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matrix-ts

Decomposition overview

Decompose matricies into various forms, such as LUP. Can be used in various applications like solving a system of equations, or calculating the principle components of a multivariate distribution (SVD)

Added in v1.0.0

Table of contents

Constructors

LUP

Decomposition of a square matrix into a lower triangular matrix L and an upper triangular matrix U, with a permutation matrix P, such that:

PA = LU

Signature

export declare const LUP: <M extends number>(
  m: M.Mat<M, M, number>
) => C.Computation<
  string,
  Decomposition<
    M,
    M,
    number,
    [
      MatTypes.LowerTriangularMatrix<M, number>,
      MatTypes.UpperTriangularMatrix<M, number>,
      MatTypes.OrthogonalMatrix<M, number>
    ]
  > &
    Solvable<M, number> &
    Determinant
>

Added in v1.0.0

QR

QR decomposition with column pivoting using Householder Reflections. Based on an algorithm described in Fundamentals of Matrix Computations, David S. Watkins.

Can be used to calculate a matrix’s:

  • Singularity
  • Rank
  • Least Square Solution for overdetermined systems

Efficiency: O(n^3)

Returns a tuple with:

  • A': The assembled matrix R with lower triangular components being orthogonal vectors collectively used to construct the reflector Q.
  • Q: An IO that returns a constructed reflector Q. This has the same efficiency as QR decomposition itself, so evaluation is deferred.
  • R: The upper triangular matrix extracted from A'.
  • P: The permutation matrix used to permute the columns of A

QR decomposes a matrix A into a matrix Q, a matrix R, and a matrix P such that:

AP = QR

Signature

export declare function QR<N extends number, M extends number>(
  mat: M.Mat<N, M, number>
): C.Computation<
  string,
  Decomposition<
    N,
    M,
    number,
    [
      M.Mat<N, M, number>,
      IO.IO<MatTypes.OrthogonalMatrix<N, number>>,
      M.Mat<N, M, number>,
      MatTypes.OrthogonalMatrix<M, number>
    ]
  > &
    IsSingular &
    Rank &
    LeastSquares<N, M, number>
>

Added in v1.1.0

Model

Decomposition (interface)

Signature

export interface Decomposition<N, M, R, A> {
  input: M.Mat<N, M, R>
  result: A
}

Added in v1.0.0

Determinant (interface)

Represents the result of a computation that can be used to calculate the determinant

Signature

export interface Determinant {
  det: IO.IO<number>
}

Added in v1.0.4

IsSingular (interface)

Determines if A is singular

Signature

export interface IsSingular {
  isSingular: boolean
}

Added in v1.1.0

LeastSquares (interface)

Represents a solution to an overdetermined system. Returns O.none if the rank(A) < m (the number of columns). Returns a tuple with the residual, and solution vector

Signature

export interface LeastSquares<N, M, R> {
  solve: (b: V.Vec<N, R>) => C.Computation<string, [number, V.Vec<M, R>]>
}

Added in v1.1.0

Rank (interface)

Returns the Rank of A

Signature

export interface Rank {
  rank: number
}

Added in v1.1.0

Solvable (interface)

Represents the result of a computation that can solve:

Ax = b

Signature

export interface Solvable<M, R> {
  solve: (b: V.Vec<M, R>) => V.Vec<M, R>
}

Added in v1.0.4