Safe Haskell	None
Language	Haskell2010

Numeric.LinAlg

Description

This module provides an interface for specifying immutable linear algebra computations.

This is done with the Matr typeclass.

Synopsis

data Dim
- = V Nat
- | M Nat Nat
class Solve n dim | dim -> n where
- (<\>) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k
- (^\) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k
- (.\) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k
- (\\) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k
class Product a b where
- type Prod a b :: Dim
- (><) :: Matr k arr => arr a k -> arr b k -> arr (Prod a b) k
class (Floating k, Scale k arr) => Matr k arr where
- fromVect :: Vect n k -> arr (V n) k
- toVect :: arr (V n) k -> Vect n k
- fromVects :: Vect m (Vect n k) -> arr (M m n) k
- toVects :: arr (M m n) k -> Vect m (Vect n k)
- toRows :: arr (M m n) k -> Vect m (arr (V n) k)
- toColumns :: arr (M m n) k -> Vect n (arr (V m) k)
- fromRows :: Vect m (arr (V n) k) -> arr (M m n) k
- fromColumns :: Vect n (arr (V m) k) -> arr (M m n) k
- asColMat :: arr (V n) k -> arr (M n 1) k
- asColVec :: arr (M n 1) k -> arr (V n) k
- fromDiag :: arr (V n) k -> arr (M n n) k
- takeDiag :: arr (M n n) k -> arr (V n) k
- dim :: arr (M m n) k -> (SNat m, SNat n)
- rows :: arr (M m n) k -> SNat m
- cols :: arr (M m n) k -> SNat n
- len :: arr (V n) k -> SNat n
- trans :: arr (M m n) k -> arr (M n m) k
- ident :: SNat n -> arr (M n n) k
- outer :: arr (V m) k -> arr (V n) k -> arr (M m n) k
- (>.<) :: arr (V n) k -> arr (V n) k -> k
- mXm :: arr (M m n) k -> arr (M n p) k -> arr (M m p) k
- elementwiseprod :: arr (M m n) k -> arr (M m n) k -> arr (M m n) k
- inv :: arr (M n n) k -> arr (M n n) k
- invL :: arr (M n n) k -> arr (M n n) k
- invU :: arr (M n n) k -> arr (M n n) k
- chol :: arr (M n n) k -> arr (M n n) k
- cholInv :: arr (M n n) k -> arr (M n n) k
- cholLnDet :: arr (M n n) k -> k
- trsymprod :: arr (M n n) k -> arr (M n n) k -> k
- constant :: k -> SNat n -> arr (V n) k
- linearSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k
- ltriSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k
- utriSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k
- posdefSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k
- cholSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k
square :: Matr k arr => arr (M m n) k -> Maybe (m :~: n)
showMat :: (Show k, Matr k arr) => arr (M m n) k -> String
toLists :: Matr k arr => arr (M m n) k -> [[k]]
class Scale k arr where
- (.*) :: k -> arr dim k -> arr dim k

Documentation

data Dim

Constructors

V Nat
M Nat Nat

Instances

Container Vector e => Scale Dim e HArr

class Solve n dim | dim -> n where

Infix, overloaded versions of functions for solving linear equations

Methods

(<\>) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k infixl 7

linearSolve

(^\) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k infixl 7

utriSolve

(.\) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k infixl 7

ltriSolve

(\\) :: Matr k arr => arr (M n n) k -> arr dim k -> arr dim k infixl 7

posdefSolve

Instances

Solve n (V n)
Solve n (M n p)

class Product a b where

Associated Types

type Prod a b :: Dim

Methods

(><) :: Matr k arr => arr a k -> arr b k -> arr (Prod a b) k infixl 7

Instances

Product (V n) (M n p)
Product (M m n) (V n)
Product (M m n) (M n p)

class (Floating k, Scale k arr) => Matr k arr where

An instance of Matr k v m means that v k is a vector type and m k is a matrix type over the field k.

Minimal complete definition

fromVect, toVect, fromVects, toVects, toRows, fromRows, fromColumns, asColMat, fromDiag, takeDiag, dim, len, trans, ident, outer, (>.<), mXm, elementwiseprod, chol, trsymprod, constant, linearSolve

Methods

fromVect :: Vect n k -> arr (V n) k

Convert a list of elements to a vector.

toVect :: arr (V n) k -> Vect n k

Convert a vector to a list of its elements.

fromVects :: Vect m (Vect n k) -> arr (M m n) k

Convert a row-major list of lists of elements (which should all have the same length) to a matrix containing those elements.

toVects :: arr (M m n) k -> Vect m (Vect n k)

Convert a matrix to a list of its rows, each given as a list of elements.

toRows :: arr (M m n) k -> Vect m (arr (V n) k)

Convert a matrix to a list of its rows.

toColumns :: arr (M m n) k -> Vect n (arr (V m) k)

Convert a matrix to a list of its columns.

fromRows :: Vect m (arr (V n) k) -> arr (M m n) k

Convert a list of vectors to a matrix having those vectors as rows.

fromColumns :: Vect n (arr (V m) k) -> arr (M m n) k

Convert a list of vectors to a matrix having those vectors as columns.

asColMat :: arr (V n) k -> arr (M n 1) k

Regard a vector as a matrix with a single column.

asColVec :: arr (M n 1) k -> arr (V n) k

Convert a matrix which has only one column to a vector. This function may have undefined behavior if the input matrix has more than one column.

fromDiag :: arr (V n) k -> arr (M n n) k

Produce a diagonal matrix with the given vector along its diagonal (and zeros elsewhere).

takeDiag :: arr (M n n) k -> arr (V n) k

Return a vector of elements along the diagonal of the matrix. Does not necessarily fail if the matrix is not square.

dim :: arr (M m n) k -> (SNat m, SNat n)

Dimension of a matrix (rows, columns).

rows :: arr (M m n) k -> SNat m

The number of rows in a matrix.

cols :: arr (M m n) k -> SNat n

The number of columns in a matrix.

len :: arr (V n) k -> SNat n

The length of a vector.

trans :: arr (M m n) k -> arr (M n m) k

Transpose a matrix.

ident :: SNat n -> arr (M n n) k

Construct the identity matrix of a given dimension.

outer :: arr (V m) k -> arr (V n) k -> arr (M m n) k

Compute the outer product of two vectors.

(>.<) :: arr (V n) k -> arr (V n) k -> k infixl 7

Compute the dot product (i.e., inner product) of two vectors.

mXm :: arr (M m n) k -> arr (M n p) k -> arr (M m p) k

elementwiseprod :: arr (M m n) k -> arr (M m n) k -> arr (M m n) k

Compute the elementwise product (i.e., Hadamard product) of two matrices.

inv :: arr (M n n) k -> arr (M n n) k

General matrix inverse.

invL :: arr (M n n) k -> arr (M n n) k

Inverse of a lower-triangular matrix.

invU :: arr (M n n) k -> arr (M n n) k

Inverse of an upper-triangular matrix.

chol :: arr (M n n) k -> arr (M n n) k

Cholesky decomposition of a positive-definite symmetric matrix. Returns lower triangular matrix of the decomposition. May not necessarily zero out the upper portion of the matrix.

cholInv :: arr (M n n) k -> arr (M n n) k

Invert a positive-definite symmetric system using a precomputed Cholesky decomposition. That is, if l == chol a and b == cholInv l , then b is the inverse of a .

cholLnDet :: arr (M n n) k -> k

Compute the log-determinant of a positive-definite symmetric matrix using its precomputed Cholesky decomposition. That is, if l == chol a and d == cholLnDet l , then d is the log-determinant of a .

trsymprod :: arr (M n n) k -> arr (M n n) k -> k

If matrices a and b are symmetric, then trsymprod a b is the trace of a >< b . Note that in this case, the trace of a >< b is just the sum of the elements in the element-wise product of a and b .

constant :: k -> SNat n -> arr (V n) k

Create a vector of a given length whose elements all have the same value.

linearSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k

Solve a general system. If there is some x such that m >< x == b , then x == 'linearSolve m b' .

ltriSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k

Solve a lower-triangular system. If l is a lower-triangular matrix, then x == ltriSolve l b means that l >< x == b .

utriSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k

Solve a upper-triangular system. If u is a upper-triangular matrix, then x == utriSolve u b means that u >< x == b .

posdefSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k

Solve a positive-definite symmetric system. That is, if a is a positive-definite symmetric matrix and x == posdefSolve a b , then a >< x == b .

cholSolve :: arr (M n n) k -> arr (M n p) k -> arr (M n p) k

Solve a positive-definite symmetric system using a precomputed Cholesky decomposition. If l == chol a and x == l `'cholSolve'\` b , then a >< x == b .

Instances

(Floating k, Num (Matrix k), Num (Vector k), Field k, Numeric k) => Matr k HArr

square :: Matr k arr => arr (M m n) k -> Maybe (m :~: n)

If the matrix is square, return Just its dimension; otherwise, Nothing.

showMat :: (Show k, Matr k arr) => arr (M m n) k -> String

Pretty-print a matrix. (Actual prettiness not guaranteed!)

toLists :: Matr k arr => arr (M m n) k -> [[k]]

class Scale k arr where

Scalar multiplication for vector spaces.

Methods

(.*) :: k -> arr dim k -> arr dim k infixl 7

Instances

Container Vector e => Scale Dim e HArr