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svd3_cuda.h
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svd3_cuda.h
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/**************************************************************************
**
** 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
**
** Identical GPU version
** Implementated by: Kui Wu
** kwu@cs.utah.edu
**
** May 2018
**
**************************************************************************/
#ifndef SVD3_CUDA_H
#define SVD3_CUDA_H
#include <cuda.h>
#include "math.h" // CUDA math library
#define gone 1065353216
#define gsine_pi_over_eight 1053028117
#define gcosine_pi_over_eight 1064076127
#define gone_half 0.5f
#define gsmall_number 1.e-12f
#define gtiny_number 1.e-20f
#define gfour_gamma_squared 5.8284273147583007813f
union un { float f; unsigned int ui; };
__device__ __forceinline__
void svd(
float a11, float a12, float a13, float a21, float a22, float a23, float a31, float a32, float a33, // input A
float &u11, float &u12, float &u13, float &u21, float &u22, float &u23, float &u31, float &u32, float &u33, // output U
float &s11,
//float &s12, float &s13, float &s21,
float &s22,
//float &s23, float &s31, float &s32,
float &s33, // output S
float &v11, float &v12, float &v13, float &v21, float &v22, float &v23, float &v31, float &v32, float &v33 // output V
)
{
un Sa11, Sa21, Sa31, Sa12, Sa22, Sa32, Sa13, Sa23, Sa33;
un Su11, Su21, Su31, Su12, Su22, Su32, Su13, Su23, Su33;
un Sv11, Sv21, Sv31, Sv12, Sv22, Sv32, Sv13, Sv23, Sv33;
un Sc, Ss, Sch, Ssh;
un Stmp1, Stmp2, Stmp3, Stmp4, Stmp5;
un Ss11, Ss21, Ss31, Ss22, Ss32, Ss33;
un Sqvs, Sqvvx, Sqvvy, Sqvvz;
Sa11.f = a11; Sa12.f = a12; Sa13.f = a13;
Sa21.f = a21; Sa22.f = a22; Sa23.f = a23;
Sa31.f = a31; Sa32.f = a32; Sa33.f = a33;
//###########################################################
// Compute normal equations matrix
//###########################################################
Ss11.f = Sa11.f*Sa11.f;
Stmp1.f = Sa21.f*Sa21.f;
Ss11.f = __fadd_rn(Stmp1.f, Ss11.f);
Stmp1.f = Sa31.f*Sa31.f;
Ss11.f = __fadd_rn(Stmp1.f, Ss11.f);
Ss21.f = Sa12.f*Sa11.f;
Stmp1.f = Sa22.f*Sa21.f;
Ss21.f = __fadd_rn(Stmp1.f, Ss21.f);
Stmp1.f = Sa32.f*Sa31.f;
Ss21.f = __fadd_rn(Stmp1.f, Ss21.f);
Ss31.f = Sa13.f*Sa11.f;
Stmp1.f = Sa23.f*Sa21.f;
Ss31.f = __fadd_rn(Stmp1.f, Ss31.f);
Stmp1.f = Sa33.f*Sa31.f;
Ss31.f = __fadd_rn(Stmp1.f, Ss31.f);
Ss22.f = Sa12.f*Sa12.f;
Stmp1.f = Sa22.f*Sa22.f;
Ss22.f = __fadd_rn(Stmp1.f, Ss22.f);
Stmp1.f = Sa32.f*Sa32.f;
Ss22.f = __fadd_rn(Stmp1.f, Ss22.f);
Ss32.f = Sa13.f*Sa12.f;
Stmp1.f = Sa23.f*Sa22.f;
Ss32.f = __fadd_rn(Stmp1.f, Ss32.f);
Stmp1.f = Sa33.f*Sa32.f;
Ss32.f = __fadd_rn(Stmp1.f, Ss32.f);
Ss33.f = Sa13.f*Sa13.f;
Stmp1.f = Sa23.f*Sa23.f;
Ss33.f = __fadd_rn(Stmp1.f, Ss33.f);
Stmp1.f = Sa33.f*Sa33.f;
Ss33.f = __fadd_rn(Stmp1.f, Ss33.f);
Sqvs.f = 1.f; Sqvvx.f = 0.f; Sqvvy.f = 0.f; Sqvvz.f = 0.f;
//###########################################################
// Solve symmetric eigenproblem using Jacobi iteration
//###########################################################
for (int i = 0; i < 4; i++)
{
Ssh.f = Ss21.f * 0.5f;
Stmp5.f = __fsub_rn(Ss11.f, Ss22.f);
Stmp2.f = Ssh.f*Ssh.f;
Stmp1.ui = (Stmp2.f >= gtiny_number) ? 0xffffffff : 0;
Ssh.ui = Stmp1.ui&Ssh.ui;
Sch.ui = Stmp1.ui&Stmp5.ui;
Stmp2.ui = ~Stmp1.ui&gone;
Sch.ui = Sch.ui | Stmp2.ui;
Stmp1.f = Ssh.f*Ssh.f;
Stmp2.f = Sch.f*Sch.f;
Stmp3.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp4.f = __frsqrt_rn(Stmp3.f);
Ssh.f = Stmp4.f*Ssh.f;
Sch.f = Stmp4.f*Sch.f;
Stmp1.f = gfour_gamma_squared*Stmp1.f;
Stmp1.ui = (Stmp2.f <= Stmp1.f) ? 0xffffffff : 0;
Stmp2.ui = gsine_pi_over_eight&Stmp1.ui;
Ssh.ui = ~Stmp1.ui&Ssh.ui;
Ssh.ui = Ssh.ui | Stmp2.ui;
Stmp2.ui = gcosine_pi_over_eight&Stmp1.ui;
Sch.ui = ~Stmp1.ui&Sch.ui;
Sch.ui = Sch.ui | Stmp2.ui;
Stmp1.f = Ssh.f * Ssh.f;
Stmp2.f = Sch.f * Sch.f;
Sc.f = __fsub_rn(Stmp2.f, Stmp1.f);
Ss.f = Sch.f * Ssh.f;
Ss.f = __fadd_rn(Ss.f, Ss.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("GPU s %.20g, c %.20g, sh %.20g, ch %.20g\n", Ss.f, Sc.f, Ssh.f, Sch.f);
#endif
//###########################################################
// Perform the actual Givens conjugation
//###########################################################
Stmp3.f = __fadd_rn(Stmp1.f, Stmp2.f);
Ss33.f = Ss33.f * Stmp3.f;
Ss31.f = Ss31.f * Stmp3.f;
Ss32.f = Ss32.f * Stmp3.f;
Ss33.f = Ss33.f * Stmp3.f;
Stmp1.f = Ss.f * Ss31.f;
Stmp2.f = Ss.f * Ss32.f;
Ss31.f = Sc.f * Ss31.f;
Ss32.f = Sc.f * Ss32.f;
Ss31.f = __fadd_rn(Stmp2.f, Ss31.f);
Ss32.f = __fsub_rn(Ss32.f, Stmp1.f);
Stmp2.f = Ss.f*Ss.f;
Stmp1.f = Ss22.f*Stmp2.f;
Stmp3.f = Ss11.f*Stmp2.f;
Stmp4.f = Sc.f*Sc.f;
Ss11.f = Ss11.f*Stmp4.f;
Ss22.f = Ss22.f*Stmp4.f;
Ss11.f = __fadd_rn(Ss11.f, Stmp1.f);
Ss22.f = __fadd_rn(Ss22.f, Stmp3.f);
Stmp4.f = __fsub_rn(Stmp4.f, Stmp2.f);
Stmp2.f = __fadd_rn(Ss21.f, Ss21.f);
Ss21.f = Ss21.f*Stmp4.f;
Stmp4.f = Sc.f*Ss.f;
Stmp2.f = Stmp2.f*Stmp4.f;
Stmp5.f = Stmp5.f*Stmp4.f;
Ss11.f = __fadd_rn(Ss11.f, Stmp2.f);
Ss21.f = __fsub_rn(Ss21.f, Stmp5.f);
Ss22.f = __fsub_rn(Ss22.f, Stmp2.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("%.20g\n", Ss11.f);
printf("%.20g %.20g\n", Ss21.f, Ss22.f);
printf("%.20g %.20g %.20g\n", Ss31.f, Ss32.f, Ss33.f);
#endif
//###########################################################
// Compute the cumulative rotation, in quaternion form
//###########################################################
Stmp1.f = Ssh.f*Sqvvx.f;
Stmp2.f = Ssh.f*Sqvvy.f;
Stmp3.f = Ssh.f*Sqvvz.f;
Ssh.f = Ssh.f*Sqvs.f;
Sqvs.f = Sch.f*Sqvs.f;
Sqvvx.f = Sch.f*Sqvvx.f;
Sqvvy.f = Sch.f*Sqvvy.f;
Sqvvz.f = Sch.f*Sqvvz.f;
Sqvvz.f = __fadd_rn(Sqvvz.f, Ssh.f);
Sqvs.f = __fsub_rn(Sqvs.f, Stmp3.f);
Sqvvx.f = __fadd_rn(Sqvvx.f, Stmp2.f);
Sqvvy.f = __fsub_rn(Sqvvy.f, Stmp1.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("GPU q %.20g %.20g %.20g %.20g\n", Sqvvx.f, Sqvvy.f, Sqvvz.f, Sqvs.f);
#endif
//////////////////////////////////////////////////////////////////////////
// (1->3)
//////////////////////////////////////////////////////////////////////////
Ssh.f = Ss32.f * 0.5f;
Stmp5.f = __fsub_rn(Ss22.f, Ss33.f);
Stmp2.f = Ssh.f * Ssh.f;
Stmp1.ui = (Stmp2.f >= gtiny_number) ? 0xffffffff : 0;
Ssh.ui = Stmp1.ui&Ssh.ui;
Sch.ui = Stmp1.ui&Stmp5.ui;
Stmp2.ui = ~Stmp1.ui&gone;
Sch.ui = Sch.ui | Stmp2.ui;
Stmp1.f = Ssh.f * Ssh.f;
Stmp2.f = Sch.f * Sch.f;
Stmp3.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp4.f = __frsqrt_rn(Stmp3.f);
Ssh.f = Stmp4.f * Ssh.f;
Sch.f = Stmp4.f * Sch.f;
Stmp1.f = gfour_gamma_squared * Stmp1.f;
Stmp1.ui = (Stmp2.f <= Stmp1.f) ? 0xffffffff : 0;
Stmp2.ui = gsine_pi_over_eight&Stmp1.ui;
Ssh.ui = ~Stmp1.ui&Ssh.ui;
Ssh.ui = Ssh.ui | Stmp2.ui;
Stmp2.ui = gcosine_pi_over_eight&Stmp1.ui;
Sch.ui = ~Stmp1.ui&Sch.ui;
Sch.ui = Sch.ui | Stmp2.ui;
Stmp1.f = Ssh.f * Ssh.f;
Stmp2.f = Sch.f * Sch.f;
Sc.f = __fsub_rn(Stmp2.f, Stmp1.f);
Ss.f = Sch.f*Ssh.f;
Ss.f = __fadd_rn(Ss.f, Ss.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("GPU s %.20g, c %.20g, sh %.20g, ch %.20g\n", Ss.f, Sc.f, Ssh.f, Sch.f);
#endif
//###########################################################
// Perform the actual Givens conjugation
//###########################################################
Stmp3.f = __fadd_rn(Stmp1.f, Stmp2.f);
Ss11.f = Ss11.f * Stmp3.f;
Ss21.f = Ss21.f * Stmp3.f;
Ss31.f = Ss31.f * Stmp3.f;
Ss11.f = Ss11.f * Stmp3.f;
Stmp1.f = Ss.f*Ss21.f;
Stmp2.f = Ss.f*Ss31.f;
Ss21.f = Sc.f*Ss21.f;
Ss31.f = Sc.f*Ss31.f;
Ss21.f = __fadd_rn(Stmp2.f, Ss21.f);
Ss31.f = __fsub_rn(Ss31.f, Stmp1.f);
Stmp2.f = Ss.f*Ss.f;
Stmp1.f = Ss33.f*Stmp2.f;
Stmp3.f = Ss22.f*Stmp2.f;
Stmp4.f = Sc.f * Sc.f;
Ss22.f = Ss22.f * Stmp4.f;
Ss33.f = Ss33.f * Stmp4.f;
Ss22.f = __fadd_rn(Ss22.f, Stmp1.f);
Ss33.f = __fadd_rn(Ss33.f, Stmp3.f);
Stmp4.f = __fsub_rn(Stmp4.f, Stmp2.f);
Stmp2.f = __fadd_rn(Ss32.f, Ss32.f);
Ss32.f = Ss32.f*Stmp4.f;
Stmp4.f = Sc.f*Ss.f;
Stmp2.f = Stmp2.f*Stmp4.f;
Stmp5.f = Stmp5.f*Stmp4.f;
Ss22.f = __fadd_rn(Ss22.f, Stmp2.f);
Ss32.f = __fsub_rn(Ss32.f, Stmp5.f);
Ss33.f = __fsub_rn(Ss33.f, Stmp2.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("%.20g\n", Ss11.f);
printf("%.20g %.20g\n", Ss21.f, Ss22.f);
printf("%.20g %.20g %.20g\n", Ss31.f, Ss32.f, Ss33.f);
#endif
//###########################################################
// Compute the cumulative rotation, in quaternion form
//###########################################################
Stmp1.f = Ssh.f*Sqvvx.f;
Stmp2.f = Ssh.f*Sqvvy.f;
Stmp3.f = Ssh.f*Sqvvz.f;
Ssh.f = Ssh.f*Sqvs.f;
Sqvs.f = Sch.f*Sqvs.f;
Sqvvx.f = Sch.f*Sqvvx.f;
Sqvvy.f = Sch.f*Sqvvy.f;
Sqvvz.f = Sch.f*Sqvvz.f;
Sqvvx.f = __fadd_rn(Sqvvx.f, Ssh.f);
Sqvs.f = __fsub_rn(Sqvs.f, Stmp1.f);
Sqvvy.f = __fadd_rn(Sqvvy.f, Stmp3.f);
Sqvvz.f = __fsub_rn(Sqvvz.f, Stmp2.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("GPU q %.20g %.20g %.20g %.20g\n", Sqvvx.f, Sqvvy.f, Sqvvz.f, Sqvs.f);
#endif
#if 1
//////////////////////////////////////////////////////////////////////////
// 1 -> 2
//////////////////////////////////////////////////////////////////////////
Ssh.f = Ss31.f * 0.5f;
Stmp5.f = __fsub_rn(Ss33.f, Ss11.f);
Stmp2.f = Ssh.f*Ssh.f;
Stmp1.ui = (Stmp2.f >= gtiny_number) ? 0xffffffff : 0;
Ssh.ui = Stmp1.ui&Ssh.ui;
Sch.ui = Stmp1.ui&Stmp5.ui;
Stmp2.ui = ~Stmp1.ui&gone;
Sch.ui = Sch.ui | Stmp2.ui;
Stmp1.f = Ssh.f*Ssh.f;
Stmp2.f = Sch.f*Sch.f;
Stmp3.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp4.f = __frsqrt_rn(Stmp3.f);
Ssh.f = Stmp4.f*Ssh.f;
Sch.f = Stmp4.f*Sch.f;
Stmp1.f = gfour_gamma_squared*Stmp1.f;
Stmp1.ui = (Stmp2.f <= Stmp1.f) ? 0xffffffff : 0;
Stmp2.ui = gsine_pi_over_eight&Stmp1.ui;
Ssh.ui = ~Stmp1.ui&Ssh.ui;
Ssh.ui = Ssh.ui | Stmp2.ui;
Stmp2.ui = gcosine_pi_over_eight&Stmp1.ui;
Sch.ui = ~Stmp1.ui&Sch.ui;
Sch.ui = Sch.ui | Stmp2.ui;
Stmp1.f = Ssh.f*Ssh.f;
Stmp2.f = Sch.f*Sch.f;
Sc.f = __fsub_rn(Stmp2.f, Stmp1.f);
Ss.f = Sch.f*Ssh.f;
Ss.f = __fadd_rn(Ss.f, Ss.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("GPU s %.20g, c %.20g, sh %.20g, ch %.20g\n", Ss.f, Sc.f, Ssh.f, Sch.f);
#endif
//###########################################################
// Perform the actual Givens conjugation
//###########################################################
Stmp3.f = __fadd_rn(Stmp1.f, Stmp2.f);
Ss22.f = Ss22.f * Stmp3.f;
Ss32.f = Ss32.f * Stmp3.f;
Ss21.f = Ss21.f * Stmp3.f;
Ss22.f = Ss22.f * Stmp3.f;
Stmp1.f = Ss.f*Ss32.f;
Stmp2.f = Ss.f*Ss21.f;
Ss32.f = Sc.f*Ss32.f;
Ss21.f = Sc.f*Ss21.f;
Ss32.f = __fadd_rn(Stmp2.f, Ss32.f);
Ss21.f = __fsub_rn(Ss21.f, Stmp1.f);
Stmp2.f = Ss.f*Ss.f;
Stmp1.f = Ss11.f*Stmp2.f;
Stmp3.f = Ss33.f*Stmp2.f;
Stmp4.f = Sc.f*Sc.f;
Ss33.f = Ss33.f*Stmp4.f;
Ss11.f = Ss11.f*Stmp4.f;
Ss33.f = __fadd_rn(Ss33.f, Stmp1.f);
Ss11.f = __fadd_rn(Ss11.f, Stmp3.f);
Stmp4.f = __fsub_rn(Stmp4.f, Stmp2.f);
Stmp2.f = __fadd_rn(Ss31.f, Ss31.f);
Ss31.f = Ss31.f*Stmp4.f;
Stmp4.f = Sc.f*Ss.f;
Stmp2.f = Stmp2.f*Stmp4.f;
Stmp5.f = Stmp5.f*Stmp4.f;
Ss33.f = __fadd_rn(Ss33.f, Stmp2.f);
Ss31.f = __fsub_rn(Ss31.f, Stmp5.f);
Ss11.f = __fsub_rn(Ss11.f, Stmp2.f);
#ifdef DEBUG_JACOBI_CONJUGATE
printf("%.20g\n", Ss11.f);
printf("%.20g %.20g\n", Ss21.f, Ss22.f);
printf("%.20g %.20g %.20g\n", Ss31.f, Ss32.f, Ss33.f);
#endif
//###########################################################
// Compute the cumulative rotation, in quaternion form
//###########################################################
Stmp1.f = Ssh.f*Sqvvx.f;
Stmp2.f = Ssh.f*Sqvvy.f;
Stmp3.f = Ssh.f*Sqvvz.f;
Ssh.f = Ssh.f*Sqvs.f;
Sqvs.f = Sch.f*Sqvs.f;
Sqvvx.f = Sch.f*Sqvvx.f;
Sqvvy.f = Sch.f*Sqvvy.f;
Sqvvz.f = Sch.f*Sqvvz.f;
Sqvvy.f = __fadd_rn(Sqvvy.f, Ssh.f);
Sqvs.f = __fsub_rn(Sqvs.f, Stmp2.f);
Sqvvz.f = __fadd_rn(Sqvvz.f, Stmp1.f);
Sqvvx.f = __fsub_rn(Sqvvx.f, Stmp3.f);
#endif
}
//###########################################################
// Normalize quaternion for matrix V
//###########################################################
Stmp2.f = Sqvs.f*Sqvs.f;
Stmp1.f = Sqvvx.f*Sqvvx.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = Sqvvy.f*Sqvvy.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = Sqvvz.f*Sqvvz.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5f;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Sqvs.f = Sqvs.f*Stmp1.f;
Sqvvx.f = Sqvvx.f*Stmp1.f;
Sqvvy.f = Sqvvy.f*Stmp1.f;
Sqvvz.f = Sqvvz.f*Stmp1.f;
//###########################################################
// Transform quaternion to matrix V
//###########################################################
Stmp1.f = Sqvvx.f*Sqvvx.f;
Stmp2.f = Sqvvy.f*Sqvvy.f;
Stmp3.f = Sqvvz.f*Sqvvz.f;
Sv11.f = Sqvs.f*Sqvs.f;
Sv22.f = __fsub_rn(Sv11.f, Stmp1.f);
Sv33.f = __fsub_rn(Sv22.f, Stmp2.f);
Sv33.f = __fadd_rn(Sv33.f, Stmp3.f);
Sv22.f = __fadd_rn(Sv22.f, Stmp2.f);
Sv22.f = __fsub_rn(Sv22.f, Stmp3.f);
Sv11.f = __fadd_rn(Sv11.f, Stmp1.f);
Sv11.f = __fsub_rn(Sv11.f, Stmp2.f);
Sv11.f = __fsub_rn(Sv11.f, Stmp3.f);
Stmp1.f = __fadd_rn(Sqvvx.f, Sqvvx.f);
Stmp2.f = __fadd_rn(Sqvvy.f, Sqvvy.f);
Stmp3.f = __fadd_rn(Sqvvz.f, Sqvvz.f);
Sv32.f = Sqvs.f*Stmp1.f;
Sv13.f = Sqvs.f*Stmp2.f;
Sv21.f = Sqvs.f*Stmp3.f;
Stmp1.f = Sqvvy.f*Stmp1.f;
Stmp2.f = Sqvvz.f*Stmp2.f;
Stmp3.f = Sqvvx.f*Stmp3.f;
Sv12.f = __fsub_rn(Stmp1.f, Sv21.f);
Sv23.f = __fsub_rn(Stmp2.f, Sv32.f);
Sv31.f = __fsub_rn(Stmp3.f, Sv13.f);
Sv21.f = __fadd_rn(Stmp1.f, Sv21.f);
Sv32.f = __fadd_rn(Stmp2.f, Sv32.f);
Sv13.f = __fadd_rn(Stmp3.f, Sv13.f);
///###########################################################
// Multiply (from the right) with V
//###########################################################
Stmp2.f = Sa12.f;
Stmp3.f = Sa13.f;
Sa12.f = Sv12.f*Sa11.f;
Sa13.f = Sv13.f*Sa11.f;
Sa11.f = Sv11.f*Sa11.f;
Stmp1.f = Sv21.f*Stmp2.f;
Sa11.f = __fadd_rn(Sa11.f, Stmp1.f);
Stmp1.f = Sv31.f*Stmp3.f;
Sa11.f = __fadd_rn(Sa11.f, Stmp1.f);
Stmp1.f = Sv22.f*Stmp2.f;
Sa12.f = __fadd_rn(Sa12.f, Stmp1.f);
Stmp1.f = Sv32.f*Stmp3.f;
Sa12.f = __fadd_rn(Sa12.f, Stmp1.f);
Stmp1.f = Sv23.f*Stmp2.f;
Sa13.f = __fadd_rn(Sa13.f, Stmp1.f);
Stmp1.f = Sv33.f*Stmp3.f;
Sa13.f = __fadd_rn(Sa13.f, Stmp1.f);
Stmp2.f = Sa22.f;
Stmp3.f = Sa23.f;
Sa22.f = Sv12.f*Sa21.f;
Sa23.f = Sv13.f*Sa21.f;
Sa21.f = Sv11.f*Sa21.f;
Stmp1.f = Sv21.f*Stmp2.f;
Sa21.f = __fadd_rn(Sa21.f, Stmp1.f);
Stmp1.f = Sv31.f*Stmp3.f;
Sa21.f = __fadd_rn(Sa21.f, Stmp1.f);
Stmp1.f = Sv22.f*Stmp2.f;
Sa22.f = __fadd_rn(Sa22.f, Stmp1.f);
Stmp1.f = Sv32.f*Stmp3.f;
Sa22.f = __fadd_rn(Sa22.f, Stmp1.f);
Stmp1.f = Sv23.f*Stmp2.f;
Sa23.f = __fadd_rn(Sa23.f, Stmp1.f);
Stmp1.f = Sv33.f*Stmp3.f;
Sa23.f = __fadd_rn(Sa23.f, Stmp1.f);
Stmp2.f = Sa32.f;
Stmp3.f = Sa33.f;
Sa32.f = Sv12.f*Sa31.f;
Sa33.f = Sv13.f*Sa31.f;
Sa31.f = Sv11.f*Sa31.f;
Stmp1.f = Sv21.f*Stmp2.f;
Sa31.f = __fadd_rn(Sa31.f, Stmp1.f);
Stmp1.f = Sv31.f*Stmp3.f;
Sa31.f = __fadd_rn(Sa31.f, Stmp1.f);
Stmp1.f = Sv22.f*Stmp2.f;
Sa32.f = __fadd_rn(Sa32.f, Stmp1.f);
Stmp1.f = Sv32.f*Stmp3.f;
Sa32.f = __fadd_rn(Sa32.f, Stmp1.f);
Stmp1.f = Sv23.f*Stmp2.f;
Sa33.f = __fadd_rn(Sa33.f, Stmp1.f);
Stmp1.f = Sv33.f*Stmp3.f;
Sa33.f = __fadd_rn(Sa33.f, Stmp1.f);
//###########################################################
// Permute columns such that the singular values are sorted
//###########################################################
Stmp1.f = Sa11.f*Sa11.f;
Stmp4.f = Sa21.f*Sa21.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp4.f = Sa31.f*Sa31.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp2.f = Sa12.f*Sa12.f;
Stmp4.f = Sa22.f*Sa22.f;
Stmp2.f = __fadd_rn(Stmp2.f, Stmp4.f);
Stmp4.f = Sa32.f*Sa32.f;
Stmp2.f = __fadd_rn(Stmp2.f, Stmp4.f);
Stmp3.f = Sa13.f*Sa13.f;
Stmp4.f = Sa23.f*Sa23.f;
Stmp3.f = __fadd_rn(Stmp3.f, Stmp4.f);
Stmp4.f = Sa33.f*Sa33.f;
Stmp3.f = __fadd_rn(Stmp3.f, Stmp4.f);
// Swap columns 1-2 if necessary
Stmp4.ui = (Stmp1.f < Stmp2.f) ? 0xffffffff : 0;
Stmp5.ui = Sa11.ui^Sa12.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa11.ui = Sa11.ui^Stmp5.ui;
Sa12.ui = Sa12.ui^Stmp5.ui;
Stmp5.ui = Sa21.ui^Sa22.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa21.ui = Sa21.ui^Stmp5.ui;
Sa22.ui = Sa22.ui^Stmp5.ui;
Stmp5.ui = Sa31.ui^Sa32.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa31.ui = Sa31.ui^Stmp5.ui;
Sa32.ui = Sa32.ui^Stmp5.ui;
Stmp5.ui = Sv11.ui^Sv12.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv11.ui = Sv11.ui^Stmp5.ui;
Sv12.ui = Sv12.ui^Stmp5.ui;
Stmp5.ui = Sv21.ui^Sv22.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv21.ui = Sv21.ui^Stmp5.ui;
Sv22.ui = Sv22.ui^Stmp5.ui;
Stmp5.ui = Sv31.ui^Sv32.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv31.ui = Sv31.ui^Stmp5.ui;
Sv32.ui = Sv32.ui^Stmp5.ui;
Stmp5.ui = Stmp1.ui^Stmp2.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Stmp1.ui = Stmp1.ui^Stmp5.ui;
Stmp2.ui = Stmp2.ui^Stmp5.ui;
// If columns 1-2 have been swapped, negate 2nd column of A and V so that V is still a rotation
Stmp5.f = -2.f;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Stmp4.f = 1.f;
Stmp4.f = __fadd_rn(Stmp4.f, Stmp5.f);
Sa12.f = Sa12.f*Stmp4.f;
Sa22.f = Sa22.f*Stmp4.f;
Sa32.f = Sa32.f*Stmp4.f;
Sv12.f = Sv12.f*Stmp4.f;
Sv22.f = Sv22.f*Stmp4.f;
Sv32.f = Sv32.f*Stmp4.f;
// Swap columns 1-3 if necessary
Stmp4.ui = (Stmp1.f < Stmp3.f) ? 0xffffffff : 0;
Stmp5.ui = Sa11.ui^Sa13.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa11.ui = Sa11.ui^Stmp5.ui;
Sa13.ui = Sa13.ui^Stmp5.ui;
Stmp5.ui = Sa21.ui^Sa23.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa21.ui = Sa21.ui^Stmp5.ui;
Sa23.ui = Sa23.ui^Stmp5.ui;
Stmp5.ui = Sa31.ui^Sa33.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa31.ui = Sa31.ui^Stmp5.ui;
Sa33.ui = Sa33.ui^Stmp5.ui;
Stmp5.ui = Sv11.ui^Sv13.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv11.ui = Sv11.ui^Stmp5.ui;
Sv13.ui = Sv13.ui^Stmp5.ui;
Stmp5.ui = Sv21.ui^Sv23.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv21.ui = Sv21.ui^Stmp5.ui;
Sv23.ui = Sv23.ui^Stmp5.ui;
Stmp5.ui = Sv31.ui^Sv33.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv31.ui = Sv31.ui^Stmp5.ui;
Sv33.ui = Sv33.ui^Stmp5.ui;
Stmp5.ui = Stmp1.ui^Stmp3.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Stmp1.ui = Stmp1.ui^Stmp5.ui;
Stmp3.ui = Stmp3.ui^Stmp5.ui;
// If columns 1-3 have been swapped, negate 1st column of A and V so that V is still a rotation
Stmp5.f = -2.f;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Stmp4.f = 1.f;
Stmp4.f = __fadd_rn(Stmp4.f, Stmp5.f);
Sa11.f = Sa11.f*Stmp4.f;
Sa21.f = Sa21.f*Stmp4.f;
Sa31.f = Sa31.f*Stmp4.f;
Sv11.f = Sv11.f*Stmp4.f;
Sv21.f = Sv21.f*Stmp4.f;
Sv31.f = Sv31.f*Stmp4.f;
// Swap columns 2-3 if necessary
Stmp4.ui = (Stmp2.f < Stmp3.f) ? 0xffffffff : 0;
Stmp5.ui = Sa12.ui^Sa13.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa12.ui = Sa12.ui^Stmp5.ui;
Sa13.ui = Sa13.ui^Stmp5.ui;
Stmp5.ui = Sa22.ui^Sa23.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa22.ui = Sa22.ui^Stmp5.ui;
Sa23.ui = Sa23.ui^Stmp5.ui;
Stmp5.ui = Sa32.ui^Sa33.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sa32.ui = Sa32.ui^Stmp5.ui;
Sa33.ui = Sa33.ui^Stmp5.ui;
Stmp5.ui = Sv12.ui^Sv13.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv12.ui = Sv12.ui^Stmp5.ui;
Sv13.ui = Sv13.ui^Stmp5.ui;
Stmp5.ui = Sv22.ui^Sv23.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv22.ui = Sv22.ui^Stmp5.ui;
Sv23.ui = Sv23.ui^Stmp5.ui;
Stmp5.ui = Sv32.ui^Sv33.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Sv32.ui = Sv32.ui^Stmp5.ui;
Sv33.ui = Sv33.ui^Stmp5.ui;
Stmp5.ui = Stmp2.ui^Stmp3.ui;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Stmp2.ui = Stmp2.ui^Stmp5.ui;
Stmp3.ui = Stmp3.ui^Stmp5.ui;
// If columns 2-3 have been swapped, negate 3rd column of A and V so that V is still a rotation
Stmp5.f = -2.f;
Stmp5.ui = Stmp5.ui&Stmp4.ui;
Stmp4.f = 1.f;
Stmp4.f = __fadd_rn(Stmp4.f, Stmp5.f);
Sa13.f = Sa13.f*Stmp4.f;
Sa23.f = Sa23.f*Stmp4.f;
Sa33.f = Sa33.f*Stmp4.f;
Sv13.f = Sv13.f*Stmp4.f;
Sv23.f = Sv23.f*Stmp4.f;
Sv33.f = Sv33.f*Stmp4.f;
//###########################################################
// Construct QR factorization of A*V (=U*D) using Givens rotations
//###########################################################
Su11.f = 1.f; Su12.f = 0.f; Su13.f = 0.f;
Su21.f = 0.f; Su22.f = 1.f; Su23.f = 0.f;
Su31.f = 0.f; Su32.f = 0.f; Su33.f = 1.f;
Ssh.f = Sa21.f*Sa21.f;
Ssh.ui = (Ssh.f >= gsmall_number) ? 0xffffffff : 0;
Ssh.ui = Ssh.ui&Sa21.ui;
Stmp5.f = 0.f;
Sch.f = __fsub_rn(Stmp5.f, Sa11.f);
Sch.f = max(Sch.f, Sa11.f);
Sch.f = max(Sch.f, gsmall_number);
Stmp5.ui = (Sa11.f >= Stmp5.f) ? 0xffffffff : 0;
Stmp1.f = Sch.f*Sch.f;
Stmp2.f = Ssh.f*Ssh.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5f;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Stmp1.f = Stmp1.f*Stmp2.f;
Sch.f = __fadd_rn(Sch.f, Stmp1.f);
Stmp1.ui = ~Stmp5.ui&Ssh.ui;
Stmp2.ui = ~Stmp5.ui&Sch.ui;
Sch.ui = Stmp5.ui&Sch.ui;
Ssh.ui = Stmp5.ui&Ssh.ui;
Sch.ui = Sch.ui | Stmp1.ui;
Ssh.ui = Ssh.ui | Stmp2.ui;
Stmp1.f = Sch.f*Sch.f;
Stmp2.f = Ssh.f*Ssh.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5f;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Sch.f = Sch.f*Stmp1.f;
Ssh.f = Ssh.f*Stmp1.f;
Sc.f = Sch.f*Sch.f;
Ss.f = Ssh.f*Ssh.f;
Sc.f = __fsub_rn(Sc.f, Ss.f);
Ss.f = Ssh.f*Sch.f;
Ss.f = __fadd_rn(Ss.f, Ss.f);
//###########################################################
// Rotate matrix A
//###########################################################
Stmp1.f = Ss.f*Sa11.f;
Stmp2.f = Ss.f*Sa21.f;
Sa11.f = Sc.f*Sa11.f;
Sa21.f = Sc.f*Sa21.f;
Sa11.f = __fadd_rn(Sa11.f, Stmp2.f);
Sa21.f = __fsub_rn(Sa21.f, Stmp1.f);
Stmp1.f = Ss.f*Sa12.f;
Stmp2.f = Ss.f*Sa22.f;
Sa12.f = Sc.f*Sa12.f;
Sa22.f = Sc.f*Sa22.f;
Sa12.f = __fadd_rn(Sa12.f, Stmp2.f);
Sa22.f = __fsub_rn(Sa22.f, Stmp1.f);
Stmp1.f = Ss.f*Sa13.f;
Stmp2.f = Ss.f*Sa23.f;
Sa13.f = Sc.f*Sa13.f;
Sa23.f = Sc.f*Sa23.f;
Sa13.f = __fadd_rn(Sa13.f, Stmp2.f);
Sa23.f = __fsub_rn(Sa23.f, Stmp1.f);
//###########################################################
// Update matrix U
//###########################################################
Stmp1.f = Ss.f*Su11.f;
Stmp2.f = Ss.f*Su12.f;
Su11.f = Sc.f*Su11.f;
Su12.f = Sc.f*Su12.f;
Su11.f = __fadd_rn(Su11.f, Stmp2.f);
Su12.f = __fsub_rn(Su12.f, Stmp1.f);
Stmp1.f = Ss.f*Su21.f;
Stmp2.f = Ss.f*Su22.f;
Su21.f = Sc.f*Su21.f;
Su22.f = Sc.f*Su22.f;
Su21.f = __fadd_rn(Su21.f, Stmp2.f);
Su22.f = __fsub_rn(Su22.f, Stmp1.f);
Stmp1.f = Ss.f*Su31.f;
Stmp2.f = Ss.f*Su32.f;
Su31.f = Sc.f*Su31.f;
Su32.f = Sc.f*Su32.f;
Su31.f = __fadd_rn(Su31.f, Stmp2.f);
Su32.f = __fsub_rn(Su32.f, Stmp1.f);
// Second Givens rotation
Ssh.f = Sa31.f*Sa31.f;
Ssh.ui = (Ssh.f >= gsmall_number) ? 0xffffffff : 0;
Ssh.ui = Ssh.ui&Sa31.ui;
Stmp5.f = 0.f;
Sch.f = __fsub_rn(Stmp5.f, Sa11.f);
Sch.f = max(Sch.f, Sa11.f);
Sch.f = max(Sch.f, gsmall_number);
Stmp5.ui = (Sa11.f >= Stmp5.f) ? 0xffffffff : 0;
Stmp1.f = Sch.f*Sch.f;
Stmp2.f = Ssh.f*Ssh.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Stmp1.f = Stmp1.f*Stmp2.f;
Sch.f = __fadd_rn(Sch.f, Stmp1.f);
Stmp1.ui = ~Stmp5.ui&Ssh.ui;
Stmp2.ui = ~Stmp5.ui&Sch.ui;
Sch.ui = Stmp5.ui&Sch.ui;
Ssh.ui = Stmp5.ui&Ssh.ui;
Sch.ui = Sch.ui | Stmp1.ui;
Ssh.ui = Ssh.ui | Stmp2.ui;
Stmp1.f = Sch.f*Sch.f;
Stmp2.f = Ssh.f*Ssh.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5f;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Sch.f = Sch.f*Stmp1.f;
Ssh.f = Ssh.f*Stmp1.f;
Sc.f = Sch.f*Sch.f;
Ss.f = Ssh.f*Ssh.f;
Sc.f = __fsub_rn(Sc.f, Ss.f);
Ss.f = Ssh.f*Sch.f;
Ss.f = __fadd_rn(Ss.f, Ss.f);
//###########################################################
// Rotate matrix A
//###########################################################
Stmp1.f = Ss.f*Sa11.f;
Stmp2.f = Ss.f*Sa31.f;
Sa11.f = Sc.f*Sa11.f;
Sa31.f = Sc.f*Sa31.f;
Sa11.f = __fadd_rn(Sa11.f, Stmp2.f);
Sa31.f = __fsub_rn(Sa31.f, Stmp1.f);
Stmp1.f = Ss.f*Sa12.f;
Stmp2.f = Ss.f*Sa32.f;
Sa12.f = Sc.f*Sa12.f;
Sa32.f = Sc.f*Sa32.f;
Sa12.f = __fadd_rn(Sa12.f, Stmp2.f);
Sa32.f = __fsub_rn(Sa32.f, Stmp1.f);
Stmp1.f = Ss.f*Sa13.f;
Stmp2.f = Ss.f*Sa33.f;
Sa13.f = Sc.f*Sa13.f;
Sa33.f = Sc.f*Sa33.f;
Sa13.f = __fadd_rn(Sa13.f, Stmp2.f);
Sa33.f = __fsub_rn(Sa33.f, Stmp1.f);
//###########################################################
// Update matrix U
//###########################################################
Stmp1.f = Ss.f*Su11.f;
Stmp2.f = Ss.f*Su13.f;
Su11.f = Sc.f*Su11.f;
Su13.f = Sc.f*Su13.f;
Su11.f = __fadd_rn(Su11.f, Stmp2.f);
Su13.f = __fsub_rn(Su13.f, Stmp1.f);
Stmp1.f = Ss.f*Su21.f;
Stmp2.f = Ss.f*Su23.f;
Su21.f = Sc.f*Su21.f;
Su23.f = Sc.f*Su23.f;
Su21.f = __fadd_rn(Su21.f, Stmp2.f);
Su23.f = __fsub_rn(Su23.f, Stmp1.f);
Stmp1.f = Ss.f*Su31.f;
Stmp2.f = Ss.f*Su33.f;
Su31.f = Sc.f*Su31.f;
Su33.f = Sc.f*Su33.f;
Su31.f = __fadd_rn(Su31.f, Stmp2.f);
Su33.f = __fsub_rn(Su33.f, Stmp1.f);
// Third Givens Rotation
Ssh.f = Sa32.f*Sa32.f;
Ssh.ui = (Ssh.f >= gsmall_number) ? 0xffffffff : 0;
Ssh.ui = Ssh.ui&Sa32.ui;
Stmp5.f = 0.f;
Sch.f = __fsub_rn(Stmp5.f, Sa22.f);
Sch.f = max(Sch.f, Sa22.f);
Sch.f = max(Sch.f, gsmall_number);
Stmp5.ui = (Sa22.f >= Stmp5.f) ? 0xffffffff : 0;
Stmp1.f = Sch.f*Sch.f;
Stmp2.f = Ssh.f*Ssh.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5f;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Stmp1.f = Stmp1.f*Stmp2.f;
Sch.f = __fadd_rn(Sch.f, Stmp1.f);
Stmp1.ui = ~Stmp5.ui&Ssh.ui;
Stmp2.ui = ~Stmp5.ui&Sch.ui;
Sch.ui = Stmp5.ui&Sch.ui;
Ssh.ui = Stmp5.ui&Ssh.ui;
Sch.ui = Sch.ui | Stmp1.ui;
Ssh.ui = Ssh.ui | Stmp2.ui;
Stmp1.f = Sch.f*Sch.f;
Stmp2.f = Ssh.f*Ssh.f;
Stmp2.f = __fadd_rn(Stmp1.f, Stmp2.f);
Stmp1.f = __frsqrt_rn(Stmp2.f);
Stmp4.f = Stmp1.f*0.5f;
Stmp3.f = Stmp1.f*Stmp4.f;
Stmp3.f = Stmp1.f*Stmp3.f;
Stmp3.f = Stmp2.f*Stmp3.f;
Stmp1.f = __fadd_rn(Stmp1.f, Stmp4.f);
Stmp1.f = __fsub_rn(Stmp1.f, Stmp3.f);
Sch.f = Sch.f*Stmp1.f;
Ssh.f = Ssh.f*Stmp1.f;
Sc.f = Sch.f*Sch.f;
Ss.f = Ssh.f*Ssh.f;
Sc.f = __fsub_rn(Sc.f, Ss.f);
Ss.f = Ssh.f*Sch.f;
Ss.f = __fadd_rn(Ss.f, Ss.f);
//###########################################################
// Rotate matrix A
//###########################################################
Stmp1.f = Ss.f*Sa21.f;
Stmp2.f = Ss.f*Sa31.f;
Sa21.f = Sc.f*Sa21.f;
Sa31.f = Sc.f*Sa31.f;
Sa21.f = __fadd_rn(Sa21.f, Stmp2.f);
Sa31.f = __fsub_rn(Sa31.f, Stmp1.f);
Stmp1.f = Ss.f*Sa22.f;
Stmp2.f = Ss.f*Sa32.f;
Sa22.f = Sc.f*Sa22.f;
Sa32.f = Sc.f*Sa32.f;
Sa22.f = __fadd_rn(Sa22.f, Stmp2.f);
Sa32.f = __fsub_rn(Sa32.f, Stmp1.f);
Stmp1.f = Ss.f*Sa23.f;
Stmp2.f = Ss.f*Sa33.f;
Sa23.f = Sc.f*Sa23.f;