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Clifford Algebras with Numeric and Symbolic Computations |
The following files contain FORTRAN programs whose theoretical background is described in "Orthonormal Basis Sets in Clifford Algebras".
- IIPR.TXT- Computes the expansion of ordered products in projectors (matrix units).
- IIMT.TXT- Computes matrices isomorphic to ordered products of isotropic vectors.
- ONBS.TXT- Determines independant orthonormal basis sets for Clifford algebras.
- readme.txt- a readme file to accompany the above three files.
Copyright is reserved but permission is given to download the files for individual use.
Foreword:Real vector spaces of neutral signature (p=q) admit bases formed by dual sets of isotropic vectors. The Clifford algebras over these spaces admit bases formed by ordered products of isotropic vectors. Projectors are elements of the Clifford algebra satisfying the product relations of matrix units. A set of projectors can be constructed forming a basis of the algebra. The programs IIPR and IIMT compute the expansion of ordered products in projectors. This amounts to compute a matrix realisation of the neutral Clifford algebra. Since any simple real Clifford algebra is isomorphic to the tensor product of a neutral Clifford algebra and a Clifford algebra of type Cl(0,0),Cl(0,1) or Cl(0,2) isomorphic to the fields R,C or H, the program allows the construction of matrix algebras with entries in R,C or H isomorphic to any real simple Clifford algebra. The program ONBS computes orthogonal basis sets of Clifford algebras. Orthogonal basis sets are sets of multivectors satisfying the scalar product relations with a given signature. Orthonormal basis sets define isomorphisms and automorphisms in real Clifford algebras.
Usage notes:The files are Fortran VS programs using the bit handling subroutines IAND, IEOR, ISHFT, BTEST and IBSET( VS Fortran Language Reference, Appendix B, IBM). The DATA IMP/32/ instruction in the programs sets the logical unit label corresponding to the output file. In the VM operating system used there is a corresponding Filedef instruction. In the program IIPR the input data is MD defining the signature MD=p=q. Md is introduced by the DATA instruction DATA MD/2/ and can be changed. The Format instructions allow values MD less or equal to 13, but the computing time and output volume sets much lower limits. The program IIMT performs the same computations but the result is presented in matrix form, hence the program requires an array of 4**MD. The input data MD is introduced via a PARAMETER instruction. In the output file the ordered products are defined by twoo labels as described in the paper.
The input for the program ONBS is ND the dimension of the underlying vector space introduced by a DATA instruction. Two flags KFA and KFB are also set by DATA instructions. If KFA is set to 1 the inverse of the ONB matrices are printed, if KFB is set to 1 the squares of the multivectors are printed for all sign combination of the squares of basis vectors. The sign combinations of the latter are labelled by an index and printed under the trivial ONB formed by the basis vectors.