clifford (version 1.0-2)

Ops.clifford: Arithmetic Ops Group Methods for clifford objects

Description

Allows arithmetic operators to be used for multivariate polynomials such as addition, multiplication, integer powers, etc.

Usage

# S3 method for clifford
Ops(e1, e2)
clifford_negative(C)
geoprod(C1,C2)
clifford_times_scalar(C,x)
clifford_plus_clifford(C1,C2)
clifford_eq_clifford(C1,C2)
clifford_inverse(C)
cliffdotprod(C1,C2)
fatdot(C1,C2)
lefttick(C1,C2)
righttick(C1,C2)
wedge(C1,C2)
scalprod(C1,C2=rev(C1),drop=TRUE)
eucprod(C1,C2=C1,drop=TRUE)
maxyterm(C1,C2=as.clifford(0))
C1 %.% C2
C1 %^% C2
C1 %X% C2
C1 %star% C2
C1 % % C2
C1 %euc% C2
C1 %o% C2
C1 %_|% C2
C1 %|_% C2

Arguments

e1,e2,C,C1,C2

Objects of class clifford

x

Scalar, length one numeric vector

drop

Boolean, with default TRUE meaning to return the constant coerced to numeric, and FALSE meaning to return a (constant) Clifford object

Value

The high-level functions documented here return an object of clifford. But don't use the low-level functions.

Details

The function Ops.clifford() passes unary and binary arithmetic operators “+”, “-”, “*”, “/” and “^” to the appropriate specialist function.

Functions c_foo() are low-level helper functions that wrap the C code; function maxyterm() returns the maximum index in the terms of its arguments.

The package has several binary operators:

Geometric product	A*B = geoprod(A,B)	\(\displaystyle AB=\sum_{r,s}\left\langle A\right\rangle_r\left\langle B\right\rangle_s\)
Inner product	A %.% B = cliffdotprod(A,B)	\(\displaystyle A\cdot B=\sum_{r\neq 0\atop s\ne 0}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{\left|s-r\right|}\)
Outer product	A %^% B = wedge(A,B)	\(\displaystyle A\wedge B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{s+r}\)
Fat dot product	A %o% B = fatdot(A,B)	\(\displaystyle A\bullet B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{\left|s-r\right|}\)
Left contraction	A %_|% B = lefttick(A,B)	\(\displaystyle A\rfloor B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{s-r}\)
Right contraction	A %|_% B = righttick(A,B)	\(\displaystyle A\lfloor B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{r-s}\)
Cross product	A %X% B = cross(A,B)	\(\displaystyle A\times B=\frac{1}{2_{\vphantom{j}}}\left(AB-BA\right)\)
Scalar product	A %star% B = star(A,B)	\(\displaystyle A\ast B=\sum_{r,s}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_0\)
Euclidean product	A %euc% B = eucprod(A,B)	\(\displaystyle A\star B= A\ast B^\dag\)

In R idiom, the geometric product geoprod(.,.) has to be indicated with a “*” (as in A*B) and so the binary operator must be %*%: we need a different idiom for scalar product, which is why %star% is used.

Because geometric product is often denoted by juxtaposition, package idiom includes a % % b for geometric product.

Function clifford_inverse() is problematic as nonnull blades always have an inverse; but function is.blade() is not yet implemented. Blades (including null blades) have a pseudoinverse, but this is not implemented yet either.

The scalar product of two clifford objects is defined as the zero-grade component of their geometric product:

$$ A\ast B=\left<AB\right>_0\qquad{\mbox{NB: notation used by both Perwass and Hestenes}} $$

In package idiom the scalar product is given by A %star% B or scalprod(A,B). Hestenes and Perwass both use an asterisk for scalar product as in “\(A*B\)”, but in package idiom, the asterisk is reserved for geometric product.

Note: in the package, A*B is the geometric product.

The Euclidean product (or Euclidean scalar product) of two clifford objects is defined as

$$ A\star B=A\ast B^\dag=\left<AB^\dag\right>_0\qquad{\mbox{Perwass}} $$

where \(B^\dag\) denotes Conjugate [as in Conj(a)]. In package idiom the Euclidean scalar product is given by eucprod(A,B) or A %euc% B, both of which return A * Conj(B).

Note that the scalar product \(A\ast A\) can be positive or negative [that is, A %star% A may be any sign], but the Euclidean product is guaranteed to be non-negative [that is, A %euc% A is always positive or zero].

Dorst defines the left and right contraction (Chisholm calls these the left and right inner product) as \(A\rfloor B\) and \(A\lfloor B\). See the vignette for more details.

See Also

scalprod

Examples

# NOT RUN {
u <- rcliff(5)
v <- rcliff(5)
w <- rcliff(5)

u*v

u^3


u+(v+w) == (u+v)+w              # should be TRUE 
u*(v*w) == (u*v)*w              # should be TRUE 
u %^% v == (u*v-v*u)/2        # should be TRUE 

# Now if x,y,z are _vectors_ we would have:

x <- as.1vector(5)
y <- as.1vector(5)
x*y == x%.%y + x%^%y        # should be TRUE
x %^% y == x %^% (y + 3*x)  # should be TRUE 

## Inner product is "%.%" is not associative:
 rcliff(5,g=2) -> x
 rcliff(5,g=2) -> y
 rcliff(5,g=2) -> z
x %.% (y %.% z)
(x %.% y) %.% z

## Geometric product *is* associative:
x * (y * z)
(x * y) * z


# }