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| | == Numeric Quest modules hosted on haskell.org == | | == Numeric Quest modules hosted on haskell.org == |
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| − | === Rational numbers ===
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| − |
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| − | ;[http://darcs.haskell.org/numeric-quest/Fraction.hs Fraction.hs]
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| − |
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| − | :This is a generalized Rational in disguise. Rational, as a type
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| − | synonim, could not be directly made an instance of any new class
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| − | at all.
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| − | But we would like it to be an instance of Transcendental, where
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| − | trigonometry, hyperbolics, logarithms, etc. are defined.
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| − | So here we are tiptoe-ing around, re-defining everything from
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| − | scratch, before designing the transcendental functions -- which
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| − | is the main motivation for this module.
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| − |
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| − | :Aside from its ability to compute transcendentals, Fraction
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| − | allows for denominators zero. Unlike Rational, Fraction does
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| − | not produce run-time errors for zero denominators, but use such
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| − | entities as indicators of invalid results -- plus or minus
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| − | infinities. Operations on fractions never fail in principle.
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| − |
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| − | === Polynomials ===
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| − |
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| − | ;[http://darcs.haskell.org/numeric-quest/Roots.hs Roots.hs]
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| − |
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| − | :List of complex roots of a polynomial
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| − | a0 + a1*x + a2*x^2...
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| − |
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| − | === General linear algebra ===
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| − |
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| − | ;[http://darcs.haskell.org/numeric-quest/Eigensystem.hs Eigensystem.hs]
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| − |
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| − | :This module extends the QuantumVector module by providing functions
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| − | to calculate eigenvalues and eigenvectors of Hermitian operators.
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| − | Such toolkit is of primary importance due to pervasiveness of
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| − | eigenproblems in Quantum Mechanics.
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| − |
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| − | ;[http://darcs.haskell.org/numeric-quest/EigensystemNum.hs EigensystemNum.hs]
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| − |
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| − | === Tensors ===
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| − |
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| − | ;[http://darcs.haskell.org/numeric-quest/Tensor.lhs Tensor.lhs]
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| − |
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| − | :This is a quick sketch of what might be a basis of a real
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| − | Tensor module.
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| − |
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| − | :Datatype Tensor defined here is an instance
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| − | of class Eq, Show and Num.
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| − | In addition, several
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| − | customized operations are defined for
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| − | variety of inner products.
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| − |
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| − | :Tensor components are Doubles.
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| − |
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| − | :The shape of tensors defined here involves two parameters:
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| − | dimension and rank. Rank is associated with the
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| − | depth of the tensor tree and corresponds to a total number
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| − | of indices by which you can access the individual components.
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| − |
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| − | === Quantum mechanics ===
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| − |
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| − | ;[http://darcs.haskell.org/numeric-quest/QuantumVector.lhs QuantumVector.lhs]
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| − |
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| − | :This is our attempt to model the abstract Dirac's formalism
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| − | of Quantum Mechanics in Haskell.
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| − |
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| − | :We recognize a quantum state as an abstract vector | x >,
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| − | which can be represented in one of many possible bases -- similar
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| − | to many alternative representations of a 3D vector in rotated systems
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| − | of coordinates. A choice of a particular basis is controlled
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| − | by a generic type variable, which can be any Haskell object
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| − | -- providing that it supports a notion of equality and ordering.
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| − | A state which is composed of many quantum subsystems, not
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| − | necessarily of the same type, can be represented in a vector
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| − | space considered to be a tensor product of the subspaces.
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| − |
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| − | :With this abstract notion we proceed with Haskell definition of two
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| − | vector spaces: Ket and its dual Bra. We demonstrate
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| − | that both are properly defined according to the abstract
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| − | mathematical definition of vector spaces. We then introduce inner
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| − | product and show that our Bra and Ket can be indeed
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| − | considered the vector spaces with inner product.
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| − |
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| | [[Category:Mathematics]] | | [[Category:Mathematics]] |
| | {{LibrariesPage}} | | {{LibrariesPage}} |