Numeric.LAPACK.Matrix.Function

Documentation

class Property property => SqRt property

Minimal complete definition

sqrt

Instances

SqRt Symmetric

Defined in Numeric.LAPACK.Matrix.Function

Methods

sqrt :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack Symmetric lower upper sh a -> Quadratic pack Symmetric lower upper sh a

SqRt Arbitrary

For Full matrices: Explicit solution for matrices up to size 2. Solution via sqrtDenmanBeavers for larger sizes.

Defined in Numeric.LAPACK.Matrix.Function

Methods

sqrt :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack Arbitrary lower upper sh a -> Quadratic pack Arbitrary lower upper sh a

SqRt Unit

Defined in Numeric.LAPACK.Matrix.Function

Methods

sqrt :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack Unit lower upper sh a -> Quadratic pack Unit lower upper sh a

(neg ~ False, C zero, C pos) => SqRt (Hermitian neg zero pos)

Defined in Numeric.LAPACK.Matrix.Function

Methods

sqrt :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack (Hermitian neg zero pos) lower upper sh a -> Quadratic pack (Hermitian neg zero pos) lower upper sh a

sqrt :: (SqRt property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a

sqrtSchur :: (C sh, Real a) => Square sh a -> Square sh a

Square root solver that works on the Schur decomposition.

Schur decomposition enables computing the square root of (some) singular matrices like ((1,0),(0,0)). However, the Schur decomposition might emit small negative values on the diagonal, where exact computation would yield zeros. This would let the square root solver fail. And there are singular matrices that have no square root, at all, e.g. ((0,1),(0,0)).

The solver is restricted to a real triangular Schur matrix. The check for non-real eigenvalues may exclude matrices that actually have a real-valued square root. E.g. sqrt ((0,-2),(2,0)) = ((1,-1),(1,-1))

In the future we might fix this by solving 2x2 blocks at the diagonal using sqrt2.

sqrtDenmanBeavers :: (Homogeneous prop, Additive prop) => (Packing pack, PowerStrip lower, PowerStrip upper) => (C sh, Floating a, RealOf a ~ ar, Real ar) => Quadratic pack prop lower upper sh a -> Quadratic pack prop lower upper sh a

Iterative square root solver, similar to Newton iteration.

Eigenvalues must all be positive, otherwise, the iteration might loop forever, or if an eigenvalue is zero, the computation of matrix inverse will fail.

class Property property => Exp property

Minimal complete definition

exp

Instances

Exp Symmetric

Defined in Numeric.LAPACK.Matrix.Function

Methods

exp :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Floating a) => Quadratic pack Symmetric lower upper sh a -> Quadratic pack Symmetric lower upper sh a

Exp Arbitrary

Defined in Numeric.LAPACK.Matrix.Function

Methods

exp :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Floating a) => Quadratic pack Arbitrary lower upper sh a -> Quadratic pack Arbitrary lower upper sh a

(neg ~ True, zero ~ True, pos ~ True) => Exp (Hermitian neg zero pos)

Defined in Numeric.LAPACK.Matrix.Function

Methods

exp :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Floating a) => Quadratic pack (Hermitian neg zero pos) lower upper sh a -> Quadratic pack (Hermitian neg zero pos) lower upper sh a

exp :: (Exp property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Floating a) => Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a

expRealHermitian :: (Packing pack, C sh, Real a) => HermitianP pack sh a -> HermitianPosSemidefP pack sh a

Mathematically the name expRealSymmetric would be more common, but we support definiteness tags only for the Hermitian type. Formally the result is always positive definite, but negative eigenvalues easily yield numerically singular matrices as result.

class Property property => Log property

Minimal complete definition

log

Instances

Log Symmetric

Defined in Numeric.LAPACK.Matrix.Function

Methods

log :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack Symmetric lower upper sh a -> Quadratic pack Symmetric lower upper sh a

Log Arbitrary

Defined in Numeric.LAPACK.Matrix.Function

Methods

log :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack Arbitrary lower upper sh a -> Quadratic pack Arbitrary lower upper sh a

(neg ~ True, zero ~ True, pos ~ True) => Log (Hermitian neg zero pos)

Defined in Numeric.LAPACK.Matrix.Function

Methods

log :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack (Hermitian neg zero pos) lower upper sh a -> Quadratic pack (Hermitian neg zero pos) lower upper sh a

log :: (Log property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a

logUnipotentUpper :: (Packing pack, C sh, Floating a) => UnitUpperP pack sh a -> UpperP pack sh a

class Property property => LiftReal property

Minimal complete definition

liftReal

Instances

LiftReal Symmetric

Defined in Numeric.LAPACK.Matrix.Function

Methods

liftReal :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => (a -> a) -> Quadratic pack Symmetric lower upper sh a -> Quadratic pack Symmetric lower upper sh a

LiftReal Arbitrary

Generic algorithm that applies a scalar function to the elements of the diagonal factor of a full, triangular or diagonal matrix with distinct eigenvalues. It is not checked whether the matrix has distinct eigenvalues.

Defined in Numeric.LAPACK.Matrix.Function

Methods

liftReal :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => (a -> a) -> Quadratic pack Arbitrary lower upper sh a -> Quadratic pack Arbitrary lower upper sh a

(neg ~ True, zero ~ True, pos ~ True) => LiftReal (Hermitian neg zero pos)

Defined in Numeric.LAPACK.Matrix.Function

Methods

liftReal :: (Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => (a -> a) -> Quadratic pack (Hermitian neg zero pos) lower upper sh a -> Quadratic pack (Hermitian neg zero pos) lower upper sh a

liftReal :: (LiftReal property, Packing pack, PowerStrip lower, PowerStrip upper, C sh, Real a) => (a -> a) -> Quadratic pack property lower upper sh a -> Quadratic pack property lower upper sh a

Lift any function with a Taylor expansion to a diagonalizable matrix.