path: root/redist/svd.h

/**************************************************************************
**
**  svd3
**
** Quick singular value decomposition as described by:
** A. McAdams, A. Selle, R. Tamstorf, J. Teran and E. Sifakis,
** "Computing the Singular Value Decomposition of 3x3 matrices
** with minimal branching and elementary floating point operations",
**  University of Wisconsin - Madison technical report TR1690, May 2011
**
**	OPTIMIZED CPU VERSION
** 	Implementation by: Eric Jang
**
**  13 Apr 2014
**
** This file originally retrieved from:
** https://github.com/ericjang/svd3/blob/master/svd3.h  3/26/2017
**
** Original licesnse is MIT per:
** https://github.com/ericjang/svd3/blob/master/LICENSE.md
**
** Ported from C++ to C by Mike Turvey.  All modifications also released 
** under an MIT license.
**************************************************************************/


#ifndef SVD3_H
#define SVD3_H

#define _gamma 5.828427124 // FOUR_GAMMA_SQUARED = sqrt(8)+3;
#define _cstar 0.923879532 // cos(pi/8)
#define _sstar 0.3826834323 // sin(p/8)
#define EPSILON 1e-6

#include <math.h>

/* This is a novel and fast routine for the reciprocal square root of an
IEEE float (single precision).
http://www.lomont.org/Math/Papers/2003/InvSqrt.pdf
http://playstation2-linux.com/download/p2lsd/fastrsqrt.pdf
http://www.beyond3d.com/content/articles/8/
*/
inline float rsqrt(float x) {
	// int ihalf = *(int *)&x - 0x00800000; // Alternative to next line,
	// float xhalf = *(float *)&ihalf;      // for sufficiently large nos.
	float xhalf = 0.5f*x;
	int i = *(int *)&x;          // View x as an int.
								 // i = 0x5f3759df - (i >> 1);   // Initial guess (traditional).
	i = 0x5f375a82 - (i >> 1);   // Initial guess (slightly better).
	x = *(float *)&i;            // View i as float.
	x = x*(1.5f - xhalf*x*x);    // Newton step.
								 // x = x*(1.5008908 - xhalf*x*x);  // Newton step for a balanced error.
	return x;
}

/* This is rsqrt with an additional step of the Newton iteration, for
increased accuracy. The constant 0x5f37599e makes the relative error
range from 0 to -0.00000463.
You can't balance the error by adjusting the constant. */
inline float rsqrt1(float x) {
	float xhalf = 0.5f*x;
	int i = *(int *)&x;          // View x as an int.
	i = 0x5f37599e - (i >> 1);   // Initial guess.
	x = *(float *)&i;            // View i as float.
	x = x*(1.5f - xhalf*x*x);    // Newton step.
	x = x*(1.5f - xhalf*x*x);    // Newton step again.
	return x;
}

inline float accurateSqrt(float x)
{
	return x * rsqrt1(x);
}

inline void condSwap(bool c, float *X, float *Y)
{
	// used in step 2
	float Z = *X;
	*X = c ? *Y : *X;
	*Y = c ? Z : *Y;
}

inline void condNegSwap(bool c, float *X, float *Y)
{
	// used in step 2 and 3
	float Z = -*X;
	*X = c ? *Y : *X;
	*Y = c ? Z : *Y;
}

// matrix multiplication M = A * B
inline void multAB(float a11, float a12, float a13,
	float a21, float a22, float a23,
	float a31, float a32, float a33,
	//
	float b11, float b12, float b13,
	float b21, float b22, float b23,
	float b31, float b32, float b33,
	//
	float *m11, float *m12, float *m13,
	float *m21, float *m22, float *m23,
	float *m31, float *m32, float *m33)
{

	*m11 = a11*b11 + a12*b21 + a13*b31; *m12 = a11*b12 + a12*b22 + a13*b32; *m13 = a11*b13 + a12*b23 + a13*b33;
	*m21 = a21*b11 + a22*b21 + a23*b31; *m22 = a21*b12 + a22*b22 + a23*b32; *m23 = a21*b13 + a22*b23 + a23*b33;
	*m31 = a31*b11 + a32*b21 + a33*b31; *m32 = a31*b12 + a32*b22 + a33*b32; *m33 = a31*b13 + a32*b23 + a33*b33;
}

// matrix multiplication M = Transpose[A] * B
inline void multAtB(float a11, float a12, float a13,
	float a21, float a22, float a23,
	float a31, float a32, float a33,
	//
	float b11, float b12, float b13,
	float b21, float b22, float b23,
	float b31, float b32, float b33,
	//
	float *m11, float *m12, float *m13,
	float *m21, float *m22, float *m23,
	float *m31, float *m32, float *m33)
{
	*m11 = a11*b11 + a21*b21 + a31*b31; *m12 = a11*b12 + a21*b22 + a31*b32; *m13 = a11*b13 + a21*b23 + a31*b33;
	*m21 = a12*b11 + a22*b21 + a32*b31; *m22 = a12*b12 + a22*b22 + a32*b32; *m23 = a12*b13 + a22*b23 + a32*b33;
	*m31 = a13*b11 + a23*b21 + a33*b31; *m32 = a13*b12 + a23*b22 + a33*b32; *m33 = a13*b13 + a23*b23 + a33*b33;
}

inline void quatToMat3(const float * qV,
	float *m11, float *m12, float *m13,
	float *m21, float *m22, float *m23,
	float *m31, float *m32, float *m33
)
{
	float w = qV[3];
	float x = qV[0];
	float y = qV[1];
	float z = qV[2];

	float qxx = x*x;
	float qyy = y*y;
	float qzz = z*z;
	float qxz = x*z;
	float qxy = x*y;
	float qyz = y*z;
	float qwx = w*x;
	float qwy = w*y;
	float qwz = w*z;

	*m11 = 1 - 2 * (qyy + qzz); *m12 = 2 * (qxy - qwz); *m13 = 2 * (qxz + qwy);
	*m21 = 2 * (qxy + qwz); *m22 = 1 - 2 * (qxx + qzz); *m23 = 2 * (qyz - qwx);
	*m31 = 2 * (qxz - qwy); *m32 = 2 * (qyz + qwx); *m33 = 1 - 2 * (qxx + qyy);
}

inline void approximateGivensQuaternion(float a11, float a12, float a22, float *ch, float *sh)
{
	/*
	* Given givens angle computed by approximateGivensAngles,
	* compute the corresponding rotation quaternion.
	*/
	*ch = 2 * (a11 - a22);
	*sh = a12;
	bool b = _gamma* (*sh)*(*sh) < (*ch)*(*ch);
	// fast rsqrt function suffices
	// rsqrt2 (https://code.google.com/p/lppython/source/browse/algorithm/HDcode/newCode/rsqrt.c?r=26)
	// is even faster but results in too much error
	float w = rsqrt((*ch)*(*ch) + (*sh)*(*sh));
	*ch = b ? w*(*ch) : (float)_cstar;
	*sh = b ? w*(*sh) : (float)_sstar;
}

inline void jacobiConjugation(const int x, const int y, const int z,
	float *s11,
	float *s21, float *s22,
	float *s31, float *s32, float *s33,
	float * qV)
{
	float ch, sh;
	approximateGivensQuaternion(*s11, *s21, *s22, &ch, &sh);

	float scale = ch*ch + sh*sh;
	float a = (ch*ch - sh*sh) / scale;
	float b = (2 * sh*ch) / scale;

	// make temp copy of S
	float _s11 = *s11;
	float _s21 = *s21; float _s22 = *s22;
	float _s31 = *s31; float _s32 = *s32; float _s33 = *s33;

	// perform conjugation S = Q'*S*Q
	// Q already implicitly solved from a, b
	*s11 = a*(a*_s11 + b*_s21) + b*(a*_s21 + b*_s22);
	*s21 = a*(-b*_s11 + a*_s21) + b*(-b*_s21 + a*_s22);	*s22 = -b*(-b*_s11 + a*_s21) + a*(-b*_s21 + a*_s22);
	*s31 = a*_s31 + b*_s32;								*s32 = -b*_s31 + a*_s32; *s33 = _s33;

	// update cumulative rotation qV
	float tmp[3];
	tmp[0] = qV[0] * sh;
	tmp[1] = qV[1] * sh;
	tmp[2] = qV[2] * sh;
	sh *= qV[3];

	qV[0] *= ch;
	qV[1] *= ch;
	qV[2] *= ch;
	qV[3] *= ch;

	// (x,y,z) corresponds to ((0,1,2),(1,2,0),(2,0,1))
	// for (p,q) = ((0,1),(1,2),(0,2))
	qV[z] += sh;
	qV[3] -= tmp[z]; // w
	qV[x] += tmp[y];
	qV[y] -= tmp[x];

	// re-arrange matrix for next iteration
	_s11 = *s22;
	_s21 = *s32; _s22 = *s33;
	_s31 = *s21; _s32 = *s31; _s33 = *s11;
	*s11 = _s11;
	*s21 = _s21; *s22 = _s22;
	*s31 = _s31; *s32 = _s32; *s33 = _s33;

}

inline float dist2(float x, float y, float z)
{
	return x*x + y*y + z*z;
}

// finds transformation that diagonalizes a symmetric matrix
inline void jacobiEigenanlysis( // symmetric matrix
	float *s11,
	float *s21, float *s22,
	float *s31, float *s32, float *s33,
	// quaternion representation of V
	float * qV)
{
	qV[3] = 1; qV[0] = 0; qV[1] = 0; qV[2] = 0; // follow same indexing convention as GLM
	for (int i = 0; i<4; i++)
	{
		// we wish to eliminate the maximum off-diagonal element
		// on every iteration, but cycling over all 3 possible rotations
		// in fixed order (p,q) = (1,2) , (2,3), (1,3) still retains
		//  asymptotic convergence
		jacobiConjugation(0, 1, 2, s11, s21, s22, s31, s32, s33, qV); // p,q = 0,1
		jacobiConjugation(1, 2, 0, s11, s21, s22, s31, s32, s33, qV); // p,q = 1,2
		jacobiConjugation(2, 0, 1, s11, s21, s22, s31, s32, s33, qV); // p,q = 0,2
	}
}


inline void sortSingularValues(// matrix that we want to decompose
	float *b11, float *b12, float *b13,
	float *b21, float *b22, float *b23,
	float *b31, float *b32, float *b33,
	// sort V simultaneously
	float *v11, float *v12, float *v13,
	float *v21, float *v22, float *v23,
	float *v31, float *v32, float *v33)
{
	float rho1 = dist2(*b11, *b21, *b31);
	float rho2 = dist2(*b12, *b22, *b32);
	float rho3 = dist2(*b13, *b23, *b33);
	bool c;
	c = rho1 < rho2;
	condNegSwap(c, b11, b12); condNegSwap(c, v11, v12);
	condNegSwap(c, b21, b22); condNegSwap(c, v21, v22);
	condNegSwap(c, b31, b32); condNegSwap(c, v31, v32);
	condSwap(c, &rho1, &rho2);
	c = rho1 < rho3;
	condNegSwap(c, b11, b13); condNegSwap(c, v11, v13);
	condNegSwap(c, b21, b23); condNegSwap(c, v21, v23);
	condNegSwap(c, b31, b33); condNegSwap(c, v31, v33);
	condSwap(c, &rho1, &rho3);
	c = rho2 < rho3;
	condNegSwap(c, b12, b13); condNegSwap(c, v12, v13);
	condNegSwap(c, b22, b23); condNegSwap(c, v22, v23);
	condNegSwap(c, b32, b33); condNegSwap(c, v32, v33);
}


void QRGivensQuaternion(float a1, float a2, float *ch, float *sh)
{
	// a1 = pivot point on diagonal
	// a2 = lower triangular entry we want to annihilate
	float epsilon = (float)EPSILON;
	float rho = accurateSqrt(a1*a1 + a2*a2);

	*sh = rho > epsilon ? a2 : 0;
	*ch = fabsf(a1) + fmaxf(rho, epsilon);
	bool b = a1 < 0;
	condSwap(b, sh, ch);
	float w = rsqrt((*ch)*(*ch) + (*sh)*(*sh));
	*ch *= w;
	*sh *= w;
}


inline void QRDecomposition(// matrix that we want to decompose
	float b11, float b12, float b13,
	float b21, float b22, float b23,
	float b31, float b32, float b33,
	// output Q
	float *q11, float *q12, float *q13,
	float *q21, float *q22, float *q23,
	float *q31, float *q32, float *q33,
	// output R
	float *r11, float *r12, float *r13,
	float *r21, float *r22, float *r23,
	float *r31, float *r32, float *r33)
{
	float ch1, sh1, ch2, sh2, ch3, sh3;
	float a, b;

	// first givens rotation (ch,0,0,sh)
	QRGivensQuaternion(b11, b21, &ch1, &sh1);
	a = 1 - 2 * sh1*sh1;
	b = 2 * ch1*sh1;
	// apply B = Q' * B
	*r11 = a*b11 + b*b21;  *r12 = a*b12 + b*b22;  *r13 = a*b13 + b*b23;
	*r21 = -b*b11 + a*b21; *r22 = -b*b12 + a*b22; *r23 = -b*b13 + a*b23;
	*r31 = b31;          *r32 = b32;          *r33 = b33;

	// second givens rotation (ch,0,-sh,0)
	QRGivensQuaternion(*r11, *r31, &ch2, &sh2);
	a = 1 - 2 * sh2*sh2;
	b = 2 * ch2*sh2;
	// apply B = Q' * B;
	b11 = a*(*r11) + b*(*r31);  b12 = a*(*r12) + b*(*r32);  b13 = a*(*r13) + b*(*r33);
	b21 = *r21;           b22 = *r22;           b23 = *r23;
	b31 = -b*(*r11) + a*(*r31); b32 = -b*(*r12) + a*(*r32); b33 = -b*(*r13) + a*(*r33);

	// third givens rotation (ch,sh,0,0)
	QRGivensQuaternion(b22, b32, &ch3, &sh3);
	a = 1 - 2 * sh3*sh3;
	b = 2 * ch3*sh3;
	// R is now set to desired value
	*r11 = b11;               *r12 = b12;             *r13 = b13;
	*r21 = a*b21 + b*b31;     *r22 = a*b22 + b*b32;   *r23 = a*b23 + b*b33;
	*r31 = -b*b21 + a*b31;    *r32 = -b*b22 + a*b32;  *r33 = -b*b23 + a*b33;

	// construct the cumulative rotation Q=Q1 * Q2 * Q3
	// the number of floating point operations for three quaternion multiplications
	// is more or less comparable to the explicit form of the joined matrix.
	// certainly more memory-efficient!
	float sh12 = sh1*sh1;
	float sh22 = sh2*sh2;
	float sh32 = sh3*sh3;

	*q11 = (-1 + 2 * sh12)*(-1 + 2 * sh22);
	*q12 = 4 * ch2*ch3*(-1 + 2 * sh12)*sh2*sh3 + 2 * ch1*sh1*(-1 + 2 * sh32);
	*q13 = 4 * ch1*ch3*sh1*sh3 - 2 * ch2*(-1 + 2 * sh12)*sh2*(-1 + 2 * sh32);

	*q21 = 2 * ch1*sh1*(1 - 2 * sh22);
	*q22 = -8 * ch1*ch2*ch3*sh1*sh2*sh3 + (-1 + 2 * sh12)*(-1 + 2 * sh32);
	*q23 = -2 * ch3*sh3 + 4 * sh1*(ch3*sh1*sh3 + ch1*ch2*sh2*(-1 + 2 * sh32));

	*q31 = 2 * ch2*sh2;
	*q32 = 2 * ch3*(1 - 2 * sh22)*sh3;
	*q33 = (-1 + 2 * sh22)*(-1 + 2 * sh32);
}

void svd(// input A
	float a11, float a12, float a13,
	float a21, float a22, float a23,
	float a31, float a32, float a33,
	// output U
	float *u11, float *u12, float *u13,
	float *u21, float *u22, float *u23,
	float *u31, float *u32, float *u33,
	// output S
	float *s11, float *s12, float *s13,
	float *s21, float *s22, float *s23,
	float *s31, float *s32, float *s33,
	// output V
	float *v11, float *v12, float *v13,
	float *v21, float *v22, float *v23,
	float *v31, float *v32, float *v33)
{
	// normal equations matrix
	float ATA11, ATA12, ATA13;
	float ATA21, ATA22, ATA23;
	float ATA31, ATA32, ATA33;

	multAtB(a11, a12, a13, a21, a22, a23, a31, a32, a33,
		a11, a12, a13, a21, a22, a23, a31, a32, a33,
		&ATA11, &ATA12, &ATA13, &ATA21, &ATA22, &ATA23, &ATA31, &ATA32, &ATA33);

	// symmetric eigenalysis
	float qV[4];
	jacobiEigenanlysis(&ATA11, &ATA21, &ATA22, &ATA31, &ATA32, &ATA33, qV);
	quatToMat3(qV, v11, v12, v13, v21, v22, v23, v31, v32, v33);

	float b11, b12, b13;
	float b21, b22, b23;
	float b31, b32, b33;
	multAB(a11, a12, a13, a21, a22, a23, a31, a32, a33,
		*v11, *v12, *v13, *v21, *v22, *v23, *v31, *v32, *v33,
		&b11, &b12, &b13, &b21, &b22, &b23, &b31, &b32, &b33);

	// sort singular values and find V
	sortSingularValues(&b11, &b12, &b13, &b21, &b22, &b23, &b31, &b32, &b33,
		v11, v12, v13, v21, v22, v23, v31, v32, v33);

	// QR decomposition
	QRDecomposition(b11, b12, b13, b21, b22, b23, b31, b32, b33,
		u11, u12, u13, u21, u22, u23, u31, u32, u33,
		s11, s12, s13, s21, s22, s23, s31, s32, s33
	);
}

/// polar decomposition can be reconstructed trivially from SVD result
// A = UP
void pd(float a11, float a12, float a13,
	float a21, float a22, float a23,
	float a31, float a32, float a33,
	// output U
	float *u11, float *u12, float *u13,
	float *u21, float *u22, float *u23,
	float *u31, float *u32, float *u33,
	// output P
	float *p11, float *p12, float *p13,
	float *p21, float *p22, float *p23,
	float *p31, float *p32, float *p33)
{
	float w11, w12, w13, w21, w22, w23, w31, w32, w33;
	float s11, s12, s13, s21, s22, s23, s31, s32, s33;
	float v11, v12, v13, v21, v22, v23, v31, v32, v33;

	svd(a11, a12, a13, a21, a22, a23, a31, a32, a33,
		&w11, &w12, &w13, &w21, &w22, &w23, &w31, &w32, &w33,
		&s11, &s12, &s13, &s21, &s22, &s23, &s31, &s32, &s33,
		&v11, &v12, &v13, &v21, &v22, &v23, &v31, &v32, &v33);

	// P = VSV'
	float t11, t12, t13, t21, t22, t23, t31, t32, t33;
	multAB(v11, v12, v13, v21, v22, v23, v31, v32, v33,
		s11, s12, s13, s21, s22, s23, s31, s32, s33,
		&t11, &t12, &t13, &t21, &t22, &t23, &t31, &t32, &t33);

	multAB(t11, t12, t13, t21, t22, t23, t31, t32, t33,
		v11, v21, v31, v12, v22, v32, v13, v23, v33,
		p11, p12, p13, p21, p22, p23, p31, p32, p33);

	// U = WV'
	multAB(w11, w12, w13, w21, w22, w23, w31, w32, w33,
		v11, v21, v31, v12, v22, v32, v13, v23, v33,
		u11, u12, u13, u21, u22, u23, u31, u32, u33);
}

#endif