Welcome to Cascade v2.1. Cascade is a library designed to store and solve differential equations to arbitrary precision. This is accomplished through the use of Arblib's acb_t data type, which uses ball arithmetic to store arbitrary precision floating point numbers with error bounds. Solutions are computed as power series expansions about the origin using a recursion relation between the coefficients. With the help of analytic continuation this can be turned into a solution anywhere in the complex plane.

For a similar library using p-adic arithmetic, see Implode.

Author: Matthias Gessinger

This library can be build from source as a shared object library through CMake. For a default installation, run the following commands:

mkdir build
cd build
make
make test

If all tests pass, finally install the library by running

sudo make install

#include <acb_poly.h>
#include <cascade.h>

int main ()
{
    acb_t z;
    acb_poly_t pol;
    acb_ode_t ODE;

    acb_init(z);
    acb_poly_init(pol);
    acb_ode_legendre(ODE, 4);

    acb_set_d(z, 0.375);
    acb_poly_set_coeff_acb(pol, 0, z);
    acb_ode_solve_fuchs(pol, ODE, 5, 1024);
    acb_ode_dump(ODE, NULL);
    acb_poly_printd(pol, 10);

    acb_clear(z);
    acb_poly_clear(pol);
    acb_ode_clear(ODE);
    flint_cleanup();
    return 0;
}

To compile the program, run

gcc test.c -lcascade -larb -lflint

The output of time ./a.out should then look something like this:

Order: 2
Degree: 2
diff_eq_poly(ODE,0) = (20 + 0j)  +/-  (0, 0j)	(0 + 0j)  +/-  (0, 0j)	(0 + 0j)  +/-  (0, 0j)
diff_eq_poly(ODE,1) = (0 + 0j)  +/-  (0, 0j)	(-2 + 0j)  +/-  (0, 0j)	(0 + 0j)  +/-  (0, 0j)
diff_eq_poly(ODE,2) = (1 + 0j)  +/-  (0, 0j)	(0 + 0j)  +/-  (0, 0j)	(-1 + 0j)  +/-  (0, 0j)

Solution:
[(0.375 + 0j)  +/-  (0, 0j)
(0 + 0j)  +/-  (0, 0j)
(-3.75 + 0j)  +/-  (2.8e-308, 0j)
(0 + 0j)  +/-  (0, 0j)
(4.375 + 0j)  +/-  (3.26e-313, 0j)]
real	0m0,009s
user	0m0,005s
sys	0m0,005s

Because Arb caches some constants internally, it is recommended to call flint_cleanup() at the end of your main program. This will clear Arb's internal cache and guarantee a clean output of Valgrind.

Cascade uses Arb to store complex numbers and Flint to handle memory management. Therefore both of these libraries have to be installed in order to build Cascade, and also to build programs using Cascade!