// ------------------------------------------------------------------------
//
// Init function - call prior to start of iterations
//
// ------------------------------------------------------------------------
void Init(const Matrix<T>& A,
DenseMatrix<T>& W,
DenseMatrix<T>& H)
{
WtW.Resize(W.Width(), W.Width());
WtA.Resize(W.Width(), A.Width());
HHt.Resize(H.Height(), H.Height());
AHt.Resize(A.Height(), H.Height());
ScaleFactors.Resize(H.Height(), 1);
// compute the kxk matrix WtW = W' * W
Gemm(TRANSPOSE, NORMAL, T(1), W, W, T(0), WtW);
// compute the kxn matrix WtA = W' * A
Gemm(TRANSPOSE, NORMAL, T(1), W, A, T(0), WtA);
}
/// Construct a constant matrix coefficient times a scalar Coefficient
MatrixFunctionCoefficient(const DenseMatrix &m, Coefficient &q)
: MatrixCoefficient(m.Height(), m.Width()), Q(&q)
{
Function = NULL;
TDFunction = NULL;
mat = m;
}
void OverwriteCols(DenseMatrix<T>& A,
const DenseMatrix<T>& B,
const std::vector<unsigned int>& col_indices,
const unsigned int num_cols)
{
const unsigned int height = A.Height();
// Overwrite columns of A with B.
if (B.Height() != A.Height())
throw std::logic_error("OverwriteCols: height mismatch");
if (num_cols > static_cast<unsigned int>(A.Width()))
throw std::logic_error("OverwriteCols: col indices out of bounds");
T* buf_a = A.Buffer();
const unsigned int ldim_a = A.LDim();
const T* buf_b = B.LockedBuffer();
const unsigned int ldim_b = B.LDim();
for (unsigned int c=0; c<num_cols; ++c)
{
unsigned int col_a = col_indices[c];
unsigned int offset_a = col_a*ldim_a;
unsigned int offset_b = c*ldim_b;
memcpy(&buf_a[offset_a], &buf_b[offset_b], height * sizeof(T));
}
}
bool SolveNormalEq(const DenseMatrix<T>& LHS, // kxk
const DenseMatrix<T>& RHS, // kxn
DenseMatrix<T>& X) // kxn
{
// Solve LHS * X = RHS for X, where LHS is assumed SPD.
if (LHS.Width() != LHS.Height())
throw std::logic_error("SolveNormalEq: expected square matrix on LHS");
if ( (X.Height() != LHS.Height()) || (X.Width() != RHS.Width()))
throw std::logic_error("SolveNormalEq: non-conformant matrix X");
// copy the input, since the solver overwrites it
DenseMatrix<T> M(LHS);
X = RHS;
return SolveNormalEq(M, X);
}
void IsoparametricTransformation::Transform (const DenseMatrix &matrix,
DenseMatrix &result)
{
MFEM_ASSERT(matrix.Height() == GetDimension(), "invalid input");
result.SetSize(PointMat.Height(), matrix.Width());
IntegrationPoint ip;
Vector col;
for (int j = 0; j < matrix.Width(); j++)
{
ip.Set(matrix.GetColumn(j), matrix.Height());
result.GetColumnReference(j, col);
Transform(ip, col);
}
}
void VisualizationSceneVector3d::PrepareFlat()
{
int i, j;
glNewList (displlist, GL_COMPILE);
int nbe = mesh -> GetNBE();
DenseMatrix pointmat;
Array<int> vertices;
double p[4][3], c[4];
for (i = 0; i < nbe; i++)
{
if (!bdr_attr_to_show[mesh->GetBdrAttribute(i)-1]) continue;
mesh->GetBdrPointMatrix(i, pointmat);
mesh->GetBdrElementVertices(i, vertices);
for (j = 0; j < pointmat.Width(); j++)
{
pointmat(0, j) += (*solx)(vertices[j])*(ianim)/ianimmax;
pointmat(1, j) += (*soly)(vertices[j])*(ianim)/ianimmax;
pointmat(2, j) += (*solz)(vertices[j])*(ianim)/ianimmax;
}
for (j = 0; j < pointmat.Width(); j++)
{
p[j][0] = pointmat(0, j);
p[j][1] = pointmat(1, j);
p[j][2] = pointmat(2, j);
c[j] = (*sol)(vertices[j]);
}
if (j == 3)
DrawTriangle(p, c, minv, maxv);
else
DrawQuad(p, c, minv, maxv);
}
glEndList();
}
void MakeDiagonallyDominant(DenseMatrix<T>& M)
{
// Make the diagonal element larger than the row sum, to ensure that
// the matrix is nonsingular. All entries in the matrix are nonnegative,
// so no absolute values are needed.
for (int r=0; r<M.Height(); ++r)
{
T row_sum = 0.0;
for (int c=0; c<M.Width(); ++c)
row_sum += M.Get(r, c);
M.Set(r, r, row_sum + T(1));
}
}
void OptimalActiveSetH(DenseMatrix<T>& H, // 2 x n
const DenseMatrix<T>& WtW, // 2 x 2
const DenseMatrix<T>& WtA) // 2 x n
{
// Remove negative entries in each of the n columns of matrix H, but
// do so in a manner that minimizes the overall objective function.
// Each column can be considered in isolation.
//
// The problem for each column i of H, h = H(:,i), is as follows:
//
// min_{h>=0} |Wh - a|^2
//
// This is minimized when h* = w'a / (w'w). Expressed in terms of the
// individual elements, this becomes:
//
// h*[0] = w[0]'a / (w[0]'w[0])
// h*[1] = w[1]'a / (w[1]'w[1])
// If both elements of h* are nonnegative, this is the optimal solution.
// If any elements are negative, the Rank2 algorithm is used to adjust
// their values.
int width = H.Width();
const T wtw_00 = WtW.Get(0,0);
const T wtw_11 = WtW.Get(1,1);
const T inv_wtw_00 = T(1.0) / wtw_00;
const T inv_wtw_11 = T(1.0) / wtw_11;
const T sqrt_wtw_00 = sqrt(wtw_00);
const T sqrt_wtw_11 = sqrt(wtw_11);
for (int i=0; i<width; ++i)
{
T v1 = WtA.Get(0,i) * inv_wtw_00;
T v2 = WtA.Get(1,i) * inv_wtw_11;
T vv1 = v1 * sqrt_wtw_00;
T vv2 = v2 * sqrt_wtw_11;
if (vv1 >= vv2)
v2 = T(0);
else
v1 = T(0);
if ( (H.Get(0,i) <= T(0)) || (H.Get(1,i) <= T(0)))
{
H.Set(0, i, v1);
H.Set(1, i, v2);
}
}
}
void IsoparametricTransformation::Transform (const DenseMatrix &matrix,
DenseMatrix &result)
{
result.SetSize(PointMat.Height(), matrix.Width());
IntegrationPoint ip;
Vector col;
for (int j = 0; j < matrix.Width(); j++)
{
ip.x = matrix(0, j);
if (matrix.Height() > 1)
{
ip.y = matrix(1, j);
if (matrix.Height() > 2)
{
ip.z = matrix(2, j);
}
}
result.GetColumnReference(j, col);
Transform(ip, col);
}
}
bool HasNaNs(const DenseMatrix<T>& M)
{
// returns true if any matrix element is NaN
int height = M.Height();
int width = M.Width();
for (int c=0; c<width; ++c)
{
for (int r=0; r<height; ++r)
{
T elt = M.Get(r, c);
if (std::isnan(elt))
return true;
}
}
return false;
}
// helper to set submatrix of A repeated vdim times
static void SetVDofSubMatrixTranspose(SparseMatrix& A,
Array<int>& rows, Array<int>& cols,
const DenseMatrix& subm, int vdim)
{
if (vdim == 1)
{
A.SetSubMatrixTranspose(rows, cols, subm, 1);
}
else
{
int nr = subm.Width(), nc = subm.Height();
for (int d = 0; d < vdim; d++)
{
Array<int> rows_sub(rows.GetData() + d*nr, nr); // (not owner)
Array<int> cols_sub(cols.GetData() + d*nc, nc); // (not owner)
A.SetSubMatrixTranspose(rows_sub, cols_sub, subm, 1);
}
}
}
void Overwrite(DenseMatrix<T>& A,
const DenseMatrix<T>& B,
const std::vector<unsigned int>& row_indices,
const std::vector<unsigned int>& col_indices,
const unsigned int num_rows,
const unsigned int num_cols)
{
// Overwrite entries in A with entries in B. The row and column
// index arrays contain the destination indices to be overwritten.
// Matrix B has size num_rows x num_cols.
if (num_rows > static_cast<unsigned int>(A.Height()))
throw std::logic_error("Overwrite: row indices out of bounds");
if (num_cols > static_cast<unsigned int>(A.Width()))
throw std::logic_error("Overwrite: col indices out of bounds");
const T* buf_b = B.LockedBuffer();
const unsigned int ldim_b = B.LDim();
T* buf_a = A.Buffer();
const unsigned int ldim_a = A.LDim();
for (unsigned int c=0; c<num_cols; ++c)
{
unsigned int col_a = col_indices[c];
unsigned int offset_a = ldim_a * col_a;
unsigned int offset_b = ldim_b * c;
for (unsigned int r=0; r<num_rows; ++r)
{
unsigned int row_a = row_indices[r];
//T val = B.Get(r, c);
//A.Set(row_a, col_a, val);
buf_a[offset_a + row_a] = buf_b[offset_b + r];
}
}
}
void ZeroizeSmallValues(DenseMatrix<T>& A, const T tol)
{
// set Aij to zero if |Aij| < tol
T* buf = A.Buffer();
const unsigned int ldim = A.LDim();
const unsigned int height = A.Height();
const unsigned int width = A.Width();
OPENMP_PRAGMA(omp parallel for)
for (unsigned int c=0; c<width; ++c)
{
unsigned int col_offset = c*ldim;
for (unsigned int r=0; r<height; ++r)
{
T val = buf[col_offset + r];
if (std::abs(val) < tol)
{
buf[col_offset + r] = T(0);
}
}
}
}
bool SystemSolveH(DenseMatrix<T>& X, // H 2 x n
DenseMatrix<T>& A, // WtW 2 x 2
DenseMatrix<T>& B) // WtA 2 x n
{
// This function solves the system A*X = B for X. The columns of X and B
// are treated as separate subproblems. Each subproblem can be expressed
// as follows, for column i of X and B (the ' means transpose):
//
// A * [x0 x1]' = [b0 b1]'
//
// The solution proceeds by transforming matrix A to upper triangular form
// via a fast Givens rotation. The resulting triangular system for the
// two unknowns in each column of X is then solved by backsubstitution.
//
// The fast Givens method relies on the fact that matrix A can be
// expressed as the product of a diagonal matrix D and another matrix Y,
// and that this form is preserved under the rotation. If J is the
// Givens rotation matrix, then:
//
// AX = B
// JAX = JB J is the Givens rotation matrix
// (JA)X = JB letting A = D1Y1
// (JD1Y1)X = JB
// (D2Y2)X = JB
//
// The matrix D1 is the identity matrix, and the matrix Y1 is the
// original matrix A. The matrix J has the form [c -s; c s] in Matlab
// notation. NOTE: with this convention, the tangent has a minus sign:
//
// t = -A(1, 0) / A(0, 0)
//
// Matrix A has the form [a b; c d], and after the rotation it is
// transformed to [a2 b2; 0 d2].
//
// There are two forms for the D2 and Y2 matrices, depending on whether
// cosines or sines are 'factored out'. If |A00| >= |A01|, then the
// upper left element is factored out and the 'cosine' formulation is
// used. If |A00| < |A01|, then the upper right element is factored
// out and the 'sine' formulation is used.
//
// A general reference for this code is the paper 'Fast Plane Rotations
// with Dynamic Scaling', by A. Anda and H. Park, SIAM Journal on Matrix
// Analysis and Applications, Vol 15, no. 1, pp. 162-174, Jan. 1994.
int n = B.Width();
T abs_A00 = std::abs(A.Get(0, 0));
T abs_A01 = std::abs(A.Get(0, 1));
const T epsilon = std::numeric_limits<T>::epsilon();
if ( (abs_A00 < epsilon) && (abs_A01 < epsilon))
{
std::cerr << "SystemSolveH: singular matrix" << std::endl;
return false;
}
T a2, b2, d2, e2, f2, inv_a2, inv_d2, x_1;
if (abs_A00 >= abs_A01)
{
// use 'cosine' formulation; t is the tangent
T t = -A.Get(1,0) / A.Get(0,0);
a2 = A.Get(0,0) - t*A.Get(1,0);
b2 = A.Get(0,1) - t*A.Get(1,1);
d2 = A.Get(1,1) + t*A.Get(0,1);
// precompute 1/a2 and 1/d2 to avoid repeated division
inv_a2 = T(1.0) / a2;
inv_d2 = T(1.0) / d2;
// a2 is guaranteed to be positive
if (std::abs(d2/a2) < epsilon)
return false;
// solve the upper triangular systems by backsubstitution
for (int i=0; i<n; ++i)
{
e2 = B.Get(0,i) - t*B.Get(1,i);
f2 = B.Get(1,i) + t*B.Get(0,i);
x_1 = f2 * inv_d2;
X.Set(1, i, x_1);
X.Set(0, i, (e2 - b2*x_1)*inv_a2);
}
}
else
{
// use 'sine' formulation; ct is the cotangent
T ct = -A.Get(0,0) / A.Get(1,0);
a2 = -A.Get(1,0) + ct*A.Get(0,0);
b2 = -A.Get(1,1) + ct*A.Get(0,1);
d2 = A.Get(0,1) + ct*A.Get(1,1);
// precompute 1/a2 and 1/d2 to avoid repeated division
inv_a2 = T(1.0) / a2;
inv_d2 = T(1.0) / d2;
// a2 is guaranteed to be positive
if (std::abs(d2/a2) < epsilon)
return false;
// solve the upper triangular systems by backsubstitution
for (int i=0; i<n; ++i)
{
e2 = -B.Get(1,i) + ct*B.Get(0,i);
f2 = B.Get(0,i) + ct*B.Get(1,i);
x_1 = f2 * inv_d2;
X.Set(1, i, x_1);
X.Set(0, i, (e2 - b2*x_1) * inv_a2);
}
}
return true;
}
void BppUpdateSets(BitMatrix& nonopt_set,
BitMatrix& infeas_set,
const BitMatrix& not_opt_mask,
const DenseMatrix<T>& X,
const DenseMatrix<T>& Y,
const BitMatrix& passive_set)
{
// This function performs the equivalent of these operations:
//
// nonopt_set = not_opt_mask & (Y < T(0)) & ~passive_set;
// infeas_set = not_opt_mask & (X < T(0)) & passive_set;
const unsigned int height = not_opt_mask.Height();
const unsigned int width = not_opt_mask.Width();
if ( (static_cast<unsigned int>(X.Height()) != height) ||
(static_cast<unsigned int>(Y.Height()) != height) ||
(passive_set.Height() != height))
throw std::logic_error("BppUpdateSets: height mismatch");
if ( (static_cast<unsigned int>(X.Width()) != width) ||
(static_cast<unsigned int>(Y.Width()) != width) ||
(passive_set.Width() != width))
throw std::logic_error("BppUpdateSets: width mismatch");
nonopt_set.Resize(height, width);
infeas_set.Resize(height, width);
const unsigned int BITS = BitMatrix::BITS_PER_WORD;
const unsigned int MASK = nonopt_set.Mask();
assert(infeas_set.Mask() == MASK);
unsigned int* buf_r = nonopt_set.Buffer();
const unsigned int ldim_r = nonopt_set.LDim();
unsigned int* buf_i = infeas_set.Buffer();
const unsigned int ldim_i = infeas_set.LDim();
const unsigned int* buf_m = not_opt_mask.LockedBuffer();
const unsigned int ldim_m = not_opt_mask.LDim();
const T* buf_x = X.LockedBuffer();
const unsigned int ldim_x = X.LDim();
const T* buf_y = Y.LockedBuffer();
const unsigned int ldim_y = Y.LDim();
const unsigned int* buf_p = passive_set.LockedBuffer();
const unsigned int ldim_p = passive_set.LDim();
const unsigned int full_wds = height / BITS;
const unsigned int extra = height - BITS*full_wds;
assert(ldim_r >= ldim_m);
assert(ldim_r >= ldim_p);
OPENMP_PRAGMA(omp parallel for)
for (unsigned int c=0; c<width; ++c)
{
unsigned int offset_r = c*ldim_r;
unsigned int offset_i = c*ldim_i;
unsigned int offset_m = c*ldim_m;
unsigned int offset_x = c*ldim_x;
unsigned int offset_y = c*ldim_y;
unsigned int offset_p = c*ldim_p;
unsigned int r_wd = 0, r=0;
for (; r_wd<full_wds; ++r_wd)
{
unsigned int x_wd = 0, y_wd = 0;
for (unsigned int q=0; q<BITS; ++q, ++r)
{
if (buf_x[offset_x + r] < T(0))
x_wd |= (1 << q);
if (buf_y[offset_y + r] < T(0))
y_wd |= (1 << q);
}
buf_r[offset_r + r_wd] = buf_m[offset_m + r_wd] & y_wd & ~buf_p[offset_p + r_wd];
buf_i[offset_i + r_wd] = buf_m[offset_m + r_wd] & x_wd & buf_p[offset_p + r_wd];
}
if (extra > 0)
{
unsigned int x_wd = 0, y_wd = 0;
for (unsigned int q=0; q<extra; ++q, ++r)
{
if (buf_x[offset_x + r] < T(0))
x_wd |= (1 << q);
if (buf_y[offset_y + r] < T(0))
y_wd |= (1 << q);
}
buf_r[offset_r + r_wd] = MASK & buf_m[offset_m + r_wd] & y_wd & ~buf_p[offset_p + r_wd];
buf_i[offset_i + r_wd] = MASK & buf_m[offset_m + r_wd] & x_wd & buf_p[offset_p + r_wd];
}
}
}
bool NnlsHals(const MatrixType<T>& A,
DenseMatrix<T>& W,
DenseMatrix<T>& H,
const T tol,
const bool verbose,
const unsigned int max_iter)
{
unsigned int n = A.Width();
unsigned int k = W.Width();
if (static_cast<unsigned int>(W.Height()) != static_cast<unsigned int>(A.Height()))
throw std::logic_error("NnlsHals: W and A must have identical height");
if (static_cast<unsigned int>(H.Width()) != static_cast<unsigned int>(A.Width()))
throw std::logic_error("NnlsHals: H and A must have identical width");
if (H.Height() != W.Width())
throw std::logic_error("NnlsHals: non-conformant W and H");
DenseMatrix<T> WtW(k, k), WtA(k, n), WtWH_r(1, n), gradH(k, n);
if (verbose)
std::cout << "\nRunning NNLS solver..." << std::endl;
// compute W'W and W'A for the normal equations
Gemm(TRANSPOSE, NORMAL, T(1.0), W, W, T(0.0), WtW);
Gemm(TRANSPOSE, NORMAL, T(1.0), W, A, T(0.0), WtA);
bool success = false;
T pg0 = T(0), pg;
for (unsigned int i=0; i<max_iter; ++i)
{
// compute the new matrix H
UpdateH_Hals(H, WtWH_r, WtW, WtA);
// compute gradH = WtW*H - WtA
Gemm(NORMAL, NORMAL, T(1.0), WtW, H, T(0.0), gradH);
Axpy( T(-1.0), WtA, gradH);
// compute progress metric
if (0 == i)
{
pg0 = ProjectedGradientNorm(gradH, H);
if (verbose)
ReportProgress(i+1, T(1.0));
continue;
}
else
{
pg = ProjectedGradientNorm(gradH, H);
}
if (verbose)
ReportProgress(i+1, pg/pg0);
// check progress vs. desired tolerance
if (pg < tol * pg0)
{
success = true;
NormalizeAndScale<T>(W, H);
break;
}
}
if (!success)
std::cerr << "NNLS solver reached iteration limit." << std::endl;
return success;
}
bool NnlsBlockpivot(const DenseMatrix<T>& LHS,
const DenseMatrix<T>& RHS,
DenseMatrix<T>& X, // input as xinit
DenseMatrix<T>& Y) // gradX
{
// Solve (LHS)*X = RHS for X by block principal pivoting. Matrix LHS
// is assumed to be symmetric positive definite.
const int PBAR = 3;
const unsigned int WIDTH = RHS.Width();
const unsigned int HEIGHT = RHS.Height();
const unsigned int MAX_ITER = HEIGHT*5;
BitMatrix passive_set = (X > T(0));
std::vector<unsigned int> tmp_indices(WIDTH);
for (unsigned int i=0; i<WIDTH; ++i)
tmp_indices[i] = i;
MakeZeros(X);
if (!BppSolveNormalEqNoGroup(tmp_indices, passive_set, LHS, RHS, X))
return false;
// Y = LHS * X - RHS
Gemm(NORMAL, NORMAL, T(1), LHS, X, T(0), Y);
Axpy( T(-1), RHS, Y);
std::vector<int> P(WIDTH, PBAR), Ninf(WIDTH, HEIGHT+1);
BitMatrix nonopt_set = (Y < T(0)) & ~passive_set;
BitMatrix infeas_set = (X < T(0)) & passive_set;
std::vector<int> col_sums(WIDTH);
std::vector<int> not_good(WIDTH);
nonopt_set.SumColumns(not_good);
infeas_set.SumColumns(col_sums);
not_good += col_sums;
BitMatrix not_opt_cols = (not_good > 0);
BitMatrix not_opt_mask;
std::vector<unsigned int> non_opt_col_indices(WIDTH);
not_opt_cols.Find(non_opt_col_indices);
DenseMatrix<double> RHSsub(HEIGHT, WIDTH);
DenseMatrix<double> Xsub(HEIGHT, WIDTH);
DenseMatrix<double> Ysub(HEIGHT, WIDTH);
unsigned int iter = 0;
while (!non_opt_col_indices.empty())
{
// exit if not getting anywhere
if (iter >= MAX_ITER)
return false;
UpdatePassiveSet(passive_set, PBAR, HEIGHT,
not_opt_cols, nonopt_set, infeas_set,
not_good, P, Ninf);
// equivalent of repmat(NotOptCols, HEIGHT, 1)
not_opt_mask = MatrixFromColumnMask(not_opt_cols, HEIGHT);
// Setup for the normal equation solver by extracting submatrices
// from RHS and X. The normal equation solver will extract further
// subproblems from RHSsub and Xsub and write all updated values
// back into RHSsub and Xsub.
RHS.SubmatrixFromCols(RHSsub, non_opt_col_indices);
X.SubmatrixFromCols(Xsub, non_opt_col_indices);
if (!BppSolveNormalEqNoGroup(non_opt_col_indices, passive_set, LHS, RHSsub, Xsub))
return false;
ZeroizeSmallValues(Xsub, 1.0e-12);
// compute Ysub = LHS * Xsub - RHSsub
Ysub.Resize(RHSsub.Height(), RHSsub.Width());
Gemm(NORMAL, NORMAL, T(1), LHS, Xsub, T(0), Ysub);
Axpy( T(-1), RHSsub, Ysub);
// update Y and X using the new values in Ysub and Xsub
OverwriteCols(Y, Ysub, non_opt_col_indices, non_opt_col_indices.size());
OverwriteCols(X, Xsub, non_opt_col_indices, non_opt_col_indices.size());
ZeroizeSmallValues(X, 1.0e-12);
ZeroizeSmallValues(Y, 1.0e-12);
// Check optimality - BppUpdateSets does the equivalent of the next two lines.
// nonopt_set = not_opt_mask & (Y < T(0)) & ~passive_set;
// infeas_set = not_opt_mask & (X < T(0)) & passive_set;
BppUpdateSets(nonopt_set, infeas_set, not_opt_mask, X, Y, passive_set);
nonopt_set.SumColumns(not_good);
infeas_set.SumColumns(col_sums);
not_good += col_sums;
not_opt_cols = (not_good > 0);
not_opt_cols.Find(non_opt_col_indices);
++iter;
}
return true;
}
void VisualizationSceneVector3d::Prepare()
{
int i,j;
switch (shading)
{
case 0:
PrepareFlat();
return;
case 2:
PrepareFlat2();
return;
default:
break;
}
glNewList(displlist, GL_COMPILE);
int ne = mesh -> GetNBE();
int nv = mesh -> GetNV();
DenseMatrix pointmat;
Array<int> vertices;
double nor[3];
Vector nx(nv);
Vector ny(nv);
Vector nz(nv);
for (int d = 0; d < mesh -> bdr_attributes.Size(); d++)
{
if (!bdr_attr_to_show[mesh -> bdr_attributes[d]-1]) continue;
nx=0.;
ny=0.;
nz=0.;
for (i = 0; i < ne; i++)
if (mesh -> GetBdrAttribute(i) == mesh -> bdr_attributes[d])
{
mesh->GetBdrPointMatrix (i, pointmat);
mesh->GetBdrElementVertices (i, vertices);
for (j = 0; j < pointmat.Size(); j++)
{
pointmat(0, j) += (*solx)(vertices[j])*(ianim)/ianimmax;
pointmat(1, j) += (*soly)(vertices[j])*(ianim)/ianimmax;
pointmat(2, j) += (*solz)(vertices[j])*(ianim)/ianimmax;
}
if (pointmat.Width() == 3)
j = Compute3DUnitNormal(&pointmat(0,0), &pointmat(0,1),
&pointmat(0,2), nor);
else
j = Compute3DUnitNormal(&pointmat(0,0), &pointmat(0,1),
&pointmat(0,2), &pointmat(0,3), nor);
if (j == 0)
for (j = 0; j < pointmat.Size(); j++)
{
nx(vertices[j]) += nor[0];
ny(vertices[j]) += nor[1];
nz(vertices[j]) += nor[2];
}
}
for (i = 0; i < ne; i++)
if (mesh -> GetBdrAttribute(i) == mesh -> bdr_attributes[d])
{
switch (mesh->GetBdrElementType(i))
{
case Element::TRIANGLE:
glBegin (GL_TRIANGLES);
break;
case Element::QUADRILATERAL:
glBegin (GL_QUADS);
break;
}
mesh->GetBdrPointMatrix (i, pointmat);
mesh->GetBdrElementVertices (i, vertices);
for (j = 0; j < pointmat.Size(); j++)
{
pointmat(0, j) += (*solx)(vertices[j])*(ianim)/ianimmax;
pointmat(1, j) += (*soly)(vertices[j])*(ianim)/ianimmax;
pointmat(2, j) += (*solz)(vertices[j])*(ianim)/ianimmax;
}
for (j = 0; j < pointmat.Size(); j++)
{
MySetColor((*sol)(vertices[j]), minv, maxv);
glNormal3d(nx(vertices[j]), ny(vertices[j]), nz(vertices[j]));
glVertex3dv(&pointmat(0, j));
}
glEnd();
}
}
glEndList();
}
bool SolveNormalEqLeft(const DenseMatrix<T>& LHS, // kxk
const DenseMatrix<T>& RHS, // mxk
DenseMatrix<T>& X) // mxk
{
// Solve X * LHS = RHS for X, where LHS is symmetric positive-definite.
if (LHS.Width() != LHS.Height())
throw std::logic_error("SolveNormalEqLeft: expected square matrix on LHS");
if ( (X.Width() != LHS.Height()) || (X.Height() != RHS.Height()))
throw std::logic_error("SolveNormalEqLeft: non-conformant matrix X");
// // copy the input, since the solver overwrites it
DenseMatrix<T> U(LHS);
// Compute the Cholesky factor LHS = U'U
bool success = true;
try
{
Cholesky(UPPER, U);
}
catch (EL::NonHPSDMatrixException& e)
{
std::cerr << "Cholesky factorization failure - ";
std::cerr << "matrix was not symmetric positive-definite." << std::endl;
success = false;
}
catch (std::logic_error& e)
{
std::cerr << "Cholesky factorization failure - ";
std::cerr << "matrix was not symmetric positive-definite." << std::endl;
success = false;
}
if (!success)
return false;
// Solve X (U'U) = RHS as follows:
//
// Group the two left matrices:
//
// (XU') U = RHS
//
// Let Y = XU', so the equation becomes:
//
// YU = RHS
//
// Do a triangular solve: Y = (RHS) * inverse(U)
X = RHS;
Trsm(RIGHT, UPPER, NORMAL, NON_UNIT, T(1), U, X);
// The solution Y is stored in X.
//
// Now solve XU' = Y, so X = Y * inverse(U')
Trsm(RIGHT, UPPER, TRANSPOSE, NON_UNIT, T(1), U, X);
// check
// DenseMatrix<T> temp(m, k);
// Gemm(NORMAL, NORMAL, T(1), X, LHS, T(0), temp);
// Axpy(-1.0, RHS, temp);
// double norm = Norm(temp, FROBENIUS_NORM);
// std::cout << "\tSolveNormalEqLeft: norm = " << norm << std::endl;
return true;
}
MatrixConstantCoefficient(const DenseMatrix &m)
: MatrixCoefficient(m.Height(), m.Width()), mat(m) { }