LinearAlgebra (FSharp.Stats)

LinearAlgebra Module

Namespace: FSharp.Stats.Algebra
Assembly: FSharp.Stats.dll

Types

Type	Description
Factorization

Functions and values

Function or value	Description
`Cholesky a` Full Usage: `Cholesky a` Parameters: a : `matrix` - Returns: `Matrix<float>`	Given A[n,n] real symmetric positive definite. Finds the cholesky decomposition L such that L' * L = A. May fail if not positive definite. a : `matrix` Returns: `Matrix<float>` Example
`Determinant A` Full Usage: `Determinant A` Parameters: A : `Matrix<float>` - Returns: `float`	Compute the determinant of a matrix by performing an LU decomposition since if A = P'LU, then det(A) = det(P') * det(L) * det(U). A : `Matrix<float>` Returns: `float` Example
`EVD m` Full Usage: `EVD m` Parameters: m : `'a` Returns: `'b`	Compoutes for an N-by-N real nonsymmetric matrix A, the eigenvalue decomposition eigenvalues and right eigenvectors. The right eigenvector v(j) of A satisfies A * v(j) = lambda(j) * v(j) where lambda(j) is its eigenvalue. The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real. Uses the LAPACK subroutine dgeev with arguments JOBVR = 'V' and JOBVL = 'N' Returns the real (first array) and imaginary (second array) parts of the eigenvalues and a matrix containing the corresponding eigenvectors m : `'a` Returns: `'b`
`EigenSpectrumWhenSymmetric a` Full Usage: `EigenSpectrumWhenSymmetric a` Parameters: a : `Matrix<float>` - Returns: `Matrix<float> * Vector<float>`	Compute eigenvalues and eigenvectors for a real symmetric matrix. Returns arrays of the values and vectors (both based on reals). This call may fail. a : `Matrix<float>` Returns: `Matrix<float> * Vector<float>` Example
`EigenValues m` Full Usage: `EigenValues m` Parameters: m : `'a` - Returns: `'b`	Compute eigenvalues of a square real matrix. Returns arrays containing the eigenvalues which may be complex. This call may fail. m : `'a` Returns: `'b` Example
`EigenValuesWhenSymmetric a` Full Usage: `EigenValuesWhenSymmetric a` Parameters: a : `'a` - Returns: `'b`	Compute eigenvalues for a real symmetric matrix. Returns array of real eigenvalues. This call may fail. a : `'a` Returns: `'b` Example
`Hessenberg A` Full Usage: `Hessenberg A` Parameters: A : `matrix` Returns: `matrix * Matrix<float>`	A : `matrix` Returns: `matrix * Matrix<float>`
`Inverse a` Full Usage: `Inverse a` Parameters: a : `Matrix<float>` - Returns: `Matrix<float>`	Given A[n,n] find it's inverse. This call may fail. a : `Matrix<float>` Returns: `Matrix<float>` Example
`LU a` Full Usage: `LU a` Parameters: a : `Matrix<float>` - Returns: `(int -> int) * Matrix<float> * Matrix<float>`	Given A[n,n] real matrix. Finds P,L,U such that LU = PA with L,U lower/upper triangular. a : `Matrix<float>` Returns: `(int -> int) * Matrix<float> * Matrix<float>` Example
`LeastSquares a b` Full Usage: `LeastSquares a b` Parameters: a : `Matrix<float>` - b : `vector` - Returns: `Vector<float>`	Given A[m,n] and B[m] solves AX = B for X[n]. When m => n, have over constrained system, finds least squares solution for X. When m < n, have under constrained system, finds least norm solution for X. a : `Matrix<float>` b : `vector` Returns: `Vector<float>` Example
`LeastSquaresCholesky a b` Full Usage: `LeastSquaresCholesky a b` Parameters: a : `Matrix<float>` - b : `Vector<float>` - Returns: `Vector<float>`	Given A[m,n] and b[m] solves AX = b for X[n]. When the system is under constrained, for example when the columns of A are not linearly independent, then it will not give sensible results. a : `Matrix<float>` b : `Vector<float>` Returns: `Vector<float>` Example
`QR a` Full Usage: `QR a` Parameters: a : `matrix` - Returns: `matrix * matrix`	Given A[m,n] finds Q[m,m] and R[k,n] where k = min m n. Have A = Q.R when m < =n. Have A = Q.RX when m > n and RX[m,n] is R[n,n] row extended with (m-n) zero rows. a : `matrix` Returns: `matrix * matrix` Example
`SVD a` Full Usage: `SVD a` Parameters: a : `Matrix<float>` Returns: `vector * Matrix<float> * Matrix<float>`	Returns the full Singular Value Decomposition of the input MxN matrix A : A = U * SIGMA * VT in the tuple (S, U, VT), where S is an array containing the diagonal elements of SIGMA. uses the LAPACK routine dgesdd with the argument JOBZ = 'A' a : `Matrix<float>` Returns: `vector * Matrix<float> * Matrix<float>`
`Service ()` Full Usage: `Service ()` Parameters: () : `unit` Returns: `ILinearAlgebra`	() : `unit` Returns: `ILinearAlgebra`
`SolveLinearSystem A b` Full Usage: `SolveLinearSystem A b` Parameters: A : `matrix` b : `vector` Returns: `Vector<float>`	A : `matrix` b : `vector` Returns: `Vector<float>`
`SolveLinearSystems A B` Full Usage: `SolveLinearSystems A B` Parameters: A : `matrix` B : `matrix` Returns: `Matrix<float>`	A : `matrix` B : `matrix` Returns: `Matrix<float>`
`SolveTriangularLinearSystem A b isLower` Full Usage: `SolveTriangularLinearSystem A b isLower` Parameters: A : `matrix` b : `vector` isLower : `bool` Returns: `Vector<float>`	A : `matrix` b : `vector` isLower : `bool` Returns: `Vector<float>`
`SolveTriangularLinearSystems A B isLower` Full Usage: `SolveTriangularLinearSystems A B isLower` Parameters: A : `matrix` B : `matrix` isLower : `bool` Returns: `Matrix<float>`	A : `matrix` B : `matrix` isLower : `bool` Returns: `Matrix<float>`
`hatmatrix A` Full Usage: `hatmatrix A` Parameters: A : `Matrix<float>` - Returns: `Matrix<float>`	computes the hat matrix by the QR decomposition of the designmatrix used in ordinary least squares approaches A : `Matrix<float>` Returns: `Matrix<float>` Example
`leverage A` Full Usage: `leverage A` Parameters: A : `Matrix<float>` - Returns: `vector`	computes the leverage directly by QR decomposition of the designmatrix used in ordinary least squares approaches and computing of the diagnonal entries of the Hat matrix, known as the leverages of the regressors A : `Matrix<float>` Returns: `vector` Example
`leverageBy A` Full Usage: `leverageBy A` Parameters: A : `Matrix<float>` - Returns: `Vector<float>`	computes the hat matrix by the QR decomposition of the designmatrix used in ordinary least squares approaches A : `Matrix<float>` Returns: `Vector<float>` Example
`preDivideByMatrix a b` Full Usage: `preDivideByMatrix a b` Parameters: a : `matrix` - b : `matrix` - Returns: `Matrix<float>`	Given A[n,m] and B[n,k] solve for X[m,k] such that AX = B This call may fail. a : `matrix` b : `matrix` Returns: `Matrix<float>` Example
`preDivideByVector A b` Full Usage: `preDivideByVector A b` Parameters: A : `matrix` - b : `vector` - Returns: `Vector<float>`	Given A[n,m] and B[n] solve for x[m] such that Ax = B This call may fail. A : `matrix` b : `vector` Returns: `Vector<float>` Example
`spectralNorm a` Full Usage: `spectralNorm a` Parameters: a : `Matrix<float>` - Returns: `float`	spectral norm of a matrix (for Frobenius norm use Matrix.norm) a : `Matrix<float>` Returns: `float` Example
`thinSVD a` Full Usage: `thinSVD a` Parameters: a : `Matrix<float>` Returns: `vector * Matrix<float> * Matrix<float>`	Returns the thin Singular Value Decomposition of the input MxN matrix A A = U * SIGMA * VT in the tuple (S, U, V), where S is an array containing the diagonal elements of SIGMA. The first min(M,N) columns of U and the first min(M,N) rows of VT are returned in the arrays U and VT; uses the LAPACK routine dgesdd with the argument JOBZ = 'S' a : `Matrix<float>` Returns: `vector * Matrix<float> * Matrix<float>`

LinearAlgebra Module

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