public void TestUpdating()
{
// first row
var m = DoubleFactory2D.Dense.Make(_a.ViewColumn(0).ToArray(), _a.Rows);
var initialSVD = new SingularValueDecomposition(m);
var u = initialSVD.U;
var s = initialSVD.S;
var v = initialSVD.V;
// second row
var d = DoubleFactory2D.Dense.Make(_a.ViewColumn(1).ToArray(), _a.Rows);
var l = u.ViewDice().ZMult(d, null);
var uu = u.ZMult(u.ViewDice(), null);
var ul = uu.ZMult(d, null);
var h = d.Copy().Assign(uu.ZMult(d, null), BinaryFunctions.Minus);
var k = Math.Sqrt(d.Aggregate(BinaryFunctions.Plus, a => a * a) - (2 * l.Aggregate(BinaryFunctions.Plus, a => a * a)) + ul.Aggregate(BinaryFunctions.Plus, a => a * a));
var j1 = h.Assign(UnaryFunctions.Div(k));
Assert.AreEqual(j1.Assign(UnaryFunctions.Mult(k)), h);
var q =
DoubleFactory2D.Dense.Compose(
new[] { new[] { s, l }, new[] { null, DoubleFactory2D.Dense.Make(1, 1, k) } });
var svdq = new SingularValueDecomposition(q);
var u2 = DoubleFactory2D.Dense.AppendColumns(u, j1).ZMult(svdq.U, null);
var s2 = svdq.S;
var v2 = DoubleFactory2D.Dense.ComposeDiagonal(v, DoubleFactory2D.Dense.Identity(1)).ZMult(
svdq.V, null);
var svd = new SingularValueDecomposition(_a.ViewPart(0, 0, _a.Rows, 2));
Assert.AreEqual(svd.S, s2);
}
/// <summary>
/// Outer product of two vectors; Sets <i>A[i,j] = x[i] * y[j]</i>.
/// </summary>
/// <param name="x">the first source vector.</param>
/// <param name="y">the second source vector.</param>
/// <param name="A">the matrix to hold the resultsd Set this parameter to <i>null</i> to indicate that a new result matrix shall be constructed.</param>
/// <returns>A (for convenience only).</returns>
public static DoubleMatrix2D MultOuter(DoubleMatrix1D x, DoubleMatrix1D y, DoubleMatrix2D A)
{
int rows = x.Size;
int columns = y.Size;
if (A == null)
{
A = x.Like2D(rows, columns);
}
if (A.Rows != rows || A.Columns != columns)
{
throw new ArgumentException();
}
for (int row = rows; --row >= 0;)
{
A.ViewRow(row).Assign(y);
}
for (int column = columns; --column >= 0;)
{
A.ViewColumn(column).Assign(x, BinaryFunctions.Mult);
}
return(A);
}
/// <summary>
/// Constructs and returns the covariance matrix of the given matrix.
/// The covariance matrix is a square, symmetric matrix consisting of nothing but covariance coefficientsd
/// The rows and the columns represent the variables, the cells represent covariance coefficientsd
/// The diagonal cells (i.ed the covariance between a variable and itself) will equal the variances.
/// The covariance of two column vectors x and y is given by <i>cov(x,y) = (1/n) * Sum((x[i]-mean(x)) * (y[i]-mean(y)))</i>.
/// See the <A HREF="http://www.cquest.utoronto.ca/geog/ggr270y/notes/not05efg.html"> math definition</A>.
/// Compares two column vectors at a timed Use dice views to compare two row vectors at a time.
/// </summary>
/// <param name="matrix">any matrix; a column holds the values of a given variable.</param>
/// <returns>the covariance matrix (<i>n x n, n=matrix.Columns</i>).</returns>
public static DoubleMatrix2D Covariance(DoubleMatrix2D matrix)
{
int rows = matrix.Rows;
int columns = matrix.Columns;
DoubleMatrix2D covariance = new DenseDoubleMatrix2D(columns, columns);
double[] sums = new double[columns];
DoubleMatrix1D[] cols = new DoubleMatrix1D[columns];
for (int i = columns; --i >= 0;)
{
cols[i] = matrix.ViewColumn(i);
sums[i] = cols[i].ZSum();
}
for (int i = columns; --i >= 0;)
{
for (int j = i + 1; --j >= 0;)
{
double sumOfProducts = cols[i].ZDotProduct(cols[j]);
double cov = (sumOfProducts - sums[i] * sums[j] / rows) / rows;
covariance[i, j] = cov;
covariance[j, i] = cov; // symmetric
}
}
return(covariance);
}
/// <summary>
/// Sorts the matrix rows into ascending order, according to the <i>natural ordering</i> of the matrix values in the given column.
/// </summary>
/// <param name="matrix">
/// The matrix to be sorted.
/// </param>
/// <param name="column">
/// The index of the column inducing the order.
/// </param>
/// <returns>
/// A new matrix view having rows sorted by the given column.
/// </returns>
/// <exception cref="IndexOutOfRangeException">
/// If <i>column < 0 || column >= matrix.columns()</i>.
/// </exception>
public DoubleMatrix2D Sort(DoubleMatrix2D matrix, int column)
{
if (column < 0 || column >= matrix.Columns)
{
throw new IndexOutOfRangeException("column=" + column + ", matrix=" + AbstractFormatter.Shape(matrix));
}
var rowIndexes = new int[matrix.Rows]; // row indexes to reorder instead of matrix itself
for (int i = rowIndexes.Length; --i >= 0;)
{
rowIndexes[i] = i;
}
DoubleMatrix1D col = matrix.ViewColumn(column);
RunSort(
rowIndexes,
0,
rowIndexes.Length,
(a, b) =>
{
double av = col[a];
double bv = col[b];
if (Double.IsNaN(av) || Double.IsNaN(bv))
{
return(CompareNaN(av, bv)); // swap NaNs to the end
}
return(av < bv ? -1 : (av == bv ? 0 : 1));
});
// view the matrix according to the reordered row indexes
// take all columns in the original order
return(matrix.ViewSelection(rowIndexes, null));
}
/// <summary>
/// Returns the one-norm of matrix <i>A</i>, which is the maximum absolute column sum.
/// </summary>
/// <param name="A">The matrix A</param>
/// <returns>The one-norm of A.</returns>
public static double Norm1(DoubleMatrix2D A)
{
double max = 0;
for (int column = A.Columns; --column >= 0;)
{
max = System.Math.Max(max, Norm1(A.ViewColumn(column)));
}
return(max);
}
/// <summary>
/// Same as <see cref="Cern.Colt.Partitioning.Partition(int[], int, int, int[], int, int, int[])"/>
/// except that it <i>synchronously</i> partitions the rows of the given matrix by the values of the given matrix column;
/// This is essentially the same as partitioning a list of composite objects by some instance variable;
/// In other words, two entire rows of the matrix are swapped, whenever two column values indicate so.
/// <p>
/// Let's say, a "row" is an "object" (tuple, d-dimensional point).
/// A "column" is the list of "object" values of a given variable (field, dimension).
/// A "matrix" is a list of "objects" (tuples, points).
/// <p>
/// Now, rows (objects, tuples) are partially sorted according to their values in one given variable (dimension).
/// Two entire rows of the matrix are swapped, whenever two column values indicate so.
/// <p>
/// Note that arguments are not checked for validity.
///
/// </summary>
/// <param name="matrix">
/// the matrix to be partitioned.
/// </param>
/// <param name="rowIndexes">
/// the index of the i-th row; is modified by this method to reflect partitioned indexes.
/// </param>
/// <param name="rowFrom">
/// the index of the first row (inclusive).
/// </param>
/// <param name="rowTo">
/// the index of the last row (inclusive).
/// </param>
/// <param name="column">
/// the index of the column to partition on.
/// </param>
/// <param name="splitters">
/// the values at which the rows shall be split into intervals.
/// Must be sorted ascending and must not contain multiple identical values.
/// These preconditions are not checked; be sure that they are met.
/// </param>
/// <param name="splitFrom">
/// the index of the first splitter element to be considered.
/// </param>
/// <param name="splitTo">
/// the index of the last splitter element to be considered.
/// The method considers the splitter elements<i>splitters[splitFrom] .d splitters[splitTo]</i>.
/// </param>
/// <param name="splitIndexes">
/// a list into which this method fills the indexes of rows delimiting intervals.
/// Upon return <i>splitIndexes[splitFrom..splitTo]</i> will be set accordingly.
/// Therefore, must satisfy <i>splitIndexes.Length >= splitters.Length</i>.
/// </param>
/// <example>
/// <table border="1" cellspacing="0">
/// <tr nowrap>
/// <td valign="top"><i>8 x 3 matrix:<br>
/// 23, 22, 21<br>
/// 20, 19, 18<br>
/// 17, 16, 15<br>
/// 14, 13, 12<br>
/// 11, 10, 9<br>
/// 8, 7, 6<br>
/// 5, 4, 3<br>
/// 2, 1, 0 </i></td>
/// <td align="left" valign="top">
/// <p><i>column = 0;<br>
/// rowIndexes = {0,1,2,.d,matrix.Rows-1};
/// rowFrom = 0;<br>
/// rowTo = matrix.Rows-1;<br>
/// splitters = {5,10,12}<br>
/// c = 0; <br>
/// d = splitters.Length-1;<br>
/// partition(matrix,rowIndexes,rowFrom,rowTo,column,splitters,c,d,splitIndexes);<br>
/// ==><br>
/// splitIndexes == {0, 2, 3}<br>
/// rowIndexes == {7, 6, 5, 4, 0, 1, 2, 3}</i></p>
/// </td>
/// <td valign="top">
/// The matrix IS NOT REORDERED.<br>
/// Here is how it would look<br>
/// like, if it would be reordered<br>
/// accoring to <i>rowIndexes</i>.<br>
/// <i>8 x 3 matrix:<br>
/// 2, 1, 0<br>
/// 5, 4, 3<br>
/// 8, 7, 6<br>
/// 11, 10, 9<br>
/// 23, 22, 21<br>
/// 20, 19, 18<br>
/// 17, 16, 15<br>
/// 14, 13, 12 </i></td>
/// </tr>
/// </table>
/// </example>
public static void Partition(DoubleMatrix2D matrix, int[] rowIndexes, int rowFrom, int rowTo, int column, double[] splitters, int splitFrom, int splitTo, int[] splitIndexes)
{
if (rowFrom < 0 || rowTo >= matrix.Rows || rowTo >= rowIndexes.Length)
{
throw new ArgumentException();
}
if (column < 0 || column >= matrix.Columns)
{
throw new ArgumentException();
}
if (splitFrom < 0 || splitTo >= splitters.Length)
{
throw new ArgumentException();
}
if (splitIndexes.Length < splitters.Length)
{
throw new ArgumentException();
}
// this one knows how to swap two row indexes (a,b)
int[] g = rowIndexes;
Swapper swapper = new Swapper((b, c) =>
{
int tmp = g[b]; g[b] = g[c]; g[c] = tmp;
});
// compare splitter[a] with columnView[rowIndexes[b]]
DoubleMatrix1D columnView = matrix.ViewColumn(column);
IntComparator comp = new IntComparator((a, b) =>
{
double av = splitters[a];
double bv = columnView[g[b]];
return(av < bv ? -1 : (av == bv ? 0 : 1));
});
// compare columnView[rowIndexes[a]] with columnView[rowIndexes[b]]
IntComparator comp2 = new IntComparator((a, b) =>
{
double av = columnView[g[a]];
double bv = columnView[g[b]];
return(av < bv ? -1 : (av == bv ? 0 : 1));
});
// compare splitter[a] with splitter[b]
IntComparator comp3 = new IntComparator((a, b) =>
{
double av = splitters[a];
double bv = splitters[b];
return(av < bv ? -1 : (av == bv ? 0 : 1));
});
// generic partitioning does the main work of reordering row indexes
Cern.Colt.Partitioning.GenericPartition(rowFrom, rowTo, splitFrom, splitTo, splitIndexes, comp, comp2, comp3, swapper);
}
/// <summary>
/// A matrix <i>A</i> is <i>diagonally dominant by column</i> if the absolute value of each diagonal element is larger than the sum of the absolute values of theoff-diagonal elements in the corresponding column.
/// <i>returns true if for all i: abs(A[i,i]) > Sum(abs(A[j,i])); j != i.</i>
/// Matrix may but need not be square.
/// <p>
/// Note: Ignores tolerance.
/// <summary>
public Boolean IsDiagonallyDominantByColumn(DoubleMatrix2D A)
{
Cern.Jet.Math.Functions F = Cern.Jet.Math.Functions.functions;
double epsilon = Tolerance;
int min = System.Math.Min(A.Rows, A.Columns);
for (int i = min; --i >= 0;)
{
double diag = System.Math.Abs(A[i, i]);
diag += diag;
if (diag <= A.ViewColumn(i).Aggregate(F2.Plus, F1.Abs))
{
return(false);
}
}
return(true);
}
/// <summary>
/// Modifies the given matrix square matrix<i>A</i> such that it is diagonally dominant by row and column, hence non-singular, hence invertible.
/// For testing purposes only.
/// <summary>
///<param name = "A" > the square matrix to modify.</param>
///<exception cref = "ArgumentException" >if <i>!isSquare(A)</i>. </exception>
public void GenerateNonSingular(DoubleMatrix2D A)
{
CheckSquare(A);
var F = Cern.Jet.Math.Functions.functions;
int min = System.Math.Min(A.Rows, A.Columns);
for (int i = min; --i >= 0;)
{
A[i, i] = 0;
}
for (int i = min; --i >= 0;)
{
double rowSum = A.ViewRow(i).Aggregate(F2.Plus, F1.Abs);
double colSum = A.ViewColumn(i).Aggregate(F2.Plus, F1.Abs);
A[i, i] = System.Math.Max(rowSum, colSum) + i + 1;
}
}
/// <summary>
/// Applies the given aggregation functions to each column and stores the results in a the result matrix.
/// If matrix has shape <i>m x n</i>, then result must have shape <i>aggr.Length x n</i>.
/// Tip: To do aggregations on rows use dice views (transpositions), as in <i>aggregate(matrix.viewDice(),aggr,result.viewDice())</i>.
/// </summary>
/// <param name="matrix">any matrix; a column holds the values of a given variable.</param>
/// <param name="aggr">the aggregation functions to be applied to each column.</param>
/// <param name="result">the matrix to hold the aggregation results.</param>
/// <returns><i>result</i> (for convenience only).</returns>
/// <see cref="Formatter"/>
/// <see cref="Hep.Aida.Bin.BinFunction1D"/>
/// <see cref="Hep.Aida.Bin.BinFunctions1D"/>
public static DoubleMatrix2D Aggregate(DoubleMatrix2D matrix, Hep.Aida.Bin.BinFunction1D[] aggr, DoubleMatrix2D result)
{
var bin = new DynamicBin1D();
var elements = new double[matrix.Rows];
var values = elements;
for (int column = matrix.Columns; --column >= 0;)
{
matrix.ViewColumn(column).ToArray(ref elements); // copy column into values
bin.Clear();
bin.AddAllOf(new DoubleArrayList(values));
for (int i = aggr.Length; --i >= 0;)
{
result[i, column] = aggr[i](bin);
}
}
return(result);
}
public static DoubleMatrix2D PermuteRows(DoubleMatrix2D A, int[] indexes, int[] work)
{
// check validity
int size = A.Rows;
if (indexes.Length != size)
{
throw new IndexOutOfRangeException("invalid permutation");
}
/*
* int i=size;
* int a;
* while (--i >= 0 && (a=indexes[i])==i) if (a < 0 || a >= size) throw new IndexOutOfRangeException("invalid permutation");
* if (i<0) return; // nothing to permute
*/
int columns = A.Columns;
if (columns < size / 10)
{ // quicker
double[] doubleWork = new double[size];
for (int j = A.Columns; --j >= 0;)
{
Permute(A.ViewColumn(j), indexes, doubleWork);
}
return(A);
}
var swapper = new Cern.Colt.Swapper((a, b) =>
{
A.ViewRow(a).Swap(A.ViewRow(b));
}
);
Cern.Colt.GenericPermuting.permute(indexes, swapper, work, null);
return(A);
}
/// <summary>
/// Copies the rows of the indicated columns into a new sub matrix.
/// <i>sub[0..rowTo-rowFrom,0..columnIndexes.Length-1] = A[rowFrom..rowTo,columnIndexes(:)]</i>;
/// The returned matrix is <i>not backed</i> by this matrix, so changes in the returned matrix are <i>not reflected</i> in this matrix, and vice-versa.
/// </summary>
/// <param name="A">the source matrix to copy from.</param>
/// <param name="rowFrom">the index of the first row to copy (inclusive).</param>
/// <param name="rowTo">the index of the last row to copy (inclusive).</param>
/// <param name="columnIndexes">the indexes of the columns to copyd May be unsorted.</param>
/// <returns>a new sub matrix; with <i>sub.Rows==rowTo-rowFrom+1; sub.Columns==columnIndexes.Length</i>.</returns>
/// <exception cref="IndexOutOfRangeException">if <i>rowFrom < 0 || rowTo-rowFrom + 1 < 0 || rowTo + 1 > matrix.Rows || for any col = columnIndexes[i]: col < 0 || col >= matrix.Columns</i>.</exception>
private static DoubleMatrix2D SubMatrix(DoubleMatrix2D A, int rowFrom, int rowTo, int[] columnIndexes)
{
if (rowTo - rowFrom >= A.Rows)
{
throw new IndexOutOfRangeException("Too many rows");
}
int height = rowTo - rowFrom + 1;
int columns = A.Columns;
A = A.ViewPart(rowFrom, 0, height, columns);
DoubleMatrix2D sub = A.Like(height, columnIndexes.Length);
for (int c = columnIndexes.Length; --c >= 0;)
{
int column = columnIndexes[c];
if (column < 0 || column >= columns)
{
throw new IndexOutOfRangeException("Illegal Index");
}
sub.ViewColumn(c).Assign(A.ViewColumn(column));
}
return(sub);
}
/// <summary>
/// Constructs and returns the distance matrix of the given matrix.
/// The distance matrix is a square, symmetric matrix consisting of nothing but distance coefficientsd
/// The rows and the columns represent the variables, the cells represent distance coefficientsd
/// The diagonal cells (i.ed the distance between a variable and itself) will be zero.
/// Compares two column vectors at a timed Use dice views to compare two row vectors at a time.
/// </summary>
/// <param name="matrix">any matrix; a column holds the values of a given variable (vector).</param>
/// <param name="distanceFunction">(EUCLID, CANBERRA, ..d, or any user defined distance function operating on two vectors).</param>
/// <returns>the distance matrix (<i>n x n, n=matrix.Columns</i>).</returns>
public static DoubleMatrix2D Distance(DoubleMatrix2D matrix, VectorVectorFunction distanceFunction)
{
int columns = matrix.Columns;
DoubleMatrix2D distance = new DenseDoubleMatrix2D(columns, columns);
// cache views
DoubleMatrix1D[] cols = new DoubleMatrix1D[columns];
for (int i = columns; --i >= 0;)
{
cols[i] = matrix.ViewColumn(i);
}
// work out all permutations
for (int i = columns; --i >= 0;)
{
for (int j = i; --j >= 0;)
{
double d = distanceFunction(cols[i], cols[j]);
distance[i, j] = d;
distance[j, i] = d; // symmetric
}
}
return(distance);
}
/// <summary>
/// Constructs and returns a new QR decomposition object; computed by Householder reflections;
/// The decomposed matrices can be retrieved via instance methods of the returned decomposition object.
/// Return a decomposition object to access <i>R</i> and the Householder vectors <i>H</i>, and to compute <i>Q</i>.
/// </summary>
/// <param name="A">A rectangular matrix.</param>
/// <exception cref="ArgumentException">if <i>A.Rows < A.Columns</i>.</exception>
public QRDecomposition(DoubleMatrix2D A)
{
Property.DEFAULT.CheckRectangular(A);
Functions F = Functions.functions;
// Initialize.
QR = A.Copy();
m = A.Rows;
n = A.Columns;
Rdiag = A.Like1D(n);
//Rdiag = new double[n];
DoubleDoubleFunction hypot = Algebra.HypotFunction();
// precompute and cache some views to avoid regenerating them time and again
DoubleMatrix1D[] QRcolumns = new DoubleMatrix1D[n];
DoubleMatrix1D[] QRcolumnsPart = new DoubleMatrix1D[n];
for (int k = 0; k < n; k++)
{
QRcolumns[k] = QR.ViewColumn(k);
QRcolumnsPart[k] = QR.ViewColumn(k).ViewPart(k, m - k);
}
// Main loop.
for (int k = 0; k < n; k++)
{
//DoubleMatrix1D QRcolk = QR.ViewColumn(k).ViewPart(k,m-k);
// Compute 2-norm of k-th column without under/overflow.
double nrm = 0;
//if (k<m) nrm = QRcolumnsPart[k].aggregate(hypot,F.identity);
for (int i = k; i < m; i++)
{ // fixes bug reported by [email protected]
nrm = Algebra.Hypot(nrm, QR[i, k]);
}
if (nrm != 0.0)
{
// Form k-th Householder vector.
if (QR[k, k] < 0)
{
nrm = -nrm;
}
QRcolumnsPart[k].Assign(F2.Div(nrm));
/*
* for (int i = k; i < m; i++) {
* QR[i][k] /= nrm;
* }
*/
QR[k, k] = QR[k, k] + 1;
// Apply transformation to remaining columns.
for (int j = k + 1; j < n; j++)
{
DoubleMatrix1D QRcolj = QR.ViewColumn(j).ViewPart(k, m - k);
double s = QRcolumnsPart[k].ZDotProduct(QRcolj);
/*
* // fixes bug reported by John Chambers
* DoubleMatrix1D QRcolj = QR.ViewColumn(j).ViewPart(k,m-k);
* double s = QRcolumnsPart[k].ZDotProduct(QRcolumns[j]);
* double s = 0.0;
* for (int i = k; i < m; i++) {
* s += QR[i][k]*QR[i][j];
* }
*/
s = -s / QR[k, k];
//QRcolumnsPart[j].Assign(QRcolumns[k], F.PlusMult(s));
for (int i = k; i < m; i++)
{
QR[i, j] = QR[i, j] + s * QR[i, k];
}
}
}
Rdiag[k] = -nrm;
}
}
public void Decompose(DoubleMatrix2D A)
{
int CUT_OFF = 10;
// setup
LU = A;
int m = A.Rows;
int n = A.Columns;
// setup pivot vector
if (this.piv == null || this.piv.Length != m)
{
this.piv = new int[m];
}
for (int i = m; --i >= 0;)
{
piv[i] = i;
}
pivsign = 1;
if (m * n == 0)
{
LU = LU;
return; // nothing to do
}
//precompute and cache some views to avoid regenerating them time and again
DoubleMatrix1D[] LUrows = new DoubleMatrix1D[m];
for (int i = 0; i < m; i++)
{
LUrows[i] = LU.ViewRow(i);
}
IntArrayList nonZeroIndexes = new IntArrayList(); // sparsity
DoubleMatrix1D LUcolj = LU.ViewColumn(0).Like(); // blocked column j
Cern.Jet.Math.Mult multFunction = Cern.Jet.Math.Mult.CreateInstance(0);
// Outer loop.
for (int j = 0; j < n; j++)
{
// blocking (make copy of j-th column to localize references)
LUcolj.Assign(LU.ViewColumn(j));
// sparsity detection
int maxCardinality = m / CUT_OFF; // == heuristic depending on speedup
LUcolj.GetNonZeros(nonZeroIndexes, null, maxCardinality);
int cardinality = nonZeroIndexes.Count;
Boolean sparse = (cardinality < maxCardinality);
// Apply previous transformations.
for (int i = 0; i < m; i++)
{
int kmax = System.Math.Min(i, j);
double s;
if (sparse)
{
s = LUrows[i].ZDotProduct(LUcolj, 0, kmax, nonZeroIndexes);
}
else
{
s = LUrows[i].ZDotProduct(LUcolj, 0, kmax);
}
double before = LUcolj[i];
double after = before - s;
LUcolj[i] = after; // LUcolj is a copy
LU[i, j] = after; // this is the original
if (sparse)
{
if (before == 0 && after != 0)
{ // nasty bug fixed!
int pos = nonZeroIndexes.BinarySearch(i);
pos = -pos - 1;
nonZeroIndexes.Insert(pos, i);
}
if (before != 0 && after == 0)
{
nonZeroIndexes.Remove(nonZeroIndexes.BinarySearch(i));
}
}
}
// Find pivot and exchange if necessary.
int p = j;
if (p < m)
{
double max = System.Math.Abs(LUcolj[p]);
for (int i = j + 1; i < m; i++)
{
double v = System.Math.Abs(LUcolj[i]);
if (v > max)
{
p = i;
max = v;
}
}
}
if (p != j)
{
LUrows[p].Swap(LUrows[j]);
int k = piv[p]; piv[p] = piv[j]; piv[j] = k;
pivsign = -pivsign;
}
// Compute multipliers.
double jj;
if (j < m && (jj = LU[j, j]) != 0.0)
{
multFunction.Multiplicator = 1 / jj;
LU.ViewColumn(j).ViewPart(j + 1, m - (j + 1)).Assign(multFunction);
}
}
LU = LU;
}
