### 7.10.6 ncrat_lib

Status: experimental

Library:
  ncrat.lib
Purpose:
  Framework for working with non-commutative rational functions

Author:
  Ricardo Schnur, email: ricardo.schnur@math.uni-sb.de

Support:
  This project has been funded by the SFB-TRR 195
  'Symbolic Tools in Mathematics and their Application'.

Overview:
  This library provides a framework for working with
  non-commutative rational functions (or rather, expressions) and their linearized representations

References:
  T. Mai: On the analytic theory of non-commutative distributions in free probability. Universitaet des Saarlandes, Dissertation, 2017

Note:
  an almost self-explaining introduction to the possibilities of the framework can be achieved by running the example for the procedure ncrepGetRegularMinimal.

Procedures:

> 7.10.6.1 ncInit |  | Set up framework, list contains nc variables
> 7.10.6.2 ncVarsGet |  | List nc variables that are in use
> 7.10.6.3 ncVarsAdd |  | Add variables from list to 'NCRING'
> 7.10.6.4 ncratDefine |  | Define element of type ncrat
> 7.10.6.5 ncratAdd |  | Addition of two ncrat's
> 7.10.6.6 ncratSubstract |  | Subtraction of two ncrat's
> 7.10.6.7 ncratMultiply |  | Multiplication of two ncrat's
> 7.10.6.8 ncratInvert |  | Invert an ncrat
> 7.10.6.9 ncratSPrint |  | Print-to-string for ncrat
> 7.10.6.10 ncratPrint |  | Print for ncrat
> 7.10.6.11 ncratFromString |  | Reads string into ncrat
> 7.10.6.12 ncratFromPoly |  | Converts poly to ncrat
> 7.10.6.13 ncratPower |  | Raises ncrat to an integer power
> 7.10.6.14 ncratEvaluateAt |  | Evaluate ncrat at scalar or matrix point
> 7.10.6.15 ncrepGet |  | Calculate representation of ncrat
> 7.10.6.16 ncrepAdd |  | Addition of two ncrep's
> 7.10.6.17 ncrepSubstract |  | Subtraction of two ncrep's
> 7.10.6.18 ncrepMultiply |  | Multiplication of two ncrep's
> 7.10.6.19 ncrepInvert |  | Invert an ncrep
> 7.10.6.20 ncrepPrint |  | Print for ncrep
> 7.10.6.21 ncrepDim |  | Return the size of ncrep
> 7.10.6.22 ncrepSubstitute |  | Plug matrices into nc variables in ncrep
> 7.10.6.23 ncrepEvaluate |  | Given (u, Q, v) calculate -u*Q^(-1)*v
> 7.10.6.24 ncrepEvaluateAt |  | Evaluate ncrep at scalar or matrix point
> 7.10.6.25 ncrepIsDefinedDim |  | Random matrix test if ncrep can be evaluated at size dim
> 7.10.6.26 ncrepIsDefined |  | Random matrix test if domain of ncrep is not empty
> 7.10.6.27 ncrepIsRegular |  | Random matrix test if ncrep can be evaluated at scalar point
> 7.10.6.28 ncrepRegularZeroMinimize |  | Yields a minimal representation if regular at zero
> 7.10.6.29 ncrepRegularMinimize |  | Yields a minimal representation if regular at scalar point
> 7.10.6.30 ncrepGetRegularZeroMinimal |  | Get a minimal representation of ncrat regular at zero
> 7.10.6.31 ncrepGetRegularMinimal |  | Get a minimal representation of ncrat regular at scalar point
> 7.10.6.32 ncrepPencilGet |  | Given representation decompose its matrix in linear pencil
> 7.10.6.33 ncrepPencilCombine |  | Given linear pencil add up its elements to single matrix

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