numeric
::odesolve2
Numerical solution of an ordinary differential equation
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numeric::odesolve2(f
, t_{0}
, Y_{0}
, <method
>, <RememberLast>, <RelativeError = rtol
>, <AbsoluteError = atol
>, <Stepsize = h
>, <MaxStepsize = h_{max}
>)
numeric::odesolve2( f, t_{0}, Y_{0},
… )
returns a function representing the numerical
solution Y(t) of
the first order differential equation (dynamical system) , Y(t_{0})
= Y_{0} with and .
The utility function numeric::ode2vectorfield
may
be used to produce the input parameters f, t0, Y0
from
a set of differential expressions representing the ODE. Cf. Example 1.
The function generated by Y := numeric::odesolve2(f,
t_{0}, Y_{0})
is
essentially
Y := t > numeric::odesolve(t_0..t, f, Y_0)
.
Numerical integration is launched, when Y
is
called with a real numerical argument. The call Y(t)
returns
the solution vector in a format corresponding to the type of the initial
condition Y_{0} with
which Y
was defined: Y(t)
either
yields a list or a 1dimensional array.
If t
is not a real numerical value, then Y(t)
returns
a symbolic function call.
See the help page of numeric::odesolve
for
details on the parameters and the options.
The options Alldata = n
and Symbolic
accepted
by numeric::odesolve
have
no effect: numeric::odesolve2
ignores these options.
Without RememberLast
, the function Y remembers
all values it has computed. When calling Y(T),
it searches its remember table for the time t0 < T < t closest to t,
and integrates from T to t using
the previously computed Y(T) as
initial value. Here, t0 is the time for which the
initial value of the ODE is given. This reduces the costs of a call
considerably, if Y
must be evaluated many times,
for example, when plotting the ODE solution. The best approach is
to call Y only with a monotonically increasing
(or decreasing) sequence of values t starting from t0.
Further, the function must be reinitialized whenever DIGITS
is
increased. See Example 3.
The function returned by numeric::odesolve2
is
sensitive to the environment variable DIGITS
, which determines
the numerical working precision.
Without RememberLast
, the function returned
by numeric::odesolve2
uses option remember
.
The numerical solution of the initial value problem , y(0) = 2 is represented by the following function Y = [y]:
f := (t, Y) > [t*sin(Y[1])]:
Alternatively, the utility function numeric::ode2vectorfield
can be used
to generate the input parameters in a more intuitive way:
[f, t0, Y0] := [numeric::ode2vectorfield({y'(t) = t*sin(y(t)), y(0) = 2}, [y(t)])]
Y := numeric::odesolve2(f, t0, Y0)
The procedure Y
starts the numerical integration
when called with a numerical argument:
Y(2), Y(0), Y(0.1), Y(PI + sqrt(2))
Calling Y
with a symbolic argument yields
a symbolic call:
Y(t), Y(t + 5), Y(t^2  4)
eval(subs(%, t = PI))
The numerical solution can be plotted. Note that Y(t)
returns
a list, so we plot the list element Y(t)[1]
:
plotfunc2d(Y(t)[1], t = 5..5):
delete f, t0, Y0, Y:
We consider the differential equation with initial conditions y(0) = 0, . The second order equation is converted to a first order system for :
.
f := (t, Y) > [Y[2], Y[1]^2]: t0 := 0: Y0 := [0, 1]: Y := numeric::odesolve2(f, t0, Y0): Y(1), Y(PI)
delete f, t0, Y0, Y:
We consider the system
:
f := (t, Y) > [Y[1] + Y[2], Y[1]  Y[2]]: Y := numeric::odesolve2(f, 0, [1, I]): DIGITS := 5: Y(1)
Increasing DIGITS
does not lead to a more accurate
result because of the remember mechanism:
DIGITS := 15: Y(1)
This is the previous value computed with 5 digits, printed with
15 digits. Indeed, only 5 digits are correct. For getting a result
that is accurate to full precision, one has to erase the remember
table via Y:=subsop(Y,5=NIL)
. Alternatively, one
may create a new numerical solution with a fresh (empty) remember
table:
Y := numeric::odesolve2(f, 0, [1, I]): Y(1)
delete f, Y, DIGITS:
We demonstrate the effect of the option RememberLast
.
We consider the ODE
:
f := (t, Y) > [Y[1] + sin(t)]: Y := numeric::odesolve2(f, 0, [1]): Z := numeric::odesolve2(f, 0, [1], RememberLast):
After many calls of Y
, its remember table
has grown large. In each call, searching the remember table for input
parameters close to the present time value becomes expensive. Created
with RememberLast
, the procedure Z
does
not remember all its previously computed values apart from the last
one. Consequently, it becomes faster than Y
:
time(for i from 1 to 1000 do Y(i/100) end)*msec, time(for i from 1 to 1000 do Z(i/100) end)*msec
Apart from the efficiency, the values returned by Y
and Z
coincide:
Y(10.5), Z(10.5)
delete f, Y, Z, i:

A procedure representing the vector field of the dynamical system 

A numerical real value for the initial time 

A list or 1dimensional array of numerical values representing the initial value 

One of the RungeKutta schemes listed below. 

Option, specified as Name of the RungeKutta scheme. For details, see the documentation
of 

Modifies the internal remember mechanism: the procedure returned
by Without this option, the procedure returned by This option is highly recommended when the numerical procedure
returned by 

Option, specified as Forces internal numerical RungeKutta steps to use step sizes
with relative local discretization errors below 

Option, specified as Forces internal numerical RungeKutta steps to use step sizes
with absolute local discretization errors below 

Option, specified as Switches off the internal error control and uses a RungeKutta
iteration with constant step size 

Option, specified as Restricts adaptive step sizes to values not larger than 
Procedure.