Symbolic Representation of Matrix Determinant.

SymbolicDeterminants

The goal of SymbolicDeterminants is to provide the user with a basic tool for exploring the structure of matrix determinants. Rather than calculating the determinant of a numeric matrix, SymbolicDeterminants produces the formula for calculating it as a textual function of the elements of the matrix.

Installation

You can install the released version of SymbolicDeterminants from https://CRAN.R-project.org with:

install.packages("SymbolicDeterminants")

Use of the functions included in the package

The package can store a complete detguide if it is not too large; otherwise, it stores a minidetguide. The functions included process one or the other of these files. The following descriptions all apply to detguides and the later ones apply to minidetguides.

The first function to run is anewdetguide(p,storage). This function sets up the external storage directory, creates the detguides from a 2x2 matrix up to that of a pxp matrix, and stores the detguides in subdirectories storage/2/ through storage/p/.

The vignette Background describes the creation and use of detguides, which are the lists of indices that guide the definition of the textual output.

The function confirm.det( ) defines a random matrix of the correct size and tests its detguide determinant against that obtained from the matrix eigenvalues. The user can also specify their own test matrix.

Note that all lower order detguides will be created in order to enable calculation of the detguide for p. This can take a long time and may run into issues of object size. Maximum object size may need to be increased in the workspace. As each detguide is created (by anewdetguide( ) ), the variable max.created is defined automatically in the storage directory for ease in determining what has already been created. It may be advisable to increase p by one level at a time, certainly if p > 10.

To obtain a symbolic representation, run parsedetguide(p,storage,browser,symmetric=FALSE) for a given value of p. This is a listing of individual products that make up that determinant. A listing can be obtained for an unstructured matrix and for a symmetric matrix. Note that there are p! products in a determinant, each of which requires a line of text. Each line has a line number for ease of identification and discussion. The output of this function is also stored for further processing, if desired.

To obtain a table of files created and stored for each value of p, run detindex(storage,xmini="D"). All detguides and symbolic representations are stored in the subdirectory assigned to the input variable “storage”, including the smallest ones for consistency. To retrieve one of these files run retrieve(p,storage,browser). The file is designated interactively from the console. The value of this function is either a detguide or a display of the symbolic determinant in a text file. retrieve may be assigned to a name in the R workspace for further processing of a detguide, if desired.

In order to anticipate the impact of p on calculation time and storage, some indication can be found from predictor(p).

Saving storage space by use of minidetguides

A minidetguide consists only of the products of the determinant that have a leading plus sign (+). It also omits the first row of the detguide because this is indicated by the storage directory and is reconstituted when the minidetguide is parsed. In that way, some savings are accomplished at the cost of greater processing time. The greatest saving of storage space occurs because lower order detguides do not have to be stored when using minidetguides.

As above, the first function to run is anewminidet(p,storage). The minidet matrices are stored in subdirectories of storage/p/ named R1C1, R1C2, ..., R1Cp corresponding to the leading element of all the products in that sub- directory.

The function confirm.minidet(x,p,storage) defines a random matrix of the correct size and tests its minidetguide determinant against that obtained from the matrix eigenvalues. The user can also specify their own test matrix.

To obtain a symbolic representation, run parsemini(p, storage, symmetric, browser=NULL, cols=1:p, verbose=TRUE) for a given value of p. This is a listing of individual products that make up that determinant. A listing can be obtained for an unstructured matrix or for a symmetric matrix. Note that there are p! products in a determinant, each of which requires a line of text. Each line has a line number for ease of identification and discussion. The output of this function is also stored for further processing, if desired. Also, because minidetguides are used for larger values of p, individual columns (R1Cs) can be specified.

The function detindex(storage, xmini=c("B","D","M"), pmini=1) can be used to obtain a listing of all detindexes, minidetindexes or both along with their parsed files, if they exist. To retrieve one of these files run retrieve.mini(p,storage,browser). The file is designated interactively from the console. The value of this function is either a detguide or a display of the symbolic determinant in a text file. retrieve.mini may be assigned to a name in the R workspace for further processing of a minidetguide, if desired. Interactively, one indicates whether a single R1Cs is wanted or all of them for a given p.