Arithmetic Ops Group Methods for clifford objects
Ops.clifford.RdDifferent 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 %|_% C2Details
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) | |
| \(\displaystyle AB=\sum_{r,s}\left\langle A\right\rangle_r\left\langle B\right\rangle_s\) | Inner product | A %.% B = cliffdotprod(A,B) |
| \(\displaystyle A\cdot B=\sum_{r\neq 0\atop s\ne 0}^{\vphantom{s\neq 0}}\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_{\left|s-r\right|}\) | 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) |
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 \(\operatorname{Cl}(p,q)\) 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:
$$ A\ast B=\left\langle AB\right\rangle_0\qquad{\mbox{NB: notation used by both Perwass and Hestenes}} $$
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
$$ A\star B= A\ast B^\dagger= \left\langle AB^\dagger\right\rangle_0\qquad{\mbox{Perwass}} $$
where \(B^\dagger\) 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.
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, \(\left[x,y\right]\) 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.
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