svd22.c File Reference

#include <math.h>

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Functions
void	svd22 (const double A[4], double U[4], double S[2], double V[4])
void	svd_sym_singular_values (double A00, double A01, double A11, double *Lmin, double *Lmax)

Function Documentation

svd22()

void svd22	(	const double	A[4],
		double	U[4],
		double	S[2],
		double	V[4]
	)

SVD 2x2.

Computes singular values and vectors without squaring the input matrix. With double precision math, results are accurate to about 1E-16.

U = [ cos(theta) -sin(theta) ] [ sin(theta) cos(theta) ]

S = [ e 0 ] [ 0 f ]

V = [ cos(phi) -sin(phi) ] [ sin(phi) cos(phi) ]

Our strategy is basically to analytically multiply everything out
and then rearrange so that we can solve for theta, phi, e, and
f. (Derivation by ebolson@umich.edu 5/2016)

V' = [ CP SP ] [ -SP CP ]

USV' = [ CT -ST ][ e*CP e*SP ] [ ST CT ][ -f*SP f*CP ]

= [e*CT*CP + f*ST*SP e*CT*SP - f*ST*CP ] [e*ST*CP - f*SP*CT e*SP*ST + f*CP*CT ]

A00+A11 = e*CT*CP + f*ST*SP + e*SP*ST + f*CP*CT = e*(CP*CT + SP*ST) + f*(SP*ST + CP*CT) = (e+f)(CP*CT + SP*ST) B0 = (e+f)*cos(P-T)

A00-A11 = e*CT*CP + f*ST*SP - e*SP*ST - f*CP*CT = e*(CP*CT - SP*ST) - f*(-ST*SP + CP*CT) = (e-f)(CP*CT - SP*ST) B1 = (e-f)*cos(P+T)

A01+A10 = e*CT*SP - f*ST*CP + e*ST*CP - f*SP*CT = e(CT*SP + ST*CP) - f*(ST*CP + SP*CT) = (e-f)*(CT*SP + ST*CP) B2 = (e-f)*sin(P+T)

A01-A10 = e*CT*SP - f*ST*CP - e*ST*CP + f*SP*CT = e*(CT*SP - ST*CP) + f(SP*CT - ST*CP) = (e+f)*(CT*SP - ST*CP) B3 = (e+f)*sin(P-T)

B0 = (e+f)*cos(P-T) B1 = (e-f)*cos(P+T) B2 = (e-f)*sin(P+T) B3 = (e+f)*sin(P-T)

B3/B0 = tan(P-T)

B2/B1 = tan(P+T)

Definition at line 88 of file svd22.c.

svd_sym_singular_values()

void svd_sym_singular_values	(	double	A00,
		double	A01,
		double	A11,
		double *	Lmin,
		double *	Lmax
	)

Definition at line 218 of file svd22.c.