/**************************************************************************\
 *
 *  This file is part of the Coin 3D visualization library.
 *  Copyright (C) 1998-2006 by Systems in Motion.  All rights reserved.
 *
 *  This library is free software; you can redistribute it and/or
 *  modify it under the terms of the GNU General Public License
 *  ("GPL") version 2 as published by the Free Software Foundation.
 *  See the file LICENSE.GPL at the root directory of this source
 *  distribution for additional information about the GNU GPL.
 *
 *  For using Coin with software that can not be combined with the GNU
 *  GPL, and for taking advantage of the additional benefits of our
 *  support services, please contact Systems in Motion about acquiring
 *  a Coin Professional Edition License.
 *
 *  See http://www.coin3d.org/ for more information.
 *
 *  Systems in Motion, Postboks 1283, Pirsenteret, 7462 Trondheim, NORWAY.
 *  http://www.sim.no/  sales@sim.no  coin-support@coin3d.org
 *
\**************************************************************************/

/*!
  \class SbMatrix SbLinear.h Inventor/SbLinear.h
  \brief The SbMatrix class is a 4x4 dimensional representation of a matrix.
  \ingroup base

  SbMatrix is used by many other classes in Coin.  It provides storage
  for a 4x4 matrix of single-precision floating point values.

  By definition, matrices in Coin should be set up in column-order
  mode. This is the same order as used by e.g. OpenGL, but note that
  books on geometry often uses the opposite row-order mode, which can
  confuse new-comers to the API.

  Another way to think of column-order matrices is that they use
  post-order multiplications: that is, to concatenate a transformation
  from a second matrix with your current matrix, it should be
  multiplied on the right-hand side, i.e. with the
  SbMatrix::multRight() method.

  If you have a matrix in row-order from some other source, it can be
  "converted" to column-order by transposing it with
  SbMatrix::transpose(). A simple example will help to explain
  this.

  With row-order matrices, a transformation matrix with position,
  rotation and scale looks like this:

  \code
  M = T * R * S
  \endcode

  Where T is translation, R is rotation and S is the scale. What this
  means is that scale is applied first. The scaled matrix is then
  rotated, and finally the scaled and rotated matrix is
  translated. When using column-order matrices, as done in Coin,
  matrices are represented slightly differently; the order of
  multiplication is reversed:

  \code
  M = S * R * T
  \endcode

  The transformation is just the same as the row-order matrix. The
  only difference being the order of multiplication. To understand why
  this is so, consider the sample transformation:

  \code
  M = T * R * S
  \endcode

  Converting M from a row-order matrix to a column-order matrix is
  done as follows:

  \code
  M^t = (T * R * S)^t
  M^t = ((T * R) * S)^t
  M^t = S^t * (T * R)^t
  M^t = S^t * R^t * T^t
  \endcode

  All left to be done is to remove the transpose symbols, and the
  matrices have been converted to column-order matrices:

  \code
  M = S * R * T
  \endcode

  This was done using the fact that:

  \code
  A^t = (B * C)^t = C^t * B^t
  \endcode

  Converting from column-order to row-order is done using the same
  principle.
*/

// FIXME:
//
//  * The SbMatrix::factor() function has not been implemented yet.
//
//  * The element access methods should be inlined.
//
//  * Not a lot of optimizations have been done yet, so there's a lot
//    of room for improvements.


#include <Inventor/SbMatrix.h>
#include <Inventor/SbRotation.h>
#include <Inventor/SbLine.h>
#include <coindefs.h> // COIN_STUB()
#include <assert.h>
#include <string.h>
#include <float.h>
#if COIN_DEBUG
#include <Inventor/errors/SoDebugError.h>
#endif // COIN_DEBUG

class SbMatrixP {
public:
  // FIXME: should merge all the PD code we're using from GGIV into
  // SbMatrix, SbRotation and SbVec3f proper (for two reasons: 1)
  // there's a lot of duplicated code here (like for instance the
  // matrix->quaternion decomposition, which also exists in
  // SbRotation::setValue(SbMatrix&)), and 2) the remaining code
  // snippets look generally useful outside the purpose of breaking down
  // a matrix into it's transformation components). 20010114 mortene.

  /*
   * declarations for polar_decomp algorithm from Graphics Gems IV,
   * by Ken Shoemake <shoemake@graphics.cis.upenn.edu>
   */
  enum QuatPart {X, Y, Z, W};
  typedef float HMatrix[4][4]; /* Right-handed, for column vectors */
  typedef struct {
    SbVec4f t;    /* Translation components */
    SbRotation  q;        /* Essential rotation     */
    SbRotation  u;        /* Stretch rotation       */
    SbVec4f k;    /* Stretch factors        */
    float f;      /* Sign of determinant    */
  } AffineParts;
  static void decomp_affine(HMatrix A, AffineParts * parts);
  static SbRotation snuggle(SbRotation q, SbVec4f & k);
  static SbVec4f spect_decomp(SbMatrixP::HMatrix S, SbMatrixP::HMatrix U);
  static float polar_decomp(SbMatrixP::HMatrix M, SbMatrixP::HMatrix Q, SbMatrixP::HMatrix S);

  static const SbMat IDENTITYMATRIX;

  static SbBool isIdentity(const float fm[][4]) {
#if 0 // I would assume that the memcmp() version is faster..? Should run some profile checks.
    return ((fm[0][0] == 1.0f) && (fm[0][1] == 0.0f) && (fm[0][2] == 0.0f) && (fm[0][3] == 0.0f) &&
            (fm[1][0] == 0.0f) && (fm[1][1] == 1.0f) && (fm[1][2] == 0.0f) && (fm[1][3] == 0.0f) &&
            (fm[2][0] == 0.0f) && (fm[2][1] == 0.0f) && (fm[2][2] == 1.0f) && (fm[2][3] == 0.0f) &&
            (fm[3][0] == 0.0f) && (fm[3][1] == 0.0f) && (fm[3][2] == 0.0f) && (fm[3][3] == 1.0f));
#else
    // Note: as far as I know, memcmp() only compares bytes until
    // there's a mismatch (and does *not* run over the full array and
    // adds up a total, as it sometimes seems from documentation). So
    // this should be very quick for non-identity matrices.
    //
    // Also, we check the first value on it's own, to avoid the function
    // call for the most common case.
    return (fm[0][0]==1.0f) && memcmp(&fm[0][1], &IDENTITYMATRIX[0][1], (4 * 3 + 3) * sizeof(float)) == 0;
#endif
  }

  static HMatrix mat_id;

private:
  static void mat_mult(SbMatrixP::HMatrix A, SbMatrixP::HMatrix B, SbMatrixP::HMatrix AB);
  static float vdot(float * va, float * vb);
  static void vcross(float * va, float * vb, float * v);
  static void adjoint_transpose(SbMatrixP::HMatrix M, SbMatrixP::HMatrix MadjT);
  static float mat_norm(SbMatrixP::HMatrix M, int tpose);
  static float norm_inf(SbMatrixP::HMatrix M) {return mat_norm(M, 0);}
  static float norm_one(SbMatrixP::HMatrix M) {return mat_norm(M, 1);}
  static int find_max_col(SbMatrixP::HMatrix M);
  static void make_reflector(float * v, float * u);
  static void reflect_cols(SbMatrixP::HMatrix M, float * u);
  static void reflect_rows(SbMatrixP::HMatrix M, float * u);
  static void do_rank1(SbMatrixP::HMatrix M, SbMatrixP::HMatrix Q);
  static void do_rank2(SbMatrixP::HMatrix M, SbMatrixP::HMatrix MadjT, SbMatrixP::HMatrix Q);
};

const SbMat SbMatrixP::IDENTITYMATRIX = {
  { 1.0f, 0.0f, 0.0f, 0.0f },
  { 0.0f, 1.0f, 0.0f, 0.0f },
  { 0.0f, 0.0f, 1.0f, 0.0f },
  { 0.0f, 0.0f, 0.0f, 1.0f }
};

SbMatrixP::HMatrix SbMatrixP::mat_id = {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}};

/*!
  The default constructor does nothing. The matrix will be uninitialized.
 */
SbMatrix::SbMatrix(void)
{
}


/*!
  Constructs a matrix instance with the given initial elements.
 */
SbMatrix::SbMatrix(const float a11, const float a12,
                   const float a13, const float a14,
                   const float a21, const float a22,
                   const float a23, const float a24,
                   const float a31, const float a32,
                   const float a33, const float a34,
                   const float a41, const float a42,
                   const float a43, const float a44)
{
  const SbMat m = { { a11, a12, a13, a14 },
                    { a21, a22, a23, a24 },
                    { a31, a32, a33, a34 },
                    { a41, a42, a43, a44 } };
  this->setValue(m);
}

/*!
  Constructs a matrix instance with the initial elements from the
  \a matrix argument.
 */
SbMatrix::SbMatrix(const SbMat & matrixref)
{
  this->setValue(matrixref);
}

/*!
  This constructor is courtesy of the Microsoft Visual C++ compiler.
*/
SbMatrix::SbMatrix(const SbMat * matrixptr)
{
  this->setValue(*matrixptr);
}

/*!
  Default destructor does nothing.
 */
SbMatrix::~SbMatrix(void)
{
}

/*!
  Returns a pointer to the 2 dimensional float array with the matrix
  elements.

  \sa setValue().
 */
const SbMat &
SbMatrix::getValue(void) const
{
  return this->matrix;
}

/*!
  Copies the elements from \a m into the matrix.

  \sa getValue().
 */
void
SbMatrix::setValue(const SbMat & m)
{
  (void)memmove(this->matrix, m, sizeof(float)*4*4);
}

/*!
  Assignment operator. Copies the elements from \a m to the matrix.
 */
SbMatrix &
SbMatrix::operator=(const SbMat & m)
{
  this->setValue(m);
  return *this;
}

/*!
  Assignment operator. Copies the elements from \a m to the matrix.
 */
SbMatrix &
SbMatrix::operator=(const SbMatrix & m)
{
  this->setValue(m.matrix);
  return *this;
}

/*!
  Set the matrix to be the identity matrix.

  \sa identity().
 */
void
SbMatrix::makeIdentity(void)
{
  this->matrix[0][0]=this->matrix[1][1]=
    this->matrix[2][2]=this->matrix[3][3] = 1.0f;

  this->matrix[0][1]=this->matrix[0][2]=this->matrix[0][3]=
    this->matrix[1][0]=this->matrix[1][2]=this->matrix[1][3]=
    this->matrix[2][0]=this->matrix[2][1]=this->matrix[2][3]=
    this->matrix[3][0]=this->matrix[3][1]=this->matrix[3][2] = 0.0f;
}

/*!
  Set matrix to be a rotation matrix with the given rotation.

  \sa setTranslate(), setScale().
*/
void
SbMatrix::setRotate(const SbRotation & q)
{
  q.getValue(*this);
}

/*!
  Multiply all element values in the matrix with \a v.
 */
void
SbMatrix::operator*=(const float v)
{
  for (int i=0; i < 4; i++) {
    for (int j=0; j < 4; j++) {
      this->matrix[i][j] *= v;
    }
  }
}

/*!
  Divide all element values in the matrix on \a v.
 */
void
SbMatrix::operator/=(const float v)
{
#if COIN_DEBUG
  if (v==0.0f)
    SoDebugError::postWarning("SbMatrix::operator/=",
                              "Division by zero.");
#endif // COIN_DEBUG

  this->operator*=(1.0f/v);
}

/*!
  Returns the determinant of the 3x3 submatrix specified by the row and
  column indices.
 */
float
SbMatrix::det3(int r1, int r2, int r3,
               int c1, int c2, int c3) const
{
#ifdef COIN_EXTRA_DEBUG
  // Check indices.
  if (r1<0 || r1>3 || r2<0 || r2>3 || r3<0 || r3>3 ||
      c1<0 || c1>3 || c2<0 || c2>3 || c3<0 || c3>3) {
    SoDebugError::post("SbMatrix::det3",
                       "At least one idx out of bounds [0, 3]. ");
  }
  if (r1==r2 || r1==r3 || r2==r3 ||
      c1==c2 || c1==c3 || c2==c3)
    SoDebugError::post("SbMatrix::det3", "Indices should be distinct.");
#endif // COIN_EXTRA_DEBUG

  // More or less directly from "Advanced Engineering Mathematics"
  // (E. Kreyszig), 6th edition.

  float a11 = this->matrix[r1][c1];
  float a12 = this->matrix[r1][c2];
  float a13 = this->matrix[r1][c3];
  float a21 = this->matrix[r2][c1];
  float a22 = this->matrix[r2][c2];
  float a23 = this->matrix[r2][c3];
  float a31 = this->matrix[r3][c1];
  float a32 = this->matrix[r3][c2];
  float a33 = this->matrix[r3][c3];

  float M11 = a22 * a33 - a32 * a23;
  float M21 = -(a12 * a33 - a32 * a13);
  float M31 = a12 * a23 - a22 * a13;

  return (a11 * M11 + a21 * M21 + a31 * M31);
}

/*!
  Returns the determinant of the upper left 3x3 submatrix.
 */
float
SbMatrix::det3(void) const
{
  return this->det3(0, 1, 2, 0, 1, 2);
}

/*!
  Returns the determinant of the matrix.
 */
float
SbMatrix::det4(void) const
{
  float det = 0.0f;
  det += this->matrix[0][0] * det3(1, 2, 3, 1, 2, 3);
  det -= this->matrix[1][0] * det3(0, 2, 3, 1, 2, 3);
  det += this->matrix[2][0] * det3(0, 1, 3, 1, 2, 3);
  det -= this->matrix[3][0] * det3(0, 1, 2, 1, 2, 3);
  return det;
}

/*!
  Return a new matrix which is the inverse matrix of this.

  The user is responsible for checking that this is a valid operation
  to execute, by first making sure that the result of SbMatrix::det4()
  is not equal to zero.
 */
SbMatrix
SbMatrix::inverse(void) const
{
  // check for identity matrix
  if (SbMatrixP::isIdentity(this->matrix)) { return SbMatrix::identity(); }

  SbMatrix result;

  // use local pointers for speed
  float (*dst)[4];
  dst = (float (*)[4]) result.matrix[0];
  float (*src)[4];
  src = (float (*)[4]) this->matrix[0];

  // check for affine matrix (common case)
  if (src[0][3] == 0.0f && src[1][3] == 0.0f &&
      src[2][3] == 0.0f && src[3][3] == 1.0f) {
    
    // More or less directly from:
    // Kevin Wu, "Fast Matrix Inversion",  Graphics Gems II
    float det_1;
    float pos, neg, temp;

#define ACCUMULATE     \
    if (temp >= 0.0f)  \
      pos += temp;     \
    else               \
      neg += temp

    /*
     * Calculate the determinant of submatrix A and determine if the
     * the matrix is singular as limited by floating-point data 
     * representation.
     */
    pos = neg = 0.0f;
    temp =  src[0][0] * src[1][1] * src[2][2];
    ACCUMULATE;
    temp =  src[0][1] * src[1][2] * src[2][0];
    ACCUMULATE;
    temp =  src[0][2] * src[1][0] * src[2][1];
    ACCUMULATE;
    temp = -src[0][2] * src[1][1] * src[2][0];
    ACCUMULATE;
    temp = -src[0][1] * src[1][0] * src[2][2];
    ACCUMULATE;
    temp = -src[0][0] * src[1][2] * src[2][1];
    ACCUMULATE;
    det_1 = pos + neg;

#undef ACCUMULATE

    /* Is the submatrix A singular? */
    if ((det_1 == 0.0f) || (SbAbs(det_1 / (pos - neg)) < FLT_EPSILON)) {
      /* Matrix M has no inverse */
#if COIN_DEBUG
      SoDebugError::postWarning("SbMatrix::inverse",
                                "Matrix is singular.");
#endif // COIN_DEBUG
      return *this;
    }
    else {
      /* Calculate inverse(A) = adj(A) / det(A) */
      det_1 = 1.0f / det_1;
      dst[0][0] = (src[1][1] * src[2][2] -
                   src[1][2] * src[2][1]) * det_1;
      dst[1][0] = - (src[1][0] * src[2][2] -
                     src[1][2] * src[2][0]) * det_1;
      dst[2][0] = (src[1][0] * src[2][1] -
                   src[1][1] * src[2][0]) * det_1;
      dst[0][1] = - (src[0][1] * src[2][2] -
                     src[0][2] * src[2][1]) * det_1;
      dst[1][1] = (src[0][0] * src[2][2] -
                   src[0][2] * src[2][0]) * det_1;
      dst[2][1] = - (src[0][0] * src[2][1] -
                     src[0][1] * src[2][0]) * det_1;
      dst[0][2] =  (src[0][1] * src[1][2] -
                    src[0][2] * src[1][1]) * det_1;
      dst[1][2] = - (src[0][0] * src[1][2] -
                     src[0][2] * src[1][0]) * det_1;
      dst[2][2] =  (src[0][0] * src[1][1] -
                    src[0][1] * src[1][0]) * det_1;
      
      /* Calculate -C * inverse(A) */
      dst[3][0] = - (src[3][0] * dst[0][0] +
                     src[3][1] * dst[1][0] +
                     src[3][2] * dst[2][0]);
      dst[3][1] = - (src[3][0] * dst[0][1] +
                     src[3][1] * dst[1][1] +
                     src[3][2] * dst[2][1]);
      dst[3][2] = - (src[3][0] * dst[0][2] +
                     src[3][1] * dst[1][2] +
                     src[3][2] * dst[2][2]);
      
      /* Fill in last column */
      dst[0][3] = dst[1][3] = dst[2][3] = 0.0f;
      dst[3][3] = 1.0f;
    }
  }
  else { // non-affine matrix
    float max, sum, tmp, inv_pivot;
    int p[4];
    int i, j, k;

    // algorithm from: Schwarz, "Numerische Mathematik"
    result = *this;
    
    for (k = 0; k < 4; k++) { 
      max = 0.0f;
      p[k] = 0;
      
      for (i = k; i < 4; i++) { 
        sum = 0.0f;
        for (j = k; j < 4; j++)
          sum += SbAbs(dst[i][j]);
        if (sum > 0.0f) {
          tmp = SbAbs(dst[i][k]) / sum;
          if (tmp > max) { 
            max = tmp;  
            p[k] = i;
          }
        }
      }
      
      if (max == 0.0f) {
#if COIN_DEBUG
        SoDebugError::postWarning("SbMatrix::inverse",
                                  "Matrix is singular.");
#endif // COIN_DEBUG
        return *this;
      }
      
      if (p[k] != k) {
        for (j = 0; j < 4; j++) {
          tmp = dst[k][j];  
          dst[k][j] = dst[p[k]][j];  
          dst[p[k]][j] = tmp;
        }
      }
      
      inv_pivot = 1.0f / dst[k][k];
      for (j = 0; j < 4; j++) {
        if (j != k) {
          dst[k][j] = - dst[k][j] * inv_pivot;
          for (i = 0; i < 4; i++) {
            if (i != k) dst[i][j] += dst[i][k] * dst[k][j];
          }
        }
      }
      
      for (i = 0; i < 4; i++) dst[i][k] *= inv_pivot;
      dst[k][k] = inv_pivot;
    }
    
    for (k = 2; k >= 0; k--) {
      if (p[k] != k) {
        for (i = 0; i < 4; i++) {
          tmp = dst[i][k];
          dst[i][k] = dst[i][p[k]]; 
          dst[i][p[k]] = tmp;
        }
      }
    }
  }
  return result;
}

/*!
  Check if the \a m matrix is equal to this one, within the given tolerance
  value. The tolerance value is applied in the comparison on a component by
  component basis.
 */
SbBool
SbMatrix::equals(const SbMatrix & m, float tolerance) const
{
#if COIN_DEBUG
  if (tolerance<0.0f)
    SoDebugError::postWarning("SbMatrix::equals",
                              "tolerance should be >=0.0f.");
#endif // COIN_DEBUG

  for (int i=0; i < 4; i++) {
    for (int j=0;  j< 4; j++) {
      if (fabs(this->matrix[i][j] - m.matrix[i][j]) > tolerance) return FALSE;
    }
  }

  return TRUE;
}


/*!
  Return pointer to the matrix' 4x4 float array.
 */
SbMatrix::operator float*(void)
{
  return &(this->matrix[0][0]);
}


/*!
  Return pointer to the matrix' 4x4 float array.
 */
SbMatrix::operator SbMat&(void)
{
  return this->matrix;
}

/*!
  Returns pointer to the 4 element array representing a matrix row.
  \a i should be within [0, 3].

  \sa getValue(), setValue().
 */
float *
SbMatrix::operator [](int i)
{
#ifdef COIN_EXTRA_DEBUG
  if (i<0 || i>3) {
    SoDebugError::post("SbMatrix::operator[]", "Index out of bounds. ");
  }
#endif // COIN_EXTRA_DEBUG

   return this->matrix[i];
}

/*!
  Returns pointer to the 4 element array representing a matrix row.
  \a i should be within [0, 3].

  \sa getValue(), setValue().
 */
const float *
SbMatrix::operator [](int i) const
{
#ifdef COIN_EXTRA_DEBUG
  if (i<0 || i>3) {
    SoDebugError::postWarning("SbMatrix::operator[]", "Index out of bounds. ");
  }
#endif // COIN_EXTRA_DEBUG

   return this->matrix[i];
}

/*!
  Set matrix to be a rotation matrix with the given rotation.

  \sa setRotate().
*/
SbMatrix&
SbMatrix::operator =(const SbRotation & q)
{
  this->setRotate(q);
  return *this;
}

/*!
  Right-multiply with the \a m matrix.

  \sa multRight().
 */
SbMatrix&
SbMatrix::operator *=(const SbMatrix & m)
{
  return this->multRight(m);
}

/*!
  \relates SbMatrix

  Multiplies matrix \a m1 with matrix \a m2 and returns the resultant
  matrix.
*/
SbMatrix
operator *(const SbMatrix & m1, const SbMatrix & m2)
{
  SbMatrix result = m1;
  result *= m2;
  return result;
}

/*!
  \relates SbMatrix

  Compare matrices to see if they are equal. For two matrices to be equal,
  all their individual elements must be equal.

  \sa equals().
*/
int
operator ==(const SbMatrix & m1, const SbMatrix & m2)
{
  for (int i=0; i < 4; i++) {
    for (int j=0; j < 4; j++) {
      if (m1.matrix[i][j] != m2.matrix[i][j]) return FALSE;
    }
  }

  return TRUE;
}

/*!
  \relates SbMatrix

  Compare matrices to see if they are not equal. For two matrices to not be
  equal, it is enough that at least one of their elements are unequal.

  \sa equals().
*/
int
operator !=(const SbMatrix & m1, const SbMatrix & m2)
{
  return !(m1 == m2);
}

/*!
  Return matrix components in the SbMat structure.

  \sa setValue().
 */
void
SbMatrix::getValue(SbMat & m) const
{
  (void)memmove(&m[0][0], &(this->matrix[0][0]), sizeof(float)*4*4);
}

/*!
  Return the identity matrix.

  \sa makeIdentity().
 */
SbMatrix
SbMatrix::identity(void)
{
  return SbMatrix(&SbMatrixP::IDENTITYMATRIX);
}

/*!
  Set matrix to be a pure scaling matrix. Scale factors are specified
  by \a s.

  \sa setRotate(), setTranslate().
 */
void
SbMatrix::setScale(const float s)
{
  this->makeIdentity();
  this->matrix[0][0] = s;
  this->matrix[1][1] = s;
  this->matrix[2][2] = s;
}

/*!
  Set matrix to be a pure scaling matrix. Scale factors in x, y and z
  is specified by the \a s vector.

  \sa setRotate(), setTranslate().
 */
void
SbMatrix::setScale(const SbVec3f & s)
{
  this->makeIdentity();
  this->matrix[0][0] = s[0];
  this->matrix[1][1] = s[1];
  this->matrix[2][2] = s[2];
}

/*!
  Make this matrix into a pure translation matrix (no scale or rotation
  components) with the given vector \t as the translation.

  \sa setRotate(), setScale().
 */
void
SbMatrix::setTranslate(const SbVec3f & t)
{
  this->makeIdentity();
  this->matrix[3][0] = t[0];
  this->matrix[3][1] = t[1];
  this->matrix[3][2] = t[2];
}

/*!
  Set translation, rotation and scaling all at once. The resulting
  matrix gets calculated like this:

  \code
  M = S * R * T
  \endcode

  where \a S, \a R and \a T is scaling, rotation and translation
  matrices.

  \sa setTranslate(), setRotate(), setScale() and getTransform().
 */
void
SbMatrix::setTransform(const SbVec3f & t, const SbRotation & r, const SbVec3f & s)
{
  // Unoptimized version:
  //
  //  this->setScale(s);
  //  tmp.setRotate(r);
  //  this->multRight(tmp);
  //  tmp.setTranslate(t);
  //  this->multRight(tmp);

  // Optimized version
  SbMatrix tmp;
  const SbRotation identity = SbRotation::identity();

  if (s != SbVec3f(1.0f, 1.0f, 1.0f)) {
    this->setScale(s);
    if (r != identity) {
      tmp.setRotate(r);
      this->multRight(tmp);
    }
    if (t != SbVec3f(0.0f, 0.0f, 0.0f)) {
      tmp.setTranslate(t);
      this->multRight(tmp);
    }
  }
  else {
    if (r != identity) {
      this->setRotate(r);
      if (t != SbVec3f(0.f, 0.0f, 0.0f)) {
        tmp.setTranslate(t);
        this->multRight(tmp);
      }
    }
    else {
      this->setTranslate(t);
    }
  }
}

/*!
  Set translation, rotation and scaling all at once with a specified
  scale orientation. The resulting matrix gets calculated like this:

  \code
  M = Ro-¹ * S * Ro * R * T
  \endcode

  where \a Ro is the scale orientation, and \a S, \a R
  and \a T is scaling, rotation and translation.

  \sa setTranslate(), setRotate(), setScale() and getTransform().
 */
void
SbMatrix::setTransform(const SbVec3f & t, const SbRotation & r,
                       const SbVec3f & s, const SbRotation & so)
{

  //  Unoptimized version:
  //
  //  this->setRotate(so.inverse());
  //  tmp.setScale(s);
  //  this->multRight(tmp);
  //  tmp.setRotate(so);
  //  this->multRight(tmp);
  //  tmp.setRotate(r);
  //  this->multRight(tmp);
  //  tmp.setTranslate(t);
  //  this->multRight(tmp);

  // Optimized version. Much faster for the common case where only
  // zero or one of the parameters are set.
  SbMatrix tmp;
  const SbRotation identity = SbRotation::identity();

  if (s != SbVec3f(1.0f, 1.0f, 1.0f)) {
    if (so != identity) {
      this->setRotate(so.inverse());
      tmp.setScale(s);
      this->multRight(tmp);
      tmp.setRotate(so);
      this->multRight(tmp);
    }
    else {
      this->setScale(s);
    }
    if (r != identity) {
      tmp.setRotate(r);
      this->multRight(tmp);
    }
    if (t != SbVec3f(0.0f, 0.0f, 0.0f)) {
      tmp.setTranslate(t);
      this->multRight(tmp);
    }
  }
  else {
    if (r != identity) {
      this->setRotate(r);
      if (t != SbVec3f(0.0f, 0.0f, 0.0f)) {
        tmp.setTranslate(t);
        this->multRight(tmp);
      }
    }
    else {
      this->setTranslate(t);
    }
  }
}

/*!
  Set translation, rotation and scaling all at once with a specified
  scale orientation and center point. The resulting matrix gets
  calculated like this:

  \code
  M = -Tc * Ro-¹ * S * Ro * R * T * Tc
  \endcode

  where \a Tc is the center point, \a Ro the scale orientation, \a S,
  \a R and \a T is scaling, rotation and translation.

  \sa setTranslate(), setRotate(), setScale() and getTransform().
 */
void
SbMatrix::setTransform(const SbVec3f & translation,
                       const SbRotation & rotation,
                       const SbVec3f & scaleFactor,
                       const SbRotation & scaleOrientation,
                       const SbVec3f & center)
{
  //  Unoptimized version:
  //
  //  this->setTranslate(-center);
  //  tmp.setRotate(scaleOrientation.inverse());
  //  this->multRight(tmp);
  //  tmp.setScale(scaleFactor);
  //  this->multRight(tmp);
  //  tmp.setRotate(scaleOrientation);
  //  this->multRight(tmp);
  //  tmp.setRotate(rotation);
  //  this->multRight(tmp);
  //  tmp.setTranslate(translation);
  //  this->multRight(tmp);
  //  tmp.setTranslate(center);
  //  this->multRight(tmp);

  // Optimized version. Much faster for the common case where only
  // zero or one of the parameters are set.
  SbMatrix tmp;
  const SbRotation identity = SbRotation::identity();

  this->setTranslate(-center);

  if (scaleFactor != SbVec3f(1.0f, 1.0f, 1.0f)) {
    if (scaleOrientation != identity ) {
      tmp.setRotate(scaleOrientation.inverse());
      this->multRight(tmp);
    }
    tmp.setScale(scaleFactor);
    this->multRight(tmp);
    
    if (scaleOrientation != identity) {
      tmp.setRotate(scaleOrientation);
      this->multRight(tmp);
    }
  }

  if (rotation != identity) {
    tmp.setRotate(rotation);
    this->multRight(tmp);
  }

  const SbVec3f sum = center + translation;
  if (sum != SbVec3f(0.f, 0.0f, 0.0f)) {
    tmp.setTranslate(sum);
    this->multRight(tmp);
  }
}

/*!
  Factor the matrix back into its translation, rotation, scale and
  scaleorientation components.

  \sa factor()
 */
void
SbMatrix::getTransform(SbVec3f & t, SbRotation & r, SbVec3f & s,
                       SbRotation & so) const
{
  // FIXME: test if this code works with non-affine matrices.
  // pederb, 2000-01-17

  SbMatrixP::AffineParts parts;
  SbMatrixP::HMatrix hmatrix;

  // transpose-copy
  hmatrix[0][0] = this->matrix[0][0];
  hmatrix[0][1] = this->matrix[1][0];
  hmatrix[0][2] = this->matrix[2][0];
  hmatrix[0][3] = this->matrix[3][0];

  hmatrix[1][0] = this->matrix[0][1];
  hmatrix[1][1] = this->matrix[1][1];
  hmatrix[1][2] = this->matrix[2][1];
  hmatrix[1][3] = this->matrix[3][1];

  hmatrix[2][0] = this->matrix[0][2];
  hmatrix[2][1] = this->matrix[1][2];
  hmatrix[2][2] = this->matrix[2][2];
  hmatrix[2][3] = this->matrix[3][2];

  hmatrix[3][0] = this->matrix[0][3];
  hmatrix[3][1] = this->matrix[1][3];
  hmatrix[3][2] = this->matrix[2][3];
  hmatrix[3][3] = this->matrix[3][3];

  SbMatrixP::decomp_affine(hmatrix, &parts);

  float mul = 1.0f;
  if (parts.t[SbMatrixP::W] != 0.0f) mul = 1.0f / parts.t[SbMatrixP::W];
  t[0] = parts.t[SbMatrixP::X] * mul;
  t[1] = parts.t[SbMatrixP::Y] * mul;
  t[2] = parts.t[SbMatrixP::Z] * mul;

  r = parts.q;
  mul = 1.0f;
  if (parts.k[SbMatrixP::W] != 0.0f) mul = 1.0f / parts.k[SbMatrixP::W];
  // mul be sign of determinant to support negative scales.
  mul *= parts.f;
  s[0] = parts.k[SbMatrixP::X] * mul;
  s[1] = parts.k[SbMatrixP::Y] * mul;
  s[2] = parts.k[SbMatrixP::Z] * mul;

  so = parts.u;
}

/*!
  Factor the matrix back into its \a translation, \a rotation,
  \a scaleFactor and \a scaleorientation components. Will eliminate
  the \a center variable from the matrix.

  \sa factor()
 */
void
SbMatrix::getTransform(SbVec3f & translation,
                       SbRotation & rotation,
                       SbVec3f & scaleFactor,
                       SbRotation & scaleOrientation,
                       const SbVec3f & center) const
{
  SbMatrix m2 = *this;
  SbMatrix trans;
  trans.setTranslate(center);
  m2.multLeft(trans);
  trans.setTranslate(-center);
  m2.multRight(trans);

  m2.getTransform(translation, rotation, scaleFactor, scaleOrientation);
}

/*!
  This function is not implemented in Coin.

  \sa getTransform()
 */
SbBool
SbMatrix::factor(SbMatrix & r, SbVec3f & s, SbMatrix & u, SbVec3f & t,
                 SbMatrix & proj)
{
  // FIXME: not implemented, not documented. 1998MMDD mortene.
  COIN_STUB();
  return FALSE;
}

/*!
  This function produces a permuted LU decomposition of the matrix.  It
  uses the common single-row-pivoting strategy.

  \a FALSE is returned if the matrix is singular, which it never is, because
  of small adjustment values inserted if a singularity is found (as Open
  Inventor does too).

  The parity argument is always set to 1.0 or -1.0.  Don't really know what
  it's for, so it's not checked for correctness.

  The index[] argument returns the permutation that was done on the matrix
  to LU-decompose it.  index[i] is the row that row i was swapped with at
  step i in the decomposition, so index[] is not the actual permutation of
  the row indexes!

  BUGS:  The function does not produce results that are numerically identical
  with those produced by Open Inventor for the same matrices, because the
  pivoting strategy in OI was never fully understood.

  \sa SbMatrix::LUBackSubstitution
*/


SbBool
SbMatrix::LUDecomposition(int index[4], float & d)
{
    int i;
    for (i = 0; i < 4; i++) index[i] = i;
    d = 1.0f;

    const float MINIMUM_PIVOT = 1e-6f; // Inventor fix...

    for (int row = 1; row < 4; row++) {
        int swap_row = row;
        float max_pivot = 0.0f;
        for (int test_row = row; test_row < 4; test_row++) {
            const float test_pivot = SbAbs(matrix[test_row][row]);
            if (test_pivot > max_pivot) {
                swap_row = test_row;
                max_pivot = test_pivot;
            }
        }

        // swap rows
        if (swap_row != row) {
            d = -d;
            index[row] = swap_row;
            for (i = 0; i < 4; i++)
                SbSwap(matrix[row][i], matrix[swap_row][i]);
        }

        float pivot = matrix[row][row];
        if (matrix[row][row] == 0.0f) {
//            return FALSE;
            // instead of returning FALSE on singulars...
            matrix[row][row] = pivot = MINIMUM_PIVOT;
        }

        // the factorization
        for (i = row + 1; i < 4; i++) {
            const float factor = (matrix[i][row] /= pivot);
            for (int j = row + 1; j < 4; j++)
                matrix[i][j] -= factor * matrix[row][j];
        }
    }
    return TRUE;
}

/*!
  This function does a solve on the "Ax = b" system, given that the matrix
  is LU-decomposed in advance.  First, a forward substitution is done on the
  lower system (Ly = b), and then a backwards substitution is done on the
  upper triangular system (Ux = y).

  The index[] argument is the one returned from
  SbMatrix::LUDecomposition(), so see that function for an explanation.

  The b[] argument must contain the b vector in "Ax = b" when calling the
  function.  After the function has solved the system, the b[] vector contains
  the x vector.

  BUGS:  As is done by Open Inventor, unsolvable x values will not return
  NaN but 0.
*/

void
SbMatrix::LUBackSubstitution(int index[4], float b[4]) const
{
    int i;

    // permute b[] the way matrix[][] is permuted
    for (i = 0; i < 4; i++)
        if (i != index[i])
            SbSwap(b[i], b[index[i]]);

    // forward substitution on L (Ly = b)
    float y[4];
    for (i = 0; i < 4; i++) {
        float sum = 0.0f;
        for (int j = 0; j < i; j++)
            sum += matrix[i][j] * y[j];
        y[i] = b[i] - sum;
    }

    // backwards substitution on U (Ux = y)
    float x[4];
    for (i = 3; i >= 0; i--) {
        float sum = 0.0f;
        for (int j = i + 1; j < 4; j++)
             sum += matrix[i][j] * x[j];
        if (matrix[i][i] != 0.0f)
            x[i] = (y[i] - sum) / matrix[i][i];
        else
            x[i] = 0.0f;
    }

    // de-permute x[]?  doesn't look like it
//    for (i = 3; i >= 0; i--)
//        if (i != index[i])
//            SbSwap(x[i], x[index[i]]);

    // copy x[] into b[] for "return to sender"
    for (i = 0; i < 4; i++) b[i] = x[i];
}

/*!
  Returns the transpose of this matrix.
*/

SbMatrix
SbMatrix::transpose(void) const
{
  SbMatrix trans = (*this);

  for (int i=0; i < 3; i++) {
    for (int j=i+1; j < 4; j++) {
      SbSwap(trans[i][j], trans[j][i]);
    }
  }

  return trans;
}

/*!
  Let this matrix be right-multiplied by \a m. Returns reference to
  self.

  This is the most common multiplication / concatenation operation
  when using column-order matrices, as SbMatrix instances are, by
  definition.

  \sa multLeft()
*/
SbMatrix &
SbMatrix::multRight(const SbMatrix & m)
{
  // Checks if one or the other matrix is equal to the identity matrix
  // before multiplying them. We do this because it's a major
  // optimization if one of them _is_, and the isIdentity() check
  // should be very quick in the common case where a matrix is not the
  // identity matrix.
  const SbMat & mfm = m.matrix;
  if (SbMatrixP::isIdentity(mfm)) { return *this; }
  SbMat & tfm = this->matrix;
  if (SbMatrixP::isIdentity(tfm)) { *this = m; return *this; }

  SbMat tmp;
  (void)memcpy(tmp, tfm, 4*4*sizeof(float));

  for (int i=0; i < 4; i++) {
    for (int j=0; j < 4; j++) {
      tfm[i][j] =
        tmp[i][0] * mfm[0][j] +
        tmp[i][1] * mfm[1][j] +
        tmp[i][2] * mfm[2][j] +
        tmp[i][3] * mfm[3][j];
    }
  }

  return *this;
}

/*!
  Let this matrix be left-multiplied by \a m. Returns reference to
  self.

  (Be aware that it is more common to use the SbMatrix::multRight()
  operation, when doing concatenation of transformations, as SbMatrix
  instances are by definition in column-order, and uses
  post-multiplication for common geometry operations.)

  \sa multRight()
*/
SbMatrix&
SbMatrix::multLeft(const SbMatrix & m)
{
  // Checks if one or the other matrix is equal to the identity
  // matrix.  See also code comments at the start of
  // SbMatrix::multRight().
  const SbMat & mfm = m.matrix;
  if (SbMatrixP::isIdentity(mfm)) { return *this; }
  SbMat & tfm = this->matrix;
  if (SbMatrixP::isIdentity(tfm)) { *this = m; return *this; }

  SbMat tmp;
  (void)memcpy(tmp, tfm, 4*4*sizeof(float));

  for (int i=0; i < 4; i++) {
    for (int j=0; j < 4; j++) {
      tfm[i][j] =
        tmp[0][j] * mfm[i][0] +
        tmp[1][j] * mfm[i][1] +
        tmp[2][j] * mfm[i][2] +
        tmp[3][j] * mfm[i][3];
    }
  }
  return *this;
}

/*!
  Multiply \a src vector with this matrix and return the result in \a dst.
  Multiplication is done with the vector on the right side of the
  expression, i.e. dst = M * src.

  (Be aware that it is more common to use the
  SbMatrix::multVecMatrix() operation, when doing vector
  transformations, as SbMatrix instances are by definition in
  column-order, and uses post-multiplication for common geometry
  operations.)

  \sa multVecMatrix(), multDirMatrix() and multLineMatrix().
*/
void
SbMatrix::multMatrixVec(const SbVec3f & src, SbVec3f & dst) const
{
  // Checks if the "this" matrix is equal to the identity matrix.  See
  // also code comments at the start of SbMatrix::multRight().
  if (SbMatrixP::isIdentity(this->matrix)) { dst = src; return; }

  const float * t0 = (*this)[0];
  const float * t1 = (*this)[1];
  const float * t2 = (*this)[2];
  const float * t3 = (*this)[3];
  // Copy the src vector, just in case src and dst is the same vector.
  SbVec3f s = src;

  float W = s[0]*t3[0] + s[1]*t3[1] + s[2]*t3[2] + t3[3];

  dst[0] = (s[0]*t0[0] + s[1]*t0[1] + s[2]*t0[2] + t0[3])/W;
  dst[1] = (s[0]*t1[0] + s[1]*t1[1] + s[2]*t1[2] + t1[3])/W;
  dst[2] = (s[0]*t2[0] + s[1]*t2[1] + s[2]*t2[2] + t2[3])/W;
}

/*!
  Multiply \a src vector with this matrix and return the result in \a dst.
  Multiplication is done with the vector on the left side of the
  expression, i.e. dst = src * M.

  It is safe to let \a src and \a dst be the same SbVec3f instance.

  This method can be used (using the current model matrix) to
  transform a point from an object coordinate systems to the world
  coordinate system.

  This operation is what you would usually do when transforming
  vectors, as SbMatrix instances are, by definition, column-order
  matrices.

  \sa multMatrixVec(), multDirMatrix() and multLineMatrix().  
*/
void
SbMatrix::multVecMatrix(const SbVec3f & src, SbVec3f & dst) const
{
  // Checks if the "this" matrix is equal to the identity matrix.  See
  // also code comments at the start of SbMatrix::multRight().
  if (SbMatrixP::isIdentity(this->matrix)) { dst = src; return; }

  const float * t0 = this->matrix[0];
  const float * t1 = this->matrix[1];
  const float * t2 = this->matrix[2];
  const float * t3 = this->matrix[3];
  // Copy the src vector, just in case src and dst is the same vector.
  SbVec3f s = src;

  float W = s[0]*t0[3] + s[1]*t1[3] + s[2]*t2[3] + t3[3];

  dst[0] = (s[0]*t0[0] + s[1]*t1[0] + s[2]*t2[0] + t3[0])/W;
  dst[1] = (s[0]*t0[1] + s[1]*t1[1] + s[2]*t2[1] + t3[1])/W;
  dst[2] = (s[0]*t0[2] + s[1]*t1[2] + s[2]*t2[2] + t3[2])/W;
}

/*!
  \overload
*/
void
SbMatrix::multVecMatrix(const SbVec4f & src, SbVec4f & dst) const
{
  // Checks if the "this" matrix is equal to the identity matrix.  See
  // also code comments at the start of SbMatrix::multRight().
  if (SbMatrixP::isIdentity(this->matrix)) { dst = src; return; }

  const float * t0 = (*this)[0];
  const float * t1 = (*this)[1];
  const float * t2 = (*this)[2];
  const float * t3 = (*this)[3];

  SbVec4f s = src;

  dst[0] = (s[0]*t0[0] + s[1]*t1[0] + s[2]*t2[0] + t3[0]);
  dst[1] = (s[0]*t0[1] + s[1]*t1[1] + s[2]*t2[1] + t3[1]);
  dst[2] = (s[0]*t0[2] + s[1]*t1[2] + s[2]*t2[2] + t3[2]);
  dst[3] = (s[0]*t0[3] + s[1]*t1[3] + s[2]*t2[3] + t3[3]);
}

/*!
  Multiplies \a src by the matrix. \a src is assumed to be a direction
  vector, and the translation components of the matrix are therefore
  ignored.

  Multiplication is done with the vector on the left side of the
  expression, i.e. dst = src * M.

  \sa multVecMatrix(), multMatrixVec() and multLineMatrix().
 */
void
SbMatrix::multDirMatrix(const SbVec3f & src, SbVec3f & dst) const
{
  // Checks if the "this" matrix is equal to the identity matrix.  See
  // also code comments at the start of SbMatrix::multRight().
  if (SbMatrixP::isIdentity(this->matrix)) { dst = src; return; }

  const float * t0 = (*this)[0];
  const float * t1 = (*this)[1];
  const float * t2 = (*this)[2];
  // Copy the src vector, just in case src and dst is the same vector.
  SbVec3f s = src;

  dst[0] = s[0]*t0[0] + s[1]*t1[0] + s[2]*t2[0];
  dst[1] = s[0]*t0[1] + s[1]*t1[1] + s[2]*t2[1];
  dst[2] = s[0]*t0[2] + s[1]*t1[2] + s[2]*t2[2];
}

/*!
  Multiplies line point with the full matrix and multiplies the
  line direction with the matrix without the translation components.

  \sa multVecMatrix(), multMatrixVec() and multDirMatrix().
 */
void
SbMatrix::multLineMatrix(const SbLine & src, SbLine & dst) const
{
  SbVec3f newpt, newdir;
  this->multVecMatrix(src.getPosition(), newpt);
  this->multDirMatrix(src.getDirection(), newdir);

  dst.setValue(newpt, newpt + newdir);
}

/*!
  Write out the matrix contents to the given file.
 */
void
SbMatrix::print(FILE * fp) const
{
  for (int i=0; i < 4; i++) {
    fprintf(fp, "%10.5g\t%10.5g\t%10.5g\t%10.5g\n",
            this->matrix[i][0], this->matrix[i][1],
            this->matrix[i][2], this->matrix[i][3]);
  }
}

/***********************************************************************
   below is the polar_decomp implementation by Ken Shoemake
   <shoemake@graphics.cis.upenn.edu>. It was part of the
   Graphics Gems IV archive.
************************************************************************/

// FIXME: should merge all the PD code we're using from GGIV into
// SbMatrix, SbRotation and SbVec3f proper (for two reasons: 1)
// there's a lot of duplicated code here (like for instance the
// matrix->quaternion decomposition, which also exists in
// SbRotation::setValue(SbMatrix&)), and 2) the remaining code
// snippets look generally useful outside the purpose of breaking down
// a matrix into it's transformation components). 20010114 mortene.


/**** Decompose.c ****/
/* Ken Shoemake, 1993 */

/******* Matrix Preliminaries *******/

/** Fill out 3x3 matrix to 4x4 **/
#define sp_mat_pad(A) (A[W][X]=A[X][W]=A[W][Y]=A[Y][W]=A[W][Z]=A[Z][W]=0, A[W][W]=1)

/** Copy nxn matrix A to C using "gets" for assignment **/
#define sp_mat_copy(C, gets, A, n) {int i, j; for (i=0;i<n;i++) for (j=0;j<n;j++)\
    C[i][j] gets (A[i][j]);}

/** Copy transpose of nxn matrix A to C using "gets" for assignment **/
#define sp_mat_tpose(AT, gets, A, n) {int i, j; for (i=0;i<n;i++) for (j=0;j<n;j++)\
    AT[i][j] gets (A[j][i]);}

/** Assign nxn matrix C the element-wise combination of A and B using "op" **/
#define sp_mat_binop(C, gets, A, op, B, n) {int i, j; for (i=0;i<n;i++) for (j=0;j<n;j++)\
    C[i][j] gets (A[i][j]) op (B[i][j]);}

/** Multiply the upper left 3x3 parts of A and B to get AB **/
void
SbMatrixP::mat_mult(SbMatrixP::HMatrix A, SbMatrixP::HMatrix B, SbMatrixP::HMatrix AB)
{
  int i, j;
  for (i=0; i<3; i++) for (j=0; j<3; j++)
    AB[i][j] = A[i][0]*B[0][j] + A[i][1]*B[1][j] + A[i][2]*B[2][j];
}

/** Return dot product of length 3 vectors va and vb **/
float
SbMatrixP::vdot(float * va, float * vb)
{
  return (va[0]*vb[0] + va[1]*vb[1] + va[2]*vb[2]);
}

/** Set v to cross product of length 3 vectors va and vb **/
void
SbMatrixP::vcross(float * va, float * vb, float * v)
{
  v[0] = va[1]*vb[2] - va[2]*vb[1];
  v[1] = va[2]*vb[0] - va[0]*vb[2];
  v[2] = va[0]*vb[1] - va[1]*vb[0];
}

/** Set MadjT to transpose of inverse of M times determinant of M **/
void
SbMatrixP::adjoint_transpose(SbMatrixP::HMatrix M, SbMatrixP::HMatrix MadjT)
{
  vcross(M[1], M[2], MadjT[0]);
  vcross(M[2], M[0], MadjT[1]);
  vcross(M[0], M[1], MadjT[2]);
}

/******* Decomp Auxiliaries *******/

/** Compute either the 1 or infinity norm of M, depending on tpose **/
float
SbMatrixP::mat_norm(SbMatrixP::HMatrix M, int tpose)
{
  int i;
  float sum, max;
  max = 0.0;
  for (i=0; i<3; i++) {
    if (tpose) sum = (float)(fabs(M[0][i])+fabs(M[1][i])+fabs(M[2][i]));
    else sum = (float)(fabs(M[i][0])+fabs(M[i][1])+fabs(M[i][2]));
    if (max<sum) max = sum;
  }
  return max;
}

/** Return index of column of M containing maximum abs entry, or -1 if M=0 **/
int
SbMatrixP::find_max_col(SbMatrixP::HMatrix M)
{
  float abs, max;
  int i, j, col;
  max = 0.0; col = -1;
  for (i=0; i<3; i++) for (j=0; j<3; j++) {
    abs = M[i][j]; if (abs<0.0) abs = -abs;
    if (abs>max) {max = abs; col = j;}
  }
    return col;
}

/** Setup u for Household reflection to zero all v components but first **/
void
SbMatrixP::make_reflector(float * v, float * u)
{
  float s = (float)sqrt(vdot(v, v));
  u[0] = v[0]; u[1] = v[1];
  u[2] = v[2] + ((v[2]<0.0) ? -s : s);
  s = (float)sqrt(2.0/vdot(u, u));
  u[0] = u[0]*s; u[1] = u[1]*s; u[2] = u[2]*s;
}

/** Apply Householder reflection represented by u to column vectors of M **/
void
SbMatrixP::reflect_cols(SbMatrixP::HMatrix M, float * u)
{
  int i, j;
  for (i=0; i<3; i++) {
    float s = u[0]*M[0][i] + u[1]*M[1][i] + u[2]*M[2][i];
    for (j=0; j<3; j++) M[j][i] -= u[j]*s;
  }
}
/** Apply Householder reflection represented by u to row vectors of M **/
void
SbMatrixP::reflect_rows(SbMatrixP::HMatrix M, float * u)
{
  int i, j;
  for (i=0; i<3; i++) {
    float s = vdot(u, M[i]);
    for (j=0; j<3; j++) M[i][j] -= u[j]*s;
  }
}

/** Find orthogonal factor Q of rank 1 (or less) M **/
void
SbMatrixP::do_rank1(SbMatrixP::HMatrix M, SbMatrixP::HMatrix Q)
{
  float v1[3], v2[3], s;
  int col;
  sp_mat_copy(Q, =, mat_id, 4);
  /* If rank(M) is 1, we should find a non-zero column in M */
  col = find_max_col(M);
  if (col<0) return; /* Rank is 0 */
  v1[0] = M[0][col]; v1[1] = M[1][col]; v1[2] = M[2][col];
  make_reflector(v1, v1); reflect_cols(M, v1);
  v2[0] = M[2][0]; v2[1] = M[2][1]; v2[2] = M[2][2];
  make_reflector(v2, v2); reflect_rows(M, v2);
  s = M[2][2];
  if (s<0.0) Q[2][2] = -1.0;
  reflect_cols(Q, v1); reflect_rows(Q, v2);
}

/** Find orthogonal factor Q of rank 2 (or less) M using adjoint transpose **/
void
SbMatrixP::do_rank2(SbMatrixP::HMatrix M, SbMatrixP::HMatrix MadjT, SbMatrixP::HMatrix Q)
{
  float v1[3], v2[3];
  float w, x, y, z, c, s, d;
  int col;
  /* If rank(M) is 2, we should find a non-zero column in MadjT */
  col = find_max_col(MadjT);
  if (col<0) {do_rank1(M, Q); return;} /* Rank<2 */
  v1[0] = MadjT[0][col]; v1[1] = MadjT[1][col]; v1[2] = MadjT[2][col];
  make_reflector(v1, v1); reflect_cols(M, v1);
  vcross(M[0], M[1], v2);
  make_reflector(v2, v2); reflect_rows(M, v2);
  w = M[0][0]; x = M[0][1]; y = M[1][0]; z = M[1][1];
  if (w*z>x*y) {
    c = z+w; s = y-x; d = (float)sqrt(c*c+s*s); c = c/d; s = s/d;
    Q[0][0] = Q[1][1] = c; Q[0][1] = -(Q[1][0] = s);
  }
  else {
    c = z-w; s = y+x; d = (float)sqrt(c*c+s*s); c = c/d; s = s/d;
    Q[0][0] = -(Q[1][1] = c); Q[0][1] = Q[1][0] = s;
  }
  Q[0][2] = Q[2][0] = Q[1][2] = Q[2][1] = 0.0; Q[2][2] = 1.0;
  reflect_cols(Q, v1); reflect_rows(Q, v2);
}


/******* Polar Decomposition *******/

/* Polar Decomposition of 3x3 matrix in 4x4,
 * M = QS.  See Nicholas Higham and Robert S. Schreiber,
 * Fast Polar Decomposition of An Arbitrary Matrix,
 * Technical Report 88-942, October 1988,
 * Department of Computer Science, Cornell University.
 */
float
SbMatrixP::polar_decomp(SbMatrixP::HMatrix M, SbMatrixP::HMatrix Q, SbMatrixP::HMatrix S)
{
#define TOL 1.0e-6
  SbMatrixP::HMatrix Mk, MadjTk, Ek;
  float det, M_one, M_inf, MadjT_one, MadjT_inf, E_one, gamma, g1, g2;
  int i, j;
  sp_mat_tpose(Mk, =, M, 3);
  M_one = norm_one(Mk);  M_inf = norm_inf(Mk);
  do {
    adjoint_transpose(Mk, MadjTk);
    det = vdot(Mk[0], MadjTk[0]);
    if (det==0.0) {do_rank2(Mk, MadjTk, Mk); break;}
    MadjT_one = norm_one(MadjTk); MadjT_inf = norm_inf(MadjTk);
    gamma = (float)sqrt(sqrt((MadjT_one*MadjT_inf)/(M_one*M_inf))/fabs(det));
    g1 = gamma*0.5f;
    g2 = 0.5f/(gamma*det);
    sp_mat_copy(Ek, =, Mk, 3);
    sp_mat_binop(Mk, =, g1*Mk, +, g2*MadjTk, 3);
    sp_mat_copy(Ek, -=, Mk, 3);
    E_one = norm_one(Ek);
    M_one = norm_one(Mk);  M_inf = norm_inf(Mk);
  } while (E_one>(M_one*TOL));
  sp_mat_tpose(Q, =, Mk, 3); sp_mat_pad(Q);
  SbMatrixP::mat_mult(Mk, M, S);    sp_mat_pad(S);
  for (i=0; i<3; i++) for (j=i; j<3; j++)
    S[i][j] = S[j][i] = 0.5f*(S[i][j]+S[j][i]);
  return (det);

#undef TOL
}

/******* Spectral Decomposition *******/

/* Compute the spectral decomposition of symmetric positive semi-definite S.
 * Returns rotation in U and scale factors in result, so that if K is a diagonal
 * matrix of the scale factors, then S = U K (U transpose). Uses Jacobi method.
 * See Gene H. Golub and Charles F. Van Loan. Matrix Computations. Hopkins 1983.
 */
SbVec4f
SbMatrixP::spect_decomp(SbMatrixP::HMatrix S, SbMatrixP::HMatrix U)
{
  SbVec4f kv;
  double Diag[3], OffD[3]; /* OffD is off-diag (by omitted index) */
  double g, h, fabsh, fabsOffDi, t, theta, c, s, tau, ta, OffDq;
  static char nxt[] = {Y, Z, X};
  int sweep, i, j;
  sp_mat_copy(U, =, mat_id, 4);
  Diag[X] = S[X][X]; Diag[Y] = S[Y][Y]; Diag[Z] = S[Z][Z];
  OffD[X] = S[Y][Z]; OffD[Y] = S[Z][X]; OffD[Z] = S[X][Y];
  for (sweep=20; sweep>0; sweep--) {
    float sm = (float)(fabs(OffD[X])+fabs(OffD[Y])+fabs(OffD[Z]));
    if (sm==0.0) break;
    for (i=Z; i>=X; i--) {
      int p = nxt[i]; int q = nxt[p];
      fabsOffDi = fabs(OffD[i]);
      g = 100.0*fabsOffDi;
      if (fabsOffDi>0.0) {
        h = Diag[q] - Diag[p];
        fabsh = fabs(h);
        if (fabsh+g==fabsh) {
          t = OffD[i]/h;
        }
        else {
          theta = 0.5*h/OffD[i];
          t = 1.0/(fabs(theta)+sqrt(theta*theta+1.0));
          if (theta<0.0) t = -t;
        }
        c = 1.0/sqrt(t*t+1.0); s = t*c;
        tau = s/(c+1.0);
        ta = t*OffD[i]; OffD[i] = 0.0;
        Diag[p] -= ta; Diag[q] += ta;
        OffDq = OffD[q];
        OffD[q] -= s*(OffD[p] + tau*OffD[q]);
        OffD[p] += s*(OffDq   - tau*OffD[p]);
        for (j=Z; j>=X; j--) {
          // FIXME: this is a very peculiar problem.. if the code in
          // the #else part is activated, Valgrind 1.9.3 reports "Use
          // of uninitialised value of size 8" in this code, when
          // built with g++ 2.95.4. The code under the #if runs
          // without any warnings, even though the only change is some
          // *seemingly* irrelevant casting.
          //
          // (And even more strange: by inserting any call to external
          // code (for instance a line of ''(void)atoi("0");'') below,
          // the Valgrind error vanishes. Could it be some sort of
          // optimalization bug related to stack overwriting or some
          // such?)
          //
          // We also have a report about a crash assumed to be in this
          // code with Coin built with MSVC6.0 with /O2 optimalization
          // (very suspect that there's a problem with both g++ and
          // MSVC++), for the following stand-alone example:
          //
          // ----8<----- [snip] ---------8<----- [snip] -----
          //  #include <Inventor/SoDB.h>
          //  #include <Inventor/SbLinear.h>
          //
          //  int
          //  main( int argc, char** argv )
          //  { 
          //    SoDB::init();
          //
          //    SbVec3f translation, scale, axis;
          //    SbRotation rot, scaleRot;
          //    SbMatrix matrix (5.96046e-008f,  1.00000f,       -2.98023e-008f, 0.000000f,
          //                     -2.98023e-008f, 5.96046e-008f,  1.00000f,       0.000000f,
          //                     1.00000f,       -2.98023e-008f, 5.96046e-008f,  0.000000f,
          //                     -162.929f,      -56.2217f,      197.110f,       1.00000f 
          //                     );
          //    matrix.getTransform ( translation, rot, scale, scaleRot );
          //    translation.print(stdout); printf(" <- translation\n");
          //    scale.print(stdout); printf(" <- scale\n");
          //    rot.print(stdout); printf(" <- rot\n");
          //    scaleRot.print(stdout); printf(" <- scaleRot\n");
          //
          //    return 0;
          //  }
          // ----8<----- [snip] ---------8<----- [snip] -----
          //
          // I was not able to reproduce the crash, so I got stuck in
          // debugging this.
          //
          // Mailed a patch with the casting changes back to the
          // original reporter of the problem (Tore K at HitecO), and
          // he reports that the crash bug is then also gone! That's
          // the damndest thing.. really weird.
          //
          // It does seem extremely likely to be a compiler problem,
          // although it's very fishy that it mucks something up for
          // both GNU GCC (with -O2) and MS's VisualC++ (with /O2).
          //
          // I'll buy a beer for anyone who can figure out this one
          // and explain it to me.
          //
          // 20030214 mortene.
          float a = U[j][p], b = U[j][q];
#if 1
          U[j][p] -= float(s)*(b + float(tau)*a);
          U[j][q] += float(s)*(a - float(tau)*b);
#else
          U[j][p] -= (float)(s*(b + tau*a));
          U[j][q] += (float)(s*(a - tau*b));
#endif
        }
      }
    }
  }
  kv[X] = (float)Diag[X]; kv[Y] = (float)Diag[Y]; kv[Z] = (float)Diag[Z];
  kv[W] = 1.0f;
  return (kv);
}

/* Helper function for the snuggle() function below. */
static inline void
cycle(float * a, SbBool flip)
{
  if (flip) {
    a[3]=a[0]; a[0]=a[1]; a[1]=a[2]; a[2]=a[3];
  }
  else {
    a[3]=a[2]; a[2]=a[1]; a[1]=a[0]; a[0]=a[3];
  }
}

/******* Spectral Axis Adjustment *******/

/* Given a unit quaternion, q, and a scale vector, k, find a unit quaternion, p,
 * which permutes the axes and turns freely in the plane of duplicate scale
 * factors, such that q p has the largest possible w component, i.e. the
 * smallest possible angle. Permutes k's components to go with q p instead of q.
 * See Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
 * Proceedings of Graphics Interface 1992. Details on p. 262-263.
 */
SbRotation
SbMatrixP::snuggle(SbRotation q, SbVec4f & k)
{
#define SQRTHALF (0.7071067811865475244f)
#define sgn(n, v)    ((n)?-(v):(v))
#define swap(a, i, j) {a[3]=a[i]; a[i]=a[j]; a[j]=a[3];}

  SbRotation p;
  float ka[4];
  int i, turn = -1;
  ka[X] = k[X]; ka[Y] = k[Y]; ka[Z] = k[Z];
  if (ka[X]==ka[Y]) {if (ka[X]==ka[Z]) turn = W; else turn = Z;}
  else {if (ka[X]==ka[Z]) turn = Y; else if (ka[Y]==ka[Z]) turn = X;}
  if (turn>=0) {
    SbRotation qtoz, qp;
    unsigned neg[3], win;
    double mag[3], t;
    static SbRotation qxtoz(0.0f, SQRTHALF, 0.0f, SQRTHALF);
    static SbRotation qytoz(SQRTHALF, 0.0f, 0.0f, SQRTHALF);
    static SbRotation qppmm(0.5f, 0.5f, -0.5f, -0.5f);
    static SbRotation qpppp(0.5f, 0.5f, 0.5f, 0.5f);
    static SbRotation qmpmm(-0.5f, 0.5f, -0.5f, -0.5f);
    static SbRotation qpppm(0.5f, 0.5f, 0.5f, -0.5f);
    static SbRotation q0001(0.0f, 0.0f, 0.0f, 1.0f);
    static SbRotation q1000(1.0f, 0.0f, 0.0f, 0.0f);
    switch (turn) {
    default: return SbRotation(q).invert();
    case X: q = (qtoz = qxtoz) * q; swap(ka, X, Z) break;
    case Y: q = (qtoz = qytoz) * q; swap(ka, Y, Z) break;
    case Z: qtoz = q0001; break;
    }
    q.invert();
    mag[0] = (double)q.getValue()[Z]*q.getValue()[Z]+(double)q.getValue()[W]*q.getValue()[W]-0.5;
    mag[1] = (double)q.getValue()[X]*q.getValue()[Z]-(double)q.getValue()[Y]*q.getValue()[W];
    mag[2] = (double)q.getValue()[Y]*q.getValue()[Z]+(double)q.getValue()[X]*q.getValue()[W];
    for (i=0; i<3; i++) if ((neg[i] = (mag[i] < 0.0))) mag[i] = -mag[i];
    if (mag[0]>mag[1]) {if (mag[0]>mag[2]) win = 0; else win = 2;}
    else {if (mag[1]>mag[2]) win = 1; else win = 2;}
    switch (win) {
    case 0: if (neg[0]) p = q1000; else p = q0001; break;
    case 1: if (neg[1]) p = qppmm; else p = qpppp; cycle(ka, FALSE); break;
    case 2: if (neg[2]) p = qmpmm; else p = qpppm; cycle(ka, TRUE); break;
    }
    qp = p * q;
    t = sqrt(mag[win]+0.5);
    p = SbRotation(0.0, 0.0, -qp.getValue()[Z]/(float)t, qp.getValue()[W]/(float)t) * p;
    p = SbRotation(p).invert() * qtoz;
  }
  else {
    float qa[4], pa[4];
    unsigned lo, hi, neg[4], par = 0;
    double all, big, two;
    qa[0] = q.getValue()[X]; qa[1] = q.getValue()[Y]; qa[2] = q.getValue()[Z]; qa[3] = q.getValue()[W];
    for (i=0; i<4; i++) {
      pa[i] = 0.0;
      if ((neg[i] = (qa[i]<0.0))) qa[i] = -qa[i];
      par ^= neg[i];
    }
    /* Find two largest components, indices in hi and lo */
    if (qa[0]>qa[1]) lo = 0; else lo = 1;
    if (qa[2]>qa[3]) hi = 2; else hi = 3;
    if (qa[lo]>qa[hi]) {
      if (qa[lo^1]>qa[hi]) {hi = lo; lo ^= 1;}
      else {hi ^= lo; lo ^= hi; hi ^= lo;}
    } else {if (qa[hi^1]>qa[lo]) lo = hi^1;}
    all = (qa[0]+qa[1]+qa[2]+qa[3])*0.5;
    two = (qa[hi]+qa[lo])*SQRTHALF;
    big = qa[hi];
    if (all>two) {
      if (all>big) {/*all*/
        {int i; for (i=0; i<4; i++) pa[i] = sgn(neg[i], 0.5f);}
        cycle(ka, par);
      }
      else {/*big*/ pa[hi] = sgn(neg[hi], 1.0f);}
    }
    else {
      if (two>big) {/*two*/
        pa[hi] = sgn(neg[hi], SQRTHALF); pa[lo] = sgn(neg[lo], SQRTHALF);
        if (lo>hi) {hi ^= lo; lo ^= hi; hi ^= lo;}
        if (hi==W) {hi = "\001\002\000"[lo]; lo = 3-hi-lo;}
        swap(ka, hi, lo)
          } else {/*big*/ pa[hi] = sgn(neg[hi], 1.0f);}
    }
    // FIXME: p = conjugate(pa)? 20010114 mortene.
    p.setValue(-pa[0], -pa[1], -pa[2], pa[3]);
  }
  k[X] = ka[X]; k[Y] = ka[Y]; k[Z] = ka[Z];
  return (p);

#undef SQRTHALF
#undef sgn
#undef swap
#undef cycle
}

/******* Decompose Affine Matrix *******/

/* Decompose 4x4 affine matrix A as TFRUK(U transpose), where t contains the
 * translation components, q contains the rotation R, u contains U, k contains
 * scale factors, and f contains the sign of the determinant.
 * Assumes A transforms column vectors in right-handed coordinates.
 * See Ken Shoemake and Tom Duff. Matrix Animation and Polar Decomposition.
 * Proceedings of Graphics Interface 1992.
 */
void
SbMatrixP::decomp_affine(SbMatrixP::HMatrix A, SbMatrixP::AffineParts * parts)
{
  SbMatrixP::HMatrix Q, S, U;
  SbRotation p;
  parts->t = SbVec4f(A[X][W], A[Y][W], A[Z][W], 0);
  float det = polar_decomp(A, Q, S);
  if (det<0.0) {
    sp_mat_copy(Q, =, -Q, 3);
    parts->f = -1;
  }
  else parts->f = 1;

  // Transpose for our code (we use OpenGL's convention for numbering
  // rows and columns).
  SbMatrix TQ((const SbMat *) &Q);
  parts->q = SbRotation(TQ.transpose());
  parts->k = spect_decomp(S, U);
  // Transpose for our code (we use OpenGL's convention for numbering
  // rows and columns).
  SbMatrix TU((const SbMat *) &U);
  parts->u = SbRotation(TU.transpose());
  p = snuggle(parts->u, parts->k);
  parts->u = p * parts->u;
}

#undef sp_mat_pad
#undef sp_mat_copy
#undef sp_mat_tpose
#undef sp_mat_binop
