mXrap - XLAMath

XLAMath

This library provides functions for linear algebra transformations

This is very much still a work in progress, the functionality here has only been tested with a few sample cases.

QuickRef

Filter
Filter.SavitzkyGolayParameters
Matrix
Matrix.copyMatrix
Matrix.determinant
Matrix.fillMatrix
Matrix.identityMatrix
Matrix.inverse
Matrix.isSymmetric
Matrix.multiplyMatrix
Matrix.transposeMatrix
Solver
Solver.cholSolve
Solver.luSolve
Solver.qrSolve
Solver.svdSolve
Transformation
Transformation.ATA
Transformation.chol
Transformation.eigen
Transformation.hessenberg
Transformation.lu
Transformation.qr
Transformation.schur
Transformation.svd
Transformation.svdPseudoInverse
Transformation.symmetricEigen
Transformation.tridiagonal

Status	Name
	Filter.SavitzkyGolayParameters ( `Vector` ↑v, `Number` ↓fm, `Matrix` ↓dim, `Number` ↓dn, `Vector` ↓w, `Vector` ↑v ) Computes the coefficients for a Savitzky Golay filter given the parameters
	`Matrix` Matrix.copyMatrix ( `Matrix` ↓A, `Number` ↓m, `Number` ↓n, `Matrix` ↑B ) Copies the contents of a m x n matrix A into another matrix B.
	`Number` Matrix.determinant ( `Matrix` ↓A, `Number` ↓am ) Computes the determinant of square matrix A, using LU decomposition
	`Matrix` Matrix.fillMatrix ( `Matrix` ⇅A, `Number` ↓m, `Number` ↓n, `Number` ↓val ) Given a m x n matrix A, fills all elements of the matrix with the specified value. This is a convenience method to initialise a matrix (e.g. fill with 0).
	`Matrix` Matrix.identityMatrix ( `Number` ↓m, `Matrix` ↑mat ) Generates an identity matrix of size m x m.
	`Boolean` Matrix.inverse ( `Matrix` ↓A, `Number` ↓am, `Matrix` ↑inverse ) Computes the inverse of square matrix A.
	`Boolean` Matrix.isSymmetric ( `Matrix` ↓A, `Number` ↓m ) Given a m x m square matrix A, returns whether it is symmetrical, within tolerance
	`Matrix` Matrix.multiplyMatrix ( `Matrix` ↓A, `Number` ↓am, `Number` ↓an, `Matrix` ↓B, `Number` ↓bm, `Number` ↓bn, `Matrix` ↑C ) Multiplies matrices A and B to output C.
	`Matrix` Matrix.transposeMatrix ( `Matrix` ↓A, `Number` ↓m, `Number` ↓n, `Matrix` ↑AT ) Given a m x n matrix A, returns the transpose, n x m matrix A ^T.
	Solver.cholSolve ( `Matrix` ↓A, `Number` ↓am, `Matrix` ↓B, `Number` ↓bm, `Number` ↓bn, `Matrix` ↑X ) For the symmetric positive definite matrix A, and matrices X and B, where AX = B, solve for X given A and B, using Cholesky Decomposition.
	`Boolean` Solver.luSolve ( `Matrix` ↓A, `Number` ↓am, `Matrix` ↓B, `Number` ↓bm, `Number` ↓bn, `Matrix` ↑X ) For the square matrix A, and matrices X and B, where AX = B, solve for X given A and B, using LU Decomposition
	`Boolean` Solver.qrSolve ( `Matrix` ↓A, `Number` ↓am, `Number` ↓an, `Matrix` ↓B, `Number` ↓bm, `Number` ↓bn, `Matrix` ↑X, `Number` ↓threshold ) For the matrices A, X and B, where AX = B, solve for X given A and B, using QR Decomposition
	Solver.svdSolve ( `Matrix` ↓A, `Number` ↓am, `Number` ↓an, `Matrix` ↓B, `Number` ↓bm, `Number` ↓bn, `Matrix` ↑X ) For the matrix A, X and B, where AX = B, solve for X given A and B, using Singular Value Decomposition
	`Matrix` Transformation.ATA ( `Matrix` ↓A, `Number` ↓m, `Number` ↓n, `Matrix` ↑ATA ) Given a m × n matrix A, computes and returns the n × n matrix, A ^T × A.
	Transformation.chol ( `Matrix` ↓A, `Number` ↓m, `Matrix` ↑L, `Matrix` ↑LT, `Number` ↓relativeSymmetryThreshold, `Number` ↓absolutePositivityThreshold, `Object` ↑details ) Calculates the Cholesky decomposition of a matrix. The Cholesky decomposition of a real symmetric positive-definite matrix A consists of a lower triangular matrix L with same size such that: A = LL ^T. In a sense, this is the square root of A.
	Transformation.eigen ( `Matrix` ↓A, `Number` ↓m, `Array` ↑realEigenValues, `Array` ↑imagEigenValues, `Matrix` ↑eigenVectors ) Transforms the matrix A of size m × m to derive its eigenvalues and eigenvectors.
	Transformation.hessenberg ( `Matrix` ↓A, `Number` ↓m, `Matrix` ↑P, `Matrix` ↑H ) Transforms the matrix A of size m × m to its Hessenberg form. The m × m matrix A can be written as the product of three matrices: A = P × H × P ^T with P an orthogonal matrix and H a Hessenberg matrix. Both P and H are m × m matrices. P is an orthogonal matrix, i.e. its inverse is also its transpose.
	`Boolean` Transformation.lu ( `Matrix` ↓A, `Number` ↓m, `Matrix` ↑L, `Matrix` ↑U, `Matrix` ↑P, `Array` ↑pivot, `Matrix` ↑LU, `Object` ↑details ) Calculates the LUP-decomposition of a square matrix. The LUP-decomposition of a matrix A consists of three matrices L, U and P that satisfy: P×A = L×U. L is lower triangular (with unit diagonal terms), U is upper triangular and P is a permutation matrix. All matrices are m×m. As shown by the presence of the P matrix, this decomposition is implemented using partial pivoting. This function uses 1e-11 as the singularity threshold.
	`Boolean` Transformation.qr ( `Matrix` ↓A, `Number` ↓m, `Number` ↓n, `Matrix` ↑Q, `Matrix` ↑R, `Matrix` ↑H, `Matrix` ↑QT, `Number` ↓threshold ) Calculates the QR-decomposition of a matrix. The QR-decomposition of a matrix A consists of two matrices Q and R that satisfy: A = QR, Q is orthogonal (Q ^TQ = I), and R is upper triangular. If A is m×n, Q is m×m and R m×n. This function compute the decomposition using Householder reflectors.
	Transformation.schur ( `Matrix` ↓A, `Number` ↓m, `Matrix` ↑P, `Matrix` ↑T ) Transforms the matrix A of size m × m to its Schur form. The m × m matrix A can be written as the product of three matrices: A = P × T × P ^T with P an orthogonal matrix and T an quasi-triangular matrix. Both P and T are m × m matrices. P is an orthogonal matrix, i.e. its inverse is also its transpose.
	Transformation.svd ( `Matrix` ↓A, `Number` ↓m, `Number` ↓n, `Matrix` ↑U, `Matrix` ↑V, `Array` ↑singularValues, `Array` ↑S, `Object` ↑details ) Calculates the compact Singular Value Decomposition of a matrix. The Singular Value Decomposition of matrix A is a set of three matrices: U, Σ and V such that A = U × Σ × V ^T. Let A be a m × n matrix, then U is a m × p orthogonal matrix, Σ is a p × p diagonal matrix with positive or null elements, V is a p × n orthogonal matrix (hence V ^T is also orthogonal) where p=min(m,n). This implementation requires m > n. If that is not the case, instead pass in the transpose of A. The V and U matrices are swapped in this case.
	`Matrix` Transformation.svdPseudoInverse ( `Number` ↓m, `Number` ↓n, `Array` ↓singularValues, `Matrix` ↓U, `Matrix` ↓V, `Number` ↓tol, `Matrix` ↑inv ) Calculates the pseudoinverse of a m × n matrix A based on the output of its SVD transformation.
	Transformation.symmetricEigen ( `Matrix` ↓A, `Number` ↓m, `Array` ↑realEigenValues, `Array` ↑imagEigenValues, `Matrix` ↑eigenVectors ) Transforms the symmetric matrix A of size m × m to derive its eigenvalues and eigenvectors.
	Transformation.tridiagonal ( `Matrix` ↓A, `Number` ↓m, `Matrix` ↑P, `Matrix` ↑T, `Array` ↑main, `Array` ↑secondary ) Transforms the symmetric matrix A of size m × m to a tridiagonal form. The m × m matrix A can be written as the product of three matrices: A = Q × T × Q ^T with Q an orthogonal matrix and T a symmetrical tridiagonal matrix. Both Q and T are m × m matrices. Q is an orthogonal matrix, i.e. its inverse is also its transpose. The function assumes that A is symmetric, only the upper part of the matrix is accessed.

Category: Filter

Methods for matrix filters

Lib.Filter.SavitzkyGolayParameters ( ↑v, ↓fm, ↓dim, ↓dn, ↓w, ↑v ) Alpha

Computes the coefficients for a Savitzky Golay filter given the parameters

Parameters:

Vector ↑v - - output vector v, which is a vector sized frame size.
Number ↓fm - - frame size - must be a positive odd integer.
Matrix ↓dim - - order of polynomial for filter. Must be less than frame size - 1.
Number ↓dn - - derivative to compute. 0 = smoothing only. Must be less than or equal to order of polynomial.
Vector ↓w - - optional, weighting vector. If provided, must be sized frame size.
Vector ↑v - - output vector v, which is a vector sized frame size.

Category: Matrix

Methods for working with matrices The matrices in this library are represented by an array, with the elements of the matrix within the array in row-major format, e.g.:

[ 1 2 3 4 5 6 7 8 9 ]

↑B = Lib.Matrix.copyMatrix ( ↓A, ↓m, ↓n, ↑B ) Alpha

Copies the contents of a m x n matrix A into another matrix B.

Parameters:

Matrix ↓A - - input matrix A to transpose
Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A.
Matrix ↑B - - output matrix B. Optional - if not provided a new array will be created and returned.

Returns: Matrix ↑B - - output matrix

↑determinant = Lib.Matrix.determinant ( ↓A, ↓am ) Alpha

Computes the determinant of square matrix A, using LU decomposition

Parameters:

Matrix ↓A - - input square matrix A to multiply
Number ↓am - - row/col dimension of input matrix A.

Returns: Number ↑determinant - - determinant of matrix A

⇅A = Lib.Matrix.fillMatrix ( ⇅A, ↓m, ↓n, ↓val ) Alpha

Given a m x n matrix A, fills all elements of the matrix with the specified value. This is a convenience method to initialise a matrix (e.g. fill with 0).

Parameters:

Matrix ⇅A - - input matrix A to fill
Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A.
Number ↓val - - value to fill input matrix A.

Returns: Matrix ⇅A - - filled matrix A

↑mat = Lib.Matrix.identityMatrix ( ↓m, ↑mat ) Alpha

Generates an identity matrix of size m x m.

Parameters:

Number ↓m - - row/col dimension of identity matrix to generate.
Matrix ↑mat - - identity matrix to generate. Optional - if not specified, a new array will be created and returned.

Returns: Matrix ↑mat - - identity matrix to generate.

↑solved = Lib.Matrix.inverse ( ↓A, ↓am, ↑inverse ) Alpha

Computes the inverse of square matrix A.

Parameters:

Matrix ↓A - - input matrix A to compute inverse
Number ↓am - - row/col dimension of input matrix A.
Matrix ↑inverse - - output matrix inverse of A.

Returns: Boolean ↑solved - - true if solution is valid. false if solution not valid (i.e. A is singular).

↑isSymmetric = Lib.Matrix.isSymmetric ( ↓A, ↓m ) Alpha

Given a m x m square matrix A, returns whether it is symmetrical, within tolerance

Parameters:

Matrix ↓A - - input matrix A to transpose
Number ↓m - - row/col dimension of input matrix A.

Returns: Boolean ↑isSymmetric - - whether the matrix A is symmetrical

↑C = Lib.Matrix.multiplyMatrix ( ↓A, ↓am, ↓an, ↓B, ↓bm, ↓bn, ↑C ) Alpha

Multiplies matrices A and B to output C.

Parameters:

Matrix ↓A - - input matrix A to multiply
Number ↓am - - row dimension of input matrix A.
Number ↓an - - col dimension of input matrix A.
Matrix ↓B - - input matrix B to multiply
Number ↓bm - - row dimension of input matrix B.
Number ↓bn - - col dimension of input matrix B.
Matrix ↑C - - output matrix C. Optional - if not provided a new array will be created and returned. This is a am x bn matrix.

Returns: Matrix ↑C - - output matrix

↑AT = Lib.Matrix.transposeMatrix ( ↓A, ↓m, ↓n, ↑AT ) Alpha

Given a m x n matrix A, returns the transpose, n x m matrix A ^T.

Parameters:

Matrix ↓A - - input matrix A to transpose
Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A.
Matrix ↑AT - - output matrix A^T. Optional - if not provided a new array will be created and returned.

Returns: Matrix ↑AT - - output matrix

Category: Solver

Methods for solving linear systems

Lib.Solver.cholSolve ( ↓A, ↓am, ↓B, ↓bm, ↓bn, ↑X ) Alpha

For the symmetric positive definite matrix A, and matrices X and B, where AX = B, solve for X given A and B, using Cholesky Decomposition.

Parameters:

Matrix ↓A - - input matrix A
Number ↓am - - row/col dimension of input matrix A.
Matrix ↓B - - input matrix B
Number ↓bm - - row dimension of input matrix B.
Number ↓bn - - col dimension of input matrix B.
Matrix ↑X - - output matrix X (same size as B).

↑solved = Lib.Solver.luSolve ( ↓A, ↓am, ↓B, ↓bm, ↓bn, ↑X ) Alpha

For the square matrix A, and matrices X and B, where AX = B, solve for X given A and B, using LU Decomposition

Parameters:

Matrix ↓A - - input matrix A
Number ↓am - - row/col dimension of input matrix A.
Matrix ↓B - - input matrix B
Number ↓bm - - row dimension of input matrix B.
Number ↓bn - - col dimension of input matrix B.
Matrix ↑X - - output matrix X (same size as B).

Returns: Boolean ↑solved - - returns true if solution is valid. Returns false if solution is not valid (i.e. A is singular).

↑solved = Lib.Solver.qrSolve ( ↓A, ↓am, ↓an, ↓B, ↓bm, ↓bn, ↑X, ↓threshold ) Alpha

For the matrices A, X and B, where AX = B, solve for X given A and B, using QR Decomposition

Parameters:

Matrix ↓A - - input matrix A
Number ↓am - - row dimension of input matrix A.
Number ↓an - - col dimension of input matrix A.
Matrix ↓B - - input matrix B
Number ↓bm - - row dimension of input matrix B.
Number ↓bn - - col dimension of input matrix B.
Matrix ↑X - - output matrix X (same size as B).
Number ↓threshold - - singularity threshold. Optional - defaults to zero if not specified.

Returns: Boolean ↑solved - - returns true if solution is valid. Returns false if solution is not valid (i.e. A is singular).

Lib.Solver.svdSolve ( ↓A, ↓am, ↓an, ↓B, ↓bm, ↓bn, ↑X ) Alpha

For the matrix A, X and B, where AX = B, solve for X given A and B, using Singular Value Decomposition

Parameters:

Matrix ↓A - - input matrix A
Number ↓am - - row dimension of input matrix A.
Number ↓an - - col dimension of input matrix A.
Matrix ↓B - - input matrix B
Number ↓bm - - row dimension of input matrix B.
Number ↓bn - - col dimension of input matrix B.
Matrix ↑X - - output matrix X

Category: Transformation

Methods for working with transformations

↑ATA = Lib.Transformation.ATA ( ↓A, ↓m, ↓n, ↑ATA ) Alpha

Given a m × n matrix A, computes and returns the n × n matrix, A ^T × A.

Parameters:

Matrix ↓A - - input matrix A.
Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A.
Matrix ↑ATA - - output matrix. This is a n × n matrix. Specifying an output array is optional but recommended. If no output array is given, an array will be created and returned.

Returns: Matrix ↑ATA - - the output array.

Lib.Transformation.chol ( ↓A, ↓m, ↑L, ↑LT, ↓relativeSymmetryThreshold, ↓absolutePositivityThreshold, ↑details ) Alpha

Calculates the Cholesky decomposition of a matrix.

The Cholesky decomposition of a real symmetric positive-definite matrix A consists of a lower triangular matrix L with same size such that: A = LL ^T. In a sense, this is the square root of A.

Parameters:

Matrix ↓A - - input matrix A.
Number ↓m - - row/col dimension of input matrix A.
Matrix ↑L - - output matrix L. L is a lower triangular, m × m matrix. Optional - this matrix is only computed if an output array is specified.
Matrix ↑LT - - output matrix LT. LT is the transpose of L. Optional - this matrix is only computed if an output array is specified.
Number ↓relativeSymmetryThreshold - - relative symmetry threshold. Optional - defaults to 1.0e-15 if not specified.
Number ↓absolutePositivityThreshold - - positive definite threshold. Optional - defaults to 1.0e-10 if not specified.
Object ↑details - - an output details object with the following element: determinant. Optional - this object is populated if it exists.

Lib.Transformation.eigen ( ↓A, ↓m, ↑realEigenValues, ↑imagEigenValues, ↑eigenVectors ) Alpha

Transforms the matrix A of size m × m to derive its eigenvalues and eigenvectors.

Parameters:

Matrix ↓A - - input square matrix
Number ↓m - - row/col dimension of input square matrix A.
Array ↑realEigenValues - - output array of size m, containing the real components of the eigenvalues of A
Array ↑imagEigenValues - - output array of size m, containing the imaginary components of the eigenvalues of A
Matrix ↑eigenVectors - - output matrix, where the eigenvectors of A are the rows of this matrix. This is an matrix of size m x m.

Lib.Transformation.hessenberg ( ↓A, ↓m, ↑P, ↑H ) Alpha

Transforms the matrix A of size m × m to its Hessenberg form.

The m × m matrix A can be written as the product of three matrices: A = P × H × P ^T with P an orthogonal matrix and H a Hessenberg matrix. Both P and H are m × m matrices.

P is an orthogonal matrix, i.e. its inverse is also its transpose.

Parameters:

Matrix ↓A - - input square matrix
Number ↓m - - row/col dimension of input square matrix A.
Matrix ↑P - - output matrix
Matrix ↑H - - output matrix

↑singular = Lib.Transformation.lu ( ↓A, ↓m, ↑L, ↑U, ↑P, ↑pivot, ↑LU, ↑details ) Alpha

Calculates the LUP-decomposition of a square matrix.

The LUP-decomposition of a matrix A consists of three matrices L, U and P that satisfy: P×A = L×U. L is lower triangular (with unit diagonal terms), U is upper triangular and P is a permutation matrix. All matrices are m×m.

As shown by the presence of the P matrix, this decomposition is implemented using partial pivoting.

This function uses 1e-11 as the singularity threshold.

Parameters:

Matrix ↓A - - input matrix A.
Number ↓m - - row/col dimension of input matrix A.
Matrix ↑L - - output matrix L. L is a lower-triangular, m × m matrix. Optional - this matrix is only computed if an output array is specified.
Matrix ↑U - - output matrix U. U is an upper-triangular matrix, m × m matrix. Optional - this matrix is only computed if an output array is specified.
Matrix ↑P - - output matrix P - rows permutation matrix. This is a m × m matrix. P is a sparse matrix with exactly one element set to 1.0 in each row and each column, all other elements being set to 0.0. The positions of the 1 elements are given by the pivot permutation vector. Optional - this matrix is only computed if an output array is specified.
Array ↑pivot - - output pivot permutation vector of size m. Optional - this vector is only returned if an output array is specified.
Matrix ↑LU - - output matrix LU. This is a m × m matrix combining aspects of L and U. Optional - this matrix is only computed if an output array is specified.
Object ↑details - - an output details object with the following elements: singular, even, determinant. Optional - this object is populated if it exists.

Returns: Boolean ↑singular - - whether the specified matrix is singular or not.

↑singular = Lib.Transformation.qr ( ↓A, ↓m, ↓n, ↑Q, ↑R, ↑H, ↑QT, ↓threshold ) Alpha

Calculates the QR-decomposition of a matrix.

The QR-decomposition of a matrix A consists of two matrices Q and R that satisfy: A = QR, Q is orthogonal (Q ^TQ = I), and R is upper triangular. If A is m×n, Q is m×m and R m×n.

This function compute the decomposition using Householder reflectors.

Parameters:

Matrix ↓A - - input matrix A.
Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A.
Matrix ↑Q - - output matrix Q. Q is an orthogonal, m × m matrix. Optional - this matrix is only computed if an output array is specified.
Matrix ↑R - - output matrix R. R is an upper-triangular matrix, m × n matrix. Optional - this matrix is only computed if an output array is specified.
Matrix ↑H - - output matrix H - Householder reflector vectors. H is a lower trapezoidal matrix, m × n whose columns represent each successive Householder reflector vector. This matrix is used to compute Q. Optional - this matrix is only computed if an output array is specified.
Matrix ↑QT - - output matrix QT. This is the transpose of Q. Optional - this matrix is only computed if an output array is specified.
Number ↓threshold - - singularity threshold. Optional - defaults to zero if not specified.

Returns: Boolean ↑singular - - whether the specified matrix is singular or not.

Lib.Transformation.schur ( ↓A, ↓m, ↑P, ↑T ) Alpha

Transforms the matrix A of size m × m to its Schur form.

The m × m matrix A can be written as the product of three matrices: A = P × T × P ^T with P an orthogonal matrix and T an quasi-triangular matrix. Both P and T are m × m matrices.

P is an orthogonal matrix, i.e. its inverse is also its transpose.

Parameters:

Matrix ↓A - - input square matrix
Number ↓m - - row/col dimension of input square matrix A.
Matrix ↑P - - output matrix
Matrix ↑T - - output matrix

Lib.Transformation.svd ( ↓A, ↓m, ↓n, ↑U, ↑V, ↑singularValues, ↑S, ↑details ) Alpha

Calculates the compact Singular Value Decomposition of a matrix.

The Singular Value Decomposition of matrix A is a set of three matrices: U, Σ and V such that A = U × Σ × V ^T. Let A be a m × n matrix, then U is a m × p orthogonal matrix, Σ is a p × p diagonal matrix with positive or null elements, V is a p × n orthogonal matrix (hence V ^T is also orthogonal) where p=min(m,n).

This implementation requires m > n. If that is not the case, instead pass in the transpose of A. The V and U matrices are swapped in this case.

Parameters:

Matrix ↓A - - input matrix
Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A. This implementation requires that m > n.
Matrix ↑U - - output array U
Matrix ↑V - - output array V
Array ↑singularValues - - output singular values. They are ranked in descending order.
Array ↑S - - optional. If supplied, the Σ matrix is returned (a matrix with the singular values as its main diagonal).
Object ↑details - - optional. If supplied as a hash or array, the method populates two properties: 'rank' denoting rank of the matrix, and 'tol' denoting the tolerance of the output matrices.

↑inv = Lib.Transformation.svdPseudoInverse ( ↓m, ↓n, ↓singularValues, ↓U, ↓V, ↓tol, ↑inv ) Alpha

Calculates the pseudoinverse of a m × n matrix A based on the output of its SVD transformation.

Parameters:

Number ↓m - - row dimension of input matrix A.
Number ↓n - - col dimension of input matrix A.
Array ↓singularValues - - singular values of A as returned by svd()
Matrix ↓U - - matrix U as returned by svd()
Matrix ↓V - - matrix V as returned by svd()
Number ↓tol - - tolerance as returned by svd(). Set to 0 if the input does not come from svd().
Matrix ↑inv - - output pseudo inverse of A. This is a n × m matrix. Specifying an output array is optional but recommended. If no output array is given, an array will be created and returned.

Returns: Matrix ↑inv - - the output array.

Lib.Transformation.symmetricEigen ( ↓A, ↓m, ↑realEigenValues, ↑imagEigenValues, ↑eigenVectors ) Alpha

Transforms the symmetric matrix A of size m × m to derive its eigenvalues and eigenvectors.

Parameters:

Matrix ↓A - - input square matrix
Number ↓m - - row/col dimension of input square matrix A.
Array ↑realEigenValues - - output array of size m, containing the real components of the eigenvalues of A
Array ↑imagEigenValues - - output array of size m, containing the imaginary components of the eigenvalues of A. A symmetric matrix A yields only real eigen values, so this should be an all-zero output.
Matrix ↑eigenVectors - - output matrix, where the eigenvectors of A are the rows of this matrix. This is an matrix of size m x m.

Lib.Transformation.tridiagonal ( ↓A, ↓m, ↑P, ↑T, ↑main, ↑secondary ) Alpha

Transforms the symmetric matrix A of size m × m to a tridiagonal form.

The m × m matrix A can be written as the product of three matrices: A = Q × T × Q ^T with Q an orthogonal matrix and T a symmetrical tridiagonal matrix. Both Q and T are m × m matrices.

Q is an orthogonal matrix, i.e. its inverse is also its transpose. The function assumes that A is symmetric, only the upper part of the matrix is accessed.

Parameters:

Matrix ↓A - - input square matrix
Number ↓m - - row/col dimension of input square matrix A.
Matrix ↑P - - output matrix
Matrix ↑T - - output matrix
Array ↑main - - optional. output main diagonal of matrix T. This is an array of size m.
Array ↑secondary - - optional. output secondary diagonal of matrix T. This is an array of size m - 1.

Libraries

Categories

Classes

XLAMath

Category: Filter

Category: Matrix

Category: Solver

Category: Transformation