3 Linear generic interface

Values of a special type gen-zero can be used with this interface to represent a zero vector of the appropriate type, or the scalar 0.

The interface contains following methods, that a type t can implement to support the interface:

• procedure (linear-add linear other) → t?

  linear : t?

  other : (or/c t? gen-zero?)

  Addition of two vectors.

  See add.

• procedure (linear-scale linear scalar) → t?

  linear : t?

  scalar : (or/c number? gen-zero?)

  Multiplication of a vector by a scalar.

  See scale.

• procedure (linear-dot linear other) → number?

  linear : t?

  other : (or/c t? gen-zero?)

  Inner product of two vectors.

  See dot.

These methods are assumed to satisfy the axioms of a vector space (or inner-product space), but this is not checked.

In addition, the following should hold, for any v : linear?:

• (equal? (linear-add v (gen-zero)) v)

• (equal? (linear-scale v (gen-zero)) (linear-scale v 0))

• (equal? (linear-dot v (gen-zero)) (gen-zero))

Otherwise, if a is gen-zero, the result is also gen-zero, and an error is raised for any other value of a.

This function should be used for a recursive method definition of linear-scale.

Otherwise, if v is gen-zero, u is returned, and an error is raised for any other value of a.

This function should be used for a recursive method definition of linear-add.

Otherwise, if v is gen-zero, the result is gen-zero, and an error is raised for any other value of a.

This function should be used for a recursive method definition of linear-dot.

3.1 Instances

Two values conform if any of the following statements are true:

• They are both numbers

• They are both null

• They are both pairs, whose car and cdr conform

• They are both vectors, where each element of one vector conforms with the corresponding element of the other

• They are both arrays of compatible shape and with conforming elements

• They are both proc-result structs, with both slots containing conforming values

• They are both boxes, with conforming contents

• They are both hash tables with the same keys, and where the values mapped to by each key conform.

Instances of gen:linear are defined for number?, null?, pair?, vector?, array?, proc-result?, box? and hash? so that any equivalence class of conforming values forms an inner-product space, where addition and scalar multiplication defined elementwise, and an inner product as the sum of the elementwise product.

3.2 Utilities for generic zero

The same as (scale u (gen-zero)).

Returns a value like u, but where any gen-zero occuring as all or part of u is replaced with a concrete zero of the appropriate kind, such that it conforms to v.

If u is not (or does not contain) gen-zero, the result is identically u. Notice too the effect of gen-zero in the last tail of a dotted list.

When applying foldl0, an improper list argument lst* with a value of gen-zero in its final position is accepted and is treated the same as if it were a proper list (that is, as if it’s final tail were null instead).