Ops.clifford: Arithmetic Ops Group Methods for 'clifford' objects

In clifford: Arbitrary Dimensional Clifford Algebras

Arithmetic Ops Group Methods for clifford objects

Description

\loadmathjax

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

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)	\mjeqn\displaystyle AB=\sum_r,s\left\langle A\right\rangle_r\left\langle B\right\rangle_ssee PDF
Inner product	A %.% B = cliffdotprod(A,B)	\mjeqn\displaystyle A\cdot B=\sum_r\neq 0\atop s\ne 0^\vphantoms\neq 0\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_\left|s-r\right|see PDF
Outer product	A %^% B = wedge(A,B)	\mjeqn\displaystyle A\wedge B=\sum_r,s\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_s+rsee PDF
Fat dot product	A %o% B = fatdot(A,B)	\mjeqn\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|see PDF
Left contraction	A %_|% B = lefttick(A,B)	\mjeqn\displaystyle A\rfloor B=\sum_r,s\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_s-rsee PDF
Right contraction	A %|_% B = righttick(A,B)	\mjeqn\displaystyle A\lfloor B=\sum_r,s\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_r-ssee PDF
Cross product	A %X% B = cross(A,B)	\mjeqn\displaystyle A\times B=\frac12_\vphantomj\left(AB-BA\right)see PDF
Scalar product	A %star% B = star(A,B)	\mjeqn\displaystyle A\ast B=\sum_r,s\left\langle\left\langle A\right\rangle_r\left\langle B\right\rangle_s\right\rangle_0see PDF
Euclidean product	A %euc% B = eucprod(A,B)	\mjeqn\displaystyle A\star B= A\ast B^\daggersee PDF

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 \mjseqnCl(p,q) for \mjeqnp+q\leq 5p+5 <= 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:

\mjdeqn

A\ast B=\left\langle AB\right\rangle_0\qquad\mboxNB: notation used by both Perwass and Hestenes omitted; see PDF

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 “\mjeqnA*BA*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

\mjdeqn

A\star B= A\ast B^\dagger= \left\langle AB^\dagger\right\rangle_0\qquad\mboxPerwass omitted: see PDF

where \mjeqnB^\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 \mjeqnA\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 \mjeqnA\rfloor Bsee PDF and \mjeqnA\lfloor Bsee PDF. 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 \mjeqnp+q\leq 5p+q <= 5.

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

In the clifford package the caret “^” is reserved for multiplicative powers, as in A^3=A*A*A. All the different Clifford products have binary operators for convenience including the wedge product %^%. Compare the stokes package, where multiplicative powers do not really make sense and A^B is interpreted as a wedge product of differential forms \mjseqnA and \mjseqnB. 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).

Author(s)

Robin K. S. Hankin

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
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

clifford documentation built on Aug. 14, 2022, 1:05 a.m.