Numeric.LAPACK.Orthogonal

Documentation

leastSquares :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => Full meas horiz Small height width a -> Full meas vert horiz height nrhs a -> Full meas vert horiz width nrhs a

If x = leastSquares a b then x minimizes Vector.norm2 (multiply a x sub b).

Precondition: a must have full rank and height a >= width a.

minimumNorm :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => Full meas Small vert height width a -> Full meas vert horiz height nrhs a -> Full meas vert horiz width nrhs a

The vector x with x = minimumNorm a b is the vector with minimal Vector.norm2 x that satisfies multiply a x == b.

Precondition: a must have full rank and height a <= width a.

leastSquaresMinimumNormRCond :: (Measure meas, C vert, C horiz, C height, Eq height, C width, C nrhs, Floating a) => RealOf a -> Full meas horiz vert height width a -> Full meas vert horiz height nrhs a -> (Int, Full meas vert horiz width nrhs a)

If (rank,x) = leastSquaresMinimumNormRCond rcond a b then x is the vector with minimum Vector.norm2 x that minimizes Vector.norm2 (a #*| x sub b).

Matrix a can have any rank but you must specify the reciprocal condition of the rank-truncated matrix.

pseudoInverseRCond :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => RealOf a -> Full meas vert horiz height width a -> (Int, Full meas horiz vert width height a)

project :: (C height, Eq height, C width, Eq width, Floating a) => Wide height width a -> Vector height a -> Vector width a -> Vector width a

project b d x projects x on the plane described by B*x = d.

b must have full rank.

leastSquaresConstraint :: (C height, Eq height, C width, Eq width, C constraints, Eq constraints, Floating a) => General height width a -> Vector height a -> Wide constraints width a -> Vector constraints a -> Vector width a

leastSquaresConstraint a c b d computes x with minimal || c - A*x ||_2 and constraint B*x = d.

b must be wide and a===b must be tall and both matrices must have full rank.

gaussMarkovLinearModel :: (C height, Eq height, C width, Eq width, C opt, Eq opt, Floating a) => Tall height width a -> General height opt a -> Vector height a -> (Vector width a, Vector opt a)

gaussMarkovLinearModel a b d computes (x,y) with minimal || y ||_2 and constraint d = A*x + B*y.

a must be tall and a|||b must be wide and both matrices must have full rank.

determinant :: (C sh, Floating a) => Square sh a -> a

determinantAbsolute :: (Measure meas, C vert, C horiz, C height, C width, Floating a) => Full meas vert horiz height width a -> RealOf a

Gramian determinant - works also for non-square matrices, but is sensitive to transposition.

determinantAbsolute a = sqrt (Herm.determinant (Herm.gramian a))

complement :: (C height, C width, Floating a) => Tall height width a -> Tall height ShapeInt a

For an m-by-n-matrix a with m>=n this function computes an m-by-(m-n)-matrix b such that Matrix.multiply (adjoint b) a is a zero matrix. The function does not try to compensate a rank deficiency of a. That is, a|||b has full rank if and only if a has full rank.

For full-rank matrices you might also call this kernel or nullspace.

affineFrameFromFiber :: (C width, Eq width, C height, Eq height, Floating a) => Wide height width a -> Vector height a -> (Tall width ShapeInt a, Vector width a)

affineFrameFromFiber a b == (c,d)

Means: An affine subspace is given implicitly by {x : a#*|x == b}. Convert this into an explicit representation {c#*|y|+|d : y}. Matrix a must have full rank, otherwise the explicit representation will miss dimensions and we cannot easily determine the origin d as a minimum norm solution.

The computation is like

c = complement $ adjoint a
d = minimumNorm a b

but the QR decomposition of a is computed only once.

affineFiberFromFrame :: (C width, Eq width, C height, Eq height, Floating a) => Tall height width a -> Vector height a -> (Wide ShapeInt height a, Vector ShapeInt a)

This conversion is somehow inverse to affineFrameFromFiber. However, it is not precisely inverse in either direction. This is because both affineFrameFromFiber and affineFiberFromFrame accept non-orthogonal matrices but always return orthogonal ones.

In affineFiberFromFrame c d, matrix c should have full rank, otherwise the implicit representation will miss dimensions.

householder :: (Measure meas, C vert, C horiz, Permutable height, C width, Floating a) => Full meas vert horiz height width a -> (Square height a, UpperTrapezoid meas vert horiz height width a)

householderTall :: (Measure meas, C vert, C height, Permutable width, Floating a) => Full meas vert Small height width a -> (Full meas vert Small height width a, Upper width a)