bullet3/examples/ThirdPartyLibs/BussIK/MatrixRmn.cpp

/*
*
* Mathematics Subpackage (VrMath)
*
*
* Author: Samuel R. Buss, sbuss@ucsd.edu.
* Web page: http://math.ucsd.edu/~sbuss/MathCG
*
*
This software is provided 'as-is', without any express or implied warranty.
In no event will the authors be held liable for any damages arising from the use of this software.
Permission is granted to anyone to use this software for any purpose,
including commercial applications, and to alter it and redistribute it freely,
subject to the following restrictions:

1. The origin of this software must not be misrepresented; you must not claim that you wrote the original software. If you use this software in a product, an acknowledgment in the product documentation would be appreciated but is not required.
2. Altered source versions must be plainly marked as such, and must not be misrepresented as being the original software.
3. This notice may not be removed or altered from any source distribution.
*
*
*/

//
// MatrixRmn.cpp:  Matrix over reals  (Variable dimensional vector)
//
//    Not very sophisticated yet.  Needs more functionality
//		To do: better handling of resizing.
//

#include "MatrixRmn.h"

MatrixRmn MatrixRmn::WorkMatrix;  // Temporary work matrix

// Fill the diagonal entries with the value d.  The rest of the matrix is unchanged.
void MatrixRmn::SetDiagonalEntries(double d)
{
	long diagLen = Min(NumRows, NumCols);
	double* dPtr = x;
	for (; diagLen > 0; diagLen--)
	{
		*dPtr = d;
		dPtr += NumRows + 1;
	}
}

// Fill the diagonal entries with values in vector d.  The rest of the matrix is unchanged.
void MatrixRmn::SetDiagonalEntries(const VectorRn& d)
{
	long diagLen = Min(NumRows, NumCols);
	assert(d.length == diagLen);
	double* dPtr = x;
	double* from = d.x;
	for (; diagLen > 0; diagLen--)
	{
		*dPtr = *(from++);
		dPtr += NumRows + 1;
	}
}

// Fill the superdiagonal entries with the value d.  The rest of the matrix is unchanged.
void MatrixRmn::SetSuperDiagonalEntries(double d)
{
	long sDiagLen = Min(NumRows, (long)(NumCols - 1));
	double* to = x + NumRows;
	for (; sDiagLen > 0; sDiagLen--)
	{
		*to = d;
		to += NumRows + 1;
	}
}

// Fill the superdiagonal entries with values in vector d.  The rest of the matrix is unchanged.
void MatrixRmn::SetSuperDiagonalEntries(const VectorRn& d)
{
	long sDiagLen = Min((long)(NumRows - 1), NumCols);
	assert(sDiagLen == d.length);
	double* to = x + NumRows;
	double* from = d.x;
	for (; sDiagLen > 0; sDiagLen--)
	{
		*to = *(from++);
		to += NumRows + 1;
	}
}

// Fill the subdiagonal entries with the value d.  The rest of the matrix is unchanged.
void MatrixRmn::SetSubDiagonalEntries(double d)
{
	long sDiagLen = Min(NumRows, NumCols) - 1;
	double* to = x + 1;
	for (; sDiagLen > 0; sDiagLen--)
	{
		*to = d;
		to += NumRows + 1;
	}
}

// Fill the subdiagonal entries with values in vector d.  The rest of the matrix is unchanged.
void MatrixRmn::SetSubDiagonalEntries(const VectorRn& d)
{
	long sDiagLen = Min(NumRows, NumCols) - 1;
	assert(sDiagLen == d.length);
	double* to = x + 1;
	double* from = d.x;
	for (; sDiagLen > 0; sDiagLen--)
	{
		*to = *(from++);
		to += NumRows + 1;
	}
}

// Set the i-th column equal to d.
void MatrixRmn::SetColumn(long i, const VectorRn& d)
{
	assert(NumRows == d.GetLength());
	double* to = x + i * NumRows;
	const double* from = d.x;
	for (i = NumRows; i > 0; i--)
	{
		*(to++) = *(from++);
	}
}

// Set the i-th column equal to d.
void MatrixRmn::SetRow(long i, const VectorRn& d)
{
	assert(NumCols == d.GetLength());
	double* to = x + i;
	const double* from = d.x;
	for (i = NumRows; i > 0; i--)
	{
		*to = *(from++);
		to += NumRows;
	}
}

// Sets a "linear" portion of the array with the values from a vector d
// The first row and column position are given by startRow, startCol.
// Successive positions are found by using the deltaRow, deltaCol values
//	to increment the row and column indices.  There is no wrapping around.
void MatrixRmn::SetSequence(const VectorRn& d, long startRow, long startCol, long deltaRow, long deltaCol)
{
	long length = d.length;
	assert(startRow >= 0 && startRow < NumRows && startCol >= 0 && startCol < NumCols);
	assert(startRow + (length - 1) * deltaRow >= 0 && startRow + (length - 1) * deltaRow < NumRows);
	assert(startCol + (length - 1) * deltaCol >= 0 && startCol + (length - 1) * deltaCol < NumCols);
	double* to = x + startRow + NumRows * startCol;
	double* from = d.x;
	long stride = deltaRow + NumRows * deltaCol;
	for (; length > 0; length--)
	{
		*to = *(from++);
		to += stride;
	}
}

// The matrix A is loaded, in into "this" matrix, based at (0,0).
//  The size of "this" matrix must be large enough to accomodate A.
//	The rest of "this" matrix is left unchanged.  It is not filled with zeroes!

void MatrixRmn::LoadAsSubmatrix(const MatrixRmn& A)
{
	assert(A.NumRows <= NumRows && A.NumCols <= NumCols);
	int extraColStep = NumRows - A.NumRows;
	double* to = x;
	double* from = A.x;
	for (long i = A.NumCols; i > 0; i--)
	{  // Copy columns of A, one per time thru loop
		for (long j = A.NumRows; j > 0; j--)
		{  // Copy all elements of this column of A
			*(to++) = *(from++);
		}
		to += extraColStep;
	}
}

// The matrix A is loaded, in transposed order into "this" matrix, based at (0,0).
//  The size of "this" matrix must be large enough to accomodate A.
//	The rest of "this" matrix is left unchanged.  It is not filled with zeroes!
void MatrixRmn::LoadAsSubmatrixTranspose(const MatrixRmn& A)
{
	assert(A.NumRows <= NumCols && A.NumCols <= NumRows);
	double* rowPtr = x;
	double* from = A.x;
	for (long i = A.NumCols; i > 0; i--)
	{  // Copy columns of A, once per loop
		double* to = rowPtr;
		for (long j = A.NumRows; j > 0; j--)
		{  // Loop copying values from the column of A
			*to = *(from++);
			to += NumRows;
		}
		rowPtr++;
	}
}

// Calculate the Frobenius Norm (square root of sum of squares of entries of the matrix)
double MatrixRmn::FrobeniusNorm() const
{
	return sqrt(FrobeniusNormSq());
}

// Multiply this matrix by column vector v.
// Result is column vector "result"
void MatrixRmn::Multiply(const VectorRn& v, VectorRn& result) const
{
	assert(v.GetLength() == NumCols && result.GetLength() == NumRows);
	double* out = result.GetPtr();  // Points to entry in result vector
	const double* rowPtr = x;       // Points to beginning of next row in matrix
	for (long j = NumRows; j > 0; j--)
	{
		const double* in = v.GetPtr();
		const double* m = rowPtr++;
		*out = 0.0f;
		for (long i = NumCols; i > 0; i--)
		{
			*out += (*(in++)) * (*m);
			m += NumRows;
		}
		out++;
	}
}

// Multiply transpose of this matrix by column vector v.
//    Result is column vector "result"
// Equivalent to mult by row vector on left
void MatrixRmn::MultiplyTranspose(const VectorRn& v, VectorRn& result) const
{
	assert(v.GetLength() == NumRows && result.GetLength() == NumCols);
	double* out = result.GetPtr();  // Points to entry in result vector
	const double* colPtr = x;       // Points to beginning of next column in matrix
	for (long i = NumCols; i > 0; i--)
	{
		const double* in = v.GetPtr();
		*out = 0.0f;
		for (long j = NumRows; j > 0; j--)
		{
			*out += (*(in++)) * (*(colPtr++));
		}
		out++;
	}
}

// Form the dot product of a vector v with the i-th column of the array
double MatrixRmn::DotProductColumn(const VectorRn& v, long colNum) const
{
	assert(v.GetLength() == NumRows);
	double* ptrC = x + colNum * NumRows;
	double* ptrV = v.x;
	double ret = 0.0;
	for (long i = NumRows; i > 0; i--)
	{
		ret += (*(ptrC++)) * (*(ptrV++));
	}
	return ret;
}

// Add a constant to each entry on the diagonal
MatrixRmn& MatrixRmn::AddToDiagonal(double d)  // Adds d to each diagonal entry
{
	long diagLen = Min(NumRows, NumCols);
	double* dPtr = x;
	for (; diagLen > 0; diagLen--)
	{
		*dPtr += d;
		dPtr += NumRows + 1;
	}
	return *this;
}

// Add a vector to the entries on the diagonal
MatrixRmn& MatrixRmn::AddToDiagonal(const VectorRn& dVec)  // Adds dVec to the diagonal entries
{
	long diagLen = Min(NumRows, NumCols);
	double* dPtr = x;
	for (int i = 0; i < diagLen && i < dVec.GetLength(); ++i)
	{
		*dPtr += dVec[i];
		dPtr += NumRows + 1;
	}
	return *this;
}

// Multiply two MatrixRmn's
MatrixRmn& MatrixRmn::Multiply(const MatrixRmn& A, const MatrixRmn& B, MatrixRmn& dst)
{
	assert(A.NumCols == B.NumRows && A.NumRows == dst.NumRows && B.NumCols == dst.NumCols);
	long length = A.NumCols;

	double* bPtr = B.x;  // Points to beginning of column in B
	double* dPtr = dst.x;
	for (long i = dst.NumCols; i > 0; i--)
	{
		double* aPtr = A.x;  // Points to beginning of row in A
		for (long j = dst.NumRows; j > 0; j--)
		{
			*dPtr = DotArray(length, aPtr, A.NumRows, bPtr, 1);
			dPtr++;
			aPtr++;
		}
		bPtr += B.NumRows;
	}

	return dst;
}

// Multiply two MatrixRmn's,  Transpose the first matrix before multiplying
MatrixRmn& MatrixRmn::TransposeMultiply(const MatrixRmn& A, const MatrixRmn& B, MatrixRmn& dst)
{
	assert(A.NumRows == B.NumRows && A.NumCols == dst.NumRows && B.NumCols == dst.NumCols);
	long length = A.NumRows;

	double* bPtr = B.x;  // bPtr Points to beginning of column in B
	double* dPtr = dst.x;
	for (long i = dst.NumCols; i > 0; i--)
	{                        // Loop over all columns of dst
		double* aPtr = A.x;  // aPtr Points to beginning of column in A
		for (long j = dst.NumRows; j > 0; j--)
		{  // Loop over all rows of dst
			*dPtr = DotArray(length, aPtr, 1, bPtr, 1);
			dPtr++;
			aPtr += A.NumRows;
		}
		bPtr += B.NumRows;
	}

	return dst;
}

// Multiply two MatrixRmn's.  Transpose the second matrix before multiplying
MatrixRmn& MatrixRmn::MultiplyTranspose(const MatrixRmn& A, const MatrixRmn& B, MatrixRmn& dst)
{
	assert(A.NumCols == B.NumCols && A.NumRows == dst.NumRows && B.NumRows == dst.NumCols);
	long length = A.NumCols;

	double* bPtr = B.x;  // Points to beginning of row in B
	double* dPtr = dst.x;
	for (long i = dst.NumCols; i > 0; i--)
	{
		double* aPtr = A.x;  // Points to beginning of row in A
		for (long j = dst.NumRows; j > 0; j--)
		{
			*dPtr = DotArray(length, aPtr, A.NumRows, bPtr, B.NumRows);
			dPtr++;
			aPtr++;
		}
		bPtr++;
	}

	return dst;
}

// Solves the equation   (*this)*xVec = b;
// Uses row operations.  Assumes *this is square and invertible.
// No error checking for divide by zero or instability (except with asserts)
void MatrixRmn::Solve(const VectorRn& b, VectorRn* xVec) const
{
	assert(NumRows == NumCols && NumCols == xVec->GetLength() && NumRows == b.GetLength());

	// Copy this matrix and b into an Augmented Matrix
	MatrixRmn& AugMat = GetWorkMatrix(NumRows, NumCols + 1);
	AugMat.LoadAsSubmatrix(*this);
	AugMat.SetColumn(NumRows, b);

	// Put into row echelon form with row operations
	AugMat.ConvertToRefNoFree();

	// Solve for x vector values using back substitution
	double* xLast = xVec->x + NumRows - 1;              // Last entry in xVec
	double* endRow = AugMat.x + NumRows * NumCols - 1;  // Last entry in the current row of the coefficient part of Augmented Matrix
	double* bPtr = endRow + NumRows;                    // Last entry in augmented matrix (end of last column, in augmented part)
	for (long i = NumRows; i > 0; i--)
	{
		double accum = *(bPtr--);
		// Next loop computes back substitution terms
		double* rowPtr = endRow;  // Points to entries of the current row for back substitution.
		double* xPtr = xLast;     // Points to entries in the x vector (also for back substitution)
		for (long j = NumRows - i; j > 0; j--)
		{
			accum -= (*rowPtr) * (*(xPtr--));
			rowPtr -= NumCols;  // Previous entry in the row
		}
		assert(*rowPtr != 0.0);  // Are not supposed to be any free variables in this matrix
		*xPtr = accum / (*rowPtr);
		endRow--;
	}
}

// ConvertToRefNoFree
// Converts the matrix (in place) to row echelon form
// For us, row echelon form allows any non-zero values, not just 1's, in the
//		position for a lead variable.
// The "NoFree" version operates on the assumption that no free variable will be found.
// Algorithm uses row operations and row pivoting (only).
// Augmented matrix is correctly accomodated.  Only the first square part participates
//		in the main work of row operations.
void MatrixRmn::ConvertToRefNoFree()
{
	// Loop over all columns (variables)
	// Find row with most non-zero entry.
	// Swap to the highest active row
	// Subtract appropriately from all the lower rows (row op of type 3)
	long numIters = Min(NumRows, NumCols);
	double* rowPtr1 = x;
	const long diagStep = NumRows + 1;
	long lenRowLeft = NumCols;
	for (; numIters > 1; numIters--)
	{
		// Find row with most non-zero entry.
		double* rowPtr2 = rowPtr1;
		double maxAbs = fabs(*rowPtr1);
		double* rowPivot = rowPtr1;
		long i;
		for (i = numIters - 1; i > 0; i--)
		{
			const double& newMax = *(++rowPivot);
			if (newMax > maxAbs)
			{
				maxAbs = *rowPivot;
				rowPtr2 = rowPivot;
			}
			else if (-newMax > maxAbs)
			{
				maxAbs = -newMax;
				rowPtr2 = rowPivot;
			}
		}
		// Pivot step: Swap the row with highest entry to the current row
		if (rowPtr1 != rowPtr2)
		{
			double* to = rowPtr1;
			for (long i = lenRowLeft; i > 0; i--)
			{
				double temp = *to;
				*to = *rowPtr2;
				*rowPtr2 = temp;
				to += NumRows;
				rowPtr2 += NumRows;
			}
		}
		// Subtract this row appropriately from all the lower rows (row operation of type 3)
		rowPtr2 = rowPtr1;
		for (i = numIters - 1; i > 0; i--)
		{
			rowPtr2++;
			double* to = rowPtr2;
			double* from = rowPtr1;
			assert(*from != 0.0);
			double alpha = (*to) / (*from);
			*to = 0.0;
			for (long j = lenRowLeft - 1; j > 0; j--)
			{
				to += NumRows;
				from += NumRows;
				*to -= (*from) * alpha;
			}
		}
		// Update for next iteration of loop
		rowPtr1 += diagStep;
		lenRowLeft--;
	}
}

// Calculate the c=cosine and s=sine values for a Givens transformation.
// The matrix M = ( (c, -s), (s, c) ) in row order transforms the
//   column vector (a, b)^T to have y-coordinate zero.
void MatrixRmn::CalcGivensValues(double a, double b, double* c, double* s)
{
	double denomInv = sqrt(a * a + b * b);
	if (denomInv == 0.0)
	{
		*c = 1.0;
		*s = 0.0;
	}
	else
	{
		denomInv = 1.0 / denomInv;
		*c = a * denomInv;
		*s = -b * denomInv;
	}
}

// Applies Givens transform to columns i and i+1.
// Equivalent to postmultiplying by the matrix
//      ( c  -s )
//		( s   c )
// with non-zero entries in rows i and i+1 and columns i and i+1
void MatrixRmn::PostApplyGivens(double c, double s, long idx)
{
	assert(0 <= idx && idx < NumCols);
	double* colA = x + idx * NumRows;
	double* colB = colA + NumRows;
	for (long i = NumRows; i > 0; i--)
	{
		double temp = *colA;
		*colA = (*colA) * c + (*colB) * s;
		*colB = (*colB) * c - temp * s;
		colA++;
		colB++;
	}
}

// Applies Givens transform to columns idx1 and idx2.
// Equivalent to postmultiplying by the matrix
//      ( c  -s )
//		( s   c )
// with non-zero entries in rows idx1 and idx2 and columns idx1 and idx2
void MatrixRmn::PostApplyGivens(double c, double s, long idx1, long idx2)
{
	assert(idx1 != idx2 && 0 <= idx1 && idx1 < NumCols && 0 <= idx2 && idx2 < NumCols);
	double* colA = x + idx1 * NumRows;
	double* colB = x + idx2 * NumRows;
	for (long i = NumRows; i > 0; i--)
	{
		double temp = *colA;
		*colA = (*colA) * c + (*colB) * s;
		*colB = (*colB) * c - temp * s;
		colA++;
		colB++;
	}
}

// ********************************************************************************************
// Singular value decomposition.
// Return othogonal matrices U and V and diagonal matrix with diagonal w such that
//     (this) = U * Diag(w) * V^T     (V^T is V-transpose.)
// Diagonal entries have all non-zero entries before all zero entries, but are not
//		necessarily sorted.  (Someday, I will write ComputedSortedSVD that handles
//		sorting the eigenvalues by magnitude.)
// ********************************************************************************************
void MatrixRmn::ComputeSVD(MatrixRmn& U, VectorRn& w, MatrixRmn& V) const
{
	assert(U.NumRows == NumRows && V.NumCols == NumCols && U.NumRows == U.NumCols && V.NumRows == V.NumCols && w.GetLength() == Min(NumRows, NumCols));

	//	double temp=0.0;
	VectorRn& superDiag = VectorRn::GetWorkVector(w.GetLength() - 1);  // Some extra work space.  Will get passed around.

	// Choose larger of U, V to hold intermediate results
	// If U is larger than V, use U to store intermediate results
	// Otherwise use V.  In the latter case, we form the SVD of A transpose,
	//		(which is essentially identical to the SVD of A).
	MatrixRmn* leftMatrix;
	MatrixRmn* rightMatrix;
	if (NumRows >= NumCols)
	{
		U.LoadAsSubmatrix(*this);  // Copy A into U
		leftMatrix = &U;
		rightMatrix = &V;
	}
	else
	{
		V.LoadAsSubmatrixTranspose(*this);  // Copy A-transpose into V
		leftMatrix = &V;
		rightMatrix = &U;
	}

	// Do the actual work to calculate the SVD
	// Now matrix has at least as many rows as columns
	CalcBidiagonal(*leftMatrix, *rightMatrix, w, superDiag);
	ConvertBidiagToDiagonal(*leftMatrix, *rightMatrix, w, superDiag);
}

void MatrixRmn::ComputeInverse(MatrixRmn& R) const
{
	assert(this->NumRows == this->NumCols);
	MatrixRmn U(this->NumRows, this->NumCols);
	VectorRn w(this->NumRows);
	MatrixRmn V(this->NumRows, this->NumCols);

	this->ComputeSVD(U, w, V);

	assert(this->DebugCheckSVD(U, w, V));

	double PseudoInverseThresholdFactor = 0.01;
	double pseudoInverseThreshold = PseudoInverseThresholdFactor * w.MaxAbs();

	MatrixRmn VD(this->NumRows, this->NumCols);
	MatrixRmn D(this->NumRows, this->NumCols);
	D.SetZero();
	long diagLength = w.GetLength();
	double* wPtr = w.GetPtr();
	for (long i = 0; i < diagLength; ++i)
	{
		double alpha = *(wPtr++);
		if (fabs(alpha) > pseudoInverseThreshold)
		{
			D.Set(i, i, 1.0 / alpha);
		}
	}

	Multiply(V, D, VD);
	MultiplyTranspose(VD, U, R);
}

// ************************************************ CalcBidiagonal **************************
// Helper routine for SVD computation
// U is a matrix to be bidiagonalized.
// On return, U and V are orthonormal and w holds the new diagonal
//	  elements and superDiag holds the super diagonal elements.

void MatrixRmn::CalcBidiagonal(MatrixRmn& U, MatrixRmn& V, VectorRn& w, VectorRn& superDiag)
{
	assert(U.NumRows >= V.NumRows);

	// The diagonal and superdiagonal entries of the bidiagonalized
	//	  version of the U matrix
	//	  are stored in the vectors w and superDiag (temporarily).

	// Apply Householder transformations to U.
	// Householder transformations come in pairs.
	//   First, on the left, we map a portion of a column to zeros
	//   Second, on the right, we map a portion of a row to zeros
	const long rowStep = U.NumCols;
	const long diagStep = U.NumCols + 1;
	double* diagPtr = U.x;
	double* wPtr = w.x;
	double* superDiagPtr = superDiag.x;
	long colLengthLeft = U.NumRows;
	long rowLengthLeft = V.NumCols;
	while (true)
	{
		// Apply a Householder xform on left to zero part of a column
		SvdHouseholder(diagPtr, colLengthLeft, rowLengthLeft, 1, rowStep, wPtr);

		if (rowLengthLeft == 2)
		{
			*superDiagPtr = *(diagPtr + rowStep);
			break;
		}
		// Apply a Householder xform on the right to zero part of a row
		SvdHouseholder(diagPtr + rowStep, rowLengthLeft - 1, colLengthLeft, rowStep, 1, superDiagPtr);

		rowLengthLeft--;
		colLengthLeft--;
		diagPtr += diagStep;
		wPtr++;
		superDiagPtr++;
	}

	int extra = 0;
	diagPtr += diagStep;
	wPtr++;
	if (colLengthLeft > 2)
	{
		extra = 1;
		// Do one last Householder transformation when the matrix is not square
		colLengthLeft--;
		SvdHouseholder(diagPtr, colLengthLeft, 1, 1, 0, wPtr);
	}
	else
	{
		*wPtr = *diagPtr;
	}

	// Form U and V from the Householder transformations
	V.ExpandHouseholders(V.NumCols - 2, 1, U.x + U.NumRows, U.NumRows, 1);
	U.ExpandHouseholders(V.NumCols - 1 + extra, 0, U.x, 1, U.NumRows);

	// Done with bidiagonalization
	return;
}

// Helper routine for CalcBidiagonal
// Performs a series of Householder transformations on a matrix
// Stores results compactly into the matrix:   The Householder vector u (normalized)
//   is stored into the first row/column being transformed.
// The leading term of that row (= plus/minus its magnitude is returned
//	 separately into "retFirstEntry"
void MatrixRmn::SvdHouseholder(double* basePt,
							   long colLength, long numCols, long colStride, long rowStride,
							   double* retFirstEntry)
{
	// Calc norm of vector u
	double* cPtr = basePt;
	double norm = 0.0;
	long i;
	for (i = colLength; i > 0; i--)
	{
		norm += Square(*cPtr);
		cPtr += colStride;
	}
	norm = sqrt(norm);  // Norm of vector to reflect to axis  e_1

	// Handle sign issues
	double imageVal;  // Choose sign to maximize distance
	if ((*basePt) < 0.0)
	{
		imageVal = norm;
		norm = 2.0 * norm * (norm - (*basePt));
	}
	else
	{
		imageVal = -norm;
		norm = 2.0 * norm * (norm + (*basePt));
	}
	norm = sqrt(norm);  // Norm is norm of reflection vector

	if (norm == 0.0)
	{  // If the vector being transformed is equal to zero
		// Force to zero in case of roundoff errors
		cPtr = basePt;
		for (i = colLength; i > 0; i--)
		{
			*cPtr = 0.0;
			cPtr += colStride;
		}
		*retFirstEntry = 0.0;
		return;
	}

	*retFirstEntry = imageVal;

	// Set up the normalized Householder vector
	*basePt -= imageVal;  // First component changes. Rest stay the same.
	// Normalize the vector
	norm = 1.0 / norm;  // Now it is the inverse norm
	cPtr = basePt;
	for (i = colLength; i > 0; i--)
	{
		*cPtr *= norm;
		cPtr += colStride;
	}

	// Transform the rest of the U matrix with the Householder transformation
	double* rPtr = basePt;
	for (long j = numCols - 1; j > 0; j--)
	{
		rPtr += rowStride;
		// Calc dot product with Householder transformation vector
		double dotP = DotArray(colLength, basePt, colStride, rPtr, colStride);
		// Transform with I - 2*dotP*(Householder vector)
		AddArrayScale(colLength, basePt, colStride, rPtr, colStride, -2.0 * dotP);
	}
}

// ********************************* ExpandHouseholders ********************************************
// The matrix will be square.
//   numXforms = number of Householder transformations to concatenate
//		Each Householder transformation is represented by a unit vector
//		Each successive Householder transformation starts one position later
//			and has one more implied leading zero
//	 basePt = beginning of the first Householder transform
//	 colStride, rowStride: Householder xforms are stored in "columns"
//   numZerosSkipped is the number of implicit zeros on the front each
//			Householder transformation vector (only values supported are 0 and 1).
void MatrixRmn::ExpandHouseholders(long numXforms, int numZerosSkipped, const double* basePt, long colStride, long rowStride)
{
	// Number of applications of the last Householder transform
	//     (That are not trivial!)
	long numToTransform = NumCols - numXforms + 1 - numZerosSkipped;
	assert(numToTransform > 0);

	if (numXforms == 0)
	{
		SetIdentity();
		return;
	}

	// Handle the first one separately as a special case,
	// "this" matrix will be treated to simulate being preloaded with the identity
	long hDiagStride = rowStride + colStride;
	const double* hBase = basePt + hDiagStride * (numXforms - 1);       // Pointer to the last Householder vector
	const double* hDiagPtr = hBase + colStride * (numToTransform - 1);  // Pointer to last entry in that vector
	long i;
	double* diagPtr = x + NumCols * NumRows - 1;      // Last entry in matrix (points to diagonal entry)
	double* colPtr = diagPtr - (numToTransform - 1);  // Pointer to column in matrix
	for (i = numToTransform; i > 0; i--)
	{
		CopyArrayScale(numToTransform, hBase, colStride, colPtr, 1, -2.0 * (*hDiagPtr));
		*diagPtr += 1.0;           // Add back in 1 to the diagonal entry (since xforming the identity)
		diagPtr -= (NumRows + 1);  // Next diagonal entry in this matrix
		colPtr -= NumRows;         // Next column in this matrix
		hDiagPtr -= colStride;
	}

	// Now handle the general case
	// A row of zeros must be in effect added to the top of each old column (in each loop)
	double* colLastPtr = x + NumRows * NumCols - numToTransform - 1;
	for (i = numXforms - 1; i > 0; i--)
	{
		numToTransform++;      // Number of non-trivial applications of this Householder transformation
		hBase -= hDiagStride;  // Pointer to the beginning of the Householder transformation
		colPtr = colLastPtr;
		for (long j = numToTransform - 1; j > 0; j--)
		{
			// Get dot product
			double dotProd2N = -2.0 * DotArray(numToTransform - 1, hBase + colStride, colStride, colPtr + 1, 1);
			*colPtr = dotProd2N * (*hBase);  // Adding onto zero at initial point
			AddArrayScale(numToTransform - 1, hBase + colStride, colStride, colPtr + 1, 1, dotProd2N);
			colPtr -= NumRows;
		}
		// Do last one as a special case (may overwrite the Householder vector)
		CopyArrayScale(numToTransform, hBase, colStride, colPtr, 1, -2.0 * (*hBase));
		*colPtr += 1.0;  // Add back one one as identity
		// Done with this Householder transformation
		colLastPtr--;
	}

	if (numZerosSkipped != 0)
	{
		assert(numZerosSkipped == 1);
		// Fill first row and column with identity (More generally: first numZerosSkipped many rows and columns)
		double* d = x;
		*d = 1;
		double* d2 = d;
		for (i = NumRows - 1; i > 0; i--)
		{
			*(++d) = 0;
			*(d2 += NumRows) = 0;
		}
	}
}

// **************** ConvertBidiagToDiagonal ***********************************************
// Do the iterative transformation from bidiagonal form to diagonal form using
//		Givens transformation.  (Golub-Reinsch)
// U and V are square.  Size of U less than or equal to that of U.
void MatrixRmn::ConvertBidiagToDiagonal(MatrixRmn& U, MatrixRmn& V, VectorRn& w, VectorRn& superDiag) const
{
	// These two index into the last bidiagonal block  (last in the matrix, it will be
	//	first one handled.
	long lastBidiagIdx = V.NumRows - 1;
	long firstBidiagIdx = 0;
	double eps = 1.0e-15 * Max(w.MaxAbs(), superDiag.MaxAbs());

	while (true)
	{
		bool workLeft = UpdateBidiagIndices(&firstBidiagIdx, &lastBidiagIdx, w, superDiag, eps);
		if (!workLeft)
		{
			break;
		}

		// Get ready for first Givens rotation
		// Push non-zero to M[2,1] with Givens transformation
		double* wPtr = w.x + firstBidiagIdx;
		double* sdPtr = superDiag.x + firstBidiagIdx;
		double extraOffDiag = 0.0;
		if ((*wPtr) == 0.0)
		{
			ClearRowWithDiagonalZero(firstBidiagIdx, lastBidiagIdx, U, wPtr, sdPtr, eps);
			if (firstBidiagIdx > 0)
			{
				if (NearZero(*(--sdPtr), eps))
				{
					*sdPtr = 0.0;
				}
				else
				{
					ClearColumnWithDiagonalZero(firstBidiagIdx, V, wPtr, sdPtr, eps);
				}
			}
			continue;
		}

		// Estimate an eigenvalue from bottom four entries of M
		// This gives a lambda value which will shift the Givens rotations
		// Last four entries of M^T * M are  ( ( A, B ), ( B, C ) ).
		double A;
		A = (firstBidiagIdx < lastBidiagIdx - 1) ? Square(superDiag[lastBidiagIdx - 2]) : 0.0;
		double BSq = Square(w[lastBidiagIdx - 1]);
		A += BSq;  // The "A" entry of M^T * M
		double C = Square(superDiag[lastBidiagIdx - 1]);
		BSq *= C;                                    // The squared "B" entry
		C += Square(w[lastBidiagIdx]);               // The "C" entry
		double lambda;                               // lambda will hold the estimated eigenvalue
		lambda = sqrt(Square((A - C) * 0.5) + BSq);  // Use the lambda value that is closest to C.
		if (A > C)
		{
			lambda = -lambda;
		}
		lambda += (A + C) * 0.5;  // Now lambda equals the estimate for the last eigenvalue
		double t11 = Square(w[firstBidiagIdx]);
		double t12 = w[firstBidiagIdx] * superDiag[firstBidiagIdx];

		double c, s;
		CalcGivensValues(t11 - lambda, t12, &c, &s);
		ApplyGivensCBTD(c, s, wPtr, sdPtr, &extraOffDiag, wPtr + 1);
		V.PostApplyGivens(c, -s, firstBidiagIdx);
		long i;
		for (i = firstBidiagIdx; i < lastBidiagIdx - 1; i++)
		{
			// Push non-zero from M[i+1,i] to M[i,i+2]
			CalcGivensValues(*wPtr, extraOffDiag, &c, &s);
			ApplyGivensCBTD(c, s, wPtr, sdPtr, &extraOffDiag, extraOffDiag, wPtr + 1, sdPtr + 1);
			U.PostApplyGivens(c, -s, i);
			// Push non-zero from M[i,i+2] to M[1+2,i+1]
			CalcGivensValues(*sdPtr, extraOffDiag, &c, &s);
			ApplyGivensCBTD(c, s, sdPtr, wPtr + 1, &extraOffDiag, extraOffDiag, sdPtr + 1, wPtr + 2);
			V.PostApplyGivens(c, -s, i + 1);
			wPtr++;
			sdPtr++;
		}
		// Push non-zero value from M[i+1,i] to M[i,i+1] for i==lastBidiagIdx-1
		CalcGivensValues(*wPtr, extraOffDiag, &c, &s);
		ApplyGivensCBTD(c, s, wPtr, &extraOffDiag, sdPtr, wPtr + 1);
		U.PostApplyGivens(c, -s, i);

		// DEBUG
		// DebugCalcBidiagCheck( V, w, superDiag, U );
	}
}

// This is called when there is a zero diagonal entry, with a non-zero superdiagonal entry on the same row.
// We use Givens rotations to "chase" the non-zero entry across the row; when it reaches the last
//	column, it is finally zeroed away.
// wPtr points to the zero entry on the diagonal.  sdPtr points to the non-zero superdiagonal entry on the same row.
void MatrixRmn::ClearRowWithDiagonalZero(long firstBidiagIdx, long lastBidiagIdx, MatrixRmn& U, double* wPtr, double* sdPtr, double eps)
{
	double curSd = *sdPtr;  // Value being chased across the row
	*sdPtr = 0.0;
	long i = firstBidiagIdx + 1;
	while (true)
	{
		// Rotate row i and row firstBidiagIdx (Givens rotation)
		double c, s;
		CalcGivensValues(*(++wPtr), curSd, &c, &s);
		U.PostApplyGivens(c, -s, i, firstBidiagIdx);
		*wPtr = c * (*wPtr) - s * curSd;
		if (i == lastBidiagIdx)
		{
			break;
		}
		curSd = s * (*(++sdPtr));  // New value pops up one column over to the right
		*sdPtr = c * (*sdPtr);
		i++;
	}
}

// This is called when there is a zero diagonal entry, with a non-zero superdiagonal entry in the same column.
// We use Givens rotations to "chase" the non-zero entry up the column; when it reaches the last
//	column, it is finally zeroed away.
// wPtr points to the zero entry on the diagonal.  sdPtr points to the non-zero superdiagonal entry in the same column.
void MatrixRmn::ClearColumnWithDiagonalZero(long endIdx, MatrixRmn& V, double* wPtr, double* sdPtr, double eps)
{
	double curSd = *sdPtr;  // Value being chased up the column
	*sdPtr = 0.0;
	long i = endIdx - 1;
	while (true)
	{
		double c, s;
		CalcGivensValues(*(--wPtr), curSd, &c, &s);
		V.PostApplyGivens(c, -s, i, endIdx);
		*wPtr = c * (*wPtr) - s * curSd;
		if (i == 0)
		{
			break;
		}
		curSd = s * (*(--sdPtr));  // New value pops up one row above
		if (NearZero(curSd, eps))
		{
			break;
		}
		*sdPtr = c * (*sdPtr);
		i--;
	}
}

// Matrix A is  ( ( a c ) ( b d ) ), i.e., given in column order.
// Mult's G[c,s]  times  A, replaces A.
void MatrixRmn::ApplyGivensCBTD(double cosine, double sine, double* a, double* b, double* c, double* d)
{
	double temp = *a;
	*a = cosine * (*a) - sine * (*b);
	*b = sine * temp + cosine * (*b);
	temp = *c;
	*c = cosine * (*c) - sine * (*d);
	*d = sine * temp + cosine * (*d);
}

// Now matrix A given in row order, A = ( ( a b c ) ( d e f ) ).
// Return G[c,s] * A, replace A.  d becomes zero, no need to return.
//  Also, it is certain the old *c value is taken to be zero!
void MatrixRmn::ApplyGivensCBTD(double cosine, double sine, double* a, double* b, double* c,
								double d, double* e, double* f)
{
	*a = cosine * (*a) - sine * d;
	double temp = *b;
	*b = cosine * (*b) - sine * (*e);
	*e = sine * temp + cosine * (*e);
	*c = -sine * (*f);
	*f = cosine * (*f);
}

// Helper routine for SVD conversion from bidiagonal to diagonal
bool MatrixRmn::UpdateBidiagIndices(long* firstBidiagIdx, long* lastBidiagIdx, VectorRn& w, VectorRn& superDiag, double eps)
{
	long lastIdx = *lastBidiagIdx;
	double* sdPtr = superDiag.GetPtr(lastIdx - 1);  // Entry above the last diagonal entry
	while (NearZero(*sdPtr, eps))
	{
		*(sdPtr--) = 0.0;
		lastIdx--;
		if (lastIdx == 0)
		{
			return false;
		}
	}
	*lastBidiagIdx = lastIdx;
	long firstIdx = lastIdx - 1;
	double* wPtr = w.GetPtr(firstIdx);
	while (firstIdx > 0)
	{
		if (NearZero(*wPtr, eps))
		{  // If this diagonal entry (near) zero
			*wPtr = 0.0;
			break;
		}
		if (NearZero(*(--sdPtr), eps))
		{  // If the entry above the diagonal entry is (near) zero
			*sdPtr = 0.0;
			break;
		}
		wPtr--;
		firstIdx--;
	}
	*firstBidiagIdx = firstIdx;
	return true;
}

// ******************************************DEBUG STUFFF

bool MatrixRmn::DebugCheckSVD(const MatrixRmn& U, const VectorRn& w, const MatrixRmn& V) const
{
	// Special SVD test code

	MatrixRmn IV(V.GetNumRows(), V.GetNumColumns());
	IV.SetIdentity();
	MatrixRmn VTV(V.GetNumRows(), V.GetNumColumns());
	MatrixRmn::TransposeMultiply(V, V, VTV);
	IV -= VTV;
	double error = IV.FrobeniusNorm();

	MatrixRmn IU(U.GetNumRows(), U.GetNumColumns());
	IU.SetIdentity();
	MatrixRmn UTU(U.GetNumRows(), U.GetNumColumns());
	MatrixRmn::TransposeMultiply(U, U, UTU);
	IU -= UTU;
	error += IU.FrobeniusNorm();

	MatrixRmn Diag(U.GetNumRows(), V.GetNumRows());
	Diag.SetZero();
	Diag.SetDiagonalEntries(w);
	MatrixRmn B(U.GetNumRows(), V.GetNumRows());
	MatrixRmn C(U.GetNumRows(), V.GetNumRows());
	MatrixRmn::Multiply(U, Diag, B);
	MatrixRmn::MultiplyTranspose(B, V, C);
	C -= *this;
	error += C.FrobeniusNorm();

	bool ret = (fabs(error) <= 1.0e-13 * w.MaxAbs());
	assert(ret);
	return ret;
}

bool MatrixRmn::DebugCheckInverse(const MatrixRmn& MInv) const
{
	assert(this->NumRows == this->NumCols);
	assert(MInv.NumRows == MInv.NumCols);
	MatrixRmn I(this->NumRows, this->NumCols);
	I.SetIdentity();
	MatrixRmn MMInv(this->NumRows, this->NumCols);
	Multiply(*this, MInv, MMInv);
	I -= MMInv;
	double error = I.FrobeniusNorm();
	bool ret = (fabs(error) <= 1.0e-13);
	assert(ret);
	return ret;
}

bool MatrixRmn::DebugCalcBidiagCheck(const MatrixRmn& U, const VectorRn& w, const VectorRn& superDiag, const MatrixRmn& V) const
{
	// Special SVD test code

	MatrixRmn IV(V.GetNumRows(), V.GetNumColumns());
	IV.SetIdentity();
	MatrixRmn VTV(V.GetNumRows(), V.GetNumColumns());
	MatrixRmn::TransposeMultiply(V, V, VTV);
	IV -= VTV;
	double error = IV.FrobeniusNorm();

	MatrixRmn IU(U.GetNumRows(), U.GetNumColumns());
	IU.SetIdentity();
	MatrixRmn UTU(U.GetNumRows(), U.GetNumColumns());
	MatrixRmn::TransposeMultiply(U, U, UTU);
	IU -= UTU;
	error += IU.FrobeniusNorm();

	MatrixRmn DiagAndSuper(U.GetNumRows(), V.GetNumRows());
	DiagAndSuper.SetZero();
	DiagAndSuper.SetDiagonalEntries(w);
	if (this->GetNumRows() >= this->GetNumColumns())
	{
		DiagAndSuper.SetSequence(superDiag, 0, 1, 1, 1);
	}
	else
	{
		DiagAndSuper.SetSequence(superDiag, 1, 0, 1, 1);
	}
	MatrixRmn B(U.GetNumRows(), V.GetNumRows());
	MatrixRmn C(U.GetNumRows(), V.GetNumRows());
	MatrixRmn::Multiply(U, DiagAndSuper, B);
	MatrixRmn::MultiplyTranspose(B, V, C);
	C -= *this;
	error += C.FrobeniusNorm();

	bool ret = (fabs(error) < 1.0e-13 * Max(w.MaxAbs(), superDiag.MaxAbs()));
	assert(ret);
	return ret;
}