MappingInversion » History » Version 3
M. Rolf, 10/18/2011 04:02 PM
| 1 | 1 | M. Rolf | h1. Mapping Inversion |
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| 2 | 1 | M. Rolf | |
| 3 | 1 | M. Rolf | Mapping objects can (in certain limits) return their own inverse-functions. |
| 4 | 1 | M. Rolf | See for instance source:trunk/nemomath/examples/ExampleMappingArithmetics.cpp |
| 5 | 1 | M. Rolf | Custom mappings need to define the @getInverse()@ function for this functionality. |
| 6 | 1 | M. Rolf | |
| 7 | 1 | M. Rolf | |
| 8 | 1 | M. Rolf | h1. Inversion of operators |
| 9 | 1 | M. Rolf | |
| 10 | 1 | M. Rolf | A rather special aspect is the inversion of Mappings that are defined by operators such as |
| 11 | 1 | M. Rolf | <pre> |
| 12 | 1 | M. Rolf | RealMapping m = 2.0 * arg<double>(); |
| 13 | 1 | M. Rolf | </pre> |
| 14 | 1 | M. Rolf | Mappings like this are invertible: |
| 15 | 1 | M. Rolf | <pre> |
| 16 | 1 | M. Rolf | RealMapping mInv = m.inverse(); // divide by two |
| 17 | 1 | M. Rolf | </pre> |
| 18 | 1 | M. Rolf | For this to be possible, NemoMath defines *Inverse elements* for datatypes like @double@. |
| 19 | 1 | M. Rolf | Therefore the above statement does not actually evaluate to @arg<double>() / 2.0@. Instead, |
| 20 | 1 | M. Rolf | NemoMath uses a method @leftMultInverse<double>(const double&)@ to see that left-multiplication |
| 21 | 1 | M. Rolf | with 2.0 can be reverted by left-multiplication with 0.5. Hence, the concrete inversion evaluates |
| 22 | 1 | M. Rolf | to @0.5 * arg<double>()@. |
| 23 | 1 | M. Rolf | |
| 24 | 1 | M. Rolf | h3. But why shouldn't NemoMath use @arg<double>() / 2.0@? |
| 25 | 1 | M. Rolf | |
| 26 | 1 | M. Rolf | The reason for that detour is that not all |
| 27 | 1 | M. Rolf | types define operators like @*@ and @/@ in pairs: |
| 28 | 1 | M. Rolf | * NemoMath's Vector and Matrix clases only provide @*@, but not @/@. |
| 29 | 1 | M. Rolf | * @std::string@ defines @+@, but not @-@. |
| 30 | 1 | M. Rolf | * Timestamps like Boost's @ptime@ define @-@, but not @+@. |
| 31 | 1 | M. Rolf | |
| 32 | 1 | M. Rolf | If NemoMath would try to use *inverse operators*, like inverting @+@ with @-@, the Mappings |
| 33 | 1 | M. Rolf | could not be used for such types. It simply wouldn't compile. |
| 34 | 2 | M. Rolf | |
| 35 | 2 | M. Rolf | However, using inverse elements comes with some functional restrictions, since not all data-types have inverse |
| 36 | 2 | M. Rolf | elements, although an inverse might be computed with inverse operators: |
| 37 | 2 | M. Rolf | * @int@ numbers have no inverse element with respect to multiplication. There is not element @I@ in @int@ such that @I*2*x==x@. |
| 38 | 2 | M. Rolf | * The same holds for @unsigned int@ values with respect to addition. |
| 39 | 2 | M. Rolf | |
| 40 | 3 | M. Rolf | A further, general restriction is that operator Mappings can only be inverted, if the argument appears only on one side of the operator. |
| 41 | 2 | M. Rolf | Mappings like |
| 42 | 2 | M. Rolf | <pre> |
| 43 | 2 | M. Rolf | RealMapping f = arg<double>() * someOtherFunction; |
| 44 | 2 | M. Rolf | </pre> |
| 45 | 3 | M. Rolf | are not invertible in a generic way. |