public static double[] Subtract(double[] aStore, int aOffset, int aStride, double[] bStore, int bOffset, int bStride, int dimension)
{
double[] store = new double[dimension];
Blas1.dCopy(aStore, aOffset, aStride, store, 0, 1, dimension);
Blas1.dAxpy(-1.0, bStore, bOffset, bStride, store, 0, 1, dimension);
return(store);
}
public static void SortValues(double[] values, double[] utStore, double[] vStore, int rows, int cols)
{
// this is a selection sort, an O(N^2) sort which requires fewer swaps than an insertion sort or an O(N ln N) sort
// loop over ranks
for (int i = 0; i < cols; i++)
{
// find the next largest value
int j = i;
double t = values[i];
for (int k = i + 1; k < cols; k++)
{
if (values[k] > t)
{
j = k;
t = values[k];
}
}
// if necessary swap it with the current element
if (j != i)
{
Global.Swap(ref values[i], ref values[j]);
if (utStore != null)
{
Blas1.dSwap(utStore, i, rows, utStore, j, rows, rows);
}
if (vStore != null)
{
Blas1.dSwap(vStore, i * cols, 1, vStore, j * cols, 1, cols);
}
}
}
}
public static void SolveLowerLeftTriangular(double[] tStore, double[] x, int xOffset, int dimension)
{
for (int r = 0; r < dimension; r++)
{
double t = Blas1.dDot(tStore, r, dimension, x, xOffset, 1, r);
x[xOffset + r] -= t;
}
}
// ** SVD **
public static void ExtractSingularValues(double[] a, double[] b, double[] uStore, double[] vStore, int rows, int cols)
{
// n is the upper limit index
int n = cols - 1;
int count = 0;
while (count < 32)
{
// move the lower boundary up as far as we can
while ((n > 0) && (Math.Abs(b[n - 1]) <= (2.0E-16) * Math.Abs(a[n - 1] + a[n])))
{
b[n - 1] = 0.0;
if (a[n] < 0.0)
{
a[n] = -a[n];
if (vStore != null)
{
Blas1.dScal(-1.0, vStore, n * cols, 1, cols);
}
}
n--;
count = 0;
}
if (n == 0)
{
return;
}
// p is the upper limit index
// find any zeros on the diagonal and reduce the problem
int p = n;
while (p > 0)
{
if (Math.Abs(b[p - 1]) <= (2.0E-16) * (Math.Abs(a[p]) + Math.Abs(a[p - 1])))
{
break;
}
p--;
// do we need a better zero test?
if (a[p] == 0.0)
{
//
ChaseBidiagonalZero(a, b, p, n, uStore, rows);
p++;
break;
}
}
// diagonal zeros will cause the Golulb-Kahn step to reproduce the same matrix,
// and eventually we will fail with non-convergence
GolubKahn(a, b, p, n, uStore, vStore, rows, cols);
count++;
}
throw new NonconvergenceException();
}
public static void SolveUpperRightTriangular(double[] tStore, double[] x, int xOffset, int dimension)
{
for (int r = dimension - 1; r >= 0; r--)
{
int i0 = dimension * r + r;
double t = Blas1.dDot(tStore, i0 + dimension, dimension, x, xOffset + r + 1, 1, dimension - r - 1);
x[xOffset + r] = (x[xOffset + r] - t) / tStore[i0];
}
}
/// <summary>
/// Gets the U factor.
/// </summary>
/// <returns>The upper-right triangular factor U of the LU decomposition.</returns>
public SquareMatrix UMatrix()
{
double[] uStore = new double[dimension * dimension];
for (int c = 0; c < dimension; c++)
{
Blas1.dCopy(luStore, dimension * c, 1, uStore, dimension * c, 1, c + 1);
}
return(new SquareMatrix(uStore, dimension));
}
/// <inheritdoc />
public override ColumnVector Column(int c)
{
if ((c < 0) || (c >= cols))
{
throw new ArgumentOutOfRangeException(nameof(c));
}
double[] cStore = new double[rows];
Blas1.dCopy(store, offset + colStride * c, rowStride, cStore, 0, 1, rows);
return(new ColumnVector(cStore, rows));
}
/// <inheritdoc />
public override RowVector Row(int r)
{
if ((r < 0) || (r >= rows))
{
throw new ArgumentOutOfRangeException(nameof(r));
}
double[] rStore = new double[cols];
Blas1.dCopy(store, offset + rowStride * r, colStride, rStore, 0, 1, cols);
return(new RowVector(rStore, cols));
}
// Apply a Householder transfrom defined by v to the vector x (which may be
// a column of a matrix, if H is applied from the left, or a row of a matrix
// if H is applied from the right). On exit v is unchanged, x is changed.
// We have H = I - \beta v v^T, so H x = x - (\beta v^T x) v.
public static void ApplyHouseholderReflection(
double[] uStore, int uOffset, int uStride,
double[] yStore, int yOffset, int yStride,
int count
)
{
double s = Blas1.dDot(uStore, uOffset, uStride, yStore, yOffset, yStride, count);
Blas1.dAxpy(-s, uStore, uOffset, uStride, yStore, yOffset, yStride, count);
}
/// <summary>
/// Multiplies a complex column vector by a complex constant.
/// </summary>
/// <param name="alpha">The complex constant.</param>
/// <param name="v">The complex column vector.</param>
/// <returns>The product αv.</returns>
/// <exception cref="ArgumentNullException"><paramref name="v"/> is null.</exception>
public static ComplexColumnVector operator *(Complex alpha, ComplexColumnVector v)
{
if (v == null)
{
throw new ArgumentNullException(nameof(v));
}
Complex[] product = new Complex[v.dimension];
Blas1.zAxpy(alpha, v.store, v.offset, 1, product, 0, 1, v.dimension);
return(new ComplexColumnVector(product, 0, v.dimension, false));
}
/// <summary>
/// Gets the L factor.
/// </summary>
/// <returns>The lower-left triangular factor L of the LU decomposition.</returns>
/// <remarks>
/// <para>The pivoted LU decomposition algorithm guarantees that the diagonal entries of this matrix are all one, and
/// that the magnitudes of the sub-diagonal entries are all less than or equal to one.</para>
/// </remarks>
public SquareMatrix LMatrix()
{
double[] lStore = new double[dimension * dimension];
for (int c = 0; c < dimension; c++)
{
int i0 = dimension * c + c;
lStore[i0] = 1.0;
Blas1.dCopy(luStore, i0 + 1, 1, lStore, i0 + 1, 1, dimension - c - 1);
}
return(new SquareMatrix(lStore, dimension));
}
private static void SwapIndexes(double[] aStore, int[] perm, int dimension, int p, int q)
{
if (p == q)
{
return;
}
Blas1.dSwap(aStore, p, dimension, aStore, q, dimension, dimension);
Blas1.dSwap(aStore, dimension * p, 1, aStore, dimension * q, 1, dimension);
if (perm != null)
{
Global.Swap <int>(ref perm[p], ref perm[q]);
}
}
public static double OneNorm(double[] store, int offset, int rowStride, int colStride, int nRows, int nColumns)
{
double norm = 0.0;
for (int c = 0; c < nColumns; c++)
{
double csum = Blas1.dNrm1(store, offset + colStride * c, rowStride, nRows);
if (csum > norm)
{
norm = csum;
}
}
return(norm);
}
public static double InfinityNorm(double[] store, int offset, int rowStride, int colStride, int nRows, int nColumns)
{
double norm = 0.0;
for (int r = 0; r < nRows; r++)
{
double rsum = Blas1.dNrm1(store, offset + rowStride * r, colStride, nColumns);
if (rsum > norm)
{
norm = rsum;
}
}
return(norm);
}
// Solve a triangular system
// aIsUpper indicates whether A is upper/right or lower/left
// aIsUnit indicates whether A's diagonal elements are all 1 or not
// The relevent elements of xStore are overwritten by the solution vector
public static void dTrsv(
bool aIsUpper, bool aIsUnit, double[] aStore, int aOffset, int aRowStride, int aColStride,
double[] xStore, int xOffset, int xStride,
int count
)
{
int aDiagonalStride = aColStride + aRowStride;
// If A is upper triangular, start at the bottom/right and reverse the direction of progression.
// This does not affect the passed in variables because integers are pass-by-value.
if (aIsUpper)
{
aOffset = aOffset + (count - 1) * aDiagonalStride;
aRowStride = -aRowStride;
aColStride = -aColStride;
aDiagonalStride = -aDiagonalStride;
xOffset = xOffset + (count - 1) * xStride;
xStride = -xStride;
}
// The index of the first row element to be subtracted in the numerator.
int aRowIndex = aOffset;
// The index of the x element to be solved for.
int xIndex = xOffset;
// We hoist the aIsUnit test outside the loop and pay the cost of a little repeated code in
// order to avoid an unnecessary per-loop test. It's possible the compiler could do this for us.
if (aIsUnit)
{
for (int n = 0; n < count; n++)
{
xStore[xIndex] -= Blas1.dDot(aStore, aRowIndex, aColStride, xStore, xOffset, xStride, n);
aRowIndex += aRowStride;
xIndex += xStride;
}
}
else
{
int aDiagonalIndex = aOffset;
for (int n = 0; n < count; n++)
{
xStore[xIndex] -= Blas1.dDot(aStore, aRowIndex, aColStride, xStore, xOffset, xStride, n);
xStore[xIndex] /= aStore[aDiagonalIndex];
aRowIndex += aRowStride;
aDiagonalIndex += aDiagonalStride;
xIndex += xStride;
}
}
}
/// <summary>
/// Sort the eigenvalues as specified.
/// </summary>
/// <param name="order">The desired ordering.</param>
public void Sort(OrderBy order)
{
// Create an auxiluary array of indexes to sort
int[] ranks = new int[dimension];
for (int i = 0; i < ranks.Length; i++)
{
ranks[i] = i;
}
// Sort indexes in the requested order
Comparison <int> comparison;
switch (order)
{
case OrderBy.ValueAscending:
comparison = (int a, int b) => eigenvalues[a].CompareTo(eigenvalues[b]);
break;
case OrderBy.ValueDescending:
comparison = (int a, int b) => eigenvalues[b].CompareTo(eigenvalues[a]);
break;
case OrderBy.MagnitudeAscending:
comparison = (int a, int b) => Math.Abs(eigenvalues[a]).CompareTo(Math.Abs(eigenvalues[b]));
break;
case OrderBy.MagnitudeDescending:
comparison = (int a, int b) => Math.Abs(eigenvalues[b]).CompareTo(Math.Abs(eigenvalues[a]));
break;
default:
throw new NotSupportedException();
}
Array.Sort(ranks, comparison);
// Create new storage in the desired order and discard the old storage
// This is faster than moving within existing storage since we can move each column just once.
double[] newEigenvalues = new double[dimension];
double[] newEigenvectorStorage = new double[dimension * dimension];
for (int i = 0; i < ranks.Length; i++)
{
newEigenvalues[i] = eigenvalues[ranks[i]];
Blas1.dCopy(eigenvectorStorage, dimension * ranks[i], 1, newEigenvectorStorage, dimension * i, 1, dimension);
}
eigenvalues = newEigenvalues;
eigenvectorStorage = newEigenvectorStorage;
}
// y <- A x + y
public static void dGemv(
double[] aStore, int aOffset, int aRowStride, int aColStride,
double[] xStore, int xOffset, int xStride,
double[] yStore, int yOffset, int yStride,
int rows, int cols
)
{
int aIndex = aOffset;
int yIndex = yOffset;
for (int n = 0; n < rows; n++)
{
yStore[yIndex] += Blas1.dDot(aStore, aIndex, aColStride, xStore, xOffset, xStride, cols);
aIndex += aRowStride;
yIndex += yStride;
}
}
/// <summary>
/// Computes the inner (scalar or dot) product of a row and a column vector.
/// </summary>
/// <param name="v">The row vector.</param>
/// <param name="u">The column vector.</param>
/// <returns>The value of the scalar product.</returns>
public static double operator *(RowVector v, ColumnVector u)
{
if (v == null)
{
throw new ArgumentNullException(nameof(v));
}
if (u == null)
{
throw new ArgumentNullException(nameof(u));
}
if (v.dimension != u.dimension)
{
throw new DimensionMismatchException();
}
double p = Blas1.dDot(v.store, v.offset, v.stride, u.store, u.offset, u.stride, v.dimension);
return(p);
}
public static double[] Copy(double[] store, int offset, int rowStride, int colStride, int nRows, int nCols)
{
double[] copyStore = AllocateStorage(nRows, nCols);
if ((rowStride == 1) && (colStride == nRows))
{
Array.Copy(store, offset, copyStore, 0, copyStore.Length);
}
else
{
for (int c = 0; c < nCols; c++)
{
int sourceIndex = offset + colStride * c;
int copyIndex = nRows * c;
Blas1.dCopy(store, sourceIndex, rowStride, copyStore, copyIndex, 1, nRows);
}
}
return(copyStore);
}
// on input:
// store contains the matrix in column-major order (must have the length dimension^2)
// permutation contains the row permutation (typically 0, 1, 2, ..., dimension - 1; must have the length dimension^2)
// parity contains the parity of the row permutation (typically 1; must be 1 or -1)
// dimension contains the dimension of the matrix (must be non-negative)
// on output:
// store is replaced by the L and U matrices, L in the lower-left triangle (with 1s along the diagonal), U in the upper-right triangle
// permutation is replaced by the row permutation, and parity be the parity of that permutation
// A = PLU
public static void LUDecompose(double[] store, int[] permutation, ref int parity, int dimension)
{
for (int d = 0; d < dimension; d++)
{
int pivotRow = -1;
double pivotValue = 0.0;
for (int r = d; r < dimension; r++)
{
int a0 = dimension * d + r;
double t = store[a0] - Blas1.dDot(store, r, dimension, store, dimension * d, 1, d);
store[a0] = t;
if (Math.Abs(t) > Math.Abs(pivotValue))
{
pivotRow = r;
pivotValue = t;
}
}
if (pivotValue == 0.0)
{
throw new DivideByZeroException();
}
if (pivotRow != d)
{
// switch rows
Blas1.dSwap(store, d, dimension, store, pivotRow, dimension, dimension);
int t = permutation[pivotRow];
permutation[pivotRow] = permutation[d];
permutation[d] = t;
parity = -parity;
}
Blas1.dScal(1.0 / pivotValue, store, dimension * d + d + 1, 1, dimension - d - 1);
for (int c = d + 1; c < dimension; c++)
{
double t = Blas1.dDot(store, d, dimension, store, dimension * c, 1, d);
store[dimension * c + d] -= t;
}
}
}
// A Householder reflection matrix is a rank-1 update to the identity.
// P = I - b v v^T
// Unitarity requires b = 2 / |v|^2. To anihilate all but the first components of vector x,
// P x = a e_1
// we must have
// v = x +/- |x| e_1
// that is, all elements the same except the first, from which we have either added or subtracted |x|. This makes
// a = -/+ |x|
// There are two way to handle the sign. One is to simply choose the sign that avoids cancelation when calculating v_1,
// i.e. + for positive x_1 and - for negative x_1. This works fine, but makes a negative for positive x_1, which is
// weird-looking (1 0 0 gets turned into -1 0 0). An alternative is to choose the negative sign even for positive x_1,
// but to avoid cancelation write
// v_1 = x_1 - |x| = ( x_1 - |x| ) ( x_1 + |x|) / ( x_1 + |x|) = ( x_1^2 - |x|^2 ) / ( x_1 + |x| )
// = - ( x_2^2 + \cdots + x_n^2 ) / ( x_1 + |x| )
// We have now moved to the second method. Note that v is the same as x except for the first element.
public static void GenerateHouseholderReflection(double[] store, int offset, int stride, int count, out double a)
{
double x0 = store[offset];
// Compute |x| and u_0.
double xm, u0;
if (x0 < 0.0)
{
xm = Blas1.dNrm2(store, offset, stride, count);
u0 = x0 - xm;
}
else
{
// This method of computing ym and xm does incur and extra square root compared to naively computing x_2 + \cdots + x_n^2,
// but doing it this way allows us to use dNrm2's over/under-flow prevention when we have large/small elements.
double ym = Blas1.dNrm2(store, offset + stride, stride, count - 1);
xm = MoreMath.Hypot(x0, ym);
// Writing ym / (x0 + xm) * ym instead of ym * ym / (x0 + xm) prevents over/under-flow for large/small ym. Note this will
// be 0 / 0 = NaN when xm = 0.
u0 = -ym / (x0 + xm) * ym;
}
// Set result element.
a = xm;
// If |x| = 0 there is nothing to do; we could have done this a little earlier but the compiler requires us to set a before returning.
if (xm == 0.0)
{
return;
}
// Set the new value of u_0
store[offset] = u0;
// Rescale to make b = 1.
double um = Math.Sqrt(xm * Math.Abs(u0));
if (um > 0.0)
{
Blas1.dScal(1.0 / um, store, offset, stride, count);
}
}
public static void SolveUpperRightTriangular(double[] tStore, int tRows, int tCols, double[] xStore, int xOffset)
{
Debug.Assert(tRows >= tCols);
// pre-calculate some useful quantities
int trp1 = tRows + 1;
int tcm1 = tCols - 1;
// initialize the x index (to its highest value) the the diagonal t index (to its highest value)
int xIndex = xOffset + tcm1;
int tIndex = trp1 * tcm1;
// do the first, trivial back-substitution
xStore[xIndex] /= tStore[tIndex];
// do the remaining back-substitutions, each of which requires subtracting n previously computed x-values
for (int n = 1; n <= tcm1; n++)
{
xIndex -= 1;
tIndex -= trp1;
double t = Blas1.dDot(tStore, tIndex + tRows, tRows, xStore, xIndex + 1, 1, n);
xStore[xIndex] = (xStore[xIndex] - t) / tStore[tIndex];
}
}
public static double[] Copy(double[] store, int offset, int stride, int dimension)
{
double[] copy = new double[dimension];
Blas1.dCopy(store, offset, stride, copy, 0, 1, dimension);
return(copy);
}
// inverts the matrix in place
// the in-place-ness makes this a bit confusing
public static void GaussJordanInvert(double[] store, int dimension)
{
// keep track of row exchanges
int[] ps = new int[dimension];
// iterate over dimensions
for (int k = 0; k < dimension; k++)
{
// look for a pivot in the kth column on any lower row
int p = k;
double q = MatrixAlgorithms.GetEntry(store, dimension, dimension, k, k);
for (int r = k + 1; r < dimension; r++)
{
double s = MatrixAlgorithms.GetEntry(store, dimension, dimension, r, k);
if (Math.Abs(s) > Math.Abs(q))
{
p = r;
q = s;
}
}
ps[k] = p;
// if no non-zero pivot is found, the matrix is singular and cannot be inverted
if (q == 0.0)
{
throw new DivideByZeroException();
}
// if the best pivot was on a lower row, swap it into the kth row
if (p != k)
{
Blas1.dSwap(store, k, dimension, store, p, dimension, dimension);
}
// divide the pivot row by the pivot element, so the diagonal element becomes unity
MatrixAlgorithms.SetEntry(store, dimension, dimension, k, k, 1.0);
Blas1.dScal(1.0 / q, store, k, dimension, dimension);
// add factors to the pivot row to zero all off-diagonal elements in the kth column
for (int r = 0; r < dimension; r++)
{
if (r == k)
{
continue;
}
double a = MatrixAlgorithms.GetEntry(store, dimension, dimension, r, k);
MatrixAlgorithms.SetEntry(store, dimension, dimension, r, k, 0.0);
Blas1.dAxpy(-a, store, k, dimension, store, r, dimension, dimension);
}
}
// unscramble exchanges
for (int k = dimension - 1; k >= 0; k--)
{
int p = ps[k];
if (p != k)
{
Blas1.dSwap(store, dimension * p, 1, store, dimension * k, 1, dimension);
}
}
}
void ICollection <double> .CopyTo(double[] array, int arrayIndex)
{
Blas1.dCopy(store, offset, stride, array, arrayIndex, 1, dimension);
}
/// <summary>
/// Computes the magnitude of the vector.
/// </summary>
/// <returns>The Euclidean norm of the vector.</returns>
public virtual double Norm()
{
return(Blas1.dNrm2(store, offset, stride, dimension));
}
/// <summary>
/// Returns a copy of the complex column vector.
/// </summary>
/// <returns>An independent copy of the complex column vector.</returns>
public ComplexColumnVector Copy()
{
Complex[] copy = new Complex[dimension];
Blas1.Copy(store, offset, 1, copy, 0, 1, dimension);
return(new ComplexColumnVector(copy, 0, dimension, false));
}
public static double[] Multiply(double alpha, double[] store, int offset, int stride, int dimension)
{
double[] product = new double[dimension];
Blas1.dAxpy(alpha, store, offset, stride, product, 0, 1, dimension);
return(product);
}