svd22.c

/* Copyright (C) 2013-2016, The Regents of The University of Michigan.
All rights reserved.
This software was developed in the APRIL Robotics Lab under the
direction of Edwin Olson, ebolson@umich.edu. This software may be
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*/

#include <math.h>

/** 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)
 **/
void svd22(const double A[4], double U[4], double S[2], double V[4])
{
    double A00 = A[0];
    double A01 = A[1];
    double A10 = A[2];
    double A11 = A[3];

    double B0 = A00 + A11;
    double B1 = A00 - A11;
    double B2 = A01 + A10;
    double B3 = A01 - A10;

    double PminusT = atan2(B3, B0);
    double PplusT = atan2(B2, B1);

    double P = (PminusT + PplusT) / 2;
    double T = (-PminusT + PplusT) / 2;

    double CP = cos(P), SP = sin(P);
    double CT = cos(T), ST = sin(T);

    U[0] = CT;
    U[1] = -ST;
    U[2] = ST;
    U[3] = CT;

    V[0] = CP;
    V[1] = -SP;
    V[2] = SP;
    V[3] = CP;

    // C0 = e+f. There are two ways to compute C0; we pick the one
    // that is better conditioned.
    double CPmT = cos(P-T), SPmT = sin(P-T);
    double C0 = 0;
    if (fabs(CPmT) > fabs(SPmT))
        C0 = B0 / CPmT;
    else
        C0 = B3 / SPmT;

    // C1 = e-f. There are two ways to compute C1; we pick the one
    // that is better conditioned.
    double CPpT = cos(P+T), SPpT = sin(P+T);
    double C1 = 0;
    if (fabs(CPpT) > fabs(SPpT))
        C1 = B1 / CPpT;
    else
        C1 = B2 / SPpT;

    // e and f are the singular values
    double e = (C0 + C1) / 2;
    double f = (C0 - C1) / 2;

    if (e < 0) {
        e = -e;
        U[0] = -U[0];
        U[2] = -U[2];
    }

    if (f < 0) {
        f = -f;
        U[1] = -U[1];
        U[3] = -U[3];
    }

    // sort singular values.
    if (e > f) {
        // already in big-to-small order.
        S[0] = e;
        S[1] = f;
    } else {
        // Curiously, this code never seems to get invoked.  Why is it
        // that S[0] always ends up the dominant vector?  However,
        // this code has been tested (flipping the logic forces us to
        // sort the singular values in ascending order).
        //
        // P = [ 0 1 ; 1 0 ]
        // USV' = (UP)(PSP)(PV')
        //      = (UP)(PSP)(VP)'
        //      = (UP)(PSP)(P'V')'
        S[0] = f;
        S[1] = e;

        // exchange columns of U and V
        double tmp[2];
        tmp[0] = U[0];
        tmp[1] = U[2];
        U[0] = U[1];
        U[2] = U[3];
        U[1] = tmp[0];
        U[3] = tmp[1];

        tmp[0] = V[0];
        tmp[1] = V[2];
        V[0] = V[1];
        V[2] = V[3];
        V[1] = tmp[0];
        V[3] = tmp[1];
    }

    /*
    double SM[4] = { S[0], 0, 0, S[1] };

    doubles_print_mat(U, 2, 2, "%20.10g");
    doubles_print_mat(SM, 2, 2, "%20.10g");
    doubles_print_mat(V, 2, 2, "%20.10g");
    printf("A:\n");
    doubles_print_mat(A, 2, 2, "%20.10g");

    double SVt[4];
    doubles_mat_ABt(SM, 2, 2, V, 2, 2, SVt, 2, 2);
    double USVt[4];
    doubles_mat_AB(U, 2, 2, SVt, 2, 2, USVt, 2, 2);

    printf("USVt\n");
    doubles_print_mat(USVt, 2, 2, "%20.10g");

    double diff[4];
    for (int i = 0; i < 4; i++)
        diff[i] = A[i] - USVt[i];

    printf("diff\n");
    doubles_print_mat(diff, 2, 2, "%20.10g");

    */

}


// for the matrix [a b; b d]
void svd_sym_singular_values(double A00, double A01, double A11,
                             double *Lmin, double *Lmax)
{
    double A10 = A01;

    double B0 = A00 + A11;
    double B1 = A00 - A11;
    double B2 = A01 + A10;
    double B3 = A01 - A10;

    double PminusT = atan2(B3, B0);
    double PplusT = atan2(B2, B1);

    double P = (PminusT + PplusT) / 2;
    double T = (-PminusT + PplusT) / 2;

    // C0 = e+f. There are two ways to compute C0; we pick the one
    // that is better conditioned.
    double CPmT = cos(P-T), SPmT = sin(P-T);
    double C0 = 0;
    if (fabs(CPmT) > fabs(SPmT))
        C0 = B0 / CPmT;
    else
        C0 = B3 / SPmT;

    // C1 = e-f. There are two ways to compute C1; we pick the one
    // that is better conditioned.
    double CPpT = cos(P+T), SPpT = sin(P+T);
    double C1 = 0;
    if (fabs(CPpT) > fabs(SPpT))
        C1 = B1 / CPpT;
    else
        C1 = B2 / SPpT;

    // e and f are the singular values
    double e = (C0 + C1) / 2;
    double f = (C0 - C1) / 2;

    *Lmin = fmin(e, f);
    *Lmax = fmax(e, f);
}