Arithmetic Ops Group Methods for clifford objects

Different arithmetic operators for clifford objects, including many different types of multiplication.

Usage

# S3 method for class '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 %dot% 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 or coerced if needed
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

Details

The function Ops.clifford() passes unary and binary arithmetic operators “+”, “-”, “*”, “/” and “^” to the appropriate specialist function. 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)`
$A B = \sum_{r, s} {⟨ A ⟩}_{r} {⟨ B ⟩}_{s}$	Inner product	`A %.% B = cliffdotprod(A,B)`
$A \cdot B = \sum_{\binom{r \neq 0}{s \neq 0}}^{} {⟨ {⟨ A ⟩}_{r} {⟨ B ⟩}_{s} ⟩}_{\| s - r \|}$	Outer product	`A %^% B = wedge(A,B)`
$A \land B = \sum_{r, s} {⟨ {⟨ A ⟩}_{r} {⟨ B ⟩}_{s} ⟩}_{s + r}$	Fat dot product	`A %o% B = fatdot(A,B)`
$A ∙ B = \sum_{r, s} {⟨ {⟨ A ⟩}_{r} {⟨ B ⟩}_{s} ⟩}_{\| s - r \|}$	Left contraction	`A %_\|% B = lefttick(A,B)`
$A ⌋ B = \sum_{r, s} {⟨ {⟨ A ⟩}_{r} {⟨ B ⟩}_{s} ⟩}_{s - r}$	Right contraction	`A %\|_% B = righttick(A,B)`
$A ⌊ B = \sum_{r, s} {⟨ {⟨ A ⟩}_{r} {⟨ B ⟩}_{s} ⟩}_{r - s}$	Cross product	`A %X% B = cross(A,B)`
$A \times B = \frac{1}{2_{}} (A B - B A)$	Scalar product	`A %star% B = star(A,B)`
$A * B = \sum_{r, s} {⟨ {⟨ A ⟩}_{r} {⟨ B ⟩}_{s} ⟩}_{0}$	Euclidean product	`A %euc% B = eucprod(A,B)`

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.

Binary operator %dot% is a synonym for %.%, which causes problems for rmarkdown.

Function clifford_inverse() returns an inverse for nonnull Clifford objects $Cl (p, q)$ for $p + q \leq 5$ , and a few other special cases. The functionality 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:

$A * B = {⟨ A B ⟩}_{0} NB: notation used by both Perwass and Hestenes$

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

$A ⋆ B = A * B^{†} = {⟨ A B^{†} ⟩}_{0} Perwass$

where $B^{†}$ 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 $A * A$ 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 $A ⌋ B$ and $A ⌊ B$ . See the vignette for more details.

Division, as in idiom x/y, is defined as x*clifford_inverse(y). Function clifford_inverse() uses the method set out by Hitzer and Sangwine but is limited to $p + q \leq 5$ .

The Lie bracket, $[x, y]$ is implemented in the package using idiom such as .[x,y], and this is documented at dot.Rd.

Many of the functions documented here use low-level helper functions that wrap C code. For example, fatdot() uses c_fatdotprod(). These are documented at lowlevel.Rd.

Value

The high-level functions documented here return a clifford object. The low-level functions are not really intended for the end-user.

Author

Robin K. S. Hankin

Note

All the different Clifford products have binary operators for convenience including the wedge product %^%. However, as an experimental facility, the caret “^” returns either multiplicative powers [as in A^3=A*A*A], or a wedge product [as in A^B = A %^% B = wedge(A,B)] depending on the class of the second argument. I don't see that “A ^ B” is at all ambiguous but OTOH I might withdraw it if it proves unsatisfactory for some reason.

Compare the stokes package, where multiplicative powers do not really make sense and A^B is interpreted as a wedge product of differential forms $A$ and $B$ . In stokes, the wedge product is the sine qua non for the whole package and needs a terse idiomatic representation (although there A%^%B returns the wedge product too).

Using %.% causes severe and weird difficult-to-debug problems in markdown documents.

References

E. Hitzer and S. Sangwine 2017. “Multivector and multivector matrix inverses in real Clifford algebras”. Applied Mathematics and Computation 311:375-389

Examples


u <- rcliff(5)
v <- rcliff(5)
w <- rcliff(5)

u
#> Element of a Clifford algebra, equal to
#> + 6 + 5e_4 + 4e_5 - 5e_245 - 4e_36
v
#> Element of a Clifford algebra, equal to
#> + 3 - 4e_4 - 2e_125 - 1e_1236 - 3e_46 + 1e_256
u*v
#> Element of a Clifford algebra, equal to
#> - 2 - 12e_12 - 9e_4 - 10e_14 - 12e_34 + 12e_5 - 20e_25 - 12e_125 + 4e_235 +
#> 16e_45 - 15e_245 - 10e_1245 - 15e_6 - 4e_26 - 12e_36 - 6e_1236 - 23e_46 -
#> 16e_346 + 5e_12346 - 9e_256 - 4e_12356 + 12e_456 - 5e_2456 - 5e_13456

u+(v+w) == (u+v)+w            # should be TRUE by associativity of "+"
#> [1] TRUE
u*(v*w) == (u*v)*w            # should be TRUE by associativity of "*"
#> [1] TRUE
u*(v+w) == u*v + u*w          # should be TRUE by distributivity
#> [1] TRUE

# Now if x,y are _vectors_ we have:

x <- as.1vector(sample(5))
y <- as.1vector(sample(5))
x*y == x%.%y + x%^%y
#> [1] TRUE
x %^% y == x %^% (y + 3*x)  
#> [1] TRUE
x %^% y == (x*y-x*y)/2        # should be TRUE 
#> [1] FALSE

#  above are TRUE for x,y vectors (but not for multivectors, in general)


## Inner product "%.%" is not associative:
x <- rcliff(5,g=2)
y <- rcliff(5,g=2)
z <- rcliff(5,g=2)
x %.% (y %.% z) == (x %.% y) %.% z
#> [1] FALSE

## Other products should work as expected:

x %|_% y   ## left contraction
#> Element of a Clifford algebra, equal to
#> + 17 - 20e_1 + 15e_2 + 20e_12 - 4e_3 + 4e_4 + 4e_5 - 16e_45 + 5e_6 - 20e_46
x %_|% y   ## right contraction
#> Element of a Clifford algebra, equal to
#> + 17 - 12e_1 - 16e_2 + 4e_4 - 3e_5 + 12e_35
x %o% y    ## fat dot product
#> Element of a Clifford algebra, equal to
#> + 17 - 32e_1 - 1e_2 + 20e_12 - 4e_3 + 8e_4 + 1e_5 + 12e_35 - 16e_45 + 5e_6 -
#> 20e_46
x ^ y        ## Experimental wedge product idiom, plain caret
#> Element of a Clifford algebra, equal to
#> + 16 - 12e_1 - 16e_2 + 20e_12 - 4e_3 - 3e_13 - 4e_23 + 8e_4 + 3e_14 + 4e_24 +
#> 5e_124 - 1e_34 + 12e_35 + 15e_1235 - 16e_45 + 12e_145 + 16e_245 - 3e_345 -
#> 20e_46 + 15e_146 + 20e_246 + 15e_3456

Arithmetic Ops Group Methods for `clifford` objects