Matrix - Documentation

new Matrix(A)

Source:

index.js, line 4

Creates a new Matrix

Parameters:

Name	Type	Description
`A`	Array.<Array.<number>>	Two dimensional array where A[i][j] represents the i-th row and j-th column of a matrix

Methods

(static) LU(A, optimizedopt) → {Array.<Matrix>}

Source:

core/decompositions/LU.js, line 19

Calculates the LUP decomposition of the Matrix, where L is lower triangular matrix which diagonal entries are always 1, U is upper triangular matrix, and P is permutation matrix.

It is implemented using Gaussian Elimination with Partial Pivoting in order to reduce the error caused by floating-point arithmetic.

Note that if optimized is true, P is a Permutation Array and both L and U are merged into one matrix in order to improve performance.

Parameters:

Name	Type	Attributes	Default	Description
`A`	Matrix			Any matrix
`optimized`	boolean	<optional>	`false`	Returns [P, LU] if it is true, [P, L, U] if it is false

Returns:

The LUP decomposition of Matrix

Type: Array.<Matrix>

(static) QR(A) → {Array.<Matrix>}

Source:

core/decompositions/QR.js, line 14

Calculates the QR decomposition of the Matrix where Q is orthogonal matrix, R is upper triangular matrix.

The algorithm is implemented using Householder Transform instead of Gram–Schmidt process because the Householder Transform is more numerically stable.

Parameters:

Name	Type	Description
`A`	Matrix	Any matrix

Returns:

The QR decomposition of matrix in the form of [Q, R]

Type: Array.<Matrix>

(static) backward(U, y) → {Matrix}

Source:

core/linear-equations/backward.js, line 20

Solve system of linear equations Ux = y using backward substitution, where U is an upper triangular matrix. If there is no unique solutions, an error is thrown.

Parameters:

Name	Type	Description
`U`	Matrix	Any n x n upper triangular Matrix
`y`	Matrix	Any n x 1 Matrix

Returns:

n x 1 Matrix which is the solution of Ux = y

Type: Matrix

(static) forward(L, y) → {Matrix}

Source:

core/linear-equations/forward.js, line 20

Solve system of linear equations Lx = y using forward substitution, where L is a lower triangular matrix. If there is no unique solutions, an error is thrown.

Parameters:

Name	Type	Description
`L`	Matrix	Any n x n lower triangular Matrix
`y`	Matrix	Any n x 1 Matrix

Returns:

n x 1 Matrix which is the solution of Lx = y

Type: Matrix

(static) solve(L, y) → {Matrix}

Source:

core/linear-equations/solve.js, line 17

Solve system of linear equations Ax = y using LU decomposition. If there is no unique solutions, an error is thrown.

Parameters:

Name	Type	Description
`L`	Matrix	Any n x n square Matrix
`y`	Matrix	Any n x 1 Matrix

Returns:

n x 1 Matrix which is the solution of Ax = y

Type: Matrix

(static) add(A, B) → {Matrix}

Source:

core/operations/add.js, line 11

Calculates the sum of two Matrices.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix
`B`	Matrix	Any Matrix that has same size with A

Returns:

The sum of two Matrices

Type: Matrix

(static) inverse(A) → {Matrix}

Source:

core/operations/inverse.js, line 12

Find the inverse of non-singular matrix using Elementary Row Operations. If the matrix is singular, an error is thrown.

Parameters:

Name	Type	Description
`A`	Matrix	Any square Matrix

Returns:

The inverse of A

Type: Matrix

(static) multiply(A, B) → {Matrix}

Source:

core/operations/multiply.js, line 12

Calculates the product of two Matrices.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix
`B`	Matrix	Any Matrix that is size-compatible with A

Returns:

The product of two Matrices

Type: Matrix

(static) pow(A, exponent) → {Matrix}

Source:

core/operations/pow.js, line 12

Calculates the power of any square matrix. The algorithm is implemented recursively.

Parameters:

Name	Type	Description
`A`	Matrix	Any square Matrix
`exponent`	number	Any Non-negative integer

Returns:

The power of A

Type: Matrix

(static) transpose(A) → {Matrix}

Source:

core/operations/transpose.js, line 10

Find the transpose of a matrix.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix

Returns:

Returns transpose of A

Type: Matrix

cond(p) → {number}

Source:

core/properties/cond.js, line 14

Calculations the condition number of square Matrix with respect to the choice of Matrix norm. If the Matrix is singular, returns Infinity.

The condition number is not cached.

Parameters:

Name	Type	Default	Description
`p`	1 \| 2 \| Infinity \| 'F'	`2`	Type of Matrix norm

Returns:

The condition number of Matrix

Type: number

det() → {number}

Source:

core/properties/det.js, line 14

Calculates the determinant of square Matrix. If the Matrix size is larger than 3, it calculates the determinant using LU decomposition, otherwise, using Leibniz Formula.

The determinant is cached.

Returns:

Returns the determinant of square matrirx

Type: number

eigenvalues() → {Array.<Complex>}

Source:

core/properties/eigenvalues.js, line 21

Calculates the eigenvalues of any square Matrix using QR Algorithm.

The eigenvalues can be either real number or complex number. Note that all eigenvalues are instance of Complex, for more details please visit Complex.js.

The eigenvalues are cached.

Returns:

Array of eigenvalues

Type: Array.<Complex>

norm(p) → {number}

Source:

core/properties/norm.js, line 18

Calculates the Matrix norm of any Matrix with respect to the choice of norm.

1-norm: Maximum absolute column sum of the Matrix.
2-norm: The largest singular value of Matrix.
Infinity-norm: Maximum absolute row sum of the Matrix.
Frobenius-norm: Euclidean norm invloving all entries.

The norms are not cached.

Parameters:

Name	Type	Default	Description
`p`	1 \| 2 \| Infinity \| 'F'	`2`	The choice of Matrix norm

Returns:

The norm of the Matrix.

Type: number

nullity() → {number}

Source:

core/properties/nullity.js, line 10

Calculates the nullity of any Matrix, which is the dimension of the nullspace.

The nullity is cached.

Returns:

The nullity of the matrix

Type: number

rank() → {number}

Source:

core/properties/rank.js, line 12

Calculates the rank of any Matrix, which is the dimension of the row space.

The rank is cached.

Returns:

The rank of the Matrix

Type: number

size() → {Array.<number>}

Source:

core/properties/size.js, line 10

Calculates the size of any Matrix, which is in the form of [row, column].

The size of Matrix is cached.

Returns:

The number of rows and columns of a Matrix

Type: Array.<number>

trace() → {number}

Source:

core/properties/trace.js, line 12

Calculates the trace of any square Matrix, which is the sum of all entries on the main diagonal.

The trace is cached.

Returns:

The trace of the square Matrix.

Type: number

isDiagonal(digitopt) → {boolean}

Source:

core/structure/isDiagonal.js, line 13

Determines whether a Matrix is diagonal or not.

Diagonal Matrix is a Matrix in which the entries outside the main diagonal are all zero. Note that the term diagonal refers to rectangular diagonal.

The result is cached.

Parameters:

Name	Type	Attributes	Default	Description
`digit`	number	<optional>	`8`	Number of significant digits

Returns:

Returns rue if the Matrix is diagonal Matrix

Type: boolean

isLowerTriangular(digitopt) → {boolean}

Source:

core/structure/isLowerTriangular.js, line 15

Determines whether a Matrix is lower triangular Matrix or not.

Lower triangular Matrix is a Matrix in which all the entries above the main diagonal are zero. Note that it can be applied to any non-square Matrix.

The result is cached.

Parameters:

Name	Type	Attributes	Default	Description
`digit`	number	<optional>	`8`	Number of significant digits

Returns:

Returns true if the Matrix is lower triangular

Type: boolean

isOrthogonal(digitopt) → {boolean}

Source:

core/structure/isOrthogonal.js, line 13

Determines whether a square Matrix is orthogonal or not.

Orthogonal Matrix is a Matrix in which all rows and columns are orthonormal vectors.

The result is cached.

Parameters:

Name	Type	Attributes	Default	Description
`digit`	number	<optional>	`8`	Number of significant digits

Returns:

Returns true if the square Matrix is orthogonal

Type: boolean

isSkewSymmetric(digitopt) → {boolean}

Source:

core/structure/isSkewSymmetric.js, line 12

Determines whether a square Matrix is skew symmetric or not.

Skew symmetric Matrix is a square Matrix whose transpose equals its negative.

The result is cached.

Parameters:

Name	Type	Attributes	Default	Description
`digit`	number	<optional>	`8`	Number of significant digits

Returns:

Returns true if the square Matrix is skew symmetric

Type: boolean

isSquare() → {boolean}

Source:

core/structure/isSquare.js, line 11

Determines whether a Matrix is square or not.

Square Matrix is a Matrix with same number of rows and columns.

The result is cached.

Returns:

Returns true if the Matrix is square

Type: boolean

isSymmetric(digitopt) → {boolean}

Source:

core/structure/isSymmetric.js, line 12

Determines whether a square Matrix is symmetric or not.

Symmetric Matrix is a square Matrix that is equal to its transpose.

The result is cached.

Parameters:

Name	Type	Attributes	Default	Description
`digit`	number	<optional>	`8`	Number of significant digits

Returns:

Returns true if the square Matrix is symmetric

Type: boolean

isUpperTriangular(digitopt) → {boolean}

Source:

core/structure/isUpperTriangular.js, line 13

Determines whether a Matrix is upper triangular Matrix or not.

Upper triangular Matrix is a Matrix in which all the entries below the main diagonal are zero. Note that it can be applied to any non-square Matrix.

The result is cached.

Parameters:

Name	Type	Attributes	Default	Description
`digit`	number	<optional>	`8`	Number of significant digits

Returns:

Returns true if the Matrix is upper triangular

Type: boolean

(static) clone(A) → {Matrix}

Source:

core/utils/clone.js, line 10

Creates a copy of Matrix. Note that it resets the cached data.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix

Returns:

Copy of A

Type: Matrix

(static) column(A, index) → {Matrix}

Source:

core/utils/column.js, line 11

Gets the column of a Matrix with valid index.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix
`index`	number	Any valid column index

Returns:

Column of A

Type: Matrix

(static) diag(values) → {Matrix}

Source:

core/utils/diag.js, line 13

Generates diagonal Matrix if the argument is an array of numbers, generates block diagonal Matrix if the argument is an array of Matrices.

Parameters:

Name	Type	Description
`values`	Array.<number> \| Array.<Matrix>	Array of numbers or Matrices

Returns:

Block diagonal Matrix

Type: Matrix

(static) elementwise(A, cb) → {Matrix}

Source:

core/utils/elementwise.js, line 19

Applys a function over each entry of a Matrix and returns a new copy of the new Matrix.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix
`cb`	entryCallback	Callback function which applies on each entry of A

Returns:

A copy of new Matrix

Type: Matrix

entry(row, col) → {number}

Source:

core/utils/entry.js, line 11

Gets the entry of a Matrix.

Parameters:

Name	Type	Description
`row`	number	Any valid row index
`col`	number	Any valid column index

Returns:

Entry of the Matrix

Type: number

(static) fromArray(arr, row, col) → {Matrix}

Source:

core/utils/fromArray.js, line 12

Generate a matrix from an array with compatible dimensions

Parameters:

Name	Type	Description
`arr`	Array	Source array
`row`	number	Row of the matrix
`col`	number	Column of the matrix

Returns:

Matrix

Type: Matrix

(static) generate(row, col, cb) → {Matrix}

Source:

core/utils/generate.js, line 21

Generates a Matrix which entries are the returned value of callback function.

Parameters:

Name	Type	Description
`row`	number	Number of rows of Matrix
`col`	number	Number of columns of Matrix
`cb`	generateCallback	Callback function which takes row and column as arguments and generates the corresponding entry

Returns:

- Generated Matrix

Type: Matrix

(static) getDiag(A) → {Array.<number>}

Source:

core/utils/getDiag.js, line 10

Gets the entries on the main diagonal.

Parameters:

Name	Type	Description
`A`	Matrix	Any Matrix

Returns:

Array of entries of A on the main diagonal

Type: Array.<number>

(static) getRandomMatrix(row, col, min, max, toFixed) → {Matrix}

Source:

core/utils/getRandomMatrix.js, line 12

Generates a random Matrix.

Parameters:

Name	Type	Default	Description
`row`	number		Number of rows of a Matrix
`col`	number		Number of columns of a Matrix
`min`	number	`0`	Lower bound of each entry
`max`	number	`1`	Upper bound of each entry
`toFixed`	number	`0`	Number of decimal places

Returns:

Generated random Matrix

Type: Matrix

(static) identity(size) → {Matrix}

Source:

core/utils/identity.js, line 8

Generates identity Matrix with given size.

Parameters:

Name	Type	Description
`size`	number	The size of Matrix

Returns:

Identity Matrix

Type: Matrix

(static) isEqual(A, B, digit) → {boolean}

Source:

core/utils/isEqual.js, line 17

Determines whether two Matrices are considered as equal.

The test criterion is Math.abs(x - y) < 1 / (10 ** digit * 2). For default value 5, it should be 5e-5. That means if the difference of two numbers is less than 5e-5, they are considered as same value.