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# inverse2 -- inverse of a birational map using a faster algorithm for a special class of maps

- Usage:
    `inverse2 Phi`
    `inverse2(Phi,Verify=>true)`
    `inverse2(Phi,Verify=>false)`
- Inputs:
  - Phi, a multi-rational map, a birational map of so-called multi-linear type
- Outputs:
  - a multi-rational map, the inverse map of Phi

## Description

This assumes that the graph of the input map Phi is defined by a simplified system of equations, which may not be true. If the option Verify is set to true, which is the default choice, then it is verified that the left and right composition of Phi with the returned map is the identity, throwing an error if this is not the case.

    i1 : K = ZZ/10000019;

    i2 : -- map defined by the cubics through the secant variety to the rational normal curve of degree 6
         Phi = multirationalMap rationalMap(ring PP_K^6,ring GG_K(2,4),gens ideal PP_K([6],2));

    o2 : MultirationalMap (rational map from PP^6 to GG(2,4))

    i3 : time Psi = inverse2 Phi;
     -- used 0.334196s (cpu); 0.296342s (thread); 0s (gc)

    o3 : MultirationalMap (birational map from GG(2,4) to PP^6)

    i4 : assert(Phi * Psi == 1)

    i5 : Phi' = Phi || Phi;

    o5 : MultirationalMap (rational map from PP^6 x PP^6 to GG(2,4) x GG(2,4))

    i6 : time Psi' = inverse2 Phi';
     -- used 1.40617s (cpu); 1.09116s (thread); 0s (gc)

    o6 : MultirationalMap (birational map from GG(2,4) x GG(2,4) to PP^6 x PP^6)

    i7 : assert(Phi' * Psi' == 1)

## See also

- inverse(MultirationalMap) -- inverse of a birational map
- MultirationalMap <==> MultirationalMap -- equality of multi-rational maps with checks on internal data

## Ways to use inverse2:

- `inverse2(MultirationalMap)`

## For the programmer

The object inverse2 is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:3360:0.