For a <constant-function> evaluation and derivative computation are trivial.
The <function> class is an abstract class for a general function. This function will usually accept vector arguments, the dimensions of domain and image are fixed when defining the function. If the function is differentiable, the gradient matrix can be obtained by evaluating the gradient slot.
A <linear-function> is determined by a matrix A and a vector b. It represents the map x -> Ax+b.
This class implements a periodic polygon.
This class implements a function which maps the unit interval to a polygon.
A <special-function> provides its own evaluation and gradient computation.
Returns a special function drawing a polar around midpoint with distance given by the function or number radial-distance with angular velocity omega. Without arguments it yields a function mapping R^1 isometrically to S^1.
On a regular partition of the unit interval interpolating values y are given. This function returns an interpolating spline.
Degree of a polynomial
Returns t if f is differentiable or differentiable of the given degree.
Differentiate a multivariate polynomial wrt the variable given by INDEX.
Returns a matrix A suitable for describing the ellipse as (Ax,x)=1.
Generic evaluation of functions on an argument. Numbers and arrays are treated as constants. Special evaluation is defined for multivariate polynomials on vectors and for <function> objects.
Generic evaluation of gradients of differentiable functions.
The factors if the multivariate polynomial poly is an exterior product of lower-variate (e.g. univariate) polynomials.
Returns a function which uses its first coordinate as a homotopy parameter.
Finds zeros of functions in 1d by the interval method.
Calculate the Langevin function to an accuracy of about 15 digits
Calculate the function Langevinx(x):=Langevin(x)/x
Constructor which simplifies the coefficient list.
Maximal partial degree of a polynomial.
Multiple evaluations of func may be optimized.
Multiple evaluations may be optimized.
Returns n-variate monomials of degree being equal or being lower or equal than deg. Examples: (n-variate-monomials-of-degree 2 2) -> (x2^2 x1*x2 x1^2) (n-variate-monomials-of-degree 2 2 ’<=) -> (1 x2 x1 x2^2 x1*x2 x1^2)
Computes a very accurate real derivative for functions which can be applied to complex arguments.
Warning: This trick can only be applied once, i.e. derivatives of higher order cannot be computed by multiple application of this function!
Computes the numerical gradient of func at pos.
Partial degree in variable INDEX of a multivariate polynomial.
Multiplies two polynomials P1 and P2.
Raises the polynomial P to power N.
Multiply the polynomials poly1 and poly2 considered as polynomials in separate variables.
Multivariate polynomial. The coefficients are represented as nested lists. A special case are 0-variate polynomials which are simply scalars. If factors is present, the polynomial was constructed as an exterior product of polynomials of lower variance.
Returns a function which projects to the ellipsoid given by Q(x-midpoint)=1 where Q is the quadratic form associated with the matrix A.
Returns a function which projects to the sphere with given midpoint and radius.
Shifts a polynomial in dimension, i.e. variables starting from index k>=from get index k+shift.
Warning: works only for real-valued functions!
Constructs a special function between 1D-spaces from ordinary Lisp functions.
Degree of a multivariate polynomial
Number of variables on which a polynomial depends.
Returns a function which distorts the xn-coordinate by a factor f(x’). Also grad-f has to be provided.
Generates a zero of the same kind as F.