Magnitude of a clifford object

Following Perwass, the magnitude of a multivector is defined as

$$\left|\left|A\right|\right| = \sqrt{A\ast A}$$

Where \(A\ast A\) denotes the Euclidean scalar product eucprod().

Usage

# S3 method for class 'clifford'
Mod(z)

Arguments

z

Clifford objects

Author

Robin K. S. Hankin

Details

For any multivector \(A\), the Euclidean scalar product \(A\ast A\) is never negative, so the square root is always defined.

The function body of Mod.clifford() is sqrt(abs(eucprod(z))); the abs() is needed to avoid numerical roundoff errors in eucprod() giving a negative value.

Note

If you want the square, \(\left|\left|A\right|\right|^2\) and not \(\left|\left|A\right|\right|\), it is faster and more accurate to use eucprod(A) [rather than Mod(A)^2], because this avoids a needless square root.

There is a nice example of scalar product at rcliff.Rd.

See also

Ops.clifford, Conj, rcliff

Examples

Mod(rcliff())
#> [1] 12.24745


# Perwass, p68, asserts that if A is a k-blade, then (in his notation)
# AA == A*A.

# In package idiom, A*A == A %star% A:

A <- rcliff()          
Mod(A*A - A %star% A)  # meh
#> [1] 82.19489

A <- rblade()
Mod(A*A - A %star% A)  # should be small
#> [1] 0