Compute the singular value decomposition of a 2x2 matrix. More...

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Functions
void	Svd2x2 (double Theta, double Sigma1, double Sigma2, double Phi, int Sign1, int Sign2, double A[2][2])
	Compute the singular value decomposition of a 2x2 matrix. More...

Detailed Description

Compute the singular value decomposition of a 2x2 matrix.

Author: Pascal Getreuer getre.nosp@m.uer@.nosp@m.gmail.nosp@m..com

This program is free software: you can use, modify and/or redistribute it under the terms of the simplified BSD License. You should have received a copy of this license along this program. If not, see http://www.opensource.org/licenses/bsd-license.html.

Definition in file svd2x2.h.

Function Documentation

void Svd2x2	(	double *	Theta,
		double *	Sigma1,
		double *	Sigma2,
		double *	Phi,
		int *	Sign1,
		int *	Sign2,
		double	A[2][2]
	)

Compute the singular value decomposition of a 2x2 matrix.

Parameters

Theta	rotation of the U matrix
Sigma1,Sigma2	singular values
Phi	rotation of the V matrix
Sign1,Sign2	correction signs
A	input 2x2 matrix

Computes the singular value decomposition of A,

$A = U \Sigma V^T.$

U is a rotation by Theta,

$U = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}.$

Sigma is the diagonal matrix with entries Sigma1 and Sigma2, which satisfy Sigma1 >= Sigma2 >= 0. V is in general not representable by a rotation alone and is further decomposed as a rotation matrix W and a diagonal sign correction matrix C so that

$A = U \Sigma C W^T.$

W is a rotation by Phi,

$W = \begin{pmatrix} \cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{pmatrix},$

and C is

$C = \begin{pmatrix} \text{Sign1} & 0 \\ 0 & \text{Sign2} \end{pmatrix},$

where Sign1 and Sign2 are signs, values of either +1 or -1.

The matrices U and V can be computed from Theta, Phi, Sign1, Sign2 as

U[0][0] = cos(Theta);
U[1][0] = sin(Theta);
U[0][1] = -sin(Theta);
U[1][1] = code(Theta);
V[0][0] = Sign1*cos(Phi);
V[1][0] = Sign1*sin(Phi);
V[0][1] = -Sign2*sin(Phi);
V[1][1] = Sign2*cos(Phi);

The algorithm used here is based on the notes http://www.ualberta.ca/~mlipsett/ENGM541/Readings/svd_ellis.pdf. The derivation is, first, by expanding the following in terms of the components

$\begin{pmatrix} a & b \\ b & c \end{pmatrix} = A A^T = U \Sigma^2 U^T,$