lapack-0.5: Numerical Linear Algebra using LAPACK

Numeric.LAPACK.Orthogonal

Synopsis

# 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 Source #

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 Source #

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) Source #

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) Source #

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

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 Source #

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) Source #

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 Source #

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

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 Source #

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) Source #

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) Source #

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) Source #

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) Source #