Copyright	(c) Scott N. Walck 2016-2018
License	BSD3 (see LICENSE)
Maintainer	Scott N. Walck <walck@lvc.edu>
Stability	experimental
Safe Haskell	Trustworthy
Language	Haskell98

Physics.Learn.QuantumMat

Contents

Complex numbers
State Vectors
Matrices (operators)
Density matrices
Quantum Dynamics
Composition
Measurement
Vector and Matrix

Description

This module contains state vectors and matrices for quantum mechanics.

Synopsis

type C = Complex Double
xp :: Vector C
xm :: Vector C
yp :: Vector C
ym :: Vector C
zp :: Vector C
zm :: Vector C
np :: Double -> Double -> Vector C
nm :: Double -> Double -> Vector C
dim :: Vector C -> Int
scaleV :: C -> Vector C -> Vector C
inner :: Vector C -> Vector C -> C
norm :: Vector C -> Double
normalize :: Vector C -> Vector C
probVector :: Vector C -> Vector Double
gramSchmidt :: [Vector C] -> [Vector C]
conjV :: Vector C -> Vector C
fromList :: [C] -> Vector C
toList :: Vector C -> [C]
sx :: Matrix C
sy :: Matrix C
sz :: Matrix C
scaleM :: C -> Matrix C -> Matrix C
(<>) :: Matrix C -> Matrix C -> Matrix C
(#>) :: Matrix C -> Vector C -> Vector C
(<#) :: Vector C -> Matrix C -> Vector C
conjugateTranspose :: Matrix C -> Matrix C
fromLists :: [[C]] -> Matrix C
toLists :: Matrix C -> [[C]]
size :: Matrix C -> (Int, Int)
matrixFunction :: (C -> C) -> Matrix C -> Matrix C
couter :: Vector C -> Vector C -> Matrix C
dm :: Vector C -> Matrix C
trace :: Matrix C -> C
normalizeDM :: Matrix C -> Matrix C
oneQubitMixed :: Matrix C
timeEvMat :: Double -> Matrix C -> Matrix C
timeEv :: Double -> Matrix C -> Vector C -> Vector C
timeEvMatSpec :: Matrix C -> Double -> Matrix C
class Kronecker a where
possibleOutcomes :: Matrix C -> [Double]
outcomesProjectors :: Matrix C -> [(Double, Matrix C)]
outcomesProbabilities :: Matrix C -> Vector C -> [(Double, Double)]
data Vector a
data Matrix t

Complex numbers

type C = Complex Double #

State Vectors

xp :: Vector C Source #

The state resulting from a measurement of spin angular momentum in the x direction on a spin-1/2 particle when the result of the measurement is hbar/2.

xm :: Vector C Source #

The state resulting from a measurement of spin angular momentum in the x direction on a spin-1/2 particle when the result of the measurement is -hbar/2.

yp :: Vector C Source #

The state resulting from a measurement of spin angular momentum in the y direction on a spin-1/2 particle when the result of the measurement is hbar/2.

ym :: Vector C Source #

The state resulting from a measurement of spin angular momentum in the y direction on a spin-1/2 particle when the result of the measurement is -hbar/2.

zp :: Vector C Source #

The state resulting from a measurement of spin angular momentum in the z direction on a spin-1/2 particle when the result of the measurement is hbar/2.

zm :: Vector C Source #

The state resulting from a measurement of spin angular momentum in the z direction on a spin-1/2 particle when the result of the measurement is -hbar/2.

np :: Double -> Double -> Vector C Source #

The state resulting from a measurement of spin angular momentum in the direction specified by spherical angles theta (polar angle) and phi (azimuthal angle) on a spin-1/2 particle when the result of the measurement is hbar/2.

nm :: Double -> Double -> Vector C Source #

The state resulting from a measurement of spin angular momentum in the direction specified by spherical angles theta (polar angle) and phi (azimuthal angle) on a spin-1/2 particle when the result of the measurement is -hbar/2.

dim :: Vector C -> Int Source #

Dimension of a vector.

scaleV :: C -> Vector C -> Vector C Source #

Scale a complex vector by a complex number.

inner :: Vector C -> Vector C -> C Source #

Complex inner product. First vector gets conjugated.

norm :: Vector C -> Double Source #

Length of a complex vector.

normalize :: Vector C -> Vector C Source #

Return a normalized version of a given state vector.

probVector Source #

Arguments

:: Vector C	state vector
-> Vector Double	vector of probabilities

Return a vector of probabilities for a given state vector.

gramSchmidt :: [Vector C] -> [Vector C] Source #

Form an orthonormal list of complex vectors from a linearly independent list of complex vectors.

conjV :: Vector C -> Vector C Source #

Conjugate the entries of a vector.

fromList :: [C] -> Vector C Source #

Construct a vector from a list of complex numbers.

toList :: Vector C -> [C] Source #

Produce a list of complex numbers from a vector.

Matrices (operators)

sx :: Matrix C Source #

The Pauli X matrix.

sy :: Matrix C Source #

The Pauli Y matrix.

sz :: Matrix C Source #

The Pauli Z matrix.

scaleM :: C -> Matrix C -> Matrix C Source #

Scale a complex matrix by a complex number.

(<>) :: Matrix C -> Matrix C -> Matrix C Source #

Matrix product.

(#>) :: Matrix C -> Vector C -> Vector C Source #

Matrix-vector product.

(<#) :: Vector C -> Matrix C -> Vector C Source #

Vector-matrix product

conjugateTranspose :: Matrix C -> Matrix C Source #

Conjugate transpose of a matrix.

fromLists :: [[C]] -> Matrix C Source #

Construct a matrix from a list of lists of complex numbers.

toLists :: Matrix C -> [[C]] Source #

Produce a list of lists of complex numbers from a matrix.

size :: Matrix C -> (Int, Int) Source #

Size of a matrix.

matrixFunction :: (C -> C) -> Matrix C -> Matrix C Source #

Apply a function to a matrix. Assumes the matrix is a normal matrix (a matrix with an orthonormal basis of eigenvectors).

Density matrices

couter :: Vector C -> Vector C -> Matrix C Source #

Complex outer product

dm :: Vector C -> Matrix C Source #

Build a pure-state density matrix from a state vector.

trace :: Matrix C -> C Source #

Trace of a matrix.

normalizeDM :: Matrix C -> Matrix C Source #

Normalize a density matrix so that it has trace one.

oneQubitMixed :: Matrix C Source #

The one-qubit totally mixed state.

Quantum Dynamics

timeEvMat :: Double -> Matrix C -> Matrix C Source #

Given a time step and a Hamiltonian matrix, produce a unitary time evolution matrix. Unless you really need the time evolution matrix, it is better to use timeEv, which gives the same numerical results without doing an explicit matrix inversion. The function assumes hbar = 1.

timeEv :: Double -> Matrix C -> Vector C -> Vector C Source #

Given a time step and a Hamiltonian matrix, advance the state vector using the Schrodinger equation. This method should be faster than using timeEvMat since it solves a linear system rather than calculating an inverse matrix. The function assumes hbar = 1.

timeEvMatSpec :: Matrix C -> Double -> Matrix C Source #

Given a Hamiltonian matrix, return a function from time to evolution matrix. Uses spectral decomposition. Assumes hbar = 1.

Composition

class Kronecker a where Source #

Minimal complete definition

kron

Methods

kron :: a -> a -> a Source #

Instances

Product t => Kronecker (Matrix t) Source #
Instance details Defined in Physics.Learn.QuantumMat Methods kron :: Matrix t -> Matrix t -> Matrix t Source #
Product t => Kronecker (Vector t) Source #
Instance details Defined in Physics.Learn.QuantumMat Methods kron :: Vector t -> Vector t -> Vector t Source #

Measurement

possibleOutcomes :: Matrix C -> [Double] Source #

The possible outcomes of a measurement of an observable. These are the eigenvalues of the matrix of the observable.

outcomesProjectors :: Matrix C -> [(Double, Matrix C)] Source #

Given an obervable, return a list of pairs of possible outcomes and projectors for each outcome.

outcomesProbabilities :: Matrix C -> Vector C -> [(Double, Double)] Source #

Given an observable and a state vector, return a list of pairs of possible outcomes and probabilites for each outcome.

Vector and Matrix

data Vector a #

Storable-based vectors

Instances

LSDiv Vector
Instance details Defined in Internal.Container Methods linSolve :: Field t => Matrix t -> Vector t -> Vector t
Normed Vector Double
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Vector Double -> RealOf Double
Normed Vector Float
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Vector Float -> RealOf Float
Container Vector Double
Instance details Defined in Internal.Numeric Methods conj' :: Vector Double -> Vector Double size' :: Vector Double -> IndexOf Vector scalar' :: Double -> Vector Double scale' :: Double -> Vector Double -> Vector Double addConstant :: Double -> Vector Double -> Vector Double add' :: Vector Double -> Vector Double -> Vector Double sub :: Vector Double -> Vector Double -> Vector Double mul :: Vector Double -> Vector Double -> Vector Double equal :: Vector Double -> Vector Double -> Bool cmap' :: Element b => (Double -> b) -> Vector Double -> Vector b konst' :: Double -> IndexOf Vector -> Vector Double build' :: IndexOf Vector -> ArgOf Vector Double -> Vector Double atIndex' :: Vector Double -> IndexOf Vector -> Double minIndex' :: Vector Double -> IndexOf Vector maxIndex' :: Vector Double -> IndexOf Vector minElement' :: Vector Double -> Double maxElement' :: Vector Double -> Double sumElements' :: Vector Double -> Double prodElements' :: Vector Double -> Double step' :: Vector Double -> Vector Double ccompare' :: Vector Double -> Vector Double -> Vector I cselect' :: Vector I -> Vector Double -> Vector Double -> Vector Double -> Vector Double find' :: (Double -> Bool) -> Vector Double -> [IndexOf Vector] assoc' :: IndexOf Vector -> Double -> [(IndexOf Vector, Double)] -> Vector Double accum' :: Vector Double -> (Double -> Double -> Double) -> [(IndexOf Vector, Double)] -> Vector Double scaleRecip :: Double -> Vector Double -> Vector Double divide :: Vector Double -> Vector Double -> Vector Double arctan2' :: Vector Double -> Vector Double -> Vector Double cmod' :: Double -> Vector Double -> Vector Double fromInt' :: Vector I -> Vector Double toInt' :: Vector Double -> Vector I fromZ' :: Vector Z -> Vector Double toZ' :: Vector Double -> Vector Z
Container Vector Float
Instance details Defined in Internal.Numeric Methods conj' :: Vector Float -> Vector Float size' :: Vector Float -> IndexOf Vector scalar' :: Float -> Vector Float scale' :: Float -> Vector Float -> Vector Float addConstant :: Float -> Vector Float -> Vector Float add' :: Vector Float -> Vector Float -> Vector Float sub :: Vector Float -> Vector Float -> Vector Float mul :: Vector Float -> Vector Float -> Vector Float equal :: Vector Float -> Vector Float -> Bool cmap' :: Element b => (Float -> b) -> Vector Float -> Vector b konst' :: Float -> IndexOf Vector -> Vector Float build' :: IndexOf Vector -> ArgOf Vector Float -> Vector Float atIndex' :: Vector Float -> IndexOf Vector -> Float minIndex' :: Vector Float -> IndexOf Vector maxIndex' :: Vector Float -> IndexOf Vector minElement' :: Vector Float -> Float maxElement' :: Vector Float -> Float sumElements' :: Vector Float -> Float prodElements' :: Vector Float -> Float step' :: Vector Float -> Vector Float ccompare' :: Vector Float -> Vector Float -> Vector I cselect' :: Vector I -> Vector Float -> Vector Float -> Vector Float -> Vector Float find' :: (Float -> Bool) -> Vector Float -> [IndexOf Vector] assoc' :: IndexOf Vector -> Float -> [(IndexOf Vector, Float)] -> Vector Float accum' :: Vector Float -> (Float -> Float -> Float) -> [(IndexOf Vector, Float)] -> Vector Float scaleRecip :: Float -> Vector Float -> Vector Float divide :: Vector Float -> Vector Float -> Vector Float arctan2' :: Vector Float -> Vector Float -> Vector Float cmod' :: Float -> Vector Float -> Vector Float fromInt' :: Vector I -> Vector Float toInt' :: Vector Float -> Vector I fromZ' :: Vector Z -> Vector Float toZ' :: Vector Float -> Vector Z
Container Vector I
Instance details Defined in Internal.Numeric Methods conj' :: Vector I -> Vector I size' :: Vector I -> IndexOf Vector scalar' :: I -> Vector I scale' :: I -> Vector I -> Vector I addConstant :: I -> Vector I -> Vector I add' :: Vector I -> Vector I -> Vector I sub :: Vector I -> Vector I -> Vector I mul :: Vector I -> Vector I -> Vector I equal :: Vector I -> Vector I -> Bool cmap' :: Element b => (I -> b) -> Vector I -> Vector b konst' :: I -> IndexOf Vector -> Vector I build' :: IndexOf Vector -> ArgOf Vector I -> Vector I atIndex' :: Vector I -> IndexOf Vector -> I minIndex' :: Vector I -> IndexOf Vector maxIndex' :: Vector I -> IndexOf Vector minElement' :: Vector I -> I maxElement' :: Vector I -> I sumElements' :: Vector I -> I prodElements' :: Vector I -> I step' :: Vector I -> Vector I ccompare' :: Vector I -> Vector I -> Vector I cselect' :: Vector I -> Vector I -> Vector I -> Vector I -> Vector I find' :: (I -> Bool) -> Vector I -> [IndexOf Vector] assoc' :: IndexOf Vector -> I -> [(IndexOf Vector, I)] -> Vector I accum' :: Vector I -> (I -> I -> I) -> [(IndexOf Vector, I)] -> Vector I scaleRecip :: I -> Vector I -> Vector I divide :: Vector I -> Vector I -> Vector I arctan2' :: Vector I -> Vector I -> Vector I cmod' :: I -> Vector I -> Vector I fromInt' :: Vector I -> Vector I toInt' :: Vector I -> Vector I fromZ' :: Vector Z -> Vector I toZ' :: Vector I -> Vector Z
Container Vector Z
Instance details Defined in Internal.Numeric Methods conj' :: Vector Z -> Vector Z size' :: Vector Z -> IndexOf Vector scalar' :: Z -> Vector Z scale' :: Z -> Vector Z -> Vector Z addConstant :: Z -> Vector Z -> Vector Z add' :: Vector Z -> Vector Z -> Vector Z sub :: Vector Z -> Vector Z -> Vector Z mul :: Vector Z -> Vector Z -> Vector Z equal :: Vector Z -> Vector Z -> Bool cmap' :: Element b => (Z -> b) -> Vector Z -> Vector b konst' :: Z -> IndexOf Vector -> Vector Z build' :: IndexOf Vector -> ArgOf Vector Z -> Vector Z atIndex' :: Vector Z -> IndexOf Vector -> Z minIndex' :: Vector Z -> IndexOf Vector maxIndex' :: Vector Z -> IndexOf Vector minElement' :: Vector Z -> Z maxElement' :: Vector Z -> Z sumElements' :: Vector Z -> Z prodElements' :: Vector Z -> Z step' :: Vector Z -> Vector Z ccompare' :: Vector Z -> Vector Z -> Vector I cselect' :: Vector I -> Vector Z -> Vector Z -> Vector Z -> Vector Z find' :: (Z -> Bool) -> Vector Z -> [IndexOf Vector] assoc' :: IndexOf Vector -> Z -> [(IndexOf Vector, Z)] -> Vector Z accum' :: Vector Z -> (Z -> Z -> Z) -> [(IndexOf Vector, Z)] -> Vector Z scaleRecip :: Z -> Vector Z -> Vector Z divide :: Vector Z -> Vector Z -> Vector Z arctan2' :: Vector Z -> Vector Z -> Vector Z cmod' :: Z -> Vector Z -> Vector Z fromInt' :: Vector I -> Vector Z toInt' :: Vector Z -> Vector I fromZ' :: Vector Z -> Vector Z toZ' :: Vector Z -> Vector Z
Container Vector t => Linear t Vector
Instance details Defined in Internal.Numeric Methods scale :: t -> Vector t -> Vector t #
Storable a => Vector Vector a
Instance details Defined in Data.Vector.Storable Methods basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) a -> m (Vector a) # basicUnsafeThaw :: PrimMonad m => Vector a -> m (Mutable Vector (PrimState m) a) # basicLength :: Vector a -> Int # basicUnsafeSlice :: Int -> Int -> Vector a -> Vector a # basicUnsafeIndexM :: Monad m => Vector a -> Int -> m a # basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) a -> Vector a -> m () # elemseq :: Vector a -> a -> b -> b #
Mul Matrix Vector Vector
Instance details Defined in Internal.Container Methods (<>) :: Product t => Matrix t -> Vector t -> Vector t
Mul Vector Matrix Vector
Instance details Defined in Internal.Container Methods (<>) :: Product t => Vector t -> Matrix t -> Vector t
Container Vector e => Konst e Int Vector
Instance details Defined in Internal.Numeric Methods konst :: e -> Int -> Vector e #
Normed Vector (Complex Double)
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Vector (Complex Double) -> RealOf (Complex Double)
Normed Vector (Complex Float)
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Vector (Complex Float) -> RealOf (Complex Float)
Container Vector (Complex Double)
Instance details Defined in Internal.Numeric Methods conj' :: Vector (Complex Double) -> Vector (Complex Double) size' :: Vector (Complex Double) -> IndexOf Vector scalar' :: Complex Double -> Vector (Complex Double) scale' :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) addConstant :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) add' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) sub :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) mul :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) equal :: Vector (Complex Double) -> Vector (Complex Double) -> Bool cmap' :: Element b => (Complex Double -> b) -> Vector (Complex Double) -> Vector b konst' :: Complex Double -> IndexOf Vector -> Vector (Complex Double) build' :: IndexOf Vector -> ArgOf Vector (Complex Double) -> Vector (Complex Double) atIndex' :: Vector (Complex Double) -> IndexOf Vector -> Complex Double minIndex' :: Vector (Complex Double) -> IndexOf Vector maxIndex' :: Vector (Complex Double) -> IndexOf Vector minElement' :: Vector (Complex Double) -> Complex Double maxElement' :: Vector (Complex Double) -> Complex Double sumElements' :: Vector (Complex Double) -> Complex Double prodElements' :: Vector (Complex Double) -> Complex Double step' :: Vector (Complex Double) -> Vector (Complex Double) ccompare' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector I cselect' :: Vector I -> Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) find' :: (Complex Double -> Bool) -> Vector (Complex Double) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Complex Double -> [(IndexOf Vector, Complex Double)] -> Vector (Complex Double) accum' :: Vector (Complex Double) -> (Complex Double -> Complex Double -> Complex Double) -> [(IndexOf Vector, Complex Double)] -> Vector (Complex Double) scaleRecip :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) divide :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) arctan2' :: Vector (Complex Double) -> Vector (Complex Double) -> Vector (Complex Double) cmod' :: Complex Double -> Vector (Complex Double) -> Vector (Complex Double) fromInt' :: Vector I -> Vector (Complex Double) toInt' :: Vector (Complex Double) -> Vector I fromZ' :: Vector Z -> Vector (Complex Double) toZ' :: Vector (Complex Double) -> Vector Z
Container Vector (Complex Float)
Instance details Defined in Internal.Numeric Methods conj' :: Vector (Complex Float) -> Vector (Complex Float) size' :: Vector (Complex Float) -> IndexOf Vector scalar' :: Complex Float -> Vector (Complex Float) scale' :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) addConstant :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) add' :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) sub :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) mul :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) equal :: Vector (Complex Float) -> Vector (Complex Float) -> Bool cmap' :: Element b => (Complex Float -> b) -> Vector (Complex Float) -> Vector b konst' :: Complex Float -> IndexOf Vector -> Vector (Complex Float) build' :: IndexOf Vector -> ArgOf Vector (Complex Float) -> Vector (Complex Float) atIndex' :: Vector (Complex Float) -> IndexOf Vector -> Complex Float minIndex' :: Vector (Complex Float) -> IndexOf Vector maxIndex' :: Vector (Complex Float) -> IndexOf Vector minElement' :: Vector (Complex Float) -> Complex Float maxElement' :: Vector (Complex Float) -> Complex Float sumElements' :: Vector (Complex Float) -> Complex Float prodElements' :: Vector (Complex Float) -> Complex Float step' :: Vector (Complex Float) -> Vector (Complex Float) ccompare' :: Vector (Complex Float) -> Vector (Complex Float) -> Vector I cselect' :: Vector I -> Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) find' :: (Complex Float -> Bool) -> Vector (Complex Float) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Complex Float -> [(IndexOf Vector, Complex Float)] -> Vector (Complex Float) accum' :: Vector (Complex Float) -> (Complex Float -> Complex Float -> Complex Float) -> [(IndexOf Vector, Complex Float)] -> Vector (Complex Float) scaleRecip :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) divide :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) arctan2' :: Vector (Complex Float) -> Vector (Complex Float) -> Vector (Complex Float) cmod' :: Complex Float -> Vector (Complex Float) -> Vector (Complex Float) fromInt' :: Vector I -> Vector (Complex Float) toInt' :: Vector (Complex Float) -> Vector I fromZ' :: Vector Z -> Vector (Complex Float) toZ' :: Vector (Complex Float) -> Vector Z
Representable Bra (Vector C) Source #
Instance details Defined in Physics.Learn.Ket Methods rep :: OrthonormalBasis -> Bra -> Vector C Source # dim :: Bra -> Int Source #
Representable Ket (Vector C) Source #
Instance details Defined in Physics.Learn.Ket Methods rep :: OrthonormalBasis -> Ket -> Vector C Source # dim :: Ket -> Int Source #
KnownNat m => Container Vector (Mod m I)
Instance details Defined in Internal.Modular Methods conj' :: Vector (Mod m I) -> Vector (Mod m I) size' :: Vector (Mod m I) -> IndexOf Vector scalar' :: Mod m I -> Vector (Mod m I) scale' :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) addConstant :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) add' :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) sub :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) mul :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) equal :: Vector (Mod m I) -> Vector (Mod m I) -> Bool cmap' :: Element b => (Mod m I -> b) -> Vector (Mod m I) -> Vector b konst' :: Mod m I -> IndexOf Vector -> Vector (Mod m I) build' :: IndexOf Vector -> ArgOf Vector (Mod m I) -> Vector (Mod m I) atIndex' :: Vector (Mod m I) -> IndexOf Vector -> Mod m I minIndex' :: Vector (Mod m I) -> IndexOf Vector maxIndex' :: Vector (Mod m I) -> IndexOf Vector minElement' :: Vector (Mod m I) -> Mod m I maxElement' :: Vector (Mod m I) -> Mod m I sumElements' :: Vector (Mod m I) -> Mod m I prodElements' :: Vector (Mod m I) -> Mod m I step' :: Vector (Mod m I) -> Vector (Mod m I) ccompare' :: Vector (Mod m I) -> Vector (Mod m I) -> Vector I cselect' :: Vector I -> Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) find' :: (Mod m I -> Bool) -> Vector (Mod m I) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Mod m I -> [(IndexOf Vector, Mod m I)] -> Vector (Mod m I) accum' :: Vector (Mod m I) -> (Mod m I -> Mod m I -> Mod m I) -> [(IndexOf Vector, Mod m I)] -> Vector (Mod m I) scaleRecip :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) divide :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) arctan2' :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) cmod' :: Mod m I -> Vector (Mod m I) -> Vector (Mod m I) fromInt' :: Vector I -> Vector (Mod m I) toInt' :: Vector (Mod m I) -> Vector I fromZ' :: Vector Z -> Vector (Mod m I) toZ' :: Vector (Mod m I) -> Vector Z
KnownNat m => Container Vector (Mod m Z)
Instance details Defined in Internal.Modular Methods conj' :: Vector (Mod m Z) -> Vector (Mod m Z) size' :: Vector (Mod m Z) -> IndexOf Vector scalar' :: Mod m Z -> Vector (Mod m Z) scale' :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) addConstant :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) add' :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) sub :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) mul :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) equal :: Vector (Mod m Z) -> Vector (Mod m Z) -> Bool cmap' :: Element b => (Mod m Z -> b) -> Vector (Mod m Z) -> Vector b konst' :: Mod m Z -> IndexOf Vector -> Vector (Mod m Z) build' :: IndexOf Vector -> ArgOf Vector (Mod m Z) -> Vector (Mod m Z) atIndex' :: Vector (Mod m Z) -> IndexOf Vector -> Mod m Z minIndex' :: Vector (Mod m Z) -> IndexOf Vector maxIndex' :: Vector (Mod m Z) -> IndexOf Vector minElement' :: Vector (Mod m Z) -> Mod m Z maxElement' :: Vector (Mod m Z) -> Mod m Z sumElements' :: Vector (Mod m Z) -> Mod m Z prodElements' :: Vector (Mod m Z) -> Mod m Z step' :: Vector (Mod m Z) -> Vector (Mod m Z) ccompare' :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector I cselect' :: Vector I -> Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) find' :: (Mod m Z -> Bool) -> Vector (Mod m Z) -> [IndexOf Vector] assoc' :: IndexOf Vector -> Mod m Z -> [(IndexOf Vector, Mod m Z)] -> Vector (Mod m Z) accum' :: Vector (Mod m Z) -> (Mod m Z -> Mod m Z -> Mod m Z) -> [(IndexOf Vector, Mod m Z)] -> Vector (Mod m Z) scaleRecip :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) divide :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) arctan2' :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) cmod' :: Mod m Z -> Vector (Mod m Z) -> Vector (Mod m Z) fromInt' :: Vector I -> Vector (Mod m Z) toInt' :: Vector (Mod m Z) -> Vector I fromZ' :: Vector Z -> Vector (Mod m Z) toZ' :: Vector (Mod m Z) -> Vector Z
Container Vector e => Build Int (e -> e) Vector e
Instance details Defined in Internal.Container Methods build :: Int -> (e -> e) -> Vector e #
Storable a => IsList (Vector a)
Instance details Defined in Data.Vector.Storable Associated Types type Item (Vector a) :: * # Methods fromList :: [Item (Vector a)] -> Vector a # fromListN :: Int -> [Item (Vector a)] -> Vector a # toList :: Vector a -> [Item (Vector a)] #
(Storable a, Eq a) => Eq (Vector a)
Instance details Defined in Data.Vector.Storable Methods (==) :: Vector a -> Vector a -> Bool # (/=) :: Vector a -> Vector a -> Bool #
(Data a, Storable a) => Data (Vector a)
Instance details Defined in Data.Vector.Storable Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Vector a -> c (Vector a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Vector a) # toConstr :: Vector a -> Constr # dataTypeOf :: Vector a -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Vector a)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Vector a)) # gmapT :: (forall b. Data b => b -> b) -> Vector a -> Vector a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Vector a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Vector a -> r # gmapQ :: (forall d. Data d => d -> u) -> Vector a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Vector a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Vector a -> m (Vector a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Vector a -> m (Vector a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Vector a -> m (Vector a) #
KnownNat m => Num (Vector (Mod m I))
Instance details Defined in Internal.Modular Methods (+) :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) # (-) :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) # (*) :: Vector (Mod m I) -> Vector (Mod m I) -> Vector (Mod m I) # negate :: Vector (Mod m I) -> Vector (Mod m I) # abs :: Vector (Mod m I) -> Vector (Mod m I) # signum :: Vector (Mod m I) -> Vector (Mod m I) # fromInteger :: Integer -> Vector (Mod m I) #
KnownNat m => Num (Vector (Mod m Z))
Instance details Defined in Internal.Modular Methods (+) :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) # (-) :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) # (*) :: Vector (Mod m Z) -> Vector (Mod m Z) -> Vector (Mod m Z) # negate :: Vector (Mod m Z) -> Vector (Mod m Z) # abs :: Vector (Mod m Z) -> Vector (Mod m Z) # signum :: Vector (Mod m Z) -> Vector (Mod m Z) # fromInteger :: Integer -> Vector (Mod m Z) #
(Storable a, Ord a) => Ord (Vector a)
Instance details Defined in Data.Vector.Storable Methods compare :: Vector a -> Vector a -> Ordering # (<) :: Vector a -> Vector a -> Bool # (<=) :: Vector a -> Vector a -> Bool # (>) :: Vector a -> Vector a -> Bool # (>=) :: Vector a -> Vector a -> Bool # max :: Vector a -> Vector a -> Vector a # min :: Vector a -> Vector a -> Vector a #
(Read a, Storable a) => Read (Vector a)
Instance details Defined in Data.Vector.Storable Methods readsPrec :: Int -> ReadS (Vector a) # readList :: ReadS [Vector a] # readPrec :: ReadPrec (Vector a) # readListPrec :: ReadPrec [Vector a] #
(Show a, Storable a) => Show (Vector a)
Instance details Defined in Data.Vector.Storable Methods showsPrec :: Int -> Vector a -> ShowS # show :: Vector a -> String # showList :: [Vector a] -> ShowS #
Storable a => Semigroup (Vector a)
Instance details Defined in Data.Vector.Storable Methods (<>) :: Vector a -> Vector a -> Vector a # sconcat :: NonEmpty (Vector a) -> Vector a # stimes :: Integral b => b -> Vector a -> Vector a #
Storable a => Monoid (Vector a)
Instance details Defined in Data.Vector.Storable Methods mempty :: Vector a # mappend :: Vector a -> Vector a -> Vector a # mconcat :: [Vector a] -> Vector a #
NFData (Vector a)
Instance details Defined in Data.Vector.Storable Methods rnf :: Vector a -> () #
Normed (Vector Float)
Instance details Defined in Internal.Util Methods norm_0 :: Vector Float -> R # norm_1 :: Vector Float -> R # norm_2 :: Vector Float -> R # norm_Inf :: Vector Float -> R #
Normed (Vector (Complex Float))
Instance details Defined in Internal.Util Methods norm_0 :: Vector (Complex Float) -> R # norm_1 :: Vector (Complex Float) -> R # norm_2 :: Vector (Complex Float) -> R # norm_Inf :: Vector (Complex Float) -> R #
KnownNat m => Normed (Vector (Mod m I))
Instance details Defined in Internal.Modular Methods norm_0 :: Vector (Mod m I) -> R # norm_1 :: Vector (Mod m I) -> R # norm_2 :: Vector (Mod m I) -> R # norm_Inf :: Vector (Mod m I) -> R #
KnownNat m => Normed (Vector (Mod m Z))
Instance details Defined in Internal.Modular Methods norm_0 :: Vector (Mod m Z) -> R # norm_1 :: Vector (Mod m Z) -> R # norm_2 :: Vector (Mod m Z) -> R # norm_Inf :: Vector (Mod m Z) -> R #
Normed (Vector I)
Instance details Defined in Internal.Util Methods norm_0 :: Vector I -> R # norm_1 :: Vector I -> R # norm_2 :: Vector I -> R # norm_Inf :: Vector I -> R #
Normed (Vector Z)
Instance details Defined in Internal.Util Methods norm_0 :: Vector Z -> R # norm_1 :: Vector Z -> R # norm_2 :: Vector Z -> R # norm_Inf :: Vector Z -> R #
Normed (Vector R)
Instance details Defined in Internal.Util Methods norm_0 :: Vector R -> R # norm_1 :: Vector R -> R # norm_2 :: Vector R -> R # norm_Inf :: Vector R -> R #
Normed (Vector C)
Instance details Defined in Internal.Util Methods norm_0 :: Vector C -> R # norm_1 :: Vector C -> R # norm_2 :: Vector C -> R # norm_Inf :: Vector C -> R #
Container Vector t => Additive (Vector t)
Instance details Defined in Internal.Numeric Methods add :: Vector t -> Vector t -> Vector t #
Storable t => TransArray (Vector t)
Instance details Defined in Internal.Devel Associated Types type Trans (Vector t) b :: * # type TransRaw (Vector t) b :: * # Methods apply :: Vector t -> (b -> IO r) -> Trans (Vector t) b -> IO r # applyRaw :: Vector t -> (b -> IO r) -> TransRaw (Vector t) b -> IO r #
Storable a => Ixed (Vector a)
Instance details Defined in Control.Lens.At Methods ix :: Index (Vector a) -> Traversal' (Vector a) (IxValue (Vector a)) #
Storable a => Wrapped (Vector a)
Instance details Defined in Control.Lens.Wrapped Associated Types type Unwrapped (Vector a) :: * # Methods _Wrapped' :: Iso' (Vector a) (Unwrapped (Vector a)) #
Product t => Kronecker (Vector t) Source #
Instance details Defined in Physics.Learn.QuantumMat Methods kron :: Vector t -> Vector t -> Vector t Source #
Indexable (Vector Double) Double
Instance details Defined in Internal.Util Methods (!) :: Vector Double -> Int -> Double #
Indexable (Vector Float) Float
Instance details Defined in Internal.Util Methods (!) :: Vector Float -> Int -> Float #
Indexable (Vector I) I
Instance details Defined in Internal.Util Methods (!) :: Vector I -> Int -> I #
Indexable (Vector Z) Z
Instance details Defined in Internal.Util Methods (!) :: Vector Z -> Int -> Z #
(Storable a, t ~ Vector a') => Rewrapped (Vector a) t
Instance details Defined in Control.Lens.Wrapped
Element t => Indexable (Matrix t) (Vector t)
Instance details Defined in Internal.Util Methods (!) :: Matrix t -> Int -> Vector t #
Indexable (Vector (Complex Double)) (Complex Double)
Instance details Defined in Internal.Util Methods (!) :: Vector (Complex Double) -> Int -> Complex Double #
Indexable (Vector (Complex Float)) (Complex Float)
Instance details Defined in Internal.Util Methods (!) :: Vector (Complex Float) -> Int -> Complex Float #
(Storable t, Indexable (Vector t) t) => Indexable (Vector (Mod m t)) (Mod m t)
Instance details Defined in Internal.Modular Methods (!) :: Vector (Mod m t) -> Int -> Mod m t #
type IndexOf Vector
Instance details Defined in Internal.Numeric type IndexOf Vector = Int
type Mutable Vector
Instance details Defined in Data.Vector.Storable type Mutable Vector = MVector
type ArgOf Vector a
Instance details Defined in Internal.Numeric type ArgOf Vector a = a -> a
type Item (Vector a)
Instance details Defined in Data.Vector.Storable type Item (Vector a) = a
type ElementOf (Vector a)
Instance details Defined in Internal.Numeric type ElementOf (Vector a) = a
type Index (Vector a)
Instance details Defined in Control.Lens.At type Index (Vector a) = Int
type IxValue (Vector a)
Instance details Defined in Control.Lens.At type IxValue (Vector a) = a
type Unwrapped (Vector a)
Instance details Defined in Control.Lens.Wrapped type Unwrapped (Vector a) = [a]
type TransRaw (Vector t) b
Instance details Defined in Internal.Devel type TransRaw (Vector t) b = CInt -> Ptr t -> b
type Trans (Vector t) b
Instance details Defined in Internal.Devel type Trans (Vector t) b = CInt -> Ptr t -> b

data Matrix t #

Matrix representation suitable for BLAS/LAPACK computations.

Instances

LSDiv Matrix
Instance details Defined in Internal.Container Methods linSolve :: Field t => Matrix t -> Matrix t -> Matrix t
Normed Matrix Double
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Matrix Double -> RealOf Double
Normed Matrix Float
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Matrix Float -> RealOf Float
(Num a, Element a, Container Vector a) => Container Matrix a
Instance details Defined in Internal.Numeric Methods conj' :: Matrix a -> Matrix a size' :: Matrix a -> IndexOf Matrix scalar' :: a -> Matrix a scale' :: a -> Matrix a -> Matrix a addConstant :: a -> Matrix a -> Matrix a add' :: Matrix a -> Matrix a -> Matrix a sub :: Matrix a -> Matrix a -> Matrix a mul :: Matrix a -> Matrix a -> Matrix a equal :: Matrix a -> Matrix a -> Bool cmap' :: Element b => (a -> b) -> Matrix a -> Matrix b konst' :: a -> IndexOf Matrix -> Matrix a build' :: IndexOf Matrix -> ArgOf Matrix a -> Matrix a atIndex' :: Matrix a -> IndexOf Matrix -> a minIndex' :: Matrix a -> IndexOf Matrix maxIndex' :: Matrix a -> IndexOf Matrix minElement' :: Matrix a -> a maxElement' :: Matrix a -> a sumElements' :: Matrix a -> a prodElements' :: Matrix a -> a step' :: Matrix a -> Matrix a ccompare' :: Matrix a -> Matrix a -> Matrix I cselect' :: Matrix I -> Matrix a -> Matrix a -> Matrix a -> Matrix a find' :: (a -> Bool) -> Matrix a -> [IndexOf Matrix] assoc' :: IndexOf Matrix -> a -> [(IndexOf Matrix, a)] -> Matrix a accum' :: Matrix a -> (a -> a -> a) -> [(IndexOf Matrix, a)] -> Matrix a scaleRecip :: a -> Matrix a -> Matrix a divide :: Matrix a -> Matrix a -> Matrix a arctan2' :: Matrix a -> Matrix a -> Matrix a cmod' :: a -> Matrix a -> Matrix a fromInt' :: Matrix I -> Matrix a toInt' :: Matrix a -> Matrix I fromZ' :: Matrix Z -> Matrix a toZ' :: Matrix a -> Matrix Z
Container Matrix t => Linear t Matrix
Instance details Defined in Internal.Numeric Methods scale :: t -> Matrix t -> Matrix t #
Mul Matrix Matrix Matrix
Instance details Defined in Internal.Container Methods (<>) :: Product t => Matrix t -> Matrix t -> Matrix t
Mul Matrix Vector Vector
Instance details Defined in Internal.Container Methods (<>) :: Product t => Matrix t -> Vector t -> Vector t
Mul Vector Matrix Vector
Instance details Defined in Internal.Container Methods (<>) :: Product t => Vector t -> Matrix t -> Vector t
Normed Matrix (Complex Double)
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Matrix (Complex Double) -> RealOf (Complex Double)
Normed Matrix (Complex Float)
Instance details Defined in Internal.Algorithms Methods pnorm :: NormType -> Matrix (Complex Float) -> RealOf (Complex Float)
Representable Operator (Matrix C) Source #
Instance details Defined in Physics.Learn.Ket Methods rep :: OrthonormalBasis -> Operator -> Matrix C Source # dim :: Operator -> Int Source #
(Num e, Container Vector e) => Konst e (Int, Int) Matrix
Instance details Defined in Internal.Numeric Methods konst :: e -> (Int, Int) -> Matrix e #
(Storable t, NFData t) => NFData (Matrix t)
Instance details Defined in Internal.Matrix Methods rnf :: Matrix t -> () #
Normed (Matrix R)
Instance details Defined in Internal.Util Methods norm_0 :: Matrix R -> R # norm_1 :: Matrix R -> R # norm_2 :: Matrix R -> R # norm_Inf :: Matrix R -> R #
Normed (Matrix C)
Instance details Defined in Internal.Util Methods norm_0 :: Matrix C -> R # norm_1 :: Matrix C -> R # norm_2 :: Matrix C -> R # norm_Inf :: Matrix C -> R #
Container Matrix t => Additive (Matrix t)
Instance details Defined in Internal.Numeric Methods add :: Matrix t -> Matrix t -> Matrix t #
KnownNat m => Testable (Matrix (Mod m I))
Instance details Defined in Internal.Modular Methods checkT :: Matrix (Mod m I) -> (Bool, IO ()) # ioCheckT :: Matrix (Mod m I) -> IO (Bool, IO ()) #
Storable t => TransArray (Matrix t)
Instance details Defined in Internal.Matrix Associated Types type Trans (Matrix t) b :: * # type TransRaw (Matrix t) b :: * # Methods apply :: Matrix t -> (b -> IO r) -> Trans (Matrix t) b -> IO r # applyRaw :: Matrix t -> (b -> IO r) -> TransRaw (Matrix t) b -> IO r #
Product t => Kronecker (Matrix t) Source #
Instance details Defined in Physics.Learn.QuantumMat Methods kron :: Matrix t -> Matrix t -> Matrix t Source #
Element t => Indexable (Matrix t) (Vector t)
Instance details Defined in Internal.Util Methods (!) :: Matrix t -> Int -> Vector t #
(CTrans t, Container Vector t) => Transposable (Matrix t) (Matrix t)
Instance details Defined in Internal.Numeric Methods tr :: Matrix t -> Matrix t # tr' :: Matrix t -> Matrix t #
Container Matrix e => Build (Int, Int) (e -> e -> e) Matrix e
Instance details Defined in Internal.Container Methods build :: (Int, Int) -> (e -> e -> e) -> Matrix e #
type IndexOf Matrix
Instance details Defined in Internal.Numeric type IndexOf Matrix = (Int, Int)
type ArgOf Matrix a
Instance details Defined in Internal.Numeric type ArgOf Matrix a = a -> a -> a
type ElementOf (Matrix a)
Instance details Defined in Internal.Numeric type ElementOf (Matrix a) = a
type TransRaw (Matrix t) b
Instance details Defined in Internal.Matrix type TransRaw (Matrix t) b = CInt -> CInt -> Ptr t -> b
type Trans (Matrix t) b
Instance details Defined in Internal.Matrix type Trans (Matrix t) b = CInt -> CInt -> CInt -> CInt -> Ptr t -> b