7.11 Package FL.FUNCTION

NIL

Class: <CONSTANT-FUNCTION>

For a <constant-function> evaluation and derivative computation are trivial.

Superclasses: <FUNCTION>

Direct slots:

• VALUE: Self-explanatory.

Class: <FUNCTION>

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.

Direct slots:

• DOMAIN-DIMENSION: Self-explanatory.
• IMAGE-DIMENSION: Self-explanatory.

Class: <LINEAR-FUNCTION>

A <linear-function> is determined by a matrix A and a vector b. It represents the map x -> Ax+b.

Superclasses: <FUNCTION>

Direct slots:

• A: Self-explanatory.
• B: Self-explanatory.

Class: <PERIODIC-POLYGON>

This class implements a periodic polygon.

Superclasses: <POLYGON>

Class: <POLYGON>

This class implements a function which maps the unit interval to a polygon.

Superclasses: <FUNCTION>

Direct slots:

• POINTS: A vector of points for the polygon.

Class: <SPECIAL-FUNCTION>

A <special-function> provides its own evaluation and gradient computation.

Superclasses: <FUNCTION>

Direct slots:

• EVALUATOR: Self-explanatory.
• GRADIENT: Self-explanatory.
• JET: Self-explanatory.

Function: CIRCLE-FUNCTION &OPTIONAL (RADIAL-DISTANCE 1.0) (MIDPOINT (COERCE (QUOTE (0.0 0.0)) (QUOTE DOUBLE-VEC))) (OMEGA 1.0) (PHI0 0.0)

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.

Function: CUBIC-SPLINE Y

On a regular partition of the unit interval interpolating values y are given. This function returns an interpolating spline.

Function: DEGREE POLY

Degree of a polynomial

Function: DIFFERENTIABLE-P F &OPTIONAL K

Returns t if f is differentiable or differentiable of the given degree.

Function: DIFFERENTIATE POLY INDEX

Differentiate a multivariate polynomial wrt the variable given by INDEX.

Function: ELLIPSE-MATRIX RADIUS EXCENTRICITY PHI

Returns a matrix A suitable for describing the ellipse as (Ax,x)=1.

Function: EVALUATE F X

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.

Function: EVALUATE-GRADIENT F X

Generic evaluation of gradients of differentiable functions.

Function: FACTORS POLY

The factors if the multivariate polynomial poly is an exterior product of lower-variate (e.g. univariate) polynomials.

Function: HOMOTOPY FUNC1 FUNC2

Returns a function which uses its first coordinate as a homotopy parameter.

Function: INTERVAL-METHOD FUNC A B ACCURACY

Finds zeros of functions in 1d by the interval method.

Function: LANGEVIN X

Calculate the Langevin function to an accuracy of about 15 digits

Function: LANGEVINX X

Calculate the function Langevinx(x):=Langevin(x)/x

Function: MAKE-POLYNOMIAL COEFFS

Constructor which simplifies the coefficient list.

Function: MAXIMAL-PARTIAL-DEGREE POLY

Maximal partial degree of a polynomial.

Function: MULTIPLE-EVALUATE FUNC POSITIONS

Multiple evaluations of func may be optimized.

Function: MULTIPLE-EVALUATE-GRADIENT F POSITIONS

Multiple evaluations may be optimized.

Function: N-VARIATE-MONOMIALS-OF-DEGREE N DEGREE &OPTIONAL (TYPE (QUOTE =))

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)

Function: NUMERICAL-COMPLEX-DERIVATIVE F

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!

Function: NUMERICAL-DERIVATIVE FUNC &KEY (SHIFT 1.0e-6)

Computes the numerical derivative of func at pos.

Function: PARTIAL-DEGREE POLY INDEX

Partial degree in variable INDEX of a multivariate polynomial.

Function: POLY* P1 P2

Multiplies two polynomials P1 and P2.

Function: POLY-EXPT P N

Raises the polynomial P to power N.

Function: POLY-EXTERIOR-PRODUCT POLY1 POLY2

Multiply the polynomials poly1 and poly2 considered as polynomials in separate variables.

Class: POLYNOMIAL

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.

Superclasses: <VECTOR>

Direct slots:

• COEFFS: Self-explanatory.
• FACTORS: Self-explanatory.
• GRADIENT: The gradient of a polynomial is only calculated on demand.

Function: PROJECT-TO-ELLIPSOID MIDPOINT A

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.

Function: PROJECT-TO-SPHERE MIDPOINT RADIUS

Returns a function which projects to the sphere with given midpoint and radius.

Function: SHIFT-POLYNOMIAL POLY SHIFT &OPTIONAL FROM

Shifts a polynomial in dimension, i.e. variables starting from index k>=from get index k+shift.

Function: SPARSE-REAL-DERIVATIVE SPARSE-F

Warning: works only for real-valued functions!

Function: SPECIAL-1D-FUNCTION F &OPTIONAL DF

Constructs a special function between 1D-spaces from ordinary Lisp functions.

Function: TOTAL-DEGREE POLY

Degree of a multivariate polynomial

Function: VARIANCE POLY

Number of variables on which a polynomial depends.

Function: XN-DISTORTION-FUNCTION F GRAD-F DIM

Returns a function which distorts the xn-coordinate by a factor f(x’). Also grad-f has to be provided.

Function: ZERO F

Generates a zero of the same kind as F.