linalg::factorQR

QR-decomposition of a matrix

Use only in the MuPAD Notebook Interface.

This functionality does not run in MATLAB.

Syntax

linalg::factorQR(A)

Description

linalg::factorQR(A) computes an QR-decomposition of an m×n matrix A, i.e., a decomposition of A into an m×m unitary matrix Q and an m×n upper triangular matrix R such that QR = A.

linalg::factorQR uses Gram-Schmidt orthonormalization to compute the decomposition.

For a singular or non-square matrix A the QR-decomposition of A is not unique.

The columns of Q form an orthonormal basis with respect to the scalar product of two vectors, defined by linalg::scalarProduct, and the 2-norm of two vectors (see the method "norm" of the domain constructor Dom::Matrix).

If the component ring of A does not define the method "conjugate", then the factor Q is orthogonal instead of unitary.

If the columns of A cannot be orthonormalized then FAIL is returned.

If A is a matrix over the domain Dom::Float and the computations are based on the standard scalar product, then the use of the corresponding function from the numeric library (numeric::factorQR) is recommended.

Even if A is defined over the real or the complex numbers the call of numeric::factorQR with the option Symbolic is recommended for better efficiency.

The component ring of the matrix A must be a field, i.e., a domain of category Cat::Field.

Examples

Example 1

We compute the QR-decomposition of a real matrix:

A := Dom::Matrix(Dom::Real)(
  [[2, -3, -1], [1, 1, -1], [0, 1, -1]]
)

QR := linalg::factorQR(A)

The orthogonal matrix Q is the first element und the upper triangular matrix R is the second element of the list QR. The product of these two matrices is equal to the input matrix A:

QR[1] * QR[2]

Example 2

The QR-decomposition of the 3×2 matrix:

B := Dom::Matrix(Dom::Real)(
  [[2, -3], [1, 2], [2, 3]]
)

yields a 3×3 orthogonal matrix and a 3×2 upper triangular matrix:

QR := linalg::factorQR(B)

QR[1] * QR[2]

For this example we may call numeric::factorQR(B, Symbolic) instead, which in general is faster than linalg::factorQR:

QR := numeric::factorQR(B, Symbolic)

Parameters

A

A matrix of a domain of category Cat::Matrix

Return Values

List [Q, R] of the two matrices Q and R (of the same domain type as A), or the value FAIL.

Algorithms

The QR-decomposition can be used to generate a least square solution to an overdetermined system of linear equations. If , then can be solved via backward substitution.

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