Ops.clifford | R Documentation |
clifford
objectsDifferent arithmetic operators for clifford objects, including many different types of multiplication.
## 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
e1 , e2 , C , C1 , C2 |
Objects of class |
x |
Scalar, length one numeric vector |
drop |
Boolean, with default |
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) |
\displaystyle A\star B= A\ast B^\dagger |
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
.
The high-level functions documented here return a clifford
object. The low-level functions are not really intended for the
end-user.
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.
Robin K. S. Hankin
E. Hitzer and S. Sangwine 2017. “Multivector and multivector matrix inverses in real Clifford algebras”. Applied Mathematics and Computation 311:375-389
dot
u <- rcliff(5)
v <- rcliff(5)
w <- rcliff(5)
u
v
u*v
u+(v+w) == (u+v)+w # should be TRUE by associativity of "+"
u*(v*w) == (u*v)*w # should be TRUE by associativity of "*"
u*(v+w) == u*v + u*w # should be TRUE by distributivity
# Now if x,y are _vectors_ we have:
x <- as.1vector(sample(5))
y <- as.1vector(sample(5))
x*y == x%.%y + x%^%y
x %^% y == x %^% (y + 3*x)
x %^% y == (x*y-x*y)/2 # should be TRUE
# 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
## Other products should work as expected:
x %|_% y ## left contraction
x %_|% y ## right contraction
x %o% y ## fat dot product
x ^ y ## Experimental wedge product idiom, plain caret
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