btDantzigLCP.cpp

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/*************************************************************************
*                                                                       *
* Open Dynamics Engine, Copyright (C) 2001,2002 Russell L. Smith.       *
* All rights reserved.  Email: russ@q12.org   Web: www.q12.org          *
*                                                                       *
* This library is free software; you can redistribute it and/or         *
* modify it under the terms of EITHER:                                  *
*   (1) The GNU Lesser General Public License as published by the Free  *
*       Software Foundation; either version 2.1 of the License, or (at  *
*       your option) any later version. The text of the GNU Lesser      *
*       General Public License is included with this library in the     *
*       file LICENSE.TXT.                                               *
*   (2) The BSD-style license that is included with this library in     *
*       the file LICENSE-BSD.TXT.                                       *
*                                                                       *
* This library is distributed in the hope that it will be useful,       *
* but WITHOUT ANY WARRANTY; without even the implied warranty of        *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the files    *
* LICENSE.TXT and LICENSE-BSD.TXT for more details.                     *
*                                                                       *
*************************************************************************/
 
/*
 
 
THE ALGORITHM
-------------
 
solve A*x = b+w, with x and w subject to certain LCP conditions.
each x(i),w(i) must lie on one of the three line segments in the following
diagram. each line segment corresponds to one index set :
 
     w(i)
     /|\      |           :
      |       |           :
      |       |i in N     :
  w>0 |       |state[i]=0 :
      |       |           :
      |       |           :  i in C
  w=0 +       +-----------------------+
      |                   :           |
      |                   :           |
  w<0 |                   :           |i in N
      |                   :           |state[i]=1
      |                   :           |
      |                   :           |
      +-------|-----------|-----------|----------> x(i)
             lo           0           hi
 
the Dantzig algorithm proceeds as follows:
  for i=1:n
    * if (x(i),w(i)) is not on the line, push x(i) and w(i) positive or
      negative towards the line. as this is done, the other (x(j),w(j))
      for j<i are constrained to be on the line. if any (x,w) reaches the
      end of a line segment then it is switched between index sets.
    * i is added to the appropriate index set depending on what line segment
      it hits.
 
we restrict lo(i) <= 0 and hi(i) >= 0. this makes the algorithm a bit
simpler, because the starting point for x(i),w(i) is always on the dotted
line x=0 and x will only ever increase in one direction, so it can only hit
two out of the three line segments.
 
 
NOTES
-----
 
this is an implementation of "lcp_dantzig2_ldlt.m" and "lcp_dantzig_lohi.m".
the implementation is split into an LCP problem object (btLCP) and an LCP
driver function. most optimization occurs in the btLCP object.
 
a naive implementation of the algorithm requires either a lot of data motion
or a lot of permutation-array lookup, because we are constantly re-ordering
rows and columns. to avoid this and make a more optimized algorithm, a
non-trivial data structure is used to represent the matrix A (this is
implemented in the fast version of the btLCP object).
 
during execution of this algorithm, some indexes in A are clamped (set C),
some are non-clamped (set N), and some are "don't care" (where x=0).
A,x,b,w (and other problem vectors) are permuted such that the clamped
indexes are first, the unclamped indexes are next, and the don't-care
indexes are last. this permutation is recorded in the array `p'.
initially p = 0..n-1, and as the rows and columns of A,x,b,w are swapped,
the corresponding elements of p are swapped.
 
because the C and N elements are grouped together in the rows of A, we can do
lots of work with a fast dot product function. if A,x,etc were not permuted
and we only had a permutation array, then those dot products would be much
slower as we would have a permutation array lookup in some inner loops.
 
A is accessed through an array of row pointers, so that element (i,j) of the
permuted matrix is A[i][j]. this makes row swapping fast. for column swapping
we still have to actually move the data.
 
during execution of this algorithm we maintain an L*D*L' factorization of
the clamped submatrix of A (call it `AC') which is the top left nC*nC
submatrix of A. there are two ways we could arrange the rows/columns in AC.
 
(1) AC is always permuted such that L*D*L' = AC. this causes a problem
when a row/column is removed from C, because then all the rows/columns of A
between the deleted index and the end of C need to be rotated downward.
this results in a lot of data motion and slows things down.
(2) L*D*L' is actually a factorization of a *permutation* of AC (which is
itself a permutation of the underlying A). this is what we do - the
permutation is recorded in the vector C. call this permutation A[C,C].
when a row/column is removed from C, all we have to do is swap two
rows/columns and manipulate C.
 
*/
 
 
#include "btDantzigLCP.h"
 
#include <string.h>//memcpy
 
bool s_error = false;
 
//***************************************************************************
// code generation parameters
 
 
#define btLCP_FAST              // use fast btLCP object
 
// option 1 : matrix row pointers (less data copying)
#define BTROWPTRS
#define BTATYPE btScalar **
#define BTAROW(i) (m_A[i])
 
// option 2 : no matrix row pointers (slightly faster inner loops)
//#define NOROWPTRS
//#define BTATYPE btScalar *
//#define BTAROW(i) (m_A+(i)*m_nskip)
 
#define BTNUB_OPTIMIZATIONS
 
 
 
/* solve L*X=B, with B containing 1 right hand sides.
 * L is an n*n lower triangular matrix with ones on the diagonal.
 * L is stored by rows and its leading dimension is lskip.
 * B is an n*1 matrix that contains the right hand sides.
 * B is stored by columns and its leading dimension is also lskip.
 * B is overwritten with X.
 * this processes blocks of 2*2.
 * if this is in the factorizer source file, n must be a multiple of 2.
 */
 
static void btSolveL1_1 (const btScalar *L, btScalar *B, int n, int lskip1)
{  
  /* declare variables - Z matrix, p and q vectors, etc */
  btScalar Z11,m11,Z21,m21,p1,q1,p2,*ex;
  const btScalar *ell;
  int i,j;
  /* compute all 2 x 1 blocks of X */
  for (i=0; i < n; i+=2) {
    /* compute all 2 x 1 block of X, from rows i..i+2-1 */
    /* set the Z matrix to 0 */
    Z11=0;
    Z21=0;
    ell = L + i*lskip1;
    ex = B;
    /* the inner loop that computes outer products and adds them to Z */
    for (j=i-2; j >= 0; j -= 2) {
      /* compute outer product and add it to the Z matrix */
      p1=ell[0];
      q1=ex[0];
      m11 = p1 * q1;
      p2=ell[lskip1];
      m21 = p2 * q1;
      Z11 += m11;
      Z21 += m21;
      /* compute outer product and add it to the Z matrix */
      p1=ell[1];
      q1=ex[1];
      m11 = p1 * q1;
      p2=ell[1+lskip1];
      m21 = p2 * q1;
      /* advance pointers */
      ell += 2;
      ex += 2;
      Z11 += m11;
      Z21 += m21;
      /* end of inner loop */
    }
    /* compute left-over iterations */
    j += 2;
    for (; j > 0; j--) {
      /* compute outer product and add it to the Z matrix */
      p1=ell[0];
      q1=ex[0];
      m11 = p1 * q1;
      p2=ell[lskip1];
      m21 = p2 * q1;
      /* advance pointers */
      ell += 1;
      ex += 1;
      Z11 += m11;
      Z21 += m21;
    }
    /* finish computing the X(i) block */
    Z11 = ex[0] - Z11;
    ex[0] = Z11;
    p1 = ell[lskip1];
    Z21 = ex[1] - Z21 - p1*Z11;
    ex[1] = Z21;
    /* end of outer loop */
  }
}
 
/* solve L*X=B, with B containing 2 right hand sides.
 * L is an n*n lower triangular matrix with ones on the diagonal.
 * L is stored by rows and its leading dimension is lskip.
 * B is an n*2 matrix that contains the right hand sides.
 * B is stored by columns and its leading dimension is also lskip.
 * B is overwritten with X.
 * this processes blocks of 2*2.
 * if this is in the factorizer source file, n must be a multiple of 2.
 */
 
static void btSolveL1_2 (const btScalar *L, btScalar *B, int n, int lskip1)
{  
  /* declare variables - Z matrix, p and q vectors, etc */
  btScalar Z11,m11,Z12,m12,Z21,m21,Z22,m22,p1,q1,p2,q2,*ex;
  const btScalar *ell;
  int i,j;
  /* compute all 2 x 2 blocks of X */
  for (i=0; i < n; i+=2) {
    /* compute all 2 x 2 block of X, from rows i..i+2-1 */
    /* set the Z matrix to 0 */
    Z11=0;
    Z12=0;
    Z21=0;
    Z22=0;
    ell = L + i*lskip1;
    ex = B;
    /* the inner loop that computes outer products and adds them to Z */
    for (j=i-2; j >= 0; j -= 2) {
      /* compute outer product and add it to the Z matrix */
      p1=ell[0];
      q1=ex[0];
      m11 = p1 * q1;
      q2=ex[lskip1];
      m12 = p1 * q2;
      p2=ell[lskip1];
      m21 = p2 * q1;
      m22 = p2 * q2;
      Z11 += m11;
      Z12 += m12;
      Z21 += m21;
      Z22 += m22;
      /* compute outer product and add it to the Z matrix */
      p1=ell[1];
      q1=ex[1];
      m11 = p1 * q1;
      q2=ex[1+lskip1];
      m12 = p1 * q2;
      p2=ell[1+lskip1];
      m21 = p2 * q1;
      m22 = p2 * q2;
      /* advance pointers */
      ell += 2;
      ex += 2;
      Z11 += m11;
      Z12 += m12;
      Z21 += m21;
      Z22 += m22;
      /* end of inner loop */
    }
    /* compute left-over iterations */
    j += 2;
    for (; j > 0; j--) {
      /* compute outer product and add it to the Z matrix */
      p1=ell[0];
      q1=ex[0];
      m11 = p1 * q1;
      q2=ex[lskip1];
      m12 = p1 * q2;
      p2=ell[lskip1];
      m21 = p2 * q1;
      m22 = p2 * q2;
      /* advance pointers */
      ell += 1;
      ex += 1;
      Z11 += m11;
      Z12 += m12;
      Z21 += m21;
      Z22 += m22;
    }
    /* finish computing the X(i) block */
    Z11 = ex[0] - Z11;
    ex[0] = Z11;
    Z12 = ex[lskip1] - Z12;
    ex[lskip1] = Z12;
    p1 = ell[lskip1];
    Z21 = ex[1] - Z21 - p1*Z11;
    ex[1] = Z21;
    Z22 = ex[1+lskip1] - Z22 - p1*Z12;
    ex[1+lskip1] = Z22;
    /* end of outer loop */
  }
}
 
 
void btFactorLDLT (btScalar *A, btScalar *d, int n, int nskip1)
{  
  int i,j;
  btScalar sum,*ell,*dee,dd,p1,p2,q1,q2,Z11,m11,Z21,m21,Z22,m22;
  if (n < 1) return;
  
  for (i=0; i<=n-2; i += 2) {
    /* solve L*(D*l)=a, l is scaled elements in 2 x i block at A(i,0) */
    btSolveL1_2 (A,A+i*nskip1,i,nskip1);
    /* scale the elements in a 2 x i block at A(i,0), and also */
    /* compute Z = the outer product matrix that we'll need. */
    Z11 = 0;
    Z21 = 0;
    Z22 = 0;
    ell = A+i*nskip1;
    dee = d;
    for (j=i-6; j >= 0; j -= 6) {
      p1 = ell[0];
      p2 = ell[nskip1];
      dd = dee[0];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[0] = q1;
      ell[nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      p1 = ell[1];
      p2 = ell[1+nskip1];
      dd = dee[1];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[1] = q1;
      ell[1+nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      p1 = ell[2];
      p2 = ell[2+nskip1];
      dd = dee[2];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[2] = q1;
      ell[2+nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      p1 = ell[3];
      p2 = ell[3+nskip1];
      dd = dee[3];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[3] = q1;
      ell[3+nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      p1 = ell[4];
      p2 = ell[4+nskip1];
      dd = dee[4];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[4] = q1;
      ell[4+nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      p1 = ell[5];
      p2 = ell[5+nskip1];
      dd = dee[5];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[5] = q1;
      ell[5+nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      ell += 6;
      dee += 6;
    }
    /* compute left-over iterations */
    j += 6;
    for (; j > 0; j--) {
      p1 = ell[0];
      p2 = ell[nskip1];
      dd = dee[0];
      q1 = p1*dd;
      q2 = p2*dd;
      ell[0] = q1;
      ell[nskip1] = q2;
      m11 = p1*q1;
      m21 = p2*q1;
      m22 = p2*q2;
      Z11 += m11;
      Z21 += m21;
      Z22 += m22;
      ell++;
      dee++;
    }
    /* solve for diagonal 2 x 2 block at A(i,i) */
    Z11 = ell[0] - Z11;
    Z21 = ell[nskip1] - Z21;
    Z22 = ell[1+nskip1] - Z22;
    dee = d + i;
    /* factorize 2 x 2 block Z,dee */
    /* factorize row 1 */
    dee[0] = btRecip(Z11);
    /* factorize row 2 */
    sum = 0;
    q1 = Z21;
    q2 = q1 * dee[0];
    Z21 = q2;
    sum += q1*q2;
    dee[1] = btRecip(Z22 - sum);
    /* done factorizing 2 x 2 block */
    ell[nskip1] = Z21;
  }
  /* compute the (less than 2) rows at the bottom */
  switch (n-i) {
    case 0:
    break;
    
    case 1:
    btSolveL1_1 (A,A+i*nskip1,i,nskip1);
    /* scale the elements in a 1 x i block at A(i,0), and also */
    /* compute Z = the outer product matrix that we'll need. */
    Z11 = 0;
    ell = A+i*nskip1;
    dee = d;
    for (j=i-6; j >= 0; j -= 6) {
      p1 = ell[0];
      dd = dee[0];
      q1 = p1*dd;
      ell[0] = q1;
      m11 = p1*q1;
      Z11 += m11;
      p1 = ell[1];
      dd = dee[1];
      q1 = p1*dd;
      ell[1] = q1;
      m11 = p1*q1;
      Z11 += m11;
      p1 = ell[2];
      dd = dee[2];
      q1 = p1*dd;
      ell[2] = q1;
      m11 = p1*q1;
      Z11 += m11;
      p1 = ell[3];
      dd = dee[3];
      q1 = p1*dd;
      ell[3] = q1;
      m11 = p1*q1;
      Z11 += m11;
      p1 = ell[4];
      dd = dee[4];
      q1 = p1*dd;
      ell[4] = q1;
      m11 = p1*q1;
      Z11 += m11;
      p1 = ell[5];
      dd = dee[5];
      q1 = p1*dd;
      ell[5] = q1;
      m11 = p1*q1;
      Z11 += m11;
      ell += 6;
      dee += 6;
    }
    /* compute left-over iterations */
    j += 6;
    for (; j > 0; j--) {
      p1 = ell[0];
      dd = dee[0];
      q1 = p1*dd;
      ell[0] = q1;
      m11 = p1*q1;
      Z11 += m11;
      ell++;
      dee++;
    }
    /* solve for diagonal 1 x 1 block at A(i,i) */
    Z11 = ell[0] - Z11;
    dee = d + i;
    /* factorize 1 x 1 block Z,dee */
    /* factorize row 1 */
    dee[0] = btRecip(Z11);
    /* done factorizing 1 x 1 block */
    break;
    
    //default: *((char*)0)=0;  /* this should never happen! */
  }
}
 
/* solve L*X=B, with B containing 1 right hand sides.
 * L is an n*n lower triangular matrix with ones on the diagonal.
 * L is stored by rows and its leading dimension is lskip.
 * B is an n*1 matrix that contains the right hand sides.
 * B is stored by columns and its leading dimension is also lskip.
 * B is overwritten with X.
 * this processes blocks of 4*4.
 * if this is in the factorizer source file, n must be a multiple of 4.
 */
 
void btSolveL1 (const btScalar *L, btScalar *B, int n, int lskip1)
{  
  /* declare variables - Z matrix, p and q vectors, etc */
  btScalar Z11,Z21,Z31,Z41,p1,q1,p2,p3,p4,*ex;
  const btScalar *ell;
  int lskip2,lskip3,i,j;
  /* compute lskip values */
  lskip2 = 2*lskip1;
  lskip3 = 3*lskip1;
  /* compute all 4 x 1 blocks of X */
  for (i=0; i <= n-4; i+=4) {
    /* compute all 4 x 1 block of X, from rows i..i+4-1 */
    /* set the Z matrix to 0 */
    Z11=0;
    Z21=0;
    Z31=0;
    Z41=0;
    ell = L + i*lskip1;
    ex = B;
    /* the inner loop that computes outer products and adds them to Z */
    for (j=i-12; j >= 0; j -= 12) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      p2=ell[lskip1];
      p3=ell[lskip2];
      p4=ell[lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[1];
      q1=ex[1];
      p2=ell[1+lskip1];
      p3=ell[1+lskip2];
      p4=ell[1+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[2];
      q1=ex[2];
      p2=ell[2+lskip1];
      p3=ell[2+lskip2];
      p4=ell[2+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[3];
      q1=ex[3];
      p2=ell[3+lskip1];
      p3=ell[3+lskip2];
      p4=ell[3+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[4];
      q1=ex[4];
      p2=ell[4+lskip1];
      p3=ell[4+lskip2];
      p4=ell[4+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[5];
      q1=ex[5];
      p2=ell[5+lskip1];
      p3=ell[5+lskip2];
      p4=ell[5+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[6];
      q1=ex[6];
      p2=ell[6+lskip1];
      p3=ell[6+lskip2];
      p4=ell[6+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[7];
      q1=ex[7];
      p2=ell[7+lskip1];
      p3=ell[7+lskip2];
      p4=ell[7+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[8];
      q1=ex[8];
      p2=ell[8+lskip1];
      p3=ell[8+lskip2];
      p4=ell[8+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[9];
      q1=ex[9];
      p2=ell[9+lskip1];
      p3=ell[9+lskip2];
      p4=ell[9+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[10];
      q1=ex[10];
      p2=ell[10+lskip1];
      p3=ell[10+lskip2];
      p4=ell[10+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* load p and q values */
      p1=ell[11];
      q1=ex[11];
      p2=ell[11+lskip1];
      p3=ell[11+lskip2];
      p4=ell[11+lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* advance pointers */
      ell += 12;
      ex += 12;
      /* end of inner loop */
    }
    /* compute left-over iterations */
    j += 12;
    for (; j > 0; j--) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      p2=ell[lskip1];
      p3=ell[lskip2];
      p4=ell[lskip3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      Z21 += p2 * q1;
      Z31 += p3 * q1;
      Z41 += p4 * q1;
      /* advance pointers */
      ell += 1;
      ex += 1;
    }
    /* finish computing the X(i) block */
    Z11 = ex[0] - Z11;
    ex[0] = Z11;
    p1 = ell[lskip1];
    Z21 = ex[1] - Z21 - p1*Z11;
    ex[1] = Z21;
    p1 = ell[lskip2];
    p2 = ell[1+lskip2];
    Z31 = ex[2] - Z31 - p1*Z11 - p2*Z21;
    ex[2] = Z31;
    p1 = ell[lskip3];
    p2 = ell[1+lskip3];
    p3 = ell[2+lskip3];
    Z41 = ex[3] - Z41 - p1*Z11 - p2*Z21 - p3*Z31;
    ex[3] = Z41;
    /* end of outer loop */
  }
  /* compute rows at end that are not a multiple of block size */
  for (; i < n; i++) {
    /* compute all 1 x 1 block of X, from rows i..i+1-1 */
    /* set the Z matrix to 0 */
    Z11=0;
    ell = L + i*lskip1;
    ex = B;
    /* the inner loop that computes outer products and adds them to Z */
    for (j=i-12; j >= 0; j -= 12) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[1];
      q1=ex[1];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[2];
      q1=ex[2];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[3];
      q1=ex[3];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[4];
      q1=ex[4];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[5];
      q1=ex[5];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[6];
      q1=ex[6];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[7];
      q1=ex[7];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[8];
      q1=ex[8];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[9];
      q1=ex[9];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[10];
      q1=ex[10];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* load p and q values */
      p1=ell[11];
      q1=ex[11];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* advance pointers */
      ell += 12;
      ex += 12;
      /* end of inner loop */
    }
    /* compute left-over iterations */
    j += 12;
    for (; j > 0; j--) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      /* compute outer product and add it to the Z matrix */
      Z11 += p1 * q1;
      /* advance pointers */
      ell += 1;
      ex += 1;
    }
    /* finish computing the X(i) block */
    Z11 = ex[0] - Z11;
    ex[0] = Z11;
  }
}
 
/* solve L^T * x=b, with b containing 1 right hand side.
 * L is an n*n lower triangular matrix with ones on the diagonal.
 * L is stored by rows and its leading dimension is lskip.
 * b is an n*1 matrix that contains the right hand side.
 * b is overwritten with x.
 * this processes blocks of 4.
 */
 
void btSolveL1T (const btScalar *L, btScalar *B, int n, int lskip1)
{  
  /* declare variables - Z matrix, p and q vectors, etc */
  btScalar Z11,m11,Z21,m21,Z31,m31,Z41,m41,p1,q1,p2,p3,p4,*ex;
  const btScalar *ell;
  int lskip2,i,j;
//  int lskip3;
  /* special handling for L and B because we're solving L1 *transpose* */
  L = L + (n-1)*(lskip1+1);
  B = B + n-1;
  lskip1 = -lskip1;
  /* compute lskip values */
  lskip2 = 2*lskip1;
  //lskip3 = 3*lskip1;
  /* compute all 4 x 1 blocks of X */
  for (i=0; i <= n-4; i+=4) {
    /* compute all 4 x 1 block of X, from rows i..i+4-1 */
    /* set the Z matrix to 0 */
    Z11=0;
    Z21=0;
    Z31=0;
    Z41=0;
    ell = L - i;
    ex = B;
    /* the inner loop that computes outer products and adds them to Z */
    for (j=i-4; j >= 0; j -= 4) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      p2=ell[-1];
      p3=ell[-2];
      p4=ell[-3];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      m21 = p2 * q1;
      m31 = p3 * q1;
      m41 = p4 * q1;
      ell += lskip1;
      Z11 += m11;
      Z21 += m21;
      Z31 += m31;
      Z41 += m41;
      /* load p and q values */
      p1=ell[0];
      q1=ex[-1];
      p2=ell[-1];
      p3=ell[-2];
      p4=ell[-3];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      m21 = p2 * q1;
      m31 = p3 * q1;
      m41 = p4 * q1;
      ell += lskip1;
      Z11 += m11;
      Z21 += m21;
      Z31 += m31;
      Z41 += m41;
      /* load p and q values */
      p1=ell[0];
      q1=ex[-2];
      p2=ell[-1];
      p3=ell[-2];
      p4=ell[-3];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      m21 = p2 * q1;
      m31 = p3 * q1;
      m41 = p4 * q1;
      ell += lskip1;
      Z11 += m11;
      Z21 += m21;
      Z31 += m31;
      Z41 += m41;
      /* load p and q values */
      p1=ell[0];
      q1=ex[-3];
      p2=ell[-1];
      p3=ell[-2];
      p4=ell[-3];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      m21 = p2 * q1;
      m31 = p3 * q1;
      m41 = p4 * q1;
      ell += lskip1;
      ex -= 4;
      Z11 += m11;
      Z21 += m21;
      Z31 += m31;
      Z41 += m41;
      /* end of inner loop */
    }
    /* compute left-over iterations */
    j += 4;
    for (; j > 0; j--) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      p2=ell[-1];
      p3=ell[-2];
      p4=ell[-3];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      m21 = p2 * q1;
      m31 = p3 * q1;
      m41 = p4 * q1;
      ell += lskip1;
      ex -= 1;
      Z11 += m11;
      Z21 += m21;
      Z31 += m31;
      Z41 += m41;
    }
    /* finish computing the X(i) block */
    Z11 = ex[0] - Z11;
    ex[0] = Z11;
    p1 = ell[-1];
    Z21 = ex[-1] - Z21 - p1*Z11;
    ex[-1] = Z21;
    p1 = ell[-2];
    p2 = ell[-2+lskip1];
    Z31 = ex[-2] - Z31 - p1*Z11 - p2*Z21;
    ex[-2] = Z31;
    p1 = ell[-3];
    p2 = ell[-3+lskip1];
    p3 = ell[-3+lskip2];
    Z41 = ex[-3] - Z41 - p1*Z11 - p2*Z21 - p3*Z31;
    ex[-3] = Z41;
    /* end of outer loop */
  }
  /* compute rows at end that are not a multiple of block size */
  for (; i < n; i++) {
    /* compute all 1 x 1 block of X, from rows i..i+1-1 */
    /* set the Z matrix to 0 */
    Z11=0;
    ell = L - i;
    ex = B;
    /* the inner loop that computes outer products and adds them to Z */
    for (j=i-4; j >= 0; j -= 4) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      ell += lskip1;
      Z11 += m11;
      /* load p and q values */
      p1=ell[0];
      q1=ex[-1];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      ell += lskip1;
      Z11 += m11;
      /* load p and q values */
      p1=ell[0];
      q1=ex[-2];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      ell += lskip1;
      Z11 += m11;
      /* load p and q values */
      p1=ell[0];
      q1=ex[-3];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      ell += lskip1;
      ex -= 4;
      Z11 += m11;
      /* end of inner loop */
    }
    /* compute left-over iterations */
    j += 4;
    for (; j > 0; j--) {
      /* load p and q values */
      p1=ell[0];
      q1=ex[0];
      /* compute outer product and add it to the Z matrix */
      m11 = p1 * q1;
      ell += lskip1;
      ex -= 1;
      Z11 += m11;
    }
    /* finish computing the X(i) block */
    Z11 = ex[0] - Z11;
    ex[0] = Z11;
  }
}
 
 
 
void btVectorScale (btScalar *a, const btScalar *d, int n)
{
  btAssert (a && d && n >= 0);
  for (int i=0; i<n; i++) {
    a[i] *= d[i];
  }
}
 
void btSolveLDLT (const btScalar *L, const btScalar *d, btScalar *b, int n, int nskip)
{
  btAssert (L && d && b && n > 0 && nskip >= n);
  btSolveL1 (L,b,n,nskip);
  btVectorScale (b,d,n);
  btSolveL1T (L,b,n,nskip);
}
 
 
 
//***************************************************************************
 
// swap row/column i1 with i2 in the n*n matrix A. the leading dimension of
// A is nskip. this only references and swaps the lower triangle.
// if `do_fast_row_swaps' is nonzero and row pointers are being used, then
// rows will be swapped by exchanging row pointers. otherwise the data will
// be copied.
 
static void btSwapRowsAndCols (BTATYPE A, int n, int i1, int i2, int nskip, 
  int do_fast_row_swaps)
{
  btAssert (A && n > 0 && i1 >= 0 && i2 >= 0 && i1 < n && i2 < n &&
    nskip >= n && i1 < i2);
 
# ifdef BTROWPTRS
  btScalar *A_i1 = A[i1];
  btScalar *A_i2 = A[i2];
  for (int i=i1+1; i<i2; ++i) {
    btScalar *A_i_i1 = A[i] + i1;
    A_i1[i] = *A_i_i1;
    *A_i_i1 = A_i2[i];
  }
  A_i1[i2] = A_i1[i1];
  A_i1[i1] = A_i2[i1];
  A_i2[i1] = A_i2[i2];
  // swap rows, by swapping row pointers
  if (do_fast_row_swaps) {
    A[i1] = A_i2;
    A[i2] = A_i1;
  }
  else {
    // Only swap till i2 column to match A plain storage variant.
    for (int k = 0; k <= i2; ++k) {
      btScalar tmp = A_i1[k];
      A_i1[k] = A_i2[k];
      A_i2[k] = tmp;
    }
  }
  // swap columns the hard way
  for (int j=i2+1; j<n; ++j) {
    btScalar *A_j = A[j];
    btScalar tmp = A_j[i1];
    A_j[i1] = A_j[i2];
    A_j[i2] = tmp;
  }
# else
  btScalar *A_i1 = A+i1*nskip;
  btScalar *A_i2 = A+i2*nskip;
  for (int k = 0; k < i1; ++k) {
    btScalar tmp = A_i1[k];
    A_i1[k] = A_i2[k];
    A_i2[k] = tmp;
  }
  btScalar *A_i = A_i1 + nskip;
  for (int i=i1+1; i<i2; A_i+=nskip, ++i) {
    btScalar tmp = A_i2[i];
    A_i2[i] = A_i[i1];
    A_i[i1] = tmp;
  }
  {
    btScalar tmp = A_i1[i1];
    A_i1[i1] = A_i2[i2];
    A_i2[i2] = tmp;
  }
  btScalar *A_j = A_i2 + nskip;
  for (int j=i2+1; j<n; A_j+=nskip, ++j) {
    btScalar tmp = A_j[i1];
    A_j[i1] = A_j[i2];
    A_j[i2] = tmp;
  }
# endif
}
 
 
// swap two indexes in the n*n LCP problem. i1 must be <= i2.
 
static void btSwapProblem (BTATYPE A, btScalar *x, btScalar *b, btScalar *w, btScalar *lo,
                         btScalar *hi, int *p, bool *state, int *findex,
                         int n, int i1, int i2, int nskip,
                         int do_fast_row_swaps)
{
  btScalar tmpr;
  int tmpi;
  bool tmpb;
  btAssert (n>0 && i1 >=0 && i2 >= 0 && i1 < n && i2 < n && nskip >= n && i1 <= i2);
  if (i1==i2) return;
  
  btSwapRowsAndCols (A,n,i1,i2,nskip,do_fast_row_swaps);
  
  tmpr = x[i1];
  x[i1] = x[i2];
  x[i2] = tmpr;
  
  tmpr = b[i1];
  b[i1] = b[i2];
  b[i2] = tmpr;
  
  tmpr = w[i1];
  w[i1] = w[i2];
  w[i2] = tmpr;
  
  tmpr = lo[i1];
  lo[i1] = lo[i2];
  lo[i2] = tmpr;
 
  tmpr = hi[i1];
  hi[i1] = hi[i2];
  hi[i2] = tmpr;
 
  tmpi = p[i1];
  p[i1] = p[i2];
  p[i2] = tmpi;
 
  tmpb = state[i1];
  state[i1] = state[i2];
  state[i2] = tmpb;
 
  if (findex) {
    tmpi = findex[i1];
    findex[i1] = findex[i2];
    findex[i2] = tmpi;
  }
}
 
 
 
 
//***************************************************************************
// btLCP manipulator object. this represents an n*n LCP problem.
//
// two index sets C and N are kept. each set holds a subset of
// the variable indexes 0..n-1. an index can only be in one set.
// initially both sets are empty.
//
// the index set C is special: solutions to A(C,C)\A(C,i) can be generated.
 
//***************************************************************************
// fast implementation of btLCP. see the above definition of btLCP for
// interface comments.
//
// `p' records the permutation of A,x,b,w,etc. p is initially 1:n and is
// permuted as the other vectors/matrices are permuted.
//
// A,x,b,w,lo,hi,state,findex,p,c are permuted such that sets C,N have
// contiguous indexes. the don't-care indexes follow N.
//
// an L*D*L' factorization is maintained of A(C,C), and whenever indexes are
// added or removed from the set C the factorization is updated.
// thus L*D*L'=A[C,C], i.e. a permuted top left nC*nC submatrix of A.
// the leading dimension of the matrix L is always `nskip'.
//
// at the start there may be other indexes that are unbounded but are not
// included in `nub'. btLCP will permute the matrix so that absolutely all
// unbounded vectors are at the start. thus there may be some initial
// permutation.
//
// the algorithms here assume certain patterns, particularly with respect to
// index transfer.
 
#ifdef btLCP_FAST
 
struct btLCP 
{
        const int m_n;
        const int m_nskip;
        int m_nub;
        int m_nC, m_nN;                         // size of each index set
        BTATYPE const m_A;                              // A rows
        btScalar *const m_x, * const m_b, *const m_w, *const m_lo,* const m_hi; // permuted LCP problem data
        btScalar *const m_L, *const m_d;                                // L*D*L' factorization of set C
        btScalar *const m_Dell, *const m_ell, *const m_tmp;
        bool *const m_state;
        int *const m_findex, *const m_p, *const m_C;
 
        btLCP (int _n, int _nskip, int _nub, btScalar *_Adata, btScalar *_x, btScalar *_b, btScalar *_w,
                btScalar *_lo, btScalar *_hi, btScalar *l, btScalar *_d,
                btScalar *_Dell, btScalar *_ell, btScalar *_tmp,
                bool *_state, int *_findex, int *p, int *c, btScalar **Arows);
        int getNub() const { return m_nub; }
        void transfer_i_to_C (int i);
        void transfer_i_to_N (int i) { m_nN++; }                        // because we can assume C and N span 1:i-1
        void transfer_i_from_N_to_C (int i);
        void transfer_i_from_C_to_N (int i, btAlignedObjectArray<btScalar>& scratch);
        int numC() const { return m_nC; }
        int numN() const { return m_nN; }
        int indexC (int i) const { return i; }
        int indexN (int i) const { return i+m_nC; }
        btScalar Aii (int i) const  { return BTAROW(i)[i]; }
        btScalar AiC_times_qC (int i, btScalar *q) const { return btLargeDot (BTAROW(i), q, m_nC); }
        btScalar AiN_times_qN (int i, btScalar *q) const { return btLargeDot (BTAROW(i)+m_nC, q+m_nC, m_nN); }
        void pN_equals_ANC_times_qC (btScalar *p, btScalar *q);
        void pN_plusequals_ANi (btScalar *p, int i, int sign=1);
        void pC_plusequals_s_times_qC (btScalar *p, btScalar s, btScalar *q);
        void pN_plusequals_s_times_qN (btScalar *p, btScalar s, btScalar *q);
        void solve1 (btScalar *a, int i, int dir=1, int only_transfer=0);
        void unpermute();
};
 
 
btLCP::btLCP (int _n, int _nskip, int _nub, btScalar *_Adata, btScalar *_x, btScalar *_b, btScalar *_w,
            btScalar *_lo, btScalar *_hi, btScalar *l, btScalar *_d,
            btScalar *_Dell, btScalar *_ell, btScalar *_tmp,
            bool *_state, int *_findex, int *p, int *c, btScalar **Arows):
  m_n(_n), m_nskip(_nskip), m_nub(_nub), m_nC(0), m_nN(0),
# ifdef BTROWPTRS
  m_A(Arows),
#else
  m_A(_Adata),
#endif
  m_x(_x), m_b(_b), m_w(_w), m_lo(_lo), m_hi(_hi),
  m_L(l), m_d(_d), m_Dell(_Dell), m_ell(_ell), m_tmp(_tmp),
  m_state(_state), m_findex(_findex), m_p(p), m_C(c)
{
  {
    btSetZero (m_x,m_n);
  }
 
  {
# ifdef BTROWPTRS
    // make matrix row pointers
    btScalar *aptr = _Adata;
    BTATYPE A = m_A;
    const int n = m_n, nskip = m_nskip;
    for (int k=0; k<n; aptr+=nskip, ++k) A[k] = aptr;
# endif
  }
 
  {
    int *p = m_p;
    const int n = m_n;
    for (int k=0; k<n; ++k) p[k]=k;             // initially unpermuted
  }
 
  /*
  // for testing, we can do some random swaps in the area i > nub
  {
    const int n = m_n;
    const int nub = m_nub;
    if (nub < n) {
    for (int k=0; k<100; k++) {
      int i1,i2;
      do {
        i1 = dRandInt(n-nub)+nub;
        i2 = dRandInt(n-nub)+nub;
      }
      while (i1 > i2); 
      //printf ("--> %d %d\n",i1,i2);
      btSwapProblem (m_A,m_x,m_b,m_w,m_lo,m_hi,m_p,m_state,m_findex,n,i1,i2,m_nskip,0);
    }
  }
  */
 
  // permute the problem so that *all* the unbounded variables are at the
  // start, i.e. look for unbounded variables not included in `nub'. we can
  // potentially push up `nub' this way and get a bigger initial factorization.
  // note that when we swap rows/cols here we must not just swap row pointers,
  // as the initial factorization relies on the data being all in one chunk.
  // variables that have findex >= 0 are *not* considered to be unbounded even
  // if lo=-inf and hi=inf - this is because these limits may change during the
  // solution process.
 
  {
    int *findex = m_findex;
    btScalar *lo = m_lo, *hi = m_hi;
    const int n = m_n;
    for (int k = m_nub; k<n; ++k) {
      if (findex && findex[k] >= 0) continue;
      if (lo[k]==-BT_INFINITY && hi[k]==BT_INFINITY) {
        btSwapProblem (m_A,m_x,m_b,m_w,lo,hi,m_p,m_state,findex,n,m_nub,k,m_nskip,0);
        m_nub++;
      }
    }
  }
 
  // if there are unbounded variables at the start, factorize A up to that
  // point and solve for x. this puts all indexes 0..nub-1 into C.
  if (m_nub > 0) {
    const int nub = m_nub;
    {
      btScalar *Lrow = m_L;
      const int nskip = m_nskip;
      for (int j=0; j<nub; Lrow+=nskip, ++j) memcpy(Lrow,BTAROW(j),(j+1)*sizeof(btScalar));
    }
    btFactorLDLT (m_L,m_d,nub,m_nskip);
    memcpy (m_x,m_b,nub*sizeof(btScalar));
    btSolveLDLT (m_L,m_d,m_x,nub,m_nskip);
    btSetZero (m_w,nub);
    {
      int *C = m_C;
      for (int k=0; k<nub; ++k) C[k] = k;
    }
    m_nC = nub;
  }
 
  // permute the indexes > nub such that all findex variables are at the end
  if (m_findex) {
    const int nub = m_nub;
    int *findex = m_findex;
    int num_at_end = 0;
    for (int k=m_n-1; k >= nub; k--) {
      if (findex[k] >= 0) {
        btSwapProblem (m_A,m_x,m_b,m_w,m_lo,m_hi,m_p,m_state,findex,m_n,k,m_n-1-num_at_end,m_nskip,1);
        num_at_end++;
      }
    }
  }
 
  // print info about indexes
  /*
  {
    const int n = m_n;
    const int nub = m_nub;
    for (int k=0; k<n; k++) {
      if (k<nub) printf ("C");
      else if (m_lo[k]==-BT_INFINITY && m_hi[k]==BT_INFINITY) printf ("c");
      else printf (".");
    }
    printf ("\n");
  }
  */
}
 
 
void btLCP::transfer_i_to_C (int i)
{
  {
    if (m_nC > 0) {
      // ell,Dell were computed by solve1(). note, ell = D \ L1solve (L,A(i,C))
      {
        const int nC = m_nC;
        btScalar *const Ltgt = m_L + nC*m_nskip, *ell = m_ell;
        for (int j=0; j<nC; ++j) Ltgt[j] = ell[j];
      }
      const int nC = m_nC;
      m_d[nC] = btRecip (BTAROW(i)[i] - btLargeDot(m_ell,m_Dell,nC));
    }
    else {
      m_d[0] = btRecip (BTAROW(i)[i]);
    }
 
    btSwapProblem (m_A,m_x,m_b,m_w,m_lo,m_hi,m_p,m_state,m_findex,m_n,m_nC,i,m_nskip,1);
 
    const int nC = m_nC;
    m_C[nC] = nC;
    m_nC = nC + 1; // nC value is outdated after this line
  }
 
}
 
 
void btLCP::transfer_i_from_N_to_C (int i)
{
  {
    if (m_nC > 0) {
      {
        btScalar *const aptr = BTAROW(i);
        btScalar *Dell = m_Dell;
        const int *C = m_C;
#   ifdef BTNUB_OPTIMIZATIONS
        // if nub>0, initial part of aptr unpermuted
        const int nub = m_nub;
        int j=0;
        for ( ; j<nub; ++j) Dell[j] = aptr[j];
        const int nC = m_nC;
        for ( ; j<nC; ++j) Dell[j] = aptr[C[j]];
#   else
        const int nC = m_nC;
        for (int j=0; j<nC; ++j) Dell[j] = aptr[C[j]];
#   endif
      }
      btSolveL1 (m_L,m_Dell,m_nC,m_nskip);
      {
        const int nC = m_nC;
        btScalar *const Ltgt = m_L + nC*m_nskip;
        btScalar *ell = m_ell, *Dell = m_Dell, *d = m_d;
        for (int j=0; j<nC; ++j) Ltgt[j] = ell[j] = Dell[j] * d[j];
      }
      const int nC = m_nC;
      m_d[nC] = btRecip (BTAROW(i)[i] - btLargeDot(m_ell,m_Dell,nC));
    }
    else {
      m_d[0] = btRecip (BTAROW(i)[i]);
    }
 
    btSwapProblem (m_A,m_x,m_b,m_w,m_lo,m_hi,m_p,m_state,m_findex,m_n,m_nC,i,m_nskip,1);
 
    const int nC = m_nC;
    m_C[nC] = nC;
    m_nN--;
    m_nC = nC + 1; // nC value is outdated after this line
  }
 
  // @@@ TO DO LATER
  // if we just finish here then we'll go back and re-solve for
  // delta_x. but actually we can be more efficient and incrementally
  // update delta_x here. but if we do this, we wont have ell and Dell
  // to use in updating the factorization later.
 
}
 
void btRemoveRowCol (btScalar *A, int n, int nskip, int r)
{
  btAssert(A && n > 0 && nskip >= n && r >= 0 && r < n);
  if (r >= n-1) return;
  if (r > 0) {
    {
      const size_t move_size = (n-r-1)*sizeof(btScalar);
      btScalar *Adst = A + r;
      for (int i=0; i<r; Adst+=nskip,++i) {
        btScalar *Asrc = Adst + 1;
        memmove (Adst,Asrc,move_size);
      }
    }
    {
      const size_t cpy_size = r*sizeof(btScalar);
      btScalar *Adst = A + r * nskip;
      for (int i=r; i<(n-1); ++i) {
        btScalar *Asrc = Adst + nskip;
        memcpy (Adst,Asrc,cpy_size);
        Adst = Asrc;
      }
    }
  }
  {
    const size_t cpy_size = (n-r-1)*sizeof(btScalar);
    btScalar *Adst = A + r * (nskip + 1);
    for (int i=r; i<(n-1); ++i) {
      btScalar *Asrc = Adst + (nskip + 1);
      memcpy (Adst,Asrc,cpy_size);
      Adst = Asrc - 1;
    }
  }
}
 
 
 
 
void btLDLTAddTL (btScalar *L, btScalar *d, const btScalar *a, int n, int nskip, btAlignedObjectArray<btScalar>& scratch)
{
  btAssert (L && d && a && n > 0 && nskip >= n);
 
  if (n < 2) return;
  scratch.resize(2*nskip);
  btScalar *W1 = &scratch[0];
  
  btScalar *W2 = W1 + nskip;
 
  W1[0] = btScalar(0.0);
  W2[0] = btScalar(0.0);
  for (int j=1; j<n; ++j) {
    W1[j] = W2[j] = (btScalar) (a[j] * SIMDSQRT12);
  }
  btScalar W11 = (btScalar) ((btScalar(0.5)*a[0]+1)*SIMDSQRT12);
  btScalar W21 = (btScalar) ((btScalar(0.5)*a[0]-1)*SIMDSQRT12);
 
  btScalar alpha1 = btScalar(1.0);
  btScalar alpha2 = btScalar(1.0);
 
  {
    btScalar dee = d[0];
    btScalar alphanew = alpha1 + (W11*W11)*dee;
    btAssert(alphanew != btScalar(0.0));
    dee /= alphanew;
    btScalar gamma1 = W11 * dee;
    dee *= alpha1;
    alpha1 = alphanew;
    alphanew = alpha2 - (W21*W21)*dee;
    dee /= alphanew;
    //btScalar gamma2 = W21 * dee;
    alpha2 = alphanew;
    btScalar k1 = btScalar(1.0) - W21*gamma1;
    btScalar k2 = W21*gamma1*W11 - W21;
    btScalar *ll = L + nskip;
    for (int p=1; p<n; ll+=nskip, ++p) {
      btScalar Wp = W1[p];
      btScalar ell = *ll;
      W1[p] =    Wp - W11*ell;
      W2[p] = k1*Wp +  k2*ell;
    }
  }
 
  btScalar *ll = L + (nskip + 1);
  for (int j=1; j<n; ll+=nskip+1, ++j) {
    btScalar k1 = W1[j];
    btScalar k2 = W2[j];
 
    btScalar dee = d[j];
    btScalar alphanew = alpha1 + (k1*k1)*dee;
    btAssert(alphanew != btScalar(0.0));
    dee /= alphanew;
    btScalar gamma1 = k1 * dee;
    dee *= alpha1;
    alpha1 = alphanew;
    alphanew = alpha2 - (k2*k2)*dee;
    dee /= alphanew;
    btScalar gamma2 = k2 * dee;
    dee *= alpha2;
    d[j] = dee;
    alpha2 = alphanew;
 
    btScalar *l = ll + nskip;
    for (int p=j+1; p<n; l+=nskip, ++p) {
      btScalar ell = *l;
      btScalar Wp = W1[p] - k1 * ell;
      ell += gamma1 * Wp;
      W1[p] = Wp;
      Wp = W2[p] - k2 * ell;
      ell -= gamma2 * Wp;
      W2[p] = Wp;
      *l = ell;
    }
  }
}
 
 
#define _BTGETA(i,j) (A[i][j])
//#define _GETA(i,j) (A[(i)*nskip+(j)])
#define BTGETA(i,j) ((i > j) ? _BTGETA(i,j) : _BTGETA(j,i))
 
inline size_t btEstimateLDLTAddTLTmpbufSize(int nskip)
{
  return nskip * 2 * sizeof(btScalar);
}
 
 
void btLDLTRemove (btScalar **A, const int *p, btScalar *L, btScalar *d,
    int n1, int n2, int r, int nskip, btAlignedObjectArray<btScalar>& scratch)
{
  btAssert(A && p && L && d && n1 > 0 && n2 > 0 && r >= 0 && r < n2 &&
           n1 >= n2 && nskip >= n1);
  #ifdef BT_DEBUG
        for (int i=0; i<n2; ++i) 
                btAssert(p[i] >= 0 && p[i] < n1);
  #endif
 
  if (r==n2-1) {
    return;             // deleting last row/col is easy
  }
  else {
    size_t LDLTAddTL_size = btEstimateLDLTAddTLTmpbufSize(nskip);
    btAssert(LDLTAddTL_size % sizeof(btScalar) == 0);
        scratch.resize(nskip * 2+n2);
    btScalar *tmp = &scratch[0];
    if (r==0) {
      btScalar *a = (btScalar *)((char *)tmp + LDLTAddTL_size);
      const int p_0 = p[0];
      for (int i=0; i<n2; ++i) {
        a[i] = -BTGETA(p[i],p_0);
      }
      a[0] += btScalar(1.0);
      btLDLTAddTL (L,d,a,n2,nskip,scratch);
    }
    else {
      btScalar *t = (btScalar *)((char *)tmp + LDLTAddTL_size);
      {
        btScalar *Lcurr = L + r*nskip;
        for (int i=0; i<r; ++Lcurr, ++i) {
          btAssert(d[i] != btScalar(0.0));
          t[i] = *Lcurr / d[i];
        }
      }
      btScalar *a = t + r;
      {
        btScalar *Lcurr = L + r*nskip;
        const int *pp_r = p + r, p_r = *pp_r;
        const int n2_minus_r = n2-r;
        for (int i=0; i<n2_minus_r; Lcurr+=nskip,++i) {
          a[i] = btLargeDot(Lcurr,t,r) - BTGETA(pp_r[i],p_r);
        }
      }
      a[0] += btScalar(1.0);
      btLDLTAddTL (L + r*nskip+r, d+r, a, n2-r, nskip, scratch);
    }
  }
 
  // snip out row/column r from L and d
  btRemoveRowCol (L,n2,nskip,r);
  if (r < (n2-1)) memmove (d+r,d+r+1,(n2-r-1)*sizeof(btScalar));
}
 
 
void btLCP::transfer_i_from_C_to_N (int i, btAlignedObjectArray<btScalar>& scratch)
{
  {
    int *C = m_C;
    // remove a row/column from the factorization, and adjust the
    // indexes (black magic!)
    int last_idx = -1;
    const int nC = m_nC;
    int j = 0;
    for ( ; j<nC; ++j) {
      if (C[j]==nC-1) {
        last_idx = j;
      }
      if (C[j]==i) {
        btLDLTRemove (m_A,C,m_L,m_d,m_n,nC,j,m_nskip,scratch);
        int k;
        if (last_idx == -1) {
          for (k=j+1 ; k<nC; ++k) {
            if (C[k]==nC-1) {
              break;
            }
          }
          btAssert (k < nC);
        }
        else {
          k = last_idx;
        }
        C[k] = C[j];
        if (j < (nC-1)) memmove (C+j,C+j+1,(nC-j-1)*sizeof(int));
        break;
      }
    }
    btAssert (j < nC);
 
    btSwapProblem (m_A,m_x,m_b,m_w,m_lo,m_hi,m_p,m_state,m_findex,m_n,i,nC-1,m_nskip,1);
 
    m_nN++;
    m_nC = nC - 1; // nC value is outdated after this line
  }
 
}
 
 
void btLCP::pN_equals_ANC_times_qC (btScalar *p, btScalar *q)
{
  // we could try to make this matrix-vector multiplication faster using
  // outer product matrix tricks, e.g. with the dMultidotX() functions.
  // but i tried it and it actually made things slower on random 100x100
  // problems because of the overhead involved. so we'll stick with the
  // simple method for now.
  const int nC = m_nC;
  btScalar *ptgt = p + nC;
  const int nN = m_nN;
  for (int i=0; i<nN; ++i) {
    ptgt[i] = btLargeDot (BTAROW(i+nC),q,nC);
  }
}
 
 
void btLCP::pN_plusequals_ANi (btScalar *p, int i, int sign)
{
  const int nC = m_nC;
  btScalar *aptr = BTAROW(i) + nC;
  btScalar *ptgt = p + nC;
  if (sign > 0) {
    const int nN = m_nN;
    for (int j=0; j<nN; ++j) ptgt[j] += aptr[j];
  }
  else {
    const int nN = m_nN;
    for (int j=0; j<nN; ++j) ptgt[j] -= aptr[j];
  }
}
 
void btLCP::pC_plusequals_s_times_qC (btScalar *p, btScalar s, btScalar *q)
{
  const int nC = m_nC;
  for (int i=0; i<nC; ++i) {
    p[i] += s*q[i];
  }
}
 
void btLCP::pN_plusequals_s_times_qN (btScalar *p, btScalar s, btScalar *q)
{
  const int nC = m_nC;
  btScalar *ptgt = p + nC, *qsrc = q + nC;
  const int nN = m_nN;
  for (int i=0; i<nN; ++i) {
    ptgt[i] += s*qsrc[i];
  }
}
 
void btLCP::solve1 (btScalar *a, int i, int dir, int only_transfer)
{
  // the `Dell' and `ell' that are computed here are saved. if index i is
  // later added to the factorization then they can be reused.
  //
  // @@@ question: do we need to solve for entire delta_x??? yes, but
  //     only if an x goes below 0 during the step.
 
  if (m_nC > 0) {
    {
      btScalar *Dell = m_Dell;
      int *C = m_C;
      btScalar *aptr = BTAROW(i);
#   ifdef BTNUB_OPTIMIZATIONS
      // if nub>0, initial part of aptr[] is guaranteed unpermuted
      const int nub = m_nub;
      int j=0;
      for ( ; j<nub; ++j) Dell[j] = aptr[j];
      const int nC = m_nC;
      for ( ; j<nC; ++j) Dell[j] = aptr[C[j]];
#   else
      const int nC = m_nC;
      for (int j=0; j<nC; ++j) Dell[j] = aptr[C[j]];
#   endif
    }
    btSolveL1 (m_L,m_Dell,m_nC,m_nskip);
    {
      btScalar *ell = m_ell, *Dell = m_Dell, *d = m_d;
      const int nC = m_nC;
      for (int j=0; j<nC; ++j) ell[j] = Dell[j] * d[j];
    }
 
    if (!only_transfer) {
      btScalar *tmp = m_tmp, *ell = m_ell;
      {
        const int nC = m_nC;
        for (int j=0; j<nC; ++j) tmp[j] = ell[j];
      }
      btSolveL1T (m_L,tmp,m_nC,m_nskip);
      if (dir > 0) {
        int *C = m_C;
        btScalar *tmp = m_tmp;
        const int nC = m_nC;
        for (int j=0; j<nC; ++j) a[C[j]] = -tmp[j];
      } else {
        int *C = m_C;
        btScalar *tmp = m_tmp;
        const int nC = m_nC;
        for (int j=0; j<nC; ++j) a[C[j]] = tmp[j];
      }
    }
  }
}
 
 
void btLCP::unpermute()
{
  // now we have to un-permute x and w
  {
    memcpy (m_tmp,m_x,m_n*sizeof(btScalar));
    btScalar *x = m_x, *tmp = m_tmp;
    const int *p = m_p;
    const int n = m_n;
    for (int j=0; j<n; ++j) x[p[j]] = tmp[j];
  }
  {
    memcpy (m_tmp,m_w,m_n*sizeof(btScalar));
    btScalar *w = m_w, *tmp = m_tmp;
    const int *p = m_p;
    const int n = m_n;
    for (int j=0; j<n; ++j) w[p[j]] = tmp[j];
  }
}
 
#endif // btLCP_FAST
 
 
//***************************************************************************
// an optimized Dantzig LCP driver routine for the lo-hi LCP problem.
 
bool btSolveDantzigLCP (int n, btScalar *A, btScalar *x, btScalar *b,
                btScalar* outer_w, int nub, btScalar *lo, btScalar *hi, int *findex, btDantzigScratchMemory& scratchMem)
{
        s_error = false;
 
//      printf("btSolveDantzigLCP n=%d\n",n);
  btAssert (n>0 && A && x && b && lo && hi && nub >= 0 && nub <= n);
  btAssert(outer_w);
 
#ifdef BT_DEBUG
  {
    // check restrictions on lo and hi
    for (int k=0; k<n; ++k) 
                btAssert (lo[k] <= 0 && hi[k] >= 0);
  }
# endif
 
 
  // if all the variables are unbounded then we can just factor, solve,
  // and return
  if (nub >= n) 
  {
   
 
    int nskip = (n);
    btFactorLDLT (A, outer_w, n, nskip);
    btSolveLDLT (A, outer_w, b, n, nskip);
    memcpy (x, b, n*sizeof(btScalar));
 
    return !s_error;
  }
 
  const int nskip = (n);
  scratchMem.L.resize(n*nskip);
 
  scratchMem.d.resize(n);
 
  btScalar *w = outer_w;
  scratchMem.delta_w.resize(n);
  scratchMem.delta_x.resize(n);
  scratchMem.Dell.resize(n);
  scratchMem.ell.resize(n);
  scratchMem.Arows.resize(n);
  scratchMem.p.resize(n);
  scratchMem.C.resize(n);
 
  // for i in N, state[i] is 0 if x(i)==lo(i) or 1 if x(i)==hi(i)
  scratchMem.state.resize(n);
 
 
  // create LCP object. note that tmp is set to delta_w to save space, this
  // optimization relies on knowledge of how tmp is used, so be careful!
  btLCP lcp(n,nskip,nub,A,x,b,w,lo,hi,&scratchMem.L[0],&scratchMem.d[0],&scratchMem.Dell[0],&scratchMem.ell[0],&scratchMem.delta_w[0],&scratchMem.state[0],findex,&scratchMem.p[0],&scratchMem.C[0],&scratchMem.Arows[0]);
  int adj_nub = lcp.getNub();
 
  // loop over all indexes adj_nub..n-1. for index i, if x(i),w(i) satisfy the
  // LCP conditions then i is added to the appropriate index set. otherwise
  // x(i),w(i) is driven either +ve or -ve to force it to the valid region.
  // as we drive x(i), x(C) is also adjusted to keep w(C) at zero.
  // while driving x(i) we maintain the LCP conditions on the other variables
  // 0..i-1. we do this by watching out for other x(i),w(i) values going
  // outside the valid region, and then switching them between index sets
  // when that happens.
 
  bool hit_first_friction_index = false;
  for (int i=adj_nub; i<n; ++i) 
  {
    s_error = false;
    // the index i is the driving index and indexes i+1..n-1 are "dont care",
    // i.e. when we make changes to the system those x's will be zero and we
    // don't care what happens to those w's. in other words, we only consider
    // an (i+1)*(i+1) sub-problem of A*x=b+w.
 
    // if we've hit the first friction index, we have to compute the lo and
    // hi values based on the values of x already computed. we have been
    // permuting the indexes, so the values stored in the findex vector are
    // no longer valid. thus we have to temporarily unpermute the x vector. 
    // for the purposes of this computation, 0*infinity = 0 ... so if the
    // contact constraint's normal force is 0, there should be no tangential
    // force applied.
 
    if (!hit_first_friction_index && findex && findex[i] >= 0) {
      // un-permute x into delta_w, which is not being used at the moment
      for (int j=0; j<n; ++j) scratchMem.delta_w[scratchMem.p[j]] = x[j];
 
      // set lo and hi values
      for (int k=i; k<n; ++k) {
        btScalar wfk = scratchMem.delta_w[findex[k]];
        if (wfk == 0) {
          hi[k] = 0;
          lo[k] = 0;
        }
        else {
          hi[k] = btFabs (hi[k] * wfk);
          lo[k] = -hi[k];
        }
      }
      hit_first_friction_index = true;
    }
 
    // thus far we have not even been computing the w values for indexes
    // greater than i, so compute w[i] now.
    w[i] = lcp.AiC_times_qC (i,x) + lcp.AiN_times_qN (i,x) - b[i];
 
    // if lo=hi=0 (which can happen for tangential friction when normals are
    // 0) then the index will be assigned to set N with some state. however,
    // set C's line has zero size, so the index will always remain in set N.
    // with the "normal" switching logic, if w changed sign then the index
    // would have to switch to set C and then back to set N with an inverted
    // state. this is pointless, and also computationally expensive. to
    // prevent this from happening, we use the rule that indexes with lo=hi=0
    // will never be checked for set changes. this means that the state for
    // these indexes may be incorrect, but that doesn't matter.
 
    // see if x(i),w(i) is in a valid region
    if (lo[i]==0 && w[i] >= 0) {
      lcp.transfer_i_to_N (i);
      scratchMem.state[i] = false;
    }
    else if (hi[i]==0 && w[i] <= 0) {
      lcp.transfer_i_to_N (i);
      scratchMem.state[i] = true;
    }
    else if (w[i]==0) {
      // this is a degenerate case. by the time we get to this test we know
      // that lo != 0, which means that lo < 0 as lo is not allowed to be +ve,
      // and similarly that hi > 0. this means that the line segment
      // corresponding to set C is at least finite in extent, and we are on it.
      // NOTE: we must call lcp.solve1() before lcp.transfer_i_to_C()
      lcp.solve1 (&scratchMem.delta_x[0],i,0,1);
 
      lcp.transfer_i_to_C (i);
    }
    else {
      // we must push x(i) and w(i)
      for (;;) {
        int dir;
        btScalar dirf;
        // find direction to push on x(i)
        if (w[i] <= 0) {
          dir = 1;
          dirf = btScalar(1.0);
        }
        else {
          dir = -1;
          dirf = btScalar(-1.0);
        }
 
        // compute: delta_x(C) = -dir*A(C,C)\A(C,i)
        lcp.solve1 (&scratchMem.delta_x[0],i,dir);
 
        // note that delta_x[i] = dirf, but we wont bother to set it
 
        // compute: delta_w = A*delta_x ... note we only care about
        // delta_w(N) and delta_w(i), the rest is ignored
        lcp.pN_equals_ANC_times_qC (&scratchMem.delta_w[0],&scratchMem.delta_x[0]);
        lcp.pN_plusequals_ANi (&scratchMem.delta_w[0],i,dir);
        scratchMem.delta_w[i] = lcp.AiC_times_qC (i,&scratchMem.delta_x[0]) + lcp.Aii(i)*dirf;
 
        // find largest step we can take (size=s), either to drive x(i),w(i)
        // to the valid LCP region or to drive an already-valid variable
        // outside the valid region.
 
        int cmd = 1;            // index switching command
        int si = 0;             // si = index to switch if cmd>3
        btScalar s = -w[i]/scratchMem.delta_w[i];
        if (dir > 0) {
          if (hi[i] < BT_INFINITY) {
            btScalar s2 = (hi[i]-x[i])*dirf;    // was (hi[i]-x[i])/dirf        // step to x(i)=hi(i)
            if (s2 < s) {
              s = s2;
              cmd = 3;
            }
          }
        }
        else {
          if (lo[i] > -BT_INFINITY) {
            btScalar s2 = (lo[i]-x[i])*dirf;    // was (lo[i]-x[i])/dirf        // step to x(i)=lo(i)
            if (s2 < s) {
              s = s2;
              cmd = 2;
            }
          }
        }
 
        {
          const int numN = lcp.numN();
          for (int k=0; k < numN; ++k) {
            const int indexN_k = lcp.indexN(k);
            if (!scratchMem.state[indexN_k] ? scratchMem.delta_w[indexN_k] < 0 : scratchMem.delta_w[indexN_k] > 0) {
                // don't bother checking if lo=hi=0
                if (lo[indexN_k] == 0 && hi[indexN_k] == 0) continue;
                btScalar s2 = -w[indexN_k] / scratchMem.delta_w[indexN_k];
                if (s2 < s) {
                  s = s2;
                  cmd = 4;
                  si = indexN_k;
                }
            }
          }
        }
 
        {
          const int numC = lcp.numC();
          for (int k=adj_nub; k < numC; ++k) {
            const int indexC_k = lcp.indexC(k);
            if (scratchMem.delta_x[indexC_k] < 0 && lo[indexC_k] > -BT_INFINITY) {
              btScalar s2 = (lo[indexC_k]-x[indexC_k]) / scratchMem.delta_x[indexC_k];
              if (s2 < s) {
                s = s2;
                cmd = 5;
                si = indexC_k;
              }
            }
            if (scratchMem.delta_x[indexC_k] > 0 && hi[indexC_k] < BT_INFINITY) {
              btScalar s2 = (hi[indexC_k]-x[indexC_k]) / scratchMem.delta_x[indexC_k];
              if (s2 < s) {
                s = s2;
                cmd = 6;
                si = indexC_k;
              }
            }
          }
        }
 
        //static char* cmdstring[8] = {0,"->C","->NL","->NH","N->C",
        //                           "C->NL","C->NH"};
        //printf ("cmd=%d (%s), si=%d\n",cmd,cmdstring[cmd],(cmd>3) ? si : i);
 
        // if s <= 0 then we've got a problem. if we just keep going then
        // we're going to get stuck in an infinite loop. instead, just cross
        // our fingers and exit with the current solution.
        if (s <= btScalar(0.0)) 
                {
//          printf("LCP internal error, s <= 0 (s=%.4e)",(double)s);
          if (i < n) {
            btSetZero (x+i,n-i);
            btSetZero (w+i,n-i);
          }
          s_error = true;
          break;
        }
 
        // apply x = x + s * delta_x
        lcp.pC_plusequals_s_times_qC (x, s, &scratchMem.delta_x[0]);
        x[i] += s * dirf;
 
        // apply w = w + s * delta_w
        lcp.pN_plusequals_s_times_qN (w, s, &scratchMem.delta_w[0]);
        w[i] += s * scratchMem.delta_w[i];
 
//        void *tmpbuf;
        // switch indexes between sets if necessary
        switch (cmd) {
        case 1:         // done
          w[i] = 0;
          lcp.transfer_i_to_C (i);
          break;
        case 2:         // done
          x[i] = lo[i];
          scratchMem.state[i] = false;
          lcp.transfer_i_to_N (i);
          break;
        case 3:         // done
          x[i] = hi[i];
          scratchMem.state[i] = true;
          lcp.transfer_i_to_N (i);
          break;
        case 4:         // keep going
          w[si] = 0;
          lcp.transfer_i_from_N_to_C (si);
          break;
        case 5:         // keep going
          x[si] = lo[si];
          scratchMem.state[si] = false;
                  lcp.transfer_i_from_C_to_N (si, scratchMem.m_scratch);
          break;
        case 6:         // keep going
          x[si] = hi[si];
          scratchMem.state[si] = true;
          lcp.transfer_i_from_C_to_N (si, scratchMem.m_scratch);
          break;
        }
 
        if (cmd <= 3) break;
      } // for (;;)
    } // else
 
    if (s_error) 
        {
      break;
    }
  } // for (int i=adj_nub; i<n; ++i)
 
  lcp.unpermute();
 
 
  return !s_error;
}