asymptote: Mathematical functions
6.11.4 Mathematical functions
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'Asymptote' has built-in versions of the standard 'libm' mathematical
real(real) functions 'sin', 'cos', 'tan', 'asin', 'acos', 'atan', 'exp',
'log', 'pow10', 'log10', 'sinh', 'cosh', 'tanh', 'asinh', 'acosh',
'atanh', 'sqrt', 'cbrt', 'fabs', 'expm1', 'log1p', as well as the
identity function 'identity'. 'Asymptote' also defines the order 'n'
Bessel functions of the first kind 'Jn(int n, real)' and second kind
'Yn(int n, real)', as well as the gamma function 'gamma', the error
function 'erf', and the complementary error function 'erfc'. The
standard real(real, real) functions 'atan2', 'hypot', 'fmod',
'remainder' are also included.
The functions 'degrees(real radians)' and 'radians(real degrees)' can
be used to convert between radians and degrees. The function
'Degrees(real radians)' returns the angle in degrees in the interval
[0,360). For convenience, 'Asymptote' defines variants 'Sin', 'Cos',
'Tan', 'aSin', 'aCos', and 'aTan' of the standard trigonometric
functions that use degrees rather than radians. We also define complex
versions of the 'sqrt', 'sin', 'cos', 'exp', 'log', and 'gamma'
functions.
The functions 'floor', 'ceil', and 'round' differ from their usual
definitions in that they all return an int value rather than a real
(since that is normally what one wants). The functions 'Floor', 'Ceil',
and 'Round' are respectively similar, except that if the result cannot
be converted to a valid int, they return 'intMax' for positive arguments
and 'intMin' for negative arguments, rather than generating an integer
overflow. We also define a function 'sgn', which returns the sign of
its real argument as an integer (-1, 0, or 1).
There is an 'abs(int)' function, as well as an 'abs(real)' function
(equivalent to 'fabs(real)'), an 'abs(pair)' function (equivalent to
'length(pair)').
Random numbers can be seeded with 'srand(int)' and generated with the
'int rand()' function, which returns a random integer between 0 and the
integer 'randMax'. The 'unitrand()' function returns a random number
uniformly distributed in the interval [0,1]. A Gaussian random number
generator 'Gaussrand' and a collection of statistics routines, including
'histogram', are provided in the base file 'stats.asy'. The functions
'factorial(int n)', which returns n!, and 'choose(int n, int k)', which
returns n!/(k!(n-k)!), are also defined.
When configured with the GNU Scientific Library (GSL), available from
<http://www.gnu.org/software/gsl/>, 'Asymptote' contains an internal
module 'gsl' that defines the airy functions 'Ai(real)', 'Bi(real)',
'Ai_deriv(real)', 'Bi_deriv(real)', 'zero_Ai(int)', 'zero_Bi(int)',
'zero_Ai_deriv(int)', 'zero_Bi_deriv(int)', the Bessel functions 'I(int,
real)', 'K(int, real)', 'j(int, real)', 'y(int, real)', 'i_scaled(int,
real)', 'k_scaled(int, real)', 'J(real, real)', 'Y(real, real)',
'I(real, real)', 'K(real, real)', 'zero_J(real, int)', the elliptic
functions 'F(real, real)', 'E(real, real)', and 'P(real, real)', the
Jacobi elliptic functions 'real[] sncndn(real,real)', the
exponential/trigonometric integrals 'Ei', 'Si', and 'Ci', the Legendre
polynomials 'Pl(int, real)', and the Riemann zeta function 'zeta(real)'.
For example, to compute the sine integral 'Si' of 1.0:
import gsl;
write(Si(1.0));
'Asymptote' also provides a few general purpose numerical routines:
'real newton(int iterations=100, real f(real), real fprime(real), real x, bool verbose=false);'
Use Newton-Raphson iteration to solve for a root of a real-valued
differentiable function 'f', given its derivative 'fprime' and an
initial guess 'x'. Diagnostics for each iteration are printed if
'verbose=true'. If the iteration fails after the maximum allowed
number of loops ('iterations'), 'realMax' is returned.
'real newton(int iterations=100, real f(real), real fprime(real), real x1, real x2, bool verbose=false);'
Use bracketed Newton-Raphson bisection to solve for a root of a
real-valued differentiable function 'f' within an interval
['x1','x2'] (on which the endpoint values of 'f' have opposite
signs), given its derivative 'fprime'. Diagnostics for each
iteration are printed if 'verbose=true'. If the iteration fails
after the maximum allowed number of loops ('iterations'), 'realMax'
is returned.
'real simpson(real f(real), real a, real b, real acc=realEpsilon, real dxmax=b-a)'
returns the integral of 'f' from 'a' to 'b' using adaptive Simpson
integration.
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