.1 sec
15.3 sec 6.9E-13
60.6 sec 1.6E-12
177.2 sec 2.1E-12
inv(m)
2E-13 1.8 sec
3.1 sec
11.4 sec
28.6 sec
Maximum residual
1E-13
1.4E-13
7E-13
6.8E-13
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