Unformatted text preview: oped by Petzold (cf. 4]) is a backward di erence code for sti di erential and di erential algebraic systems. DASSL addresses di erential equations in
the implicit form
f (t y y0) = 0:
It can also solve these systems when the Jacobian fy0 is not invertible. In these
cases, y0 cannot be determined and the problem is a system of di erential-algebraic
equations. A simple two-dimensional example is
y1 = a11 y1 + a12 y2 + b1 0 = a21 y1 + a22 y2 + b2 :
Many of these codes have been incorporated into scienti c subroutine libraries. For
example, the IVPAG procedure in the IMSL Library is a modi cation of Gear's 10]
original Adam's and BDF procedure. The procedures D02PCF and D02EJF, respectively,
are variable order, variable step Adams and BDF codes in the NAG Library.
Example 5.7.7. Hairer et al. 12], Section III.7, compare several codes on a suite of six
non-sti problems. We report their results in Figures 5.7.1 and 5.7.2. Codes that have not
already been identi ed include DOPRI8, which is the eighth-order Dormand and Prince
Runge-Kutta method 9], and D02CAF, which is similar to LSODE and EPISODE, and
contained in the NAG library. The six problems used for testing are described in Hairer
et al. 12], Section II.10. The legends and storage requirements for each code are given
in Table 5.7.1.
The Runge-Kutta code DOPRI8 generally uses more function evaluations but less
CPU time than the LMM codes. The test problems are fairly simple, but the performance
58 Figure 5.7.1: Number of function evaluations for a six-problem suite of non-sti problems
59 Figure 5.7.2: CPU times (seconds) for a six-problem suite of non-sti problems 12].
60 of the high-order Runge-Kutta method should not be ignored. Conditions, however, could
be reversed with more complicated functions. Of the LMM codes, DEABM seems to be
the most e cient and EPISODE the least. From Table 5.7.1, we see that DOPPI8
requires about half of the storage per equation as do the LMMs. Of the LMM codes,
DEBEAM has the greatest storage cost.
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