Description Usage Arguments Details Value Note Author(s) See Also Examples

A tensor can be seen as a linear mapping of a tensor to a tensor. If domain and image are the same and the tensor is definite, we can define powers.

1 | ```
power.tensor(X,i,j,p=0.5,by=NULL)
``` |

`X` |
The tensor to be decomposed |

`i` |
The image dimensions of the linear mapping |

`j` |
The domain dimensions of the linear mapping |

`p` |
the power of the tensor to be computed |

`by` |
the operation is done in parallel for these dimensions |

A tensor can be seen as a linear mapping of a tensor to a tensor. Let
denote *R_i* the space of real tensors with dimensions
*i_1...i_d*.

To compute a power `dim(X)[i]`

and `dim(X)[j]`

need to be
equal and the tensor symmetric between these dimension. Some exponents
are only valid with positive definite mappings. None of these
conditions is checked.

a tensor

symmetry of the matrix is assumed but not checked.

K. Gerald van den Boogaart

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
A <- to.tensor(rnorm(120),c(a=2,b=2,c=5,d=3,e=2))
AAt <- A %e% mark(A,"'",c("a","b"))
AAt
power.tensor(AAt,c("a","b"),c("a'","b'"),-1)
inv.tensor(AAt,c("a","b"))
power.tensor(AAt,c("a","b"),c("a'","b'"),2)
mul.tensor(AAt,c("a","b"),AAt,c("a'","b'"))
power.tensor(power.tensor(AAt,c("a","b"),c("a'","b'"),1/pi),
c("a","b"),c("a'","b'"),pi)
AAt <- einstein.tensor(A , mark(A,"'",c("a","b")),by="e")
power.tensor(AAt,c("a","b"),c("a'","b'"),-1,by="e")
inv.tensor(AAt,c("a","b"),by="e")
power.tensor(AAt,c("a","b"),c("a'","b'"),2,by="e")
mul.tensor(AAt,c("a","b"),AAt,c("a'","b'"),by="e")
power.tensor(power.tensor(AAt,c("a","b"),c("a'","b'"),1/pi,by="e"),
c("a","b"),c("a'","b'"),pi,by="e")
``` |

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